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More About This Textbook OverviewRelated Subjects Meet 6, 2002 Superb This is an excellent introduction to the mathematics of computer graphics for readers with a basic knowledge of linear algebra, written in mathematical English with standard math notation. Each chapter has a few exercises. Suitable as an assigned textbook for an introductory computer graphics math course. 1 out of 1 people found this review helpful. Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged. Anonymous Posted January 9, 2012 Swell Ive not read the hole thing yet but it seems really good so far:) Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
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calculus based physics Vs. algebra based physics calculus based physics Vs. algebra based physics in my high school physics class we are doing algebra baced physics but i have heard of calculus based physics and i wanted to know the differences between them and the different Applications that they have. Algebra based physics is learning physics that pretty much only uses algebra (some trig). Calculus based physics uses calculus. So basically your class is an intro to calc based physics. You learn all the concepts of physics without messing around with all the math. If you don't know any calculus, then it will be hard to explain the benefits of it. But if you've taken calculus, then it should be pretty obvious what it can be used for. Let me try to explain anyway, because that didn't really answer your question. Calculus lets you "add up" small contributions to get a total. For example, you have your equation for how long an object stays in the air when you throw it, right? There should be a variable there that is squared. In one of those formulas, anyway. So the only way you can do that right now is to use the formula. With calculus, you start with something more basic, and you can actually derive the formula. Okay, that was a bad explanation. Umm... just wait until someone gives you a better one. :( Adding calculus to Physics I, at least at the level of Halliday, Resnick, and Walker, results in hardly any new physics being taught. For instance in algebra based physics, you take the following formula as given: [tex]x(t)=x_0+v_0t+\frac{1}{2}at^2[/tex] In calc based physics you derive that formula by integrating [itex]\frac{d^2x}{dt^2}=a[/itex] twice. Kinematics with nonconstant acceleration is relegated to the Exercises. What really makes calc based Physics I different from algebra based Physics I is not the calculus, but the use of the dot and cross products (most algebra based physics courses don't teach this). Now when you get to Physics II, the calculus makes a huge difference, because you can finally learn Maxwell's equations. calculus based physics Vs. algebra based physics I suppose for physics I, calculus makes little difference, as what is derived by calc, can often be derived alebraicly too, but calc is probably simpler. Without calculus you can't go very far in physics. Or just about anything that uses math- business, engineering, social sciences etc. Adding calculus to Physics I, at least at the level of Halliday, Resnick, and Walker, results in hardly any new physics being taught. I just dusted off my very old Halliday and Resnick, copyright 1966, and they certainly went well beyond algebra-based physics. Here are a few topics in which calculus played an integral role: Work as a line integral; the rocket equation; coupled, damped, and forced harmonic oscillators; simple fluid dynamics and thermodynamics. Moreover, many topics which are presented as givens in pre-calculus physics are derived in that ancient version of Halliday and Resnick; e.g., Kepler's laws of motion. Has the Halliday, Resnick, and Walker text dumbed things down since I went to school back in the stone age? thharrimw, This enhancement of details is how physics education progresses. You will learn some simple aspect of a problem at one level, such as the behavior of a particle subject to a constant acceleration. Calculus-based physics throws out all those seemingly unrelated formulae you learned in algebra-based physics, replacing them with a smaller set of more abstract and more mathematically advanced equations. Junior level classical dynamics throws that simple freshman-based physics out the window. Graduate level courses throw out the simple junior level stuff. I have not yet touched on electricity, or quantum mechanics, or gravitation. The same processes occur there that occur with classical dynamics. Each step up you are learning some new physics. You are also relearning the physics you already know, but with the added twist of mathematical techniques that you presumably did not have knowledgeof the first time around. Yes, Halliday and Resnick & Co. have watered their book down considerably since the old days. I once saw an early edition of their book that showed a derivation of the differential form of Maxwell's equations from the integral form. Now, only the integral form remains and the differential form isn't even mentioned. Work as a line integral; This is presented, but in the exercises students are only asked to integrate along straight line segments. And for most exercises, no integration is required at all. the rocket equation; This is presented, but students don't actually have to do any calculus in the exercises. coupled, damped, and forced harmonic oscillators; The differential equations for these systems are presented, as well as their solutions. But they don't solve the equation in the book, and the students are never asked to do it. A curious student could plug the given solutions back into the diff eq to verify that it is indeed a correct solution, but this is never asked of the student. simple fluid dynamics and thermodynamics. The syllabus at the school where I taught as a grad student excluded these topics, as full courses in each subject were offered. So I never went through these chapters of H&R. ok all of this has made me more confused, What is derivation? What is a differential form? and What is a integral form? also in calc baced physics can you use these differntial and integral things and do more then you could if you used algebra equations? In this context, derivation is "a sequence of statements (as in logic or mathematics) showing that a result is a necessary consequence of previously accepted statements" (from Algebra-based physics is chock full of a bunch of disparate, ad-hoc formulae that must be memorized. Many of these ad-hoc formulae can be derived from a small set of seemingly simple equations. In algebra-based physics, the expression [itex]x=x_0 + v_0 t + 1/2at^2[/itex] is one of those ad-hoc forumulae. It can be derived from [itex]d^2x/dt^2 = a[/itex], which in turn is a consequence of Newton's second law. Another example is Kepler's laws. You probably had to memorize these as equations that just popped out of the blue in Kepler's mind. Kepler's laws are the result of deeper physics and more advanced math. What is a differential form? and What is a integral form? This is a differential form: [itex]\nabla \cdot \mathbf{B} = 0[/itex]. The corresponding integral form is [itex]\oint_S \mathbf{B} \cdot d\mathbf{S} = 0[/itex]. In English, there are no magnetic monopoles. If you haven't had calculus, that looks like gibberish. also in calc baced physics can you use these differntial and integral things and do more then you could if you used algebra equations? if it's a toss up between a decent teacher teaching the calc based, and a crappy teacher teaching the algebra based, take the calc based. to make a long story short, during my freshman year, I took the algebra based physics course, struggled and got a C. Then I toook the calc based physics, didn't struggle, and got a B. I was taking calc at the same time as physics, and didn't get held up by the calc...Curriculum for high school or college? Calculus is an essential mathematical tool for just about every scientific, engineering field. And why don't require it when just about every student in high school is going to have to learn calculus anyway, unless they've chosen to major in the arts and literature?
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Beecher, Penna, and Bittinger's College Algebra is known for enabling students to "see the math" through its focus on visualization and early ...Show synopsisBe
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Search Journal of Online Mathematics and its Applications: Journal of Online Mathematics and its Applications Tool Building: Web-based Linear Algebra Modules by David E. Meel and Thomas A. Hern Eigenizer Tool and Sample Activity Working with Eigenizer, similar to Transformer2D, involves coordinated actions between defining the column vectors of the matrix of transformation. The yellow box controls the column vectors defining the matrix of transformation. In particular, the green vector controls the first column vector and the blue vector controls the second column vector. By grabbing the ends of these two vectors, you can construct any 2x2 matrix. Below the yellow box is a box that controls the vector x. As you move x (the red vector) about the domain of the transformation, you can watch both x and the image T(x) (the magenta vector) change in the large area to the right of the screen, depicting the codomain of the transformation. The movement of the vector T(x) depends on the nature of the matrix of transformation. The large codomain region also displays information concerning the length of the vector x, the length of the vector T(x), the radian measure of the angle between these two vectors, and a lambda approximator. Two buttons at the bottom of this region control the display of the Eigen Equations in a red box above the codomain box. Note: The "?" at the bottom right-hand corner of the workspace is a link to Key Curriculum Press and its About JavaSketchpad web page. Sample Exploratory Activity: This sample activity provides a guided exploration of eigenvalues and eigenvectors of a particular matrix and includes a set of questions that can be asked for any other matrix, as well as some general questions about the tool and observations made from interacting with the tool. Using the vectors contained in the yellow box, move (by click dragging) the green vector to (3,-1) and the blue vector to (-2,2). This should construct the following matrix of transformation: . Move the red vector x (in the domain portion of the tool) and observe the movement of the magenta vector, T(x). If possible, move the vector x so that the vectors T(x) and x are collinear. Note: This can be aided by examining the angle measure at the bottom of the y-axis in the range portion of the tool. Click on the "Show lambda 1 equation" button, and observe whether the equation is true, i.e., do the vectors displayed on the right and left sides of the equation match each other? If they do not match, click on the "Show lambda 2 equation" button, and check if truth is found. Move the red vector x (in the domain portion of the tool) so that it and the T(x) vector are collinear in a different location. Redo step 3 for this new location. Given your exploration (and perhaps some additional ones), answer the following questions: What are the eigenvalues for this matrix? What are corresponding eigenvectors for these eigenvalues? What does the lambda approximator do? If an eigenvalue is positive, what does this mean concerning the form of collinearity between the vector x and the vector T(x)? If an eigenvalue is negative, what does this mean concerning the form of collinearlity between the vector x and the vector T(x)? Can you have a matrix with two positive eigenvalues or two negative eigenvalues? Explain why or why not For a given eigenvalue, is there a unique eigenvector or a set of corresponding eigenvectors? If unique, explain why it's unique and if there is a set, explain how to describe the set. Define a matrix that does not have any real eigenvalues? In general, what would be the nature of the column vectors of such a matrix? Is it possible to define a matrix that has eigenvalues of -3 and 2? If it is possible, state how many such matrices could be constructed, and provide a specific example of at least one. Is it possible to define a matrix that has only a single eigenvalue, say -2? Explain why or why not. The tool allows students to explore specific matrices as well as hypothesize about possible matrices with particular properties. It is with the latter type of explorations that the worlds of geometry and computation can fuse. Students need to think beyond computations that MatLab might be able to perform and ponder the possibilities of "what if?".
, media, and technology products for successful teaching and learning. Amazon.com Review About This Edition From the Publisher New Features Capstone exercise: The Capstone is a new type of exercise that appears in every section. The exercise synthesizes the main concepts of the section and presents them in one exercise. They often contain computational and non-computational parts. These exercises are excellent to work through in class to present a topic for the first time or in class homework review. The Instructor's Resource Guide offers teaching tips on how one might use the Capstone Exercises in class. Larson Join In Clicker can be found on the Instructor Companion Website. Revised exercises based on actual usage. New exercises abound in the ninth edition of Calculus. Based on analyses of actual student usage data, the exercise sets have been overhauled to improve user understanding. Many exercises were added, some were revised, and some were removed. The results are exercise sets that effectively address user learning needs. Second Order Differential Equations: Added as a new chapter to the multivariable standalone portion of the book and available online, this new chapter delves into second order differential equations. This will greatly help engineering and math majors. Additional Features Graded Homework Exercises: Online homework and tests are evaluated using powerful Maple software to ensure mathematical accuracy. Instructors control point values, weighting grades, and whether or not an item is graded. An electronic gradebook helps instructors manage course information easily and can be exported to other files, such as Excel. CAS Investigation: Many examples throughout the book are accompanied by CAS Investigations. These are collaborative investigations using a computer algebra system (e.g., Maple) to further explore the related example. CAS Investigations are located online and in the Multimedia eBook.
Algebraic Expression 0.00 (0 votes) Document Description Algebra is the most important and interesting topic of mathematical world. It starts from junior grades and goes up to college level grades. Along with this sometimes algebra problems become complex and tough. We all know that when we are moving towards higher education, things are getting tougher and more complex. To reduce this complexities student need a daily practice, which will help in getting some confidence. Lets discuss about the basic concept of Algebra. Algebra is a branch of mathematics which helps in studying the rules of operations and relations. In a simple mathematical manner we can say that It is an area of mathematics in which letters and symbols are used in place of numbers and quantities to form an equation and formula. Before proceeding further, let's talk about equation. An equation is a mathematical expression which shows the equality of two expressions. For example 2x + y = 6. Now we are going to discuss about Algebraic expressions. Algebraic expressions is basically An algebraic expression is defined as the expression of constants or numbers , variables and also the combination of two or more values that are combined with mathematical operations like addition or ... The algebraic expressions have the variables and the constants. The algebraic expressions are the finite combination of the symbols that are formed according to the rules of the context. The algebra ... A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. More than one variable could be occurred by the linear equations ... Boolean operators are defining in the Boolean algebra that has defined through the several type of meanings as follows: These operators are define as Boolean function or Boolean algebra and when we ... The basic metric units are meter (for length), gram (for mass or weight), and liter (for volume). One unit can be Converted into another one. For example, one milliliter equals one cubic centimeter ... It is a process of obtaining new data within the given range of other known data or in other words we can say that it is a process of getting an unknown price by using other related known values that ... Content Preview Algebraic Expression Algebraic Expression Algebra is the most important and interesting topic of mathematical world. It starts from junior grades and goes up to college level grades. Along with this sometimes algebra problems become complex and tough. We al know that when we are moving towards higher education, things are getting tougher and more complex. To reduce this complexities student need a daily practice, which will help in getting some confidence. Lets discuss about the basic concept of Algebra. Algebra is a branch of mathematics which helps in studying the rules of operations and relations. In a simple mathematical manner we can say that It is an area of mathematics in which letters and symbols are used in place of numbers and quantities to form an equation and formula. Before proceeding further, let's talk about equation. An equation is a mathematical expression which shows the equality of two expressions. For example 2x + y = 6. Now we are going to discuss about Algebraic expressions. Algebraic expressions is basical y Know More About Radius of a Circle Math.Tutorvista.com Page No. :- 1/4 an expression having one or more variables made up of signs and symbols of algebra. Symbols can be Arabic numerals, literal numerals, or mathematical operators. Let just take one Algebra problems to understand it better. 3x -- z = c here x and z are the variables, c is a constant and 3 is a coefficient of x. Algebra also stands high in 8th Grade Math as many other topics of mathematics like fractions, geometry, trigonometry etc. An algebraic expression can be of three types, named as Monomial, Binomial and Trinomials. Algebraic expression having single term known as Monomial and expression with two terms are known as Binomial whereas expressions with more than two terms or having three terms are known as Trinomials. Examples xy = monomial, x -- 7y : Binomial, x + 2y -- c : Trinomial Terms are of two types Like terms or Unlike terms. Terms that has the same power of the same variables are called Like terms whereas the terms that do not contain the same power of the same variables are cal ed unlike terms. Fol owing steps are used to solve an Algebraic expressions: Firstly remove all the fractions in the equation Then remove the parentheses . Combine al the like terms so that we get all the variables and terms together. Move all the variable terms by adding or subtracting on both sides of the equal sign so the variable terms are al on one side of the equal sign. And finally if there is any multiplication sign then remove it by dividing. Example 1 Compute the factors for the expression x2+ 56x+ 768. Learn More Volume of a Sphere Formula
Pages January 21, 2013 PreCalc: Semester 1 As midterms are this week, we have reached the halfway point of the school year. Hard to believe it! This seems like an ideal time to look back over first semester. In my school we start PreCalculus with trigonometry. Students only see basic triangle trig in Geometry so this is a new topic for everyone. In the past teachers found themselves sucked into review and losing the strongest students at the very beginning of the year, they decided starting with new materials is the best way to avoid this. To be sure all students were starting on equal footing when it comes to the basics of graphing and manipulating graphs we had the honors students complete a summer assignment. Then, the students who struggled came to me outside of class for extra help, and the ones who had already mastered their families of functions moved straight into Trig. The very first day of class I had students make up a random survey question and survey their classmates (getting to know you activity with ulterior motive? check). I then distributed a large circle with a point at the center to each student and instructed them to create an accurate circle graph of their data (I none too subtly pointed out where various supplies are located in the classroom). Finally, I asked students to precisely find the arc length of each arc in their circle (in centimeters). Students played right into my plan as if they knew exactly what we were doing and asked about using degrees to measure arcs like they did in geometry. Enter the radian! This lesson would have been perfect if I'd had radian protractors, but we did well enough without them (plus, they didn't even exist in September). From here we continued into your basic triangle trig, build the unit circle, graph trig function sequence. Nothing particularly spectacular to say here. I wish I'd had Fouss' awesome unit circle at the time, it's much more organized than the ones I was using. After we had finished our study of the unit circle, one of my students had the suggestion that if they could fill in an entire unit circle they should get to use one on the tests (I'd been giving them blank ones to use, but his complaint was it took too long to fill in). This was a totally valid request and so I gave everyone this option for the midterm, next year I'll do it earlier. When we got to identities, students struggled. For valid reasons I outlined before. But also because I forgot to have them ever use what they struggled to prove! Most of the time, we prove something so that we can use it, not just because proofs are fun. I realized my mistake when I was glancing over the midterm some classes used last year. Next year we will alternating proving identities and using them to solve problems. This will quiet some of the pleas for numbers (my students really missed arithmetic during this unit) and will give them a sense of purpose in amongst all of these challenging, many step problems. For this exact reason, laws of sines and cosines were received with cheers- we actually used something we proved and there were numbers again! Our final unit of first semester was inverse functions. This was part review, part extension. Most students did well and the biggest issue I encountered was students trying to solve everything in their heads. I'm hoping that second semester will be similar since we will now move into topics that should be familiar but we will build up and out and around their current knowledge. I used a version of Standards Based Grading this year where students assessed on each topic twice. We do short quizzes on one standard and tests on a few standards at a time. I'm generally happy with how that has been going. The honors students are awesome about taking advantage of the retakes and they appreciate only having to retake one section of a test (as do I- less grading!). We do a variety of in class activities for each unit (called tasks below). Some activities are longer and require out of class time, those are collected for a grade. I thought I'd done a terrible job with blogging about PreCalc but it wasn't actually so bad. I'll try to fill in the holes of how I implemented geometric proof (a post I started ages ago but never finished) and the inverse function unit. 2 comments: Neat! I find it sort of fascinating that you were able to spend several months on trigonometry. In our Pre-Calc curriculum, it's just one of four strands (Trig, Exponent/Logs, Poly/Rationals, and Function Mashup). In the Grade 11, I've got less than a month to hit Trig expectations (granted, in 75 minute periods, with, it seems, fewer overall items). Also interesting that you go in reverse to me (perhaps because radian measure isn't until Grade 12). My first go through, when I brought in the unit circle first, I lost some students that never really come back for the triangles. So the last couple times, I've done triangles first - including obtuse triangle cases - then generated the unit circle to attempt to explain the ambiguous issue with inverse sine, and on from there. Still not sure if it works, I always feel like the chicken and the egg. (The circle explains the angles, but the angles are what generate the circle...) Anyway. Yeah, an "identity crisis" isn't unusual either way, it would seem. All the best for the rest of your year! I guess we spend so long on trig because it only gets a brief introduction in geometry and then students go from here to AP Calculus so they're on a time crunch. I actually have no idea if we're pacing this well to get through the list we made at the beginning of the year, which looks similar to yours but since the other Honors PreCalc teacher is also the AP Calc teacher, I suppose whatever happens will be fine for this year. Then at the end we can reassess, which of course is my main purpose in writing these reflections. It's always a pleasant bonus when someone else finds them interesting!
The purpose of the Math Resource Center is to aid students in developing their mathematical abilities. This resource center is staffed by a director, a staff instructor, and several student tutors who are available to help students in their mathematics courses. Instruction in the Math Resource Center is very informal. Students are welcome to come to the Math Resource Center with questions whenever they need help in understanding math course work. We structure our tutoring style to meet the student's mathematical needs in order to help them to gain self-confidence to become independent learners. Our goal in this resource center is to increase each student's understanding of her or his course material. This takes time and active participation on the student's part. Please do not expect the Math Resource Center to provide quick answers for the purpose of completing homework assignments. This does you a disservice. If we can help you understand the material so you can complete your assignment, we will be glad to do so. Mr. Shane Bruno is the director of the Math Resource Center, and Ms. Mary Moore is the staff instructor. They see students by walk-in or appointment. If you would like an appointment or if you have other questions, stop in the Math Resource Center, located in room 101 in the St. Joseph Academic/Health Building or call 520- 5310. The Math Resource Center has eight computers and instructional videos to support Pre-Algebra and Pre-Calculus which students may use. Upon request, the Math Resource Center director may refer a student for private one-on-one tutoring. Prices generally range from $10-$20 per hour; dependent upon the qualifications of the tutor and the degree of difficulty of the course. The Math Resource Center assumes no responsibility for the quality of services provided by private tutors. Students are encouraged to use the Math Resource Center as a place to do homework, work in a study group, or receive assistance in math questions. Guidelines Students must sign in when they come to use the Math Resource Center. Students must be enrolled in a math course at Xavier University in order to use the Math Resource Center. The Math Resource Center's textbooks and manuals must remain in the Math Resource Center while in use. Up to two videos can be checked out for a maximum of two days for home viewing, provided there is a Math Resource Center copy. A Xavier ID card is required in order to check out videos from the Math Resource Center. Food, drinks, and smoking are not allowed in the Math Resource Center. Children are not allowed in the Math Resource Center. No student is allowed to remove or copy any Math Resource Center software or Math Resource Center videos. Talking should be at a level as not to disturb others in the Math Resource Center. The Internet is to be used for academic research only. Chat rooms are not permitted. The Computers in the Math Resource Center should be left as you found them. Do not change any settings, nor save any files to the hard drive. Failure to abide by the above guidelines will result (at minimum) in having your Math Resource Center privileges taken away. Each tutor has varying levels of math knowledge and may not be able to assist you with math topics in which they are not familiar. It is your responsibility to ensure your assignments are completed on time. If you wait until the last minute to complete your assignment or project, that is your problem, not the tutor's. Read ahead in the math textbook and prepare questions for the instructor. For each chapter, prepare your own list of math vocabulary words. Develop practice tests and time yourself while taking them. For practice, do all the example problems in the text. While doing homework, write down questions for the instructor/tutor. Be aware of the time allotted while taking a math test. Make sure you attend class regularly. Schedule a study period after your math class. For difficult topics, review the video tapes before going to class. Verbalize (silently) problems the instructor writes on the board. Solve the problem or silently verbalize each solution step. Make note cards to remind yourself how to solve various types of math problems. Get help early in the semester before you get too lost in the course. For understanding, recite back the materials you have read in the math textbook. Take notes on how to solve difficult problems. Copy all information that is written on the board. Do math homework every day. If you miss a class, ask your instructors permission to attend the same course at a different time. Remember: You are held responsible for material covered during your absence. Frequently Asked Questions Who is in charge of the Math Resource Center? Mr. Shane Bruno is the Math Resource Center Director, and Ms. Mary Moore is the staff instructor. They can be reached in office in the St. Joseph Academic/Health Building room 101A or by calling 520-5310. Is tutoring available by appointment? No. Tutoring is done on a first-come-first-serve basis. Are other services offered besides tutoring? Yes. The student can borrow text books, calculators, and solution manuals. Also, there are computer software and video tapes available to use. What should I bring with me to the Math Resource Center? If the student is interested in getting tutoring help, it is important to come in with questions ready, your text book, and any notes from your class. Remember, Math Tutors are prohibited in giving any assistance on work that receives credit. However, they can assist with any text book problems or examples presented by an instructor during class. The tutor's role is to provide assistance NOT to teach the student an entire topic. How busy does it get? It can get quite busy during the day at the Math Resource Center - especially just prior to a quiz or test. If the Math Resource Center is crowded, please be patient. Who are the tutors? They are XU students with specialized areas of math knowledge. Employment Opportunities The Math Resource Center employs qualified students as tutors on a part-time, hourly basis. Mastery of topics through those covered up to Calculus I is required for consideration. Successful completion of a small exam will also be required. Stop by the Math Resource Center and fill out an application or speak to the Math Resource Center Director, if you are interested in becoming a tutor.
Course leads to foundation level GCSE with a maximum grade C. It is modular with examinations in November, March and June (two papers for each of the three modules, calc and non-calc). Course includes arithmetic, algebra, geometry and statistics. Effective use of a simple scientific calculator is taught. Advice is given on an appropriate text book and calculator during the first lesson. Scientific calculator required (£6 – £15). Exam fees £36 payable on enrolment.
College Algebra - 4th edition Summary a...show morend summaries. ...show less Systems of Linear Equations in Two Variables. Systems of Linear Equations in Three Variables. Nonlinear Systems of Equations. Partial Fractions. Inequalities and Systems of Inequalities in Two Variables. Linear Programming. 6. Matrices and Determinants. Solving Linear Systems Using Matrices. Operations with Matrices. Multiplication of Matrices. Inverses of Matrices. Solution of Linear Systems in Two Variables. Using Determinants. Solution of Linear Systems in Three Variables Using Determinants.
Event Calendar Matlab constitutes a powerful computational environment that allows scientists and engineers to work comfortably on tasks linked to computation, simulation and the numerical analysis of a sheer unlimited number of problems. Once some fundamental concepts on how Matlab works are understood, it unfolds its strength, and expertise is acquired in the cause of time while constantly using it. This course intends to provide such an introduction to Matlab to interested doctoral candidates and early stage researchers from a generic point of view. Besides an outline of the general abilities, we discuss a fine selection of more specific functions people might wish to work with.
This is an instruction system designed for students who have successfully completed a first year of Algebra. The instruction extends all topics of Algebra while emphasizing the function concept. Topics include: graphing on the xy-plane, the use of rational number exponents, absolute values, exponential functions, and logarithm functions. Every objective is thoroughly explained and developed. Numerous examples illustrate concepts and procedures. Students are encouraged to work through partial examples. Each unit ends with an exercise specifically designed to evaluate the extent to which the objectives have been learned. The student is always informed of any skills that were not mastered. The instruction depends only upon reasonable reading skills and conscientious study habits. With those skills and attitudes, the student is assured a successful math learning experience.
Free Science & Mathematics Software Learn about the foundations of mathematics and science with these free software downloads for Microsoft Windows. Scientifical topics range from biology and physics to geometry and statistics. For mathematics, see also: Free Calculators.
(6th Edition),(a) The student does group work (labs and practice exams) throughout the course, involving both written and oral communication. (b) The student uses technology - graphing calculators and DERIVE and Excel in the computer labs - to solve problems and to be able to communicate solutions and explore options.(a) The student develops an appreciation for the intellectual honesty of mathematical reasoning.(a) The student develops an appreciation of the history of linear programming and calculus and the role played by mathematics in business problems. (b) The student learns to use the language of mathematics - symbolic notation - correctly and appropriately. Specific Course Goals: 1. Students will participate in a formal assessment of their algebra skills and do appropriate work to improve their skill level to what this course requires. 2. Students will learn mathematical concepts that apply to business (as determined by the business school). 3. Students will learn how to apply mathematics to various types of business-related problems. 4. Students will improve their problem solving skills. 5. Students will learn to use technology, specifically graphing calculators and computer software, to solve a variety of problems. 6. Students will improve their mathematical reasoning skills. 7. Students will improve their ability to communicate, primarily in writing, mathematical ideas. Attendance: Very few students seem able to learn mathematical material independently, and it is therefore important to attend class and participate actively in these class meetings. I do not use attendance in a formal way as part of the grading process, but part of the commitment to success in the course is regular attendance. I include here a detailed schedule so that if you do have to miss a class you can keep up with the material, but it's not the same – you simply miss out on a key part of the learning process if you miss a class meeting We will also make some use of the Excel spreadsheet program for the linear programming topic. Homework: The homework assignments (by the way, the listed assignments are intended to be done following the material presented on the given day – not to be due that day) are in fact the KEY to learning the material and therefore to success in the course. I cannot overstate this – you cannot learn the material unless you practice. Here's an analogy: if you paid someone to give you piano lessons you would expect to have to practice, and you know that without practice your skills will never develop. Learning mathematics is just like that – Mathematics is NOT a spectator sport! I already know how to solve these problems, but the purpose of the course is not to convince you of that, but rather to put you in a situation in which you can, IF YOU PRACTICE ENOUGH, and if you are adequately prepared, succeed in learning the content and being able to do the problems. The problems I have listed in the daily assignments should give you a basic idea of the types of problems you will be expected to solve. If you can convince yourself that you understand the particular topic well enough to do all the problems listed, you might not have to actually work them, but you should at least work enough to test yourself. The single biggest mistake a math student makes is to look at a problem and say, "I think I can do this," without actually trying it. These are the students who say, "It looks easy when you do them in class, but on the exam I 'blanked'." At each exam you will be expected to show me that you can work problems that are very much like the ones you are supposed to have practiced. I know your program requires a "C" in a support course like this, and I can tell you that while the vast majority of students do earn the credit, there are always a few who don't. You simply will not be successful unless you work at it! My teaching style includes discussion and question-answering. You will only have questions to ask if you have put in the time trying to work the problems. Only then will you specifically know what you don't yet understand. 8 (EIGHT) hours per week to study mathematics. Quizzes: It is extremely important to stay on top of the class work; learning mathematics is something like learning to play the piano – you simply have to practice. To help insure that you do this I will frequently have a little 5-point quiz, based on the homework assignment from the previous class. This also amounts to a way of taking attendance and checking to see that you are doing the homework. I am trying to create a system that places value on class attendance and homework assignments that exam – it's not worth it to cheat, and it is also unethical. The grades will then be assigned on the scale: A = 90%, B = 80%, C = 70%, D = 60%.Disability Statement: If I reserve the right to make adjustments to the schedule and the syllabus in general as we move through the course. This is a new edition of the text and it may turn out that some changes may be necessary.
Computer-based math classes are better for students: Students are disadvantaged with book-based homework Written by: Jeff Sheppard Issue date: 10/16/08 ASU is currently teaching college algebra and other math classes in two ways. Some of the classes are using a disc made by Hawkes Learning Systems while other classes are still using a book to teach students. Students using computers have a distinct advantage over the students using the book version of these classes. So if you have a choice, it is my opinion that you choose the class that will utilize the computer. Books are not as functional as the computer versions of any of the classes. These books have a limited amount of review questions and only give the students feedback by giving the answers to the odd questions. Teachers have a difficult time assigning homework because of the fact that they have to review the many pages of problems from so many students. This adds to the problem of students not receiving enough feedback for their work, putting students at a disadvantage because they don't know what they are doing wrong in some situations. The Hawkes Learning System has come up with a very nice program for students and teachers both. This system gives an infinite amount of practice problems and instant feedback for students to learn from their mistakes. Because of the certify feature available on the system, teachers can ensure that students are not only doing their homework but also learning the material. Students must complete their homework problems with a certain amount of accuracy before taking an exam. Some students find this to be a hassle because without certification, the computer will not allow them to take the test. However, it is this feature that ensures students will understand the problems. Through the implementation of an infinite amount of practice problems and interactive step-by-step tutorials, students gain an understanding of each type of problem. Students can look at practice problems that are identical to the problems they will be solving in the certification portion of each section. The step-by-step tutorial feature shows students every part of a problem and labels the formulas necessary for solving each problem. I believe that ASU should implement the Hawkes Learning System for every possible math class. Instead of using books for some sections and computers for other sections, the school should use the computer system for all classes in one area if there are classes offered using the discs. Students using the books are at a disadvantage compared to those students using the computer system, so enroll in courses that use the computer whenever possible.
Power Tools Tags: If you were an avid television watcher in the 1980s, you may remember a clever show called "Moonlighting." Known for its snappy dialogue and the romantic chemistry between its co-stars, it featured Cybill Shepherd and Bruce Willis as a couple of wisecracking private detectives named Maddie Hayes and David Addison. While investigating one particularly tough case, David asks a coroner's assistant for his best guess about possible suspects. "Beats me," says the assistant. "But you know what I don't understand?" To which David replies, "Logarithms?" Then, reacting to Maddie's look: "What? You understood those?" (Click image to play clip.) That pretty well sums up how many people feel about logarithms. Their peculiar name is just part of their image problem. Most folks never use them again after high school, at least not consciously, and are oblivious to the logarithms hiding behind the scenes of their daily lives. The same is true of many of the other functions discussed in algebra II and pre-calculus. Power functions, exponential functions — what was the point of all that? My goal in this week's column is to help you appreciate the function of all those functions, even if you never have occasion to press their buttons on your calculator. A mathematician needs functions for the same reason that a builder needs hammers and drills. Tools transform things. So do functions. In fact, mathematicians often refer to them as "transformations" because of this. But instead of wood or steel, functions pound away on numbers and shapes and, sometimes, even on other functions. To show you what I mean, let's plot the graph of the equation You may remember how this sort of activity goes: you draw a picture of the xy plane with the x-axis running horizontally and the y-axis vertically. Then for each x you compute the corresponding y and plot them together as a single point in the xy plane. For example, when x is 1, the equation says y equals 4 minus 1 squared, which is 4 minus 1, or 3. So (x,y) = (1, 3) is a point on the graph. After calculating and plotting a few more points, the following picture emerges. The droopy shape of the curve is due to the action of mathematical pliers. In the equation for y, the function that transforms x into x2 behaves a lot like the common tool for bending and pulling things. When it's applied to every point on a piece of the x-axis (which you could visualize as a straight piece of wire), the pliers bend and elongate that piece into the downward-curving arch shown above. And what role does the 4 play in the equation y = 4 – x2? It acts like a nail for hanging a picture on a wall. It lifts the bent wire arch up by 4 units. Since it raises all points by the same amount, it's known as a "constant function." This example illustrates the dual nature of functions. On the one hand, they're tools: the x2 bends the piece of the x-axis and the 4 lifts it. On the other hand, they're building blocks: the 4 and the –x2 can be regarded as component parts of a more complicated function, 4 – x2, just as wires, batteries and transistors are component parts of a radio. Once you start to look at things this way, you'll notice functions everywhere. The arching curve above — technically known as a "parabola"— is the signature of the squaring function x2 operating behind the scenes. Look for it when you're taking a sip from a water fountain or watching a basketball arc toward the hoop. And if you ever have a few minutes to spare on a layover in Detroit's International Airport, be sure to stop by the Delta terminal to enjoy the world's most breathtaking parabolas at play: Parabolas and constants are associated with a wider class of functions — "power functions" of the form xn, in which a variable x is raised to a fixed power n. For a parabola, n = 2; for a constant, n = 0. Changing the value of n yields other handy tools. For example, raising x to the first power (n = 1) gives a function that works like a ramp, a steady incline of growth or decay. It's called a "linear function" because its xy graph is a line. If you leave a bucket out in a steady rain, the water collecting at the bottom rises linearly in time. Another useful tool is the inverse square function 1/x2, corresponding to the case n = –2. It's good for describing how waves and forces attenuate as they spread out in three dimensions — for instance, how a sound softens as it moves away from its source. Power functions like these are the building blocks that scientists and engineers use to describe growth and decay in their mildest forms. But when you need mathematical dynamite, it's time to unpack the exponential functions. They describe all sorts of explosive growth, from nuclear chain reactions to the proliferation of bacteria in a Petri dish. The most familiar example is the function 10x, in which 10 is raised to the power x. Make sure not to confuse this with the earlier power functions. Here the exponent (the power x) is a variable, and the base (the number 10) is a constant — whereas in a power function like x2, it's the other way around. This switch makes a huge difference. Exponential growth is almost unimaginably rapid. That's why it's so hard to fold a piece of paper in half more than 7 or 8 times. Each folding approximately doubles the thickness of the wad, causing it to grow exponentially. Meanwhile, the wad's length shrinks in half every time, and thus decreases exponentially fast. For a standard sheet of notebook paper, after 7 folds the wad becomes thicker than it is long, so it can't be folded again. It's not a matter of the folder's strength; for a sheet to be considered legitimately folded n times, the resulting wad is required to have 2n layers in a straight line, and this can't happen if the wad is thicker than it is long. The challenge was thought to be impossible until Britney Gallivan, then a junior in high school, solved it in 2002. She began by deriving a formula that predicted the maximum number of times, n, that paper of a given thickness T and length L could be folded in one direction. Notice the forbidding presence of the exponential function 2n in two places — once to account for the doubling of the wad's thickness at each fold, and another time to account for the halving of its length. Using her formula, Britney concluded that she would need to use a special roll of toilet paper nearly three quarters of a mile long. In January 2002, she went to a shopping mall in her hometown of Pomona, Calif., and unrolled the paper. Seven hours later, and with the help of her parents, she smashed the world record by folding the paper in half 12 times! In theory, exponential growth is also supposed to grace your bank account. If your money grows at an annual interest rate of r, after one year it will be worth (1 + r) times your original deposit; after two years, (1 + r) squared; and after x years, (1 + r)x times your initial deposit. Thus the miracle of compounding that we so often hear about is caused by exponential growth in action. Which brings us back to logarithms. We need them because it's always useful to have tools that can undo one another. Just as every office worker needs both a stapler and a staple remover, every mathematician needs exponential functions and logarithms. They're "inverses." This means that if you type a number x into your calculator, and then punch the 10x button followed by the log x button, you'll get back to the number you started with. Logarithms are compressors. They're ideal for taking numbers that vary over a wide range and squeezing them together so they become more manageable. For instance, 100 and 100 million differ a million-fold, a gulf that most of us find incomprehensible. But their logarithms differ only fourfold (they are 2 and 8, because 100 = 102 and 100 million = 108). In conversation, we all use a crude version of logarithmic shorthand when we refer to any salary between $100,000 and $999,999 as being "six figures." That "six" is roughly the logarithm of these salaries, which in fact span the range from 5 to 6. As impressive as all these functions may be, a mathematician's toolbox can only do so much — which is why I still haven't assembled my Ikea bookcases. NOTES: 1. The excerpt from "Moonlighting" is from the episode "In God We Strongly Suspect." It originally aired on Feb. 11, 1986, during the show's second season. 2. Will Hoffman and Derek Paul Boyle have filmed an intriguing video of the parabolas all around us in the everyday world (along with their exponential cousins, curves called "catenaries," so-named for the shape of hanging chains). Full disclosure: the filmmakers say this video was inspired by a story I told on an episode of RadioLab. 3. For simplicity, I've referred to expressions like x2 as functions, through to be more precise I should speak of "the function that maps x into x2." I hope this sort of abbreviation won't cause confusion, since we've all seen it on calculator buttons. 4. For the story of Britney Gallivan's adventures in paper folding, see: Gallivan, B. C. "How to Fold Paper in Half Twelve Times: An 'Impossible Challenge' Solved and Explained." Pomona, CA: Historical Society of Pomona Valley, 2002. For a journalist's account, aimed at children, see Ivars Peterson, "Champion paper-folder," Muse (July/August 2004), p. 33. The Mythbusters have also attempted to replicate Britney's experiment on their television show. 5. For evidence that our innate number sense is logarithmic, see: Stanislas Dehaene, Véronique Izard, Elizabeth Spelke, and Pierre Pica, "Log or linear? Distinct intuitions of the number scale in Western and Amazonian indigene cultures," Science, Vol. 320 (2008), p. 1217. Popular accounts of this study are available at ScienceDaily and in this episode of RadioLab. Thanks to David Field, Paul Ginsparg, Jon Kleinberg, Andy Ruina and Carole Schiffman for their comments and suggestions; Diane Hopkins, Cindy Klauss and Brian Madsen for their help in finding and obtaining the "Moonlighting" clip; and Margaret Nelson, for preparing the illustration. Editors' note: Two corrections have been made to an earlier version of this column — on the formula for folding a sheet n times; and, in the compound-interest passage, on the wording of the formula for how much one's money will grow after x years. Search This Blog Follow This Blog Steven Strogatz is the Schurman Professor of applied mathematics at Cornell University. Among his honors are MIT's highest teaching prize, membership in the American Academy of Arts and Sciences, and a lifetime achievement award for communication of math to the general public, awarded by the four major American mathematical societies. A frequent guest on National Public Radio's "Radiolab," he is the author, most recently, of "The Joy of x," which grew out of his previous Opinionator series "The Elements of Math." He lives with his wife and two daughters in Ithaca, N.Y. Follow him on Twitter @stevenstrogatz
Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry The "Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry, 7/e" by Baratto/Bergman is designed for a one-semester basic ...Show synopsisThe "Streeter-Hutchison Series in Mathematics: Basic Mathematical Skills with Geometry, 7/e" by Baratto/Bergman is designed for a one-semester basic math course. This successful worktext series is appropriate for lecture, learning center, laboratory, or self-paced courses. "Basic Mathematical Skills with Geometry" continues with it's hallmark approach of encouraging the learning mathematics by focusing its coverage on mastering math through practice. The "Streeter-Hutchison" series worktexts seek to provide carefully detailed explanations and accessible pedagogy to introduce basic mathematical skills and put the content in context. With repeated exposure and consistent structure of Streeter's hallmark three-pronged approach to the introduction of basic mathematical skills, students are able to advance quickly in grasping the concepts of the mathematical skill at hand
book is designed to help bridge the gap between GCSE and AS Level Maths. It's full of clear notes and helpful practice to recap the most difficult topics from GCSE Maths that students need when going on to study AS Level Maths. Everything you need to know for all the exam boards is explained clearly and simply, in CGP's chatty straightforward style
This homeschool teacher's guide accompanies the Saxon Math 2 StudentWorkbooks. Scripted lessons are included for each chapter, with dialogue, chalkboard sketches and more. Reduced student pages are also included for easy tracking and communication between student and parent. Lesson preparation lets teachers know the materials they'll need and any beforehand preparation. Answers are lightly overlaid on the reduced student pages. 735 pages, softcover,spiral-bound Math 2, Home Study Teacher's Edition Review 1 for Math 2, Home Study Teacher's Edition Overall Rating: 4out of5 Date:April 26, 2011 momof4 Gender:female Quality: 5out of5 Value: 4out of5 Meets Expectations: 4out of5 I used Saxon I and liked it a lot so I decided to continue my kids math with Saxon II.
Basic Algebra II -- 22M:002:331 Fall 2005 This course covers the material usually found in a second-year high school algebra course. Topics include equations and inequalities, functions and graphs, exponential and logarithmic functions, and systems of linear equations. Students who do not have an adequate high school background may need to take this course before going into higher-level mathematics or even other courses in other departments. The course, together with 22M:005 (Trigonometry) provides the background necessary to enter college level calculus sequences. Requirements included weekly quizzes, two midterm exams, calculator projects, and a final exam.
Authors Why a liberal arts mathematics book with a quantitative literacy focus? How do you engage students with the study of math? Crauder, Evans, Johnson, and Noell have found the answer: Help them become intelligent consumers of the quantitative data to which they are exposed every day—in the news, on TV, and on the Internet. In an age of record credit card debt, opinion polls, and questionable statistics, too few students have mastered the basic mathematical concepts required to think about and evaluate data. Quantitative Literacy: Thinking Between the Lines develops the idea of rates of change as a key concept in helping students make good personal, financial, and political decisions. The goal of Quantitative Literacy is a more informed generation of college students who think critically about the data provided to them, the images shown to them, the facts presented to them, and the offers made to them. Quantitative Literacy shows students the mathematics that matters to them: their bank account, their medical tests, their daily news feed. It also develops their mathematical thinking, helping them to understand the difference between truthful and misleading mathematical reporting.
IDEA MATH provides in-depth enrichment in important mathematical areas, particularly in the fields from which contests problems are drawn: algebra, combinatorics, geometry, and number theory. The program provides eight major series for students with different mathematical backgrounds. Each course runs for 2 or 3 full sessions, each session is comprised of 6 class meetings, and each class meeting lasts 3 hours. All classes meet on Saturdays during the school year. We have tried our best to avoid SAT test dates, major competition (MATHCOUNTS, HMMT, etc.)dates and school vacations. [See courses schedule] Please note: () All placement requests will require approval from the IDEA Math academic staff. (*) It is highly recommended that students enroll in all sessions of a "linked" course, so that they can gain a rounded understanding of the subjects studied.
Mathematics Qualifications Introduction Mathematics is a living subject with new processes, techniques and theories constantly being devised, tested and explored. The extensive use of computers in a wide range of academic areas has led to an increasing demand for statistical and mathematical analysis in many new fields. This means that mathematicians and statisticians are being asked to develop new tools and techniques to deal with problems in areas from business management to biology. New insights are also being opened up in the more traditional areas of physical science and engineering. All this activity leads to new applications of mathematics and statistics, as well as new theoretical work on the structure of the mathematics involved. Mathematics does not just consist of formulae, it consists of ideas. To fully appreciate mathematics you must pass from the bare formulae to the ideas that lie behind them. Mathematical thought is one of the truly great human achievements. Mathematics has been around for some 4,000 years and has flourished in many countries. It has created an impressive order or structure which has a lasting quality. The study of these ideas, both past and present, contributes a great deal to your education and will enable you to gain a deeper understanding of how to work through arguments and solve problems logically. Mathematics provides skills in independent thinking and problem solving, which are of use in many fields of employment and in Engineering, Commerce, and other Science subjects. Recommended background Entry into most 100-level Mathematics courses is open to all students with entry to the University. The Department of Mathematics and Statistics offers a choice of courses designed to cater for students with a range of backgrounds and interests. Detailed entry recommendations are available on the department website (follow the link for Prospective Students). Students who have performed very well in NCEA Level 3 mathematics with statistics and/or mathematics with calculus may be eligible for direct entry into a 200-level Mathematics course. UC also offers Science Headstart summer preparatory courses in January/February for students who have not studied mathematics or statistics for some time or who lack confidence in their skills. 100-level courses The core of the 100-level (first-year) programme consists of linear algebra and calculus, found in the two courses MATH 102 and MATH 103. MATH 103 follows on from MATH 102 and has MATH 102 as a prerequisite. Together, these courses will let you into almost any 200-level Mathematics course. If you want to major in Mathematics, you should include MATH 102 and MATH 103 in your first year. In addition, anyone wanting to do a significant amount of Mathematics in their degree should take both these courses. MATH 102 is also required or recommended for people intending to major in any of several subjects, including Economics, Statistics, Physics and Management Science. Anyone planning to do Engineering will require the Engineering Mathematics courses EMTH 118 and EMTH 119. Students who have not passed a substantial amount of Year 13 mathematics, or its equivalent, are strongly advised to enrol in MATH 101 before advancing to MATH 102. MATH 120 can be taken alone or credited with any other 100-level core Mathematics course. MATH 170 is intended for students who want to progress in applied mathematics. It is recommended that students who enrol in MATH 170 either have already been credited with, or are concurrently enrolled in, MATH 103. MATH 130 is a course on logic and explores formal and informal reasoning, aspects of symbolic logic and patterns of inference, and is valuable in any undergraduate degree. 200-level and beyond We offer a wide variety of courses at 200 and 300-level. These include courses in discrete mathematics, linear algebra, calculus, differential equations, mathematical modelling and statistics. If you are majoring in Mathematics, you need 45 points from selected MATH 200-level courses and at least 60 points from MATH 302–394. In exceptional situations students can get direct entry from Year 13 into the 200-level courses. If you are unsure which papers best suit your needs, contact one of the Department's course advisors. It is good to include other subjects at 200-level. Popular choices include Statistics, Physics, Chemistry, Computer Science, Management and Economics. Further study Following a BSc or a BA you can proceed to a BSc(Hons), BA(Hons), MSc, MA, PGDipSc or PhD. If you achieve well in Mathematics, seriously consider aiming for a BSc(Hons) or BA(Hons) degree in Mathematics. This involves one year's study after your BSc or BA. To do this degree you need to do an extra two courses from MATH 310–399 or STAT 310–399, and to get a B+ average in your 300-level courses. Career opportunities Perhaps the most important quality that a Mathematics graduate develops is the ability to reason logically and in depth. Vocational courses provide expertise which seems to have an immediate usefulness, but technological change is rapid and what is learnt one year may be superseded within a decade. On the other hand, the habits of thought promoted by a study of mathematics are of permanent value. Many Mathematics graduates move into teaching and significant numbers are absorbed by computing, finance, commerce, insurance and scientific establishments, such as the Crown Research Institutes. Employment opportunities are particularly good for people who combine qualifications in Mathematics with qualifications in other disciplines such as the physical sciences, Statistics, Computer Science, Engineering, Management and Economics.
Student must have access to a high-speed (DSL or cable) Internet connection at least 4 days weekly, including a computer with sound and Java applet support, Microsoft Excel or Google Spreadsheet, a media player that can play Windows Media Player (WMA) and RealPlayer (RM) files, and a scanner or digital camera/phone. Access to a Graphing Calculator and experience using it is essential. Description: Description: This one semester Pre-Calculus/Functions course prepares students for eventual work in Calculus. The central focus of this course is functions: · linear, · exponential and logarithmic, · polynomial and rational, · discrete and continuous, · inverses, graphs, and applications. The course will include other topics from advanced mathematics such as analytic geometry and three-dimensional geometry. Students will develop skills in applying the concepts by solving real-world problems. Graphing calculators are used frequently in each lesson to familiarize students with the basics of graphing calculator use, to demonstrate concepts, to facilitate problem solving, and to verify results of problems solved algebraically. SAT practice topics and problems provide a review of the prerequisite courses.
Many students continue to struggle in high school math courses because they failed to master the basic mathematical skills. REA's new Ready, Set, Go! Workbook series takes the confusion out of math, helping students raise their grades and score higher on important exams. What makes REA's workbooks different? For starters, students will actually like using them. Here's why: Math is explained in simple language, in an easy-to-follow style The workbooks allow students to learn at their own pace and master the subject 15 lessons break down the material into the basics Each lesson is fully devoted to a key math concept and includes many step-by-step examples Paced instruction with drills and quizzes reinforces learning The innovative "Math Flash" feature offers helpful tips and strategies in each lesson—including advice on common mistakes to avoid Skill scorecard measures the student's progress and success Every answer to every question, in every test, is explained in full detail A final exam is included so students can test what they've learned When students apply the skills they've mastered in our workbooks, they can do better in class, raise their grades, and score higher on the all-important end-of-course, graduation, and exit exams. Some of the math topics covered in the Ready, Set, Go!Trigonometry Workbook include: Trigonometric Ratios Graphing Points and Angles Special Angles Inverse Trigonometric Values Solving Triangles Solving Four-Sided Figures Solving Trigonometric Equations and more! Whether used in a classroom, for home or self study, or with a tutor, this workbook gets students ready for important math tests and exams, set to take on new challenges, and helps them go forward in their studies! About the Author About the Author Author Mel Friedman is a former classroom teacher and test-item writer for Educational Testing Service and ACT, Inc
Nature of Equations Video - A video explaining the nature of equations in algebra.A video explaining the nature of equations in algebra. Math Games Multiplication - Memorizing multiplication tables is an essential part of elementary education.Memorizing multiplication tables is an essential part of elementary education. A student who has mastered multiplication gains a solid foundation for achievement in... MathAid College Algebra - Java based math course includes problem-solving lessons and testsInteractive College Algebra course designed to ensure engaging, self-paced, and self-controlled e-learning process and help students to excel in their classes. Java- and web-based... Math Games Level 1 - Undoubtedly, mathematics is one of the most important subjects taught in school.Undoubtedly, mathematics is one of the most important subjects taught in school. It is thus unfortunate that some students lack elementary mathematical skills. An... Math Homework Maker Match at Super SharewareMath Homework Help - We provide a complete array of math problemsWe provide a complete array of math problemsHaving... Science Helper For Ms Word - Easily add 1200 Scientific Graphs & Charts to your word docs. Science helper for MS Word® is a fantastic program for middle, high school or college level math, chemistry, physics or engineering teachers, students or researcher. This easy to use... StudyMinder4 - StudyMinder is the essential homework management tool.StudyMinder is the essential homework management tool. It combines the best features of a student planner and homework organizer into one, easy to use program. The latest version adds a new
Harder's Algebraic Geometry 1 is a beautiful example of explaining why an abstract subject makes sense. The book has a conversational style without wasting words, and focuses on providing intuition for the subject.
An essential element of advanced studies in mathematics, topology tends to receive a highly formal and abstract treatment, discouraging students from grasping even the simpler ideas or getting any real "feel" for the subject. This volume, on the other hand, offers students a bridge from the... Expand familiar concepts of geometry to the formalized study of topology. It begins by exploring simple transformations of familiar figures in ordinary Euclidean space and develops the idea of congruence classes. By gradually expanding the number of "permitted" transformations, these classes increase and their relationships to topological properties develop in an intuitive manner. Imaginative introductions to selected topological subjects complete the intuitive approach, and students then advance to a more conventional presentation. An invaluable initiation into the formal study of topology for prospective and first-year mathematics students.Collapse
Back in high school, when I first started taking some of the more advanced math classes, I would have loved to have had the sort of tools and resources that the Internet now offers. Instead, I had to invest in an expensive scientific graphic calculator – the famous HP 48G. It took me a full summer of washing dishes to save up for that baby, but it got me through both high school and college. Today, if you're an engineering or a science student and you want a useful and fast math computer program that can solve complicated math or algebraic equations in just seconds, you have a whole plethora of options on the Internet. One of the best free applications that I discovered for this purpose is called Deadline. Need to calculate derivatives, integrals or the roots of an equation? Not a problem. Want to save the resulting graph to a report or a document filled with your homework solutions? It's easy, just run Deadline and you can generate, manipulate and save any graph and the resulting mathematical solutions. Solve Equations & Math Problems When you first start the Deadline Wizard, you'll have an opportunity to type in your equation, enter all of the equation parameters, and also type in the x-axis interval (the range of values that will be used in the formula to generate the graph of results). When you click OK, you'll instantly see a plot of your entire graph. In the top right field, you can modify whatever parameters you used in the equation on the first screen. This is useful because you can graphically see how changing each parameter changes your entire set of results. In this example, increasing "m" shifts the entire graph up. You can also click on the roots (where the formula solution is zero), and you'll see those roots represented in the graph as large green dots. You can also change the resolution of the graph by reducing or increasing the number of "points" used in the generation of the graph. Obviously, 1000 points will create a nice and smooth graph, but you can decrease it down to only 10 points to see how this affects results. Working With Mathematical Solutions For Your Formulas The nice thing about Deadline is that it doesn't just graph results. You can actively change the formula or parameters as part of the graphic process. When you click Calculate -> Evaluate from the menu, you'll see the three derivative results (f, f' and f") for each value of x that you'd like to test. If you're familiar with derivatives, you can select Calculate -> Derive to view the differential equations based on your original formula. This may not be useful for basic math problems, but for students working on advanced calculus or engineering problems, this can be extremely useful. Even more useful to engineers and scientists is the ability to calculate and view the integrals for your equation. The integral is basically the amount of area between your graph of results and the x axis of the chart. Knowing this total area for a certain range is very valuable information, and this graphic software offers that answer with just a few keystrokes. Under the "Calculate" menu item, you'll also see the "find Extrema" tool, which gives you the peaks (extrema) of the graph in the range that you've displayed. When you're finished tweaking the chart with the parameters that you want, you can save the current graph to an image file (PNG format). This is obviously very useful for inserting the graph results in reports or homework solutions that you may be working on. In this format, you could also insert the graphs directly into a website or online posting. DeadLine also comes with a built in "calculator" for those complicated math problems that have no variables. This is great for complex math problems that you just can't do in your head. Do you have 20 apples that you sell for 10 dollars each and want to divide among 4 friends and their 2 siblings? Just kick open the calculator and type "(2010)/(42)" and you're done. No matter how complicated the problem, the calculator will always pull through. As you type in each equation, the equation and resulting solution gets logged in the top window. I find the Deadline graphic and math calculator to be one of the most useful and functional math computer programs that I've discovered so far. I would love to hear what you think of it, and whether you know of any other similar, or more useful and fast math computer programs that get the job done just as wellI remember solving parabolic and hyperbolic equations by hand back in engineering classes. Wish I had a tool like Deadline back then. Textbooks and dry professors were all we had for problem solving :)
MATH 108: Quantitative Reasoning Develops conceptual understanding and computational skills in unit analysis, uses of percentages, and dealing with quantities and their magnitudes. Includes formulas of finance for simple interest, compound interest and loan payments; functions and their graphs; linear equations; exponential growth and decay; principles of counting; fundamentals of probability; and estimation and approximation techniques to judge the reasonableness of answers. Also includes representing and analyzing data using statistical tools such as hitograms; measures of central tendency; variance and standard deviation; linear regression and scatter plots; normal distributions; and margin of error and confidence intervals. Each Semester. IAI M1 904.... more » Credits:3 Overall Rating:4.5 Stars N/A Thanks, enjoy the course! Come back and let us know how you like it by writing a review.
A lot of information, explained clearly, covers all units Disadvantages: Could have more colour ...When I was studying ALevel Maths it required me to have a lot of textbooks available for studying and most of the time to prepare for exams with. For my exams in Mathematics, I was given the textbook Edexcel AS and A Level Modular Mathematics by Keith Pledger, however if you have to purchase it, they are around as expensive as most textbooks are at £15. I was quite glad that it combined both... ... and bullet point used to outline key points. The book consists of five sections, in total 71 chapters. The aim of using so many chapters is to
The High School Assessments student's knowledge of Core Learning Goals at the indicator level. Some indicators have assessment limits which indicate more specifically what will be assessed. Assessment items and other instructional resources at the indicator level can be viewed in the CLG Toolkit. Assessment limits: The algebraic expression is a polynomial in one variable. The polynomial is not simplified. 1.1.4 The student will describe the graph of a non-linear function and discuss its appearance in terms of the basic concepts of maxima and minima, zeros (roots), rate of change, domain and range, and continuity. Assessment limits: A coordinate graph will be given with easily read coordinates. "Zeros" refers to the x-intercepts of a graph, "roots" refers to the solution of an equation in the form p(x) = 0. Problems will not involve a real-world context. Expectation 1.2 The student will model and interpret real-world situations using the language of mathematics and appropriate technology. Indicators 1.2.1 The student will determine the equation for a line, solve linear equations, and/or describe the solutions using numbers, symbols, and/or graphs. Assessment limits: Functions are to have no more than two variables with rational coefficients. Linear equations will be given in the form: Ax + By = C, Ax + By + C = 0, or y = mx + b. Vertical lines are included. The majority of these items should be in real-world context. 1.2.2 The student will solve linear inequalities and describe the solutions using numbers, symbols, and/or graphs. Assessment limits: Inequalities will have no more than two variables with rational coefficients.
Little Algebra Book The idea was to turn 'traditional' maths textbooks on their heads. Out with the dull, heavy-as-a-brick, purely functional books that have barely changed style since your parents were at school; in with something small, beautiful and focussed on helping you understand the single, most important rule of algebra: whatever you do to one side, you have to do to the other. Colin Beveridge kindly sent me a copy of his preprint book to review and comment on. Being a mathematician, I will just set out my comments as plus points and minus points, but they do not carry equal weight! Then, I'll make some final comments. Positive points The design makes clever use of the fact that it is a physical book, and the pages become something that the student can interact with instead of just turning to the next one. The central page-join becomes an analogy for the equals sign, with left-hand and right-hand pages acting as the respective sides of an algebraic equation. Pages also have fold-out and fold-over elements which help create a sense of mystery – attractive not just to younger readers but also to adults as well! This was a particularly neat approach to showing division. One book, one concept, with a simple almost minimal approach, certainly helps focus on the idea of balancing equations. The four basic arithmetic operations are each represented by their own individual images, which is useful for reinforcing the idea of different operations being carried out on each of the example equations – the bird for the subtraction is especially cute There is no attempt made to explain the order of steps for the combination of operations on the final pages, and this could be revisited with higher-level students to see if changing the order of steps makes a difference to the final result – for example, could you divide by 3 first? Negative points I'm not sure that stating "The aim of algebra is to get x on its own" on the opening page is either correct or helpful – mathematically. To me, the aim of algebra is to provide a means of notating an abstract problem, so that you can explore possible solutions algorithmically. Of course, you need something short and snappy, but x is not always the only thing you need to solve for later on in algebra! Maybe more discussion is needed to find a more appropriate statement, but I'm not going to enter into it here. The same thing applies to "In an equation…", where I would substitute "In a simple algebra equation…" Referring to numbers as "pure numbers" is also potentially confusing, I feel. Evaluation If I am being picky, I would change the precision of some of the language used, in future reprints – there is no place for "artistic licence" with terminology in mathematics! Overall, I think this book does achieve what it set out to do with a minimum of effort and an efficiency of approach which is to be commended. I can see students wanting to keep their copy, once they have learned the algebra, just because of its overall attractiveness. Anything which helps students enjoy learning algebra should be welcomed with open arms! Share this: Like this: One Response to Little Algebra Book Too many folks have already thrown away plenty of good money on nothing but useless salt tablets being shipped from South America. It's old news that tracking food intake could lead to losing a few pounds [2]. The institution has persistently offered ideal programs and services for those struggling to achieve certain levels of body weight.
Mu Alpha Theta Fosters Interest in Mathematics November 16, 1998 SIAM vice president for education Terry Herdman, who gave a talk on problems and curriculums in applied mathematics at the recent Mu Alpha Theta annual convention in Chicago, believes that SIAM members should be aware of the goals and accomplishments of this educational organization. Mu Alpha Theta, a national honor society for mathematics students attending high schools and junior colleges, was founded at the University of Oklahoma in Norman, in 1957, by Richard V. Andree (who chaired the committee of the National Council of Teachers of Mathematics that originally conceived the idea) and his wife. The goals of the society are (1) to engender keen interest in mathematics, (2) to develop sound scholarship in the subject, and (3) to promote enjoyment of mathematics among high school and junior college students. Mu Alpha Theta currently has approximately 50,000 student members in 1350 teacher-sponsored local chapters or groups; holds national, regional, and local meetings at which both students and mathematicians present mathematical material; and supports a variety of projects and contests. It also publishes The Mathematical Log, a quarterly journal that features understandable, audience-appropriate articles on mathematics as well as news about meetings, chapters, and members. Herdman believes that SIAM members could be important role models for students as sponsors and supporters of local high school chapters. In addition, members of SIAM student chapters could benefit from interaction with local chapters of Mu Alpha Theta: SIAM student members might profit from learning how to mentor and guide younger students, or they might hone their presentation skills of mathematical topics in front of a younger student audience. Members interested in learning more about Mu Alpha Theta can contact: Mu Alpha Theta, 601 Elm Avenue, Room 423, Norman, OK 73019-0315; (405) 325-4489; matheta@ou.edu;
Tuesday, 27 March 2012 Product Details: Paperback: 312 pages Publisher: Springer; Softcover reprint of hardcover 1st ed. 2007 edition (October 18, 2010) Language: English ISBN-10: 1441922326 ISBN-13: 978-1441922328 Product Dimensions: 9 x 6 x 0.7 inches Shipping Weight: 1 pounds (View shipping rates and policies) This is an undergraduate textbook on the basic aspects of personal savings and investing with a balanced mix of mathematical rigor and economic intuition lynda hardware tutorial with torrent download. It uses routine financial calculations as the motivation and basis for tools of elementary real analysis rather than taking the latter as given lynda hardware tutorial with torrent download. Proofs using induction, recurrence relations and proofs by contradiction are covered lynda hardware tutorial with torrent download. Inequalities such as the Arithmetic-Geometric Mean Inequality and the Cauchy-Schwarz Inequality are used lynda hardware tutorial with torrent download. Basic topics in probability and statistics are presented. The student is introduced to elements of saving and investing that are of life-long practical use. These include savings and checking accounts, certificates of deposit, student loans, credit cards, mortgages, buying and selling bonds, and buying and selling stocks. The book is self contained and accessible. The authors follow a systematic pattern for each chapter including a variety of examples and exercises ensuring that the student deals with realities, rather than theoretical idealizations. It is suitable for courses in mathematics, investing, banking, financial engineering, and related topics. Tags: An Introduction to the Mathematics of Money Saving and Investing (9781441922328) David Lovelock, Marilou Mendel, Arthur L. Wright , tutorials, pdf, ebook, torrent, downloads, rapidshare, filesonic, hotfile, megaupload, fileserve
Algebra And Trigonometry With Analytic Geometry - 13th edition Summary: Clear explanations, an uncluttered and appealing layout, and examples and exercises featuring a variety of real-life applications have made this book popular among students year after year. This latest edition of Swokowski and Cole's ALGEBRA AND TRIGONOMETRY WITH ANALYTIC GEOMETRY retains these features. The problems have been consistently praised for being at just the right level for precalculus students. The book also provides calculator examples, including specific keystrokes that...show more show how to use various graphing calculators to solve problems more quickly. Perhaps most important--this book effectively prepares readers for further courses in mathematics
Word Processing Business Math Courses Business Math Course Introduction The Business Math course is designed to introduce students to helpful tools that will enable them to perform quick, easy, and accurate basic math functions such as addition, subtraction, multiplication, and division. They will also learn to round numbers, calculate percentages, fractions, and decimals, as well as calculate interest, payments on loans, net present values, and returns on investments. Finally, students will compare leasing and purchasing options, look at break-even points, explore statistical terminology, and calculate and display various statistics. Course Prerequisite(s) None Course Aim To assist students in performing basic business mathematics calculations. Of Interest to Business people who wish to either learn or refresh their knowledge of basic business mathematics.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education. Chapter 2 Linear and Nonlinear WavesOur exploration of the vast mathematical continent that is partial differential equations will begin with simple first order equations. In applications, first order partial differential equations are most commonly used Chapter 5 Numerical Methods: Finite DifferencesAs you know, the differential equations that can be solved by an explicit analytic formula are few and far between. Consequently, the development of accurate numerical approximation schemes is essential for Chapter 11 Numerical Methods: Finite ElementsIn Chapter 5, we introduced the first, the oldest, and in many ways the simplest class of numerical algorithms for approximating the solutions to partial differential equations: finite differences. In the pres Chapter 10 A General Framework for Linear Partial Differential EquationsBefore pressing on to the higher dimensional forms of the heat, wave, and Laplace/ Poisson equations, it is worth taking some time to develop a general, abstract, linear algebraic fr Chapter 12 Partial Differential Equations in SpaceAt last we have reached the ultimate rung of the dimensional ladder (at least for those of us living in a three-dimensional universe): partial differential equations in physical space. As in the one- and Chapter 4 Separation of VariablesThere are three paradigmatic linear second order partial differential equations that have collectively driven the development of the entire subject. The first two we have already encountered: The wave equation describes v Chapter 1 What are Partial Differential Equations?Let us begin by specifying our object of study. A differential equation is an equation that relates the derivatives of a (scalar) function depending on one or more variables. For example, d4 u du + u2 = c Geostatic Stresses These self-weight stresses (ay, a'y, ah' a'h) are called geostatic stresses For a level surface there are no shear forces induced by the geostatic stresses, and therefore they are also principal stresses: a, =ay and a3 = ah Karl Terzag S1 Stresses Changes Due to Surface Loads (Aerv) o Stresses within a soil mass will change as a result of surface loads. The change in total stress spreads and diminishes with distance from the load. Equations and charts are available to calculate both th 84Circular Surface Load:o The stress increase at a point (A) on the axis beneath the center of a circular loaded area is "easily" determined by integrating the Boussinesq point solution over the loaded area. (same assumptions as before)dav6=30z 5 2 S11~~trlp and Square FootingsStrips footings have fixed width and infinite (or relatively long) length and are often used for smaller buildings (1-2 story) and walls. Square footings are a special case of the rectangular footing.BqzInfluence charts 813 Rectangular Surface Loads: (Boussinesq solution - the limn" methodo This method determines the vertical stress ~ay at a point P under the corner of a rectangular loaded area using footing dimensions normalized to the depth z: m = Biz n=Uz (note that S16 Embankment Load: a Another important "shape" is the embankment (e.g. highways, dams, etc.) If the embankment material is soil then the surface load (pressure) at full height is ~ q '1 H.ba=Chart on next page provides influence factors due to 1/2 p2Immediate Settlement (Pi)-=-TJ].sn~ettlementis g,y,e.to,J;otatjonal strainJ<;fLstortion) within tile soil - not a chan e in volume. (Note that if no shear strain, say a blanket loaa, then no Imme late settleiii8nt.)-QL PirOriginal DistortedImmedi p5 'Imme<llatE! Settlement Case III:~oaaing-on the-Surface of a Stiff baye Underlain oy a Less Rigl<'FEayer of Great Ihicknes.sFairly common for upper crust due to desiccation etc..HS.qE III "Table P3 gives the C"s values under the center of flex p6~_Approximate Solutions: Using various combinations of Tables P1, P2 and P3 can get "ballpark" estimate of immediate settlement for many cases not covered. Always start withclosest initial approximation, then "correct" as necessary. Examples:1.Rigi p8SupplementThese are the ,?omplete interpolation cales for the Immediate Settlement Case II example on (p. p4). Since the boundary characteristics aren't defined we need to consider both cases (i.e., and u = 0). For each case, interpolate the C's for t p27 SUMMARY OF SETTLEMENT CALCULATIONS Define Initial Stresses: a Total Stress, Pore Water Pressure, Effective Stress a Must define the state of stress prior to loading a The behavior of soils is governed by effective stress (thank you Karl Terzaghi) Def p29 ~Time Rate of Consolidation:During the consolidation process: What %Pc will have occurred at a given time? How much time is required for given %Pc to occur?TimepOne-Dimensional Consolidation: Theory presented by Karl Terzaghi in 1925 ENR.Assump op31 Using the relationship between T and U we can answer the initial questions about %Pc: 1. What %Pc will have occurred at a given time?0%Le. know t, cV, H, N => what is U?S"lv<'(.-lit -_ c., t . J-LH /rJ)u1100%+ _LT_-=:=:=L.U = % Consolidat p32o Symmetry in Time Factor Table:\-, Why do Cases 1a, 3 and 4 work for both double and single drainage, but Cases 1band 2 do not? The answer is symmetry. Think about what happens to the excess pore water pressure. If the initial excess pore pressure d p34With Z we can use a dimensionless plot of isochrones as shown below for a uniform initial pore water pressure distribution (from Perloff & Baron, 1976). Note that each isochrone at a given time t represents a time factor T and an average degree of con
This geometry rapid learning course is designed for students who have taken basic algebra (Algebra 1 or Elementary Algebra). This is often the second high school math course. This comprehensive course includes points, lines, triangles, circles quadrilaterals and coordinate geometry, with the focus on concept understanding and problem solving. The visual tutorials with expert narration provide an easier entry to the visual world of geometry. With our breakthrough 24x Rapid Learning SystemTM of smart teaching and rich media, you can now finally gain a powerful learning edge over others who are still struggling with static textbooks and online freebies. Catch up and excel in class with the host of tightly integrated learning modules, designed specifically for today's web and video savvy students and supported by a team of teaching experts. Speed up your learning one chapter one hour at a time. The entire 24-chapter rapid learning package includes: 24x Problem Drills (Interactive Games) Practice what you learn. These drills offer feedback-based quizzes, concept and word problems, summary reviews on all problems and scoring system to track your performance, with a complete solution guide at the end. 24x Review Sheets (PDF Printables) Condense what you learn. Each chapter has one-pager cheat sheet for key concepts with at-a-glance review of each chapter, printable and laminatible. It is ideal for exam prep or quick review. 24x Chapter eBooks (PDF Printables) Need something for easy skimming? The well-formatted eBooks are printable for easy quick reading and last-minute review. This optional module is available only on Certified, Platinum and Premium Editions. 24x Lecture Audiobooks (Audio MP3) Want to learn on-the-go? These mp3 audiobooks are for learning anywhere anytime and a great companion to the visual tutorials. Listen to the same content you have just learned visually and enhance your learning with your smartphone or mp3 player. This auditory module is available only on Certified and Platinum Editions.
This problem-based course presents classical topics of elementary number theory and how they pertain to teaching the elementary and Junior High School mathematics. Topics include prime numbers, GCF, LCM, division algorithm, Euclidean algorithm and the extended Euclidean algorithm. Several applications, including cryptography, will be presented using middle grade materials. The course prepares the teacher for using the CryptoClub materials with middle grade students. Required materials: The Cryptoclub: Using Mathematics to Make and Break Secret Codes by Janet Beissinger and Vera Pless. Workbook for The Cryptoclub: Using Mathematics to Make and Break Secret Codes by Janet Beissinger and Vera Pless. You may download this for free from the A.K. Peters website.
Drexel Hill SATCombinatorics studies the way in which discrete structures can be combined or arranged. Graph theory deals with the study of graphs and networks and involves terms such as edges and vertices. This is often considered a very specific branch of combinatorics.
Combining a concrete perspective with an exploration-based approach, Exploratory Galois Theory develops Galois theory at an entirely undergraduate level. The text grounds the presentation in the concept of algebraic numbers with complex approximations and assumes of its readers only a first course in abstract algebra. more... Galois theory is a mature mathematical subject of particular beauty. While Artin's book pioneered an approach to Galois theory that relies heavily on linear algebra, this book's author takes the linear algebra emphasis even further. With a chapter on transcendental extensions, it is suitable for undergraduate and beginning graduate math majors. more... Explore the foundations and modern applications of Galois theory Galois theory is widely regarded as one of the most elegant areas of mathematics. A Classical Introduction to Galois Theory develops the topic from a historical perspective, with an emphasis on the solvability of polynomials by radicals. The book provides a gradual transition from... more... This Lecture Notes volume is the fruit of two research-level summer schools jointly organized by the GTEM node at Lille University and the team of Galatasaray University (Istanbul): 'Geometry and Arithmetic of Moduli Spaces of Coverings (2008)' and 'Geometry and Arithmetic around Galois Theory (2009)'. The volume focuses... more... Praise for the First Edition ". . .will certainly fascinate anyone interested in abstract algebra: a remarkable book!" —Monatshefte fur Mathematik Galois theory is one of the most established topics in mathematics, with historical roots that led to the development of many central concepts in modern algebra, including groups and fields.... more...
In most mathematics textbooks, the most exciting part of mathematics--the process of invention and discovery--is completely hidden from the reader. The aim of Knots and Surfaces is to change all that. By means of a series of carefully selected tasks, this book leads readers to discover some real mathematics. There are no formulas to memorize; no procedures to follow. The book is a guide: its job is to start you in the right direction and to bring you back if you stray too far. Discovery is left to you. Suitable for a one-semester course at the beginning undergraduate level, there are no prerequisites for understanding the text. Any college student interested in discovering the beauty of mathematics will enjoy a course taught from this book. The book has also been used successfully with nonscience students who want to fulfill a science requirement. Undergraduate students, graduate students, and research mathematicians interested in an introduction to modern areas of mathematics. Reviews "The book is perfectly suited to a course for non-science majors in need of fulfilling a math requirement. All the sections have worked well at sparking student interest and convincing them that math is much more interesting than mere number-crunching and graphing." -- Professor William Bloch, Wheaton College "Would serve well as the basis of an independent study course in which the student would work through the tasks in a journal subject to periodic review by the instructor ... the writing is clear and engaging, and the tasks should be effective at setting a reasonable pace."
Unit specification Aims The programme unit aims to introduce quotient structures and their connection with homomorphisms in the context of rings and then again in the context of groups; present further important examples of groups and rings and develop some introduced in the context of rings, then used to construct roots of polynomials in extension fields. Factorisation in polynomial rings and rings of integers of number fields will also be studied in the first part. The second part will begin by developing further properties of key examples, such as permutation groups, and will emphasise actions of groups. Then the construction of quotient objects and the connection with hom1. Definitions and examples (partly review): domains, fields and division rings; nilpotent and idempotent elements, products of rings; (many) examples; with students gaining familiarity with the ideas and examples through attempting exercises. [3] 2. Isomorphisms and homomorphisms (of rings): what is preserved and reflected; kernel of a homomorphism, ideals; principal ideals, operations on ideals. [3] 4. Polynomial rings and unique factorisation: unique factorisation domains, principal ideal domains and euclidean domains, with emphasis on rings of polynomials and rings of integers in number fields; construction of ring of fractions of a domain; tests for irreducibility, Gauss' Lemma. [3]
Introduction to Geometric Algebra Geometric Algebra (GA) is a powerful mathematical language for expressing physical ideas. It unifies many diverse mathematical formalisms and aids physical intuition. In our various publications and lecturesyou will find many examples of the insights that geometric algebra brings to problems in physics and engineering. The links on the right are to our main educational resources. The paper "Imaginary Numbers are not Real" provides a brief, but readable introduction to geometric algebra. The most complete introduction to the subject is contained in the book "Geometric Algebra for Physicists" (CUP 2003). There are also links to the Part III Lecture Course on geometric algebra, given to final year physics undergraduates in Cambridge University. Two versions are provided - the first year's course, and the current (simplified) course.
Numerical Methods in Engineering with Python is a text for engineering students and a reference for practicing engineers. The numerous examples and applications were chosen for their relevance to real world problems, and where numerical solutions are most efficient. The Python code is available on the book web site.
MA135 - College Algebra Course Description: MA135 College Algebra:Prerequisite: MA 125, or a high school or transfer course equivalent to MA 125, or an ACT math score = 23, or an SAT math score = 510, or a COMPASS score = 66 in the Algebra placement domain, or a COMPASS score 0-45 in the College Algebra placement domain. A study of the algebra necessary for calculus. Topics include: Linear and non-linear equations, inequalities and their applications; inverse, exponential and logarithmic functions; complex numbers; systems of linear and non-linear equations; matrices and determinants. 3:0:3 (From catalog 2012-2013)
Advanced algebra students should complete both this "Algebra by Chapter Series" and the "Advanced Math Series" which would then be the equivalent of 2 years of Algebra (Algebra 1 and Algebra 2) including a complete course of Trigonometry. Learn algebra as each topic is introduced and explained with sample algebra problems followed by practice sets of algebra problems with the algebra solutions offered to the student step-by-step. Algebra 1 students will learn this material for the first time and algebra 2 students will review this very important material which is necessary for more advanced algebra studies. Set Theory is abstract, and it is possible to miss the subtlety and the beauty of certain simple ideas if you do not roll your sleeves up and work with them. In these environments, you may use the Show command to show Sets, Relations, the results of operations on sets, the results of algebraic operations (such as composition and inversion) on relations or functions, the images and preimages of relations between sets, the effects of permutations (as invertible functions) and so on, interactively. The aims are to learn the basic concepts of sets, operations over sets, how suitable sets and operations for higher-level structures such as algebras and to explore some of the basic connections between sets and logic. Sets have subsets, which are themselves sets comprising some of the elements of the original set. While this is an important tool in algebraic geometry, the author has elected to write from the point of view of commutativealgebra in order to avoid being tied to special cases from geometry. At the crossroads of representation theory, algebraic geometry and finite group theory, this book blends together many of the main concerns of modern algebra, synthesising the past 25 years of research, with full proofs of some of the most remarkable achievements in the area. Requiring only an abstract algebra course as a prerequisite, it will introduce students of mathematics to the structure and finite-dimensional representation theory of the complex classical groups and will serve as a reference for researchers in mathematics, statistics, physics and chemistry whose work involves symmetry groups, representation theory, invariant theory and algebraic group theory. This algebra text is in the form of a fantasy novel where the characters solve practical problems by using algebra. This book of engaging flline masters provides activities for algebra students to use with the graphing calculators and graphing software-technology which is rapidly becoming commonplace in the high school math classroom. Graphic Algebra was designed to be used in a variety of ways to supplement and complement the teaching of algebra. A model comprising a basic set, a set of operations whose operands and results must be members of the basic set, and a set of axioms about properties of operations. If there is a one-to-one correspondence between elements of the algebras and the operations defined in them such that the corresponding operations with corresponding operands have corresponding resluts, then the algebras are said to be isomorphic. They focus on textbooks that vary on coverage of the idea sets and on instructional Categories II and V. (d) When the examination copies of the textbooks arrive, the group browses all of the data sets for each of the books they have decided to review. The early part of the adoption process has produced a set of basic requirements for the books: strong development of algebra content, support for teachers, adaptability for diverse students, alignment with national standards, strong assessment, and a high level of student engagement. She sets up a meeting with the mathematics teachers to explore possible supplementary units that might be used, including units from some of the textbooks that were rated highly by AAAS.
Cambridge Students An essential subject for all learners, Cambridge IGCSE Mathematics is a fully examined course which encourages the development of mathematical knowledge as a key life skill, and as a basis for more advanced study. The syllabus aims to build learners' confidence by helping them develop a feel for numbers, patterns and relationships, and places a strong emphasis on solving problems and presenting and interpreting results. Learners also gain an understanding of how to communicate and reason using mathematical concepts. Display resources by: IGCSE Mathematics Revision Guide IGCSE Revision Guides contain everything you need to master the content of the Cambridge IGCSE exams. Written in a lively and easy-to-use format with an international approach, they provide a wealth of exam practice for all students. Author: Rayner, D. and Williams, P. ISBN: 9780199154876 Published in 2009. Published by Oxford University Press, UK More information on IGCSE Mathematics Revision Guide [New window] IGCSE Revision Guide for Mathematics Written for students preparing for IGCSE Mathematics (Extended syllabus) up to grade A. Author: Jeskins, J, et al ISBN: 0340815787 Published in 2004. Published by Hodder Education, UK [New window]
Fr... For courses in secondary or middle school math. This text focuses on all the complex aspects of teaching mathematics in today's classroom and the most current NCTM standards. It demonstrates how to creatively incorporate the standards into teaching along with inquiry-based instructional strateg...
Dojo Toolkit Graphing Calculator Project Readme Author: Jason Hays Trac ID: jason_hays22 Contents: Expressions Variables, Functions, and uninitializing variables toFrac in GraphPro Numbers and bases Graphing equations Substitutes for hard to type characters Making Functions Decimal points, commas, and semicolons in different languages Important mathematical functions ---------------Expressions---------------- The calculator has the ability to simplify a valid expression. With Augmented Mathematical Syntax, users are allowed to use nonstandard operators in their expressions. Those operators include ^, !, and radical. ^ is used for exponentiation. It is a binary operator, which means it needs a number on both the left and right side (like multiplication and division) 2^5 is an example of valid use of ^, and represents two to the power of five. ! is used for factorial. It supports numbers that are not whole numbers through the use of the gamma function. It uses the number on its left side. Both 2! and 2.6! are examples of valid input in America. (2,6! is valid in some nations) radicals can be used for either square root or various other roots. to use it as a square root sign, there should only be a number on its right. If you put a number on the left as well, then it will use that number as the root. To evaluate an expression, type in a valid expression, such as 2(10+5), into the input box. If you are using GraphPro, then it is the smaller text box. After you have chosen an expression, press Enter on either your keyboard or on the lower right of the calculator. If it did not evaluate, make sure you correctly closed your parentheses. In the Standard calculator, the answer will appear in the input box, in GraphPro, the answer will appear in the larger text box above. On the keyboard, you can navigate through your previous inputs with the up and down arrow keys. If you enter an operator when the textbox is empty or highlighted (like ) then Ans* should appear. That means the answer you got before will be multiplied by whatever you input next. So try Ans3. Whenever you start the calculator, Ans is set to zero. --------------Variables, Functions, and uninitializing variables---------------- A variable is basically something that stores a value. If you saw Ans in the previous example, you've also seen a variable. If you want to store your own number somewhere, you'll need to use the = operator. Valid variable and function names include cannot start with numbers, do not include spaces, but can start with the alphabet (a-z or A-Z) and can have numbers within the names "var1" is a valid name Input "myVar = 2" into the textbox and press "Enter." You've just saved a variable. Now if you ever type myVar into an expression, 2 will appear (unless you change it to something else). Variables are best used to store Ans. Ans is overridden whenever you evaluate an expression, so it is good to store the value of Ans somewhere else before it is overridden. If you want a variable (like myVar) to become empty, or undefined, you just need to set it equal to undefined. Now try "myVar = undefined" Now myVar is no longer defined. Functions are very useful for finding answers and gathering data. You can use functions by inputting their name and their arguments. For this example, I'll be using the functions named "sqrt" and "pow" sqrt is a function with one argument. That argument has a name too, its name is 'x.' x is a very common name amongst built in functions So, let's run a function. Input "sqrt" then input a left parenthesis (all arguments of a function go within parentheses). Now type a value for x, like 2. Now close the parentheses with the right parenthesis. If you used 2, you should have "sqrt(2)" in the text box. If you press enter, you should get the square root of 2 back from the calculator. Now for "pow" it has two arguments 'x' and 'y' Type in "pow(" and pick a value for 'x' (I am picking 2 again) but now, you need to separate the value you gave x with a list separator. Depending on your location, it is either a comma or a semicolon. I'm in America, and I use commas. by this point, I have "pow(2," Now we need a value for 'y' (I'm using 3). Put a ')' and now I have 'pow(2,3)' Press Enter, and, following my example, you should get 8 In this calculator, there are several ways to input arguments. You've already seen the first way, just input numbers in a specific order based on the names. The second way is with an arbitrary order, and storage. With 'pow' I can input "pow(y=3, x=2)" and get the exact answer as before. x and y will retain their assigned values, so you will need to set them to undefined it you want to try the next way. The third way is to let the calculator ask you for the values. Input "pow()" If the values have been assigned globally, then it will use those values, but otherwise, it will ask for values of x and y. They will not be stored globally this way. I'll go ahead and mention that because of the way the calculator parses, underscores should not be used to name a variable like _#_ (where # is an integer of any length) ---------------toFrac()---------------- toFrac is a function that takes one parameter, x, and converts it to a fraction for you. It is only in GraphPro, not the Standard mode. It will try to simplify pi, square roots, and rational numbers where the denominator is less than a set bound (100 right now). Immediately after the calculator starts, toFrac may seen slow, but it just needs to finish loading when the calculator starts. After that, it will respond without delay. For an example, input "toFrac(.5)" or (,5 for some). It will return "1/2" For a more complicated example, input "toFrac(atan(1))" to get back "pi/4" (atan is also known as "arc tangent" or "inverse tangent") --------------Numbers and bases---------------- This calculator supports multiple bases, and not only that, but non integer versions of multiple bases. What is a base? Well, the numbers you know and love are base 10. That means that you count to all of the numbers up until 10 before you move on to add to the tenths place. So, what about base 2? All of the numbers up until 2 are 0 and 1. If you want to type a base 2 number into the calculator, simply input "0b" (meaning base 2) followed by some number of 1's and 0's. 0b101 is 5 in base 10 Hexadecimal is 0x, and octal is 0o, but i won't go into too much detail on those here. If you want an arbitrary integer base, type the number in the correct base, insert '#' and put the radix on the end. ".1#3" is the same as 1/3 in base 10 Because there is not yet cause for it, you cannot have a base that is not a whole number. --------------Graphing Equations---------------- First thing is first, in GraphPro only, the "Graph" button in the top left corner opens the Graph Window So, now you should see a single text box adjacent to "y=" Type the right side of the equation using 'x' as the independent variable. "sin(x)" for example. To Graph it, make sure the checkbox to the left of the equation is selected, and press the Draw Selected button. You can change the color in the color tab. By default, it is black. Under window options, you can change the window size and x/y boundaries Let's add a second function. Go to the Add Function button, select the mode you want, and press Create. Another input box will appear. If you selected x= as the Mode, then y is the independent variable for the line (an example is "x=sin(y)"). If you want to erase, check the checkboxes you want to erase, and press "Erase Selected" And similarly, Delete Selected will delete the chosen functions "Close" will terminate the Graph Window completely ---------------Substitutes for hard to type characters--------------- Some characters are not simple to add in for keyboard users, so there are substitutes that are much easier to add into the text box. pi or PI can be used in place of the special character for it. For epsilon, eps or E can be used. radical has replacement functions. sqrt(x) or pow(x,y) can be used instead. --------------Making Functions----------------- My favorite part. Before we start, I'll mention that Augmented Mathematical Syntax is allowed in the Function Generator (yay). Ok, now the bad news: to prevent some security issues, keywords new and delete are forbidden. Sorry, it is a math calculator, not a game container; not that that would be so bad, but it is to keep it from being used for some evil purposes. Ok, onto function making. Most JavaScript arithmetic is supported here, but, some syntax was overlapping mathematical syntax, so ++Variable no longer increases the contents of Variable because of ++1, but Variable++ does increase it (same deal with --) Strings have incredibly limited support. Objects have near zero support So, let's make a Function: Press the "Func" button. A Function Window should pop up. Enter a name into the "functionName" box. (it must follow the name guidelines in the variables section) I'm putting myFunc Enter the variables you want into the arguments box (I'll put "x,y" so I have two arguments x and y) Now enter the giant text box. Type "return " and then the expression you want to give to the calculator. I'm putting "return x2 + y/2" Then press Save. Now your function should appear in the functionName list and you can call it in the Calculator. If you want to Delete a function you made, select it in the function Name list, and then press Delete. If you altered a previously saved function (and haven't saved over the old one) you can reset the text back to its original state with the Reset button. Clear will empty out all of the text boxes in the Function Window Close terminates the Function Window ---------------Decimal points, commas, and semicolons in different languages---------------- In America, 3.5 is three and one half. Comma is used to separate function parameters and list members. In some nations, 3,5 is three and one half. In lists, ;'s are used to separate its members. So, when you evaluate expressions, 3,5 will be valid, but in the function generator, some ambiguous texts prevent me from allowing the conversion of that format to JavaScript. So I cannot parse it in the Function Generator. And here is my example: var i = 3,5; b = 2; I cannot discern whether the semicolon between i and b are list separators or the JavaScript character for end the line. b could be intended as a global variable, and i is a local variable, but I don't know that. So language conversion isn't supported in Function Making. --------------Important mathematical Functions--------------- Here is a list of functions you may find useful and their variable arguments: sqrt(x) returns the square root of x x is in radians for all trig functions sin(x) returns the sine of x asin(x) returns the arc sine of x cos(x) returns the cosine of x acos(x) returns the arc cosine of x tan(x) returns the tangent of x atan(x) returns the arc tangent of x atan2(y, x) returns the arc tangent of y and x Round(x) returns the rounded integer form of x Int(x) Cuts off the decimal digits of x Ceil(x) If x has decimal digits, get the next highest integer ln(x) return the natural log of x log(x) return log base 10 of x pow(x, y) return x to the power of y permutations(n, r) get the permutations for n choose r P(n, r) see permutations combinations(n, r) get the combinations for n choose r C(n, r) see combinations toRadix(number, baseOut) convert a number to a different base (baseOut) toBin(number) convert number to a binary number toOct(number) convert number to an octal number toHex(number) convert number to a hexadecimal number
Search Course Communities: Course Communities Lesson 30: Exponential Functions Course Topic(s): Developmental Math \| Exponentials Beginning with a formal definition of an exponential function, the lesson then compares the graphs of increasing and decreasing exponential functions. A comparison between exponential and power functions follows, which leads to methods for determining the (p) value in the power function (h(x) = kx^p) and the value of the base( b) in the exponential function (f(x) = ab^x). A procedure for solving exponential equations is presented before a population application problem is solved. The lesson concludes with a discussion about using graphs to find approximate solutions to exponential equations.
further. In particular, the place-value numeration system used for arithmetic implicitly incorporates some of the basic concepts of algebra, and the algorithms of arithmetic rely heavily on the "laws of algebra." Nevertheless, for many students, learning algebra is an entirely different experience from learning arithmetic, and they find the transition difficult. The difficulties associated with the transition from the activities typically associated with school arithmetic to those typically associated with school algebra have been extensively studied.1 In this chapter, we review in some detail the research that examines these difficulties and describe new lines of research and development on ways that concepts and symbol use in elementary school mathematics can be made to support the development of algebraic reasoning. These recent efforts have been prompted in part by the difficulties exposed by prior research and in part by widespread dissatisfaction with student learning of mathematics in secondary school and beyond. The efforts attempt to avoid the difficulties many students now experience and to lay the foundation for a deeper set of mathematical experiences in secondary school. Before reviewing the research, we first describe and illustrate the main activities of school algebra. Previous chapters have shown how the five strands of conceptual understanding, procedural fluency, strategic competence, adaptive reasoning, and productive disposition are interwoven in achieving mathematical proficiency with number and its operations. These components of proficiency are equally important and similarly entwined in successful approaches to school algebra. The Main Activities of Algebra What is school algebra? Various authors have given different definitions, including, with "tongue in cheek, the study of the 24th letter of the alphabet [x]."2 To understand more fully the connections between elementary school mathematics and algebra, it is useful to distinguish two aspects of algebra that underlie all others: (a) algebra as a systematic way of expressing generality and abstraction, including algebra as generalized arithmetic; and (b) algebra as syntactically guided transformations of symbols.3 These two main aspects of algebra have led to various activities in school algebra, including representational activities, transformational (rule-based) activities, and generalizing and justifying activities.4 The representational activities of algebra involve translating verbal information into symbolic expressions and equations that often, but not always, involve functions. Typical examples include generating (a) equations that represent quantitative problem situations in which one or more of the quan-
Mastering the methods is more important in the long run than simply being able to do the problems sometimes. It's great if you can solve a problem in multiple ways, but most of them don't work in all cases. They might work for that one specific problem, but they might also ONLY work for that one specific problem. It's best to master the method that works in all, or the most, cases. When it's mastered, you're guaranteed to be able to solve a problem of that given form. If you practice instead a bunch of solutions that only work in a few different cases and don't learn to apply the general solution in many ways, you might eventually end up with a problem you can't solve. tl;dr The goal is to be able to solve any problem of a given type using a general solution that works with all problems of that given type, not to be able to solve specific problems of a given type with solutions that work only for specific problems of that given type.
Middle school > Course Description math Mathematics in grades 6-8 is a sequential, college preparatory program. It emphasizes the development of math concepts, computational skills, problem solving, and critical thinking. Comprehensive and appropriately challenging, this curriculum is designed to provide students with the math background necessary for subsequent math coursework. Math 6 (Grade 6) is a continuation of the Progress in Mathematics program used in the Lower School. The continuity of the program helps to ease the Middle School transition and allows the students to expand their mathematical ability. Concepts including numeration, operations, computation, algebra, functions, geometry, measurement, and probability are still presented in a variety of formats to develop higher level critical thinking. Many skills directly foreshadow pre-algebra. Math 7 (Grade 7) integrates applied arithmetic, algebra, and geometry, and connects all these areas to measurement, probability, and statistics. This course provides opportunities for students to visualize and demonstrate concepts with a focus on real-world applications. A strong algebraic influence is included to prepare students for Pre-Algebra. The text for this course is Mathematics: Course 2 by Prentice-Hall. Pre-Algebra (Grades 7-8) reviews the basic computation of real numbers while integrating skills requiring higher levels of thinking. The use of variables throughout prepares for expanded operations required in Algebra I. Algebra-thinking activity labs provide students with opportunities to dig deeper and explore algebraic concepts to build conceptual understanding. This course, normally taught to eighth graders, is also offered to seventh graders who have demonstrated above average quantitative aptitude and skill. The text for this course is Pre-Algebra by Prentice-Hall. Algebra ICP (Grades 8-9) is offered to all students who have completed Pre-Algebra. It extends the concept of set theory to include algebraic expressions, algebraic fractions, factoring, and the solution of linear and quadratic equations and inequalities. The interpretation and solution of verbal problems is incorporated within each skill area. Students are encouraged to develop precise and accurate habits of mathematical expression. The text for this course is Beginning Algebra with Applications. Algebra I Honors (Grades 8-9) is offered primarily to 8th grade students who completed Pre-Algebra in the 7th grade. This is an advanced course; therefore, the pace and rigor of this class will be significantly more challenging than Algebra I CP. Students will study linear, quadratic, absolute value, radical, and rational equations and inequalities, the graphing of linear and quadratic equations and inequalities, solving systems of equations and inequalities, multiplying and factoring polynomials, and simplifying exponential, radical, and rational expressions. Throughout the year, students will work extensively with word problems to develop their critical thinking skills. The text for this course is Algebra.
INTEGRAL tool for mastering ADVANCED CALCULUS Interested in going further in calculus but don't where to begin? No problem! With Advanced Calculus Demystified, there's no limit to how much you will learn. Beginning with an overview of functions of multiple variables and their graphs, this book covers the fundamentals, without spending too much time on rigorous proofs. Then you will move through more complex topics including partial derivatives, multiple integrals, parameterizations, vectors, and gradients, so you'll be able to solve difficult problems with ease. And, you can test yourself at the end of every chapter for calculated proof that you're mastering this subject, which is the gateway to many exciting areas of mathematics, science, and engineering. This fast and easy guide offers: Numerous detailed examples to illustrate basic concepts Geometric interpretations of vector operations such as div, grad, and curl Coverage of key integration theorems including Green's, Stokes', and Gauss' Quizzes at the end of each chapter to reinforce learning A time-saving approach to performing better on an exam or at work Simple enough for a beginner, but challenging enough for a more advanced student, Advanced Calculus Demystified is one book you won't want to function without!
Precalculus prepares students for calculus the same way as pre-algebra prepares students for Algebra I. While pre-algebra teaches students many different fundamental algebra topics, precalculus does not involve calculus, but explores topics that will be applied in calculus. Some precalculus courses might differ with others in terms of content.
Mathematica As the progress of scientific discoveries are being pushed further and further with the aid of computational power, it is becoming more and more apparent that computational and programming skill are something scientists in general have to equip themselves with. Thus, starting at the implementation of Integrated Science Curriculum in 2010, Mathematica became a compulsory lesson (taught in SP2171 Discovering Science) to all SPS students. Of course, Mathematica is not the only technical computation software available in the market. There are other softwares, e.g. Matlab, LabView, Maple, etc that you have probably heard and used before. Well, you may then ask particularly, "Why Mathematica?". Well,… First of all, any NUS Faculty of Science students are entitled a FREE licensed copy of Mathematica (available for PC, Mac or Linux user) throughout their candidature. Please follow the instruction given here, to get your copy installed. Other softwares do not come FREE. Secondly, apart from being a very good technical computation software, Mathematica supports excellent animated graphics, interactive user-interface and intuitive command lines, making it an excellent visually-engaging pedagogical tool, for instance (installation of CDF Player or Mathematica required):
JKGRAPH is a Microsoft WINDOWS program for graphing and analyzing graphs of various kinds of functions that students are likely to encounter in either a pre-calculus or first year calculus course. JKGRAPH uses four distinct function groups which include standard rectangular functions in the form Y=F(X), polar functions in the form R=F(@), parametric functions in the form X=F(T) and Y=G(T), and polar parametrized functions in the form R=F(T) and @=G(T). JKGRAPH can perform integration (numerically and graphically) by several different ways: lower, midpoint, and upper Riemann sums; Trapezoid and Simpson's rule; Gaussian quadrature and Romburg. In particular, one can view the area between two graphs in any of the four function groups. This program will also calculate (numerically and graphically) arc length, extrema, volumes (by both disks and cylinders), surface area and the user can apply Newton's Method or the Bisection Method to find the zeroes. The user can also enter a Tangent/Normal/Graph trace mode in which they can move a tangent line, normal line or a point along the graph to study the variations of these quantities along the curve. At each point on the curve the tangent line equation (or normal) and the coordinates of the point of tangency (normality) are given. A tutorial file is provided. This program combines the features of the MSDOS programs POLAR, PARAM and POLPM which are contained in jkpolpm.zip and YFUNX which is contained in jkyfunx.zip into a single program which runs under Windows 3.1 and Windows95.
This book covers the content prescribed for the New Zealand Diploma in Engineering course DE4102 Engineering Mathematics. Some foundation level material is also provided to help those students whose preparation for tertiary mathematics study is patchy, whether that be due to gaps in recent secondary... Essential Maths and Stats provides a comprehensive overview of tertiary level mathematics and statistics and is the only definitive New Zealand text for mathematics and statistics at entry level. It is also an excellent 'extension' text for secondary school students. Divided into six ke...
The algebra 1 honors course will move at a faster pace than the regular algebra 1 course, covering more material and covering that material in greater depth. In this course we will be studying properties of the real number system, variables, first degree equations and inequalities, slope, systems of equations and inequalities, graphing, factoring, quadratic equations, ratio and proportion, operations with polynomials and rational expressions, the properties of exponents and radicals, and the application of algebra in science and business. Almost every day students will have either one or two assignments. Each assignment is given a number, beginning with number 1. Each day I grade students' assignments and return them to them the next class period. Assignments are given a top score of 5. Earning a 4 is satisfactory. However, a score of 3, 2, 1 or 0 is not good. To make a 5 on a homework assignment a student must do two things. He/She must try all the problems beforehand, and then correct the paper in class as we go over it together. With these simple requirements, every student should get a 5 on every paper. Students are supposed to be keeping their graded and numbered assignments in a notebook. Parents wanting to see how their son or daughter is doing on algebra assignments can find out by looking at his or her notebook. A good notebook would consist of assignments numbered sequentially with no missing papers, and 4's or 5's on most papers. Materials Needed: Each student needs to bring a pencil, paper, an algebra notebook, and a calculator with him/her to class every day. The only additional features the calculator needs is a square root key and a +/- key. A calculator with these features can be bought at most department stores for between 5 and 10 dollars. Each student is to keep his/her Class Rules, Grade Recording Sheet, and graded homework papers in his/her algebra notebook. Grading/Evaluation: Vocabulary quizzes will be given. Focus Calendar Assessments coming from the county testing targeted benchmark material will be given. Tests and quizzes will be given on algebraic material. Almost every day students will have either one or two classwork or homework assignments. Tests and quizzes on algebraic material will be 60% of a student's grade. Vocabulary quizzes, Focus Calendar Assessments, classwork and homework will be 40% of a student's grade.
The easiest way to learn is to understand a few basic principles, and how to reason from them to solve a wide variety of problems. Homework is more fun when it can be done more quickly. Test performance is greatly improved, because, knowing the basic principles, the student can deal with whatever the teacher asks
Facilities Mathematics and computer science students have access to a variety of computers, located in academic buildings and residence halls, both for course work and recreational purposes. All computers are linked through a campus-wide network, which also provides access to the Internet. The department has one server, which provides computing infrastructure for several computer science courses. Large-screen projection system for instructional purposes, as well as a scanner and printing facilities. Variety of software, including Internet Explorer, Maple, PowerPoint, Microsoft Excel, Microsoft Word, programming languages such as Java, C++ and Scheme, and a number of course-related software packages. A number of other computer laboratories on campus offer the applications that are available in Hoffberger 149, and some of these facilities are open 24 hours a day. Software includes the programming languages C++, Java, Scheme, and several more specialized products. Many classrooms and lecture halls in the academic buildings are equipped with multimedia workstations featuring a computer, VCR, and projection system. Several rooms also have three-dimensional visualizers. All rooms in the residence halls are wired to provide access to the campus network and thereby the Internet. Students are invited to bring their own computers to campus. A member of the Computing Services Team provides technical support for students using their own computers.
College Algebra : Concepts And Contexts - 11 edition Summary: This book bridges the gap between traditional and reform approaches to algebra encouraging users to see mathematics in context. It presents fewer topics in greater depth, prioritizing data analysis as a foundation for mathematical modeling, and emphasizing the verbal, numerical, graphical and symbolic representations of mathematical concepts as well as connecting mathematics to real life situations drawn from the users' majors.44.67 +$3.99 s/h Good SellBackYourBook Aurora, IL 04953878
Maple Labs &amp Demos: Fall '12 One of the most difficult aspects of any math course, and especially calculus III, is visualizing mathematical objects and accurately calculating complex mathematical expressions. Because we study functions of multiple variables in calculus III both of these can become challenging. Because of this, we use the software program Maple in the labs and on some of the homework to make it easier to see three dimensional surfaces and structures, and to do some of the calculations connected with the homework. In most of the lab periods during the semester, you will work through a short lab exercise that will give you the Maple tools that you will need to complete the Maple problems in the written homework. These lab exercises are listed below; see syllabus for dates.
Description: UKI2490: Ready Set Learn is designed to help children practice and master a variety of skills, including beginning math, penmanship, reading comprehension, and much more. These books can be used to enrich learning, reinforce skills, and provide extra practice. ... Description: GDZ1777: Ensure algebraic success with this essential introduction to abstract mathematical thinking. Topics such as negatives, exponents, radicals, and linear equations are presented in a clear, logical manner for all students to grasp. Includes a valuable CD ROM full ...
Concrete Mathematics59.602813 FREE New: BRAND NEW 0201558025. indoo.com NJ, USA $69201558025 New Condition ~~~ Right off the Shelf-BUY NOW & INCREASE IN KNOWLEDGE... Lyric Vibes IL, USA $95.11 FREE New: New BRAND NEW BOOK! Shipped within 24-48 hours. Normal delivery time is 5-12 days. AwesomeBooksUK OXON, GBR $101.51 FREE About the Book This book introduces the mathematics that supports advanced computer programming and the analysis of algorithms. The primary aim of its well-known authors is to provide a solid and relevant base of mathematical skills - the skills needed to solve complex problems, to evaluate horrendous sums, and to discover subtle patterns in data. It is an indispensable text and reference not only for computer scientists - the authors themselves rely heavily on it! - but for serious users of mathematics in virtually every discipline. Concrete Mathematics is a blending of CONtinuous and disCRETE mathematics. "More concretely," the authors explain, "it is the controlled manipulation of mathematical formulas, using a collection of techniques for solving problems." The subject matter is primarily an expansion of the Mathematical Preliminaries section in Knuth's classic Art of Computer Programming, but the style of presentation is more leisurely, and individual topics are covered more deeply. Several new topics have been added, and the most significant ideas have been traced to their historical roots. The book includes more than 500 exercises, divided into six categories. Complete answers are provided for all exercises, except research problems, making the book particularly valuable for self-study. Major topics include: Sums Recurrences Integer functions Elementary number theory Binomial coefficients Generating functions Discrete probability Asymptotic methods This second edition includes important new material about mechanical summation. In response to the widespread use of the first edition as a reference book, the bibliography and index have also been expanded, and additional nontrivial improvements can be found on almost every page. Readers will appreciate the informal style of Concrete Mathematics. Particularly enjoyable are the marginal graffiti contributed by students who have taken courses based on this material. The authors want to convey not only the importance of the techniques presented, but some of the fun in learning and using them. 0201558025B04062001
Recom: Placement through the assessment process or MATH 075 or MATH 075SP or equivalent Course Section Days Time Room Units Start/Stop Dates MATH-080H 0922 MTWTH 9:30-10:45am MA-108 5 Units Full Term Catalog Course Description MATH-080H Elementary Algebra with Study Skills 5 Units SC Not Degree Applicable 108.00 hours Lecture Recommended: Placement through the assessment process or MATH 075 or MATH 075SP or equivalent This course is equivalent to MATH 110 and MATH 111 combined. It is an introduction to the techniques and reasoning of algebra, including linear equations and inequalities, development and use of formulas, algebraic expressions, systems of equations, graphs and introduction to quadratic equations with integrated study skills techniques and support throughout the course.
Flowcharts and Algorithms This unit from the Continuing Mathematics Project on flowcharts and Algorithms employs, three basic conventions: (i) the use of a flowchart and the appropriate symbols (ii) the use of computer statements, such as 'c = c + I1 (iii) the use of the inequality signs >, <, ≤ and ≥ Three very short programmes at the start of the unit are intended to help students if they are not familiar with the above conventions. The objectives for the unit are that students will be able: (i) to recognise in a flowchart the symbols for a START or END box, a PROCESS box, a DECISION box, and a CONNECTOR box; (ii) to follow a flowchart of elementary difficulty using these four symbols, and the convention for DECISION boxes described in the unit; (iii) to understand and use elementary statements of inequality, using the symbols >, < ≥and ≤. (iv) to substitute values for the variables in elementary computer statements such as 'c = c + 1' (v) to use algorithms, given in flowchart form, for the construction of some 'magic squares', obtaining square roots by an iterative method, and solving quadratic equations
Our users: I think this program is one of the most useful learning tools I have purchased (and believe me, I purchased a lot!), It's easy for us as parents- to work with, it saves a lot of our children's precious time. Keith Erich Johnston, KS Ive been a high-school math teacher for over sixteen years and luckily, Ive had computers aiding me in the classroom since the Apple II. But nothing has ever gotten the results or helped my students understand as many advanced equations and concepts as Algebrator has! Thats why nowadays, I make sure every student system in the school (even the notebooks) are running it! If nothing else, it makes our jobs a lot easier! Ashley Grayden, MA I like algebra but was not able to finish the homework on time. Things have changed now for me. Zoraya Christiansen Algebrator is easy to use and easy to understand and has made algebra the same for me. I am thankful that I got it. Billy Hafren, TX I started with this kind of programs as I am in an online class and there are times when "I have no clue". I am finding your program easier to follow. THANK YOU! Jeff Brooks, ID17: free math problem answers algebra solutions equations how to solve for standard form of linear equations factoring binomial calculator dividing polynomials by monomials calculator algebra with pizzazz how to do lu decomposition on ti-89 algebra tiles hardest solving polynomials kumon G14 cheats pre algebra clicking calculator using inequalities how to factor equations factor finder Inequalities Algebra Solver free algebra help free simplifying radical expressions calculator how to multiply and divide rational numbers LCD of expressions solvers examples of real story y-intercept to vertex for beginners simplify equivalent expressions learning algebra in sequence gcse level examples of QT factoring holt algebbra find the zero function for a linear equation partial fractions with nonlinear denominator Examples of Quadratic Equations algebra helper tic tac toe study guide examples algebrator word problems parabola vertex calculator free online graph holt algebra 1 book how do you work out binomial numbers synthetic division calculator online free ks3 free sats papers polynomial factoring calculator how do i write algebraic expression =100 but not anything over 100 except 999
10 Units 1000 Level Course Available in 2012 EPMATH126 Mathematical Studies prepares students for undergraduate courses requiring a basic level of mathematics. The course covers the topics of arithmetic and calculation, basic algebra, equations and descriptive statistics. The depth of treatment is similar to HSC General Mathematics course. Objectives In this course students will: 1. develop a sound knowledge and understanding of some basic skills of mathematics and statistics. 2. develop an appreciation of the concepts of mathematics and statistics as relevant, useful and transferable for those courses requiring a non-rigorous level of mathematics.
ample text 2008 carnegie learning, inc.ittsburgh, pa phone 888.851.7094 fax 412.690.2444 copyright 2008 by carnegie learning, inc. all rights reserved. carnegie learning, cognitive tutor, schoolcare, and learning by doing are all registered marks of carnegie learning, inc. all other company and product names mentioned are used for identification purposes only and may be trademarks of their respective owners. this product or portions thereof is manufactured under license from carnegie mellon university. permission is granted for photocopying rights within licensed sites only. any other usage or reproduction in any form is prohibited without the express consent of the publisher. 2008 carnegie learning, inc. isbn: 978-1-934800-98-0 fall 2008 sample text printed in the united states of americaable of contents introduction…………………………………………….p. 1 sampler table of contents • p. 1 introduction to carnegie learning tm math curricula • p. 3 mathematical representations • p. 6 what is included in a carnegie learning text set? • p. 7 collaborative classroom • p. 8 bridge to algebra …………………………………….p. 18 course description • p. 18 table of contents • p. 20 student text lesson 2.1 • p. 27 look ahead • p. 28 lesson • p. 31 teacher's implementation guide lesson 2.1 • p. 37 lesson map • p. 41 lesson with answers and teacher notes • p. 43 wrap-up • p. 48 teacher's resources and assessments (tra) • p. 51 assignment with answers lesson 2.1 • p. 53 skills practice with answers lesson 2.1 • p. 54 teacher notes on chapter 2 assessments • p. 55 pre-test chapter 2 • p.57 post test chapter 2 • p. 61 mid chapter test chapter 2 • p. 65 end of chapter test chapter 2 • p. 67 standardized practice test chapter 2 • p. 69 homework helper • p. 75 lesson 2.1 • p. 77 algebra i …………………………………………........p. 78 course description • p. 78 table of contents • p. 80 student text lesson 1.8 • p. 87 look ahead • p. 88 lesson • p. 91 teacher's implementation guide lesson 1.8 • p. 95 lesson map • p. 99 lesson with answers and teacher notes • p. 101 wrap up • p. 105 teacher's resources and assessments (tra) • p. 107 assignment with answers lesson 1.8 • p. 110 pre-test chapter 1 • p.113 post test chapter 1 • p. 119 2008 text sampler page 1mid chapter test chapter 1 • p. 125 end of chapter test chapter 1 • p. 129 standardized practice test chapter 1 • p. 135 homework helper • p. 142 lesson 1.8 • p. 143 geometry …………………………………………….p. 144 course description • p. 144 table of contents • p. 146 student text lesson 5.4 • p. 151 look ahead • p. 152 lesson • p. 155 teacher's implementation guide lesson 5.4 • p. 165 lesson map • p. 169 lesson with answers and teacher notes • p. 171 wrap-up • p. 181 teacher's resources and assessments (tra) • p. 183 assignment with answers lesson 5.4 • p. 185 pre-test chapter 5 • p.187 homework helper • p. 193 lesson 5.4 • p. 294 algebra ii …………………………………………….p. 196 course description • p. 196 table of contents • p. 198 student text lesson 1.5 • p. 204 lesson • p. 207 teacher's implementation guide lesson 1.5 • p. 216 lesson map • p. 217 lesson with answers and teacher notes • p. 219 wrap-up • p. 228 teacher's resources and assessments (tra) • p. 229 assignment with answers lesson 1.5 • p. 231 pre-test chapter 1.5 • p.235 2008 text sampler page 2ntroduction to carnegie learning tm math curricula we are excited that you are interested in exploring our unique approach to mathematics instruction. what does carnegie learning offer? x over 20 years of research into how students think and learn x blended solutions offering instructional materials in software and text x extensive use of real-world scenarios to reinforce a conceptual understanding of mathematics x instruction tailored to the individual needs of each student a research-based approach to mathematics for your student carnegie learning provides some of the only truly research-based math curricula in the country. controlled field studies have validated that carnegie learning's approach helps students improve their course grades and overall achievement. whether your student excels or struggles with mathematics, carnegie learning curricula will help strengthen their skills, content knowledge, and confidence. the curriculum uses a scientifically-researched approach incorporating the latest discoveries into how students think, learn, and apply new knowledge in mathematics. building on over 20 years of research and design at carnegie mellon university and field tests by leading mathematics educators, our approach uses students' intuitive problem solving abilities as a powerful bridge to a more formal understanding of mathematics. independent studies of carnegie learning implementations in miami-dade, fl; pittsburgh, pa; moore, ok, and kent, wa, and elsewhere demonstrate that carnegie learning curricula deliver a positive shift in standardized test scores, student attitudes toward math and problem solving, and critical thinking skills. research also indicates strong results with title i and special-needs populations, including exceptional student education students, those with limited english proficiency and students receiving free or reduced lunch. research shows that students using the carnegie learning tm algebra i curriculum: • perform 30% better on questions from the timss assessment • demonstrate 85% better performance on assessments of complex mathematical problem solving and thinking • have a 70% greater likelihood of completing subsequent geometry and algebra ii courses • achieve 15-25% better scores on the sat and iowa algebra aptitude test • experience equivalent results for both minority and nonminority students carnegie learning tm blended math curricula the carnegie learning blended solutions contain software and text components that complement one another. your students will spend about 40% of instructional time using computer-based tutorials, and 60% using a student text to collaborate with peers in the classroom. 2008 text sampler page 3the skills learned in the text are enhanced by the cognitive tutor software. the cognitive tutor software elaborates on the lessons introduced in the text. students work at their own pace in the cognitive tutor software component of the curriculum. the learning system is built on cognitive models, which represent the knowledge that a student might possess about the mathematics that they are studying. the software assesses students' prior mathematical knowledge on a step-by-step basis and presents problems tailored to their individual skill levels. using the cognitive tutor software, your student will receive the benefits of individualized instruction, ample practice, immediate feedback, and coaching. just-in-time hints, on-demand hints, and positive reinforcement will put your student in control of his or her own learning. the student text provides an opportunity for analysis, extended investigation, and the exploration of alternate solution paths. students engage in problem solving and reasoning, and communicate using multiple representations of math concepts. real-world situations are used in problems designed to emphasize conceptual understanding. the goal of the student text is to be engaging and effective so your student will have fun while learning by doing. a typical week in a carnegie learning classroom the classroom is a dynamic, adaptive environment. while no two weeks will be exactly the same, most weeks will be split between classroom activities and work in the computer lab. the number of each type of session that the teacher schedules depends on the teacher's preference and the availability of lab time. carnegie learning suggests that students spend 40% of their class time in the computer lab working with the computer and 60% with student text investigations. below is an itinerary outlining a typical mid-semester week. monday x students complete the student text investigation started on friday with group presentations. x teacher solicits questions on the completed investigation and wraps up the investigation by asking questions that lead students to reflect on the material covered. x students begin a new student text investigation. tuesday x students complete about half of the investigation started on monday. x teacher has students respond to a writing prompt to summarize their work. 2008 text sampler page 4ednesday x students work with the software in the computer lab. thursday x students complete the investigation started on tuesday. x partners present their findings of tuesday's investigation using a written format. x teacher solicits questions and comments on the completed investigation, wraps up the investigation by asking questions that lead students to apply their knowledge of the material covered. friday x students work with the software in the computer lab. carnegie learning's ongoing support teacher training to ensure that every implementation yields successful results, carnegie learning offers a comprehensive professional development program, delivered by certified implementation specialists (cis) using best practices knowledge of our curricula and years of classroom mathematics experience. our professional development programs provide teachers with the experience, insight, and support needed to grow as more reflective practitioners. our emphasis is on aligning teaching to learning using standards-based curriculum, student-centered instruction, and the integration of technology. throughout the sessions teachers learn best practices based on the latest research in the science of learning, and are provided with the opportunity to experience the student-centered classroom from the perspective of both student and teacher. the workshops emphasize our learning by doing approach, where the role of the teacher is to facilitate student interaction, communication, and problem solving. for more information on professional development, please contact your carnegie learning sales representative. carnegie learning tm resource center our online resource center provides trained teachers with access to assessments, discussion forums, technical support, and other educational materials. in addition, teachers and administrators can access software updates, installer files, and serial numbers for all of their carnegie learning products. 2008 text sampler page 5 6 the icons on this page help guide teachers in a collaborative classroom. these icons appear in the student texts and teacher's implementation guides of all carnegie learning textbooks.hat is included in a carnegie learning text set? student text the student text is a consumable textbook with room for students to take notes and work problems directly on the lesson page. each lesson contains objectives, key terms, and problems that help the students to discover and master mathematical concepts. this sample text contains selected student text pages from each of our curricula. student assignments the student assignments book contains one assignment worksheet per lesson. the student assignment book is designed to move with the student from classroom to home to computer lab, so that students can continually practice the skills taught in the lesson. this sample text contains matching assignments for the selected sample of student text lessons. homework helper the homework helper book is designed to help parents and caregivers become more informed about the concepts being covered in the student's math course. students are encouraged to keep the homework helper at home. the homework helper includes a practice page for each lesson in the student text. the page includes a worked example of the skills covered in the lesson. each page of the homework helper also has practice exercises that the student can try. the answers to the exercises are included at the back of the homework helper. this sample text contains matching homework helper pages for the selected sample of student texts lessons. teacher's implementation guide the teacher's implementation guide contains a lesson map for each student text lesson. the lesson map includes each lesson's objectives, key terms, nctm standards, essential questions, warm-up questions, open-ended questions, and closing activities. an image of each student text page, including answers, is provided in the teacher's implementation guide, along with teacher notes intended to assist with classroom practice. this sample text contains the teacher's implementation guide pages for the selected sample of student text lessons. teacher's resources and assessments the teacher's resources and assessments (tra) contains five tests per chapter of the student text. the tests are a pre-test, a post-test, a mid-chapter test, an end-of-chapter test, and a standardized test practice. the tra contains the answer keys to all assessments. editable black-line masters of the assessments are provided digitally in the carnegie learning resource center. the tra also contains answer keys for the student assignments. this sample text contains assignment answer keys for the selected sample of lessons. this sample text contains the full set of assessment answer keys for bridge to algebra chapter 2 and algebra i chapter 1. a sample pre-test is included for geometry chapter 3 and algebra ii chapter 1. finally, this sampler is printed in black and white. actual carnegie learning texts are more colorful. 2008 text sampler page 7iii collaborative classroom 2008 carnegie learning, inc. collaborative classroom collaborative classroom as you begin the process of planning for the school year , you will want to give serious consideration to how your classroom is structured. early research on teaching and learning has revealed that what happens in the classroom in the first three days determines the environment for the entire year . this insight is important as you begin to think about your classroom and the cognitive tutor algebra i curriculum. an effective implementation of the curriculum is most likely to occur in the collaborative classroom, a classroom in which knowledge is shared. carnegie learning's philosophy—learning by doing—captures the belief that students develop understanding and skill by taking an active role in their environment. furthermore, it is carnegie learning's belief that effective communication and collaboration are essential skills for the successful learner . it is through dialogue and discussion of different strategies and perspectives that students become knowledgeable independent learners. these beliefs can be realized in the collaborative classroom. defining a collaborative classroom a collaborative classroom is an environment in which knowledge and authority are shared between the teacher and the students. in a collaborative classroom, teachers are facilitators and students are active participants. all students, not segregated by ability level, interest, or achievement, benefit from the environment created in the collaborative classroom. teachers in the collaborative classroom combine their extensive knowledge about teaching and learning, content, and skills with the informal and formal knowledge, strategies, and individual experiences of their students. the collaborative classroom differs from the traditional classroom in which the teacher is seen as an information giver (tinzmann, m.b.; jones, b.f .; fennimore, t .f .; bakker , j; fine, c.; and pierce, j., 1990, rpl_esys/collab.htm). characteristics of the collaborative classroom the collaborative classroom is identified by discussion, with in-depth accountable talk and two-way interactions, whether among members of the whole class or small groups. it is a well-structured environment in which questioning and dialogue are valued and appropriate parameters are set so that active learning can occur . careful planning by the teacher ensures that students can work together to attain individual and collective goals and to develop learning strategies. in the collaborative classroom, students are encouraged to take responsibility for their learning through monitoring and reflective self-evaluation. the collaborative classroom is one in which teachers spend more time in true academic interactions as they guide students to search for information and help students to share what they know. as facilitators, teachers have the opportunity to provide the correct amount of help to individual students by providing appropriate hints, probing questions, feedback, and help in clarifying thinking or the use of a particular strategy (tinzmann, m.b.; jones, b.f .; fennimore, t .f .; bakker , j; fine, c.; and pierce, j., 1990, a1t1_fm_v1.qxd 4/11/08 10:30 am page viii 2008 text sampler page 8ollaborative classroom ix 2008 carnegie learning, inc. collaborative classroom collaborative learning versus cooperative learning two types of learning occur in the collaborative classroom; collaborative learning, which focuses on interaction, and cooperative learning, which is a structure of interaction that helps students to accomplish a goal or end product. while these two forms of learning are often described and used interchangeably, differences do exist. the significant difference between collaborative and cooperative learning environments is the amount of control that the teacher exercises in setting goals and providing choice. for instance, in a collaborative classroom, students are positioned to set their own goals and choose activities, whereas in the cooperative learning environment, the teacher directs these activities. (ten panitz, 1996, learning in the collaborative classroom critical to teaching and learning in the collaborative-cooperative environment is the ability to define the responsibilities of the teacher and students. for effective collaboration and cooperative teamwork, teachers and students must agree to certain responsibilities that support the learning process. the table below reflects the parallel responsibilities of teachers and students. effective collaboration and cooperative t eamwork teacher responsibilities student responsibilities monitor student behavior . develop the skills to work cooperatively. provide assistance when needed. learn to talk and discuss problems with each other in order to accomplish the group goal. answer questions only when they are group questions. ask for help only after each person in the group has considered the problem and the group has a question for the teacher . interrupt the process to reinforce cooperative skill or to provide direct instructions to all students. believe that all members of the group work together toward a common goal. understand that the success or failure of the group is to be shared by all members. provide closure for the lesson. reflect on the work of the group. evaluate the group process by discussing the actions of the group members. appreciate that working together is a process and encourage each group member to interact and relate to the rest of the group members. help students to become individually accountable for learning and reinforce this understanding regularly. realize that each member must contribute as much as he or she can to the group goal. understand that the success of the group is dependent on the individual work of each member of the group. understand that group members are individually accountable for their own learning. a1t1_fm_v1.qxd 4/11/08 10:30 am page ix 2008 text sampler page 9collaborative classroom 2008 carnegie learning, inc. collaborative classroom what the collaborative classroom is not it must be agreed upon by the teacher and the students that the collaborative classroom is not one in which students: work in small groups on a problem or group of problems without direction or individual responsibility. work individually while sitting in a group working on problems. work without conversation or interaction regarding the method or process being used to solve the problem. allow one member of the group to do all of the work while others sit passively. shaping the collaborative classroom to ensure that the spirit and purpose of the collaborative classroom is clear from the onset of school, you will want to engage your students in a collaborative activity on the first day. in doing so, you can accomplish two important goals. first, students immediately understand the importance and value of working together , and secondly, students quickly move into their role as active participants. the activity "facts in five," as you may have experienced in training, is an activity designed to meet these goals. another popular activity with students is known as "broken squares" (spencer kagan: cooperative learning). in this activity, members of the team are each given several pieces of a broken square. the pieces belong to different squares. students must create the whole square by taking turns giving each other one piece. no one may speak during the activity, that is, no one can ask for what he or she needs. this activity is perfect for teaching sensitivity and the importance of communication. during the first few days of class, it is extremely important that expectations and the "rules of the game" be defined. the best approach is to have the students work together in small groups to generate the guidelines for teamwork (see lesson: creating collaborative classroom guidelines on page xvi). as a guidepost for identifying the elements for successful group interactions, we suggest reviewing the "ten guidelines for students doing group work in mathematics" written by anne e. brown for the clume project ( brown developed these guidelines after viewing the video and audio tapes of more than a dozen group sessions of her students. this list reflects the apparent actions critical to the success or failure of the group. in summary, the guidelines state the following: 1. groups should be formed quickly and members of the group should sit together , facing each other , and get to work quickly. members should call each other by first name. members should not engage in "off-task" discussion. everyone should be encouraged to participate. 2. all instructions should be read aloud so that everyone is aware of the expectations of the assignment. a1t1_fm_v1.qxd 4/11/08 10:30 am page x 2008 text sampler page 10ollaborative classroom xi 2008 carnegie learning, inc. collaborative classroom 3. members of the group should listen to each other and not interrupt. comments or questions should be acknowledged and responded to by other group members. 4. members of the group should not accept being confused. if a member of the group does not understand the information that is presented, this person should ask someone to paraphrase or re-phrase what was said. 5. members of the groups should ask for clarification if a word is used in a way that is confusing. 6. the members of the group should work together on the same problem and check for agreement frequently. 7. members of the group should explain their reasoning by "thinking out loud" and ask others to do the same. this helps everyone to relate the information being presented to what they already know. 8. members of the group should monitor the group's progress and be aware of time constraints so that all members of the group meet the goals of the assignment. 9. if the group gets stuck, the members of the group should review and summarize what they have done so far . the group can then ask for questions to find errors or missing connections to help the group's work to proceed. 10. members of the group should engage in questioning, the engine that drives mathematical investigation. group work in the collaborative classroom if we expect students to work well in groups, they will need to understand what it means to learn collaboratively and how it will benefit them. a good description of collaborative learning used by many of our teachers is: collaborative learning is a process in which each individual contributes personal knowledge and skill with the intent of improving his or her learning accomplishments along with those of others. students should be aware that one of the most important goals of collaborative learning is to create a "community of learners." they should understand that the community will grow and thrive only if all members of the group are active participants. students must also understand that their role in the classroom will be different than what they may have experienced in other classes and so will the teacher's role! you will want to introduce the features of a collaborative classroom to your students. important characteristics of the collaborative classroom include: shared responsibility choice discussion about how we learn from what is right as well as what is wrong working in groups, whether as an entire class or as several small groups a1t1_fm_v1.qxd 4/11/08 10:30 am page xi 2008 text sampler page 11ii collaborative classroom 2008 carnegie learning, inc. collaborative classroom finally, you will want students to understand the goals and expectations of a collaborative classroom: students learn collaboratively to gain greater individual proficiency. groups "sink or swim" together . everyone suggests, questions, and encourages. group members are responsible for each other's learning. all group members bring valued talents and information to the task at hand. getting started in the collaborative classroom when problems and investigations in the text require that students work in groups, you will want to structure the groups. when problems and investigations in the text require that students work individually, it is possible to maintain a collaborative classroom where students are free to communicate with each other and to share information. to form groups initially, you may want to set arbitrary groups and make changes as you observe students. one suggestion for structuring groups is to think about having two types of groups, long-term groups and short-term groups. the long-term groups, or home groups, stay together for the entire school year and sit together in class. long-term groups enable students to build trust and confidence and to learn how to negotiate with each other to derive success. on the other hand, the short-term groups are randomly assigned for specific tasks. short-term groups allow students to develop the ability to work with many different people. clearly, how you arrange the groups will depend on how to best meet your students' needs. most importantly, you want to make sure that students are respectful of one another at all times. the success of the group depends on cooperation, which can be achieved only if students accept one another and value the contributions of others. if you have students who do not want to work in groups, do not force the issue. allow those students to work alone. it is important that the student who is working alone understands that the teacher is not a member of his or her group. after these students find that they cannot talk with others and that those who are sharing information are progressing more easily, they will naturally gravitate to a group. you want to structure the success of the group experience, so it is important to use guidelines and timelines. although you will want the students to come up with the operating guidelines, timelines are probably better left to you to determine. after the groups are formed, you may want to have one person from each group be designated as a facilitator . some responsibilities of the facilitator include: obtaining and returning all materials communicating information from the teacher to the group handing in the completed assignments for the group a1t1_fm_v1.qxd 4/11/08 10:30 am page xii 2008 text sampler page 12ollaborative classroom xiii 2008 carnegie learning, inc. collaborative classroom success while working collaboratively depends upon every group member working on every part of the problem, so you may find that you do not want to assign roles such as recorder or reader to group members. students working together should generate noise and movement in the room. some have defined this attribute as "controlled chaos." to ensure that the group work remains in control, you will want to monitor group interactions and check for understanding of the task at hand. you may also want to ask students to complete parts of the problem or investigation, stop and discuss the work done, summarize the main points of the task, and then continue. this works well when the problem or investigation is lengthy. because groups will work at different paces, you might want to prepare some additional tasks or extensions of the problem or investigation for those groups who finish quickly. facilitating groups in the collaborative classroom facilitating the group process is critical. as noted earlier , you should only answer a question posed by the group rather than by individual students. you may also restrict the number of questions that a group can ask, being generous the first few times that students work in groups. when a group asks questions, answer by redirecting with guiding questions such as: what does your group think? how did you arrive at that answer? how does this relate to past activities? what work have you done so far? what do you know about the problem? what do you need to figure out? what materials might help you to figure this out? are there other parts of the problem that you can do first? other tips to consider as you manage your collaborative classroom include: provide additional instruction to those struggling with a task. listen carefully and value diversity of thought that often provides instructional opportunities. balance learning with working effectively. remember that no one is on task 100% of the time. deal with conflict constructively. ask students to sign-off on other group members' papers to acknowledge that everyone understands the group's results. a1t1_fm_v1.qxd 4/11/08 10:30 am page xiii 2008 text sampler page 13iv collaborative classroom 2008 carnegie learning, inc. collaborative classroom holding the groups accountable for an end product, such as a presentation, will add further value to the learning activity. as you have surely discovered, when you truly understand a concept or idea yourself, then you are able to explain that concept or idea to someone else. presentations and discussions in the collaborative classroom to successfully close or wrap-up a problem or investigation with a presentation and discussion, students must know exactly what you expect from them. you should also make sure that students know that you will hold the entire group accountable for the presentation. (this helps to ensure that students will hold each other accountable.) some suggestions for facilitating the presentation process include: choose presenters in a group to ensure that all students have the opportunity to present. require that students defend and talk about their solutions. hold all students accountable by asking questions of group members who are not presenting. ask presenters to make connections and generalizations and extend concepts. allow groups time to process feedback and to celebrate their achievements. to bring closure to the group work and presentation process, engage students in discussion or have them keep learning journals. some suggestions for summary wrap-up questions include: what was something that you learned from this problem? what were the mathematical concepts that you applied in solving this problem? about what concepts do you still have questions? what are three things that your group did well? what is at least one thing that your group could do even better the next time? a1t1_fm_v1.qxd 4/11/08 10:30 am page xiv 2008 text sampler page 14ollaborative classroom xv 2008 carnegie learning, inc. collaborative classroom checklist of t eacher-directed and learner-centered classrooms to understand where you are in the transition process from creating a teacher-directed classroom to creating a learner-centered classroom, you may use the criteria below to evaluate your classroom (courtesy of jacquelyn snyder , jan sinopoli, and vince vernachhio, pittsburgh public schools). use your initial evaluation as a baseline measure and check yourself at regular intervals throughout the school year . teacher-directed classroom learner-centered classroom the teacher directs all classroom activity. the teacher facilitates classroom activity. each activity is dependent on the teacher . most activities require only guidance by the teacher . the teacher is in the front of the room instructing the entire class using the blackboard or over-head most of the time. the teacher walks around the classroom during all activities, watching and listening to student-to-student discourse. the teacher models examples of the lesson objective and directs students to practice similar problems found in the text or on handouts designed by the teacher . the teacher monitors the students to keep them on task, while the students actively work together on an activity. students are seated in rows, working as a class with the teacher at the front of the class or working independently. the students are typically paired or grouped to work together while the teacher facilitates the process. the teacher presents the material while students watch and take notes. the teacher systematically brings the class together on several occasions, assuring that the mathematics of the lesson is understood. the students work independently as the teacher tries to help each student individually. students are required to make presentations, explaining their progress within the activity. the teacher completely answers the problem for the student when he or she is having difficulty. if a student is having difficulty understanding something, even after consulting with his or her group members, the teacher asks the group leading questions to guide them to the desired outcome. the teacher does the thinking and the work. the students do the thinking and the work. the teacher asks low-level or fill-in-the-blank types of questions that can be answered with a single number or in a word or two. the teacher asks thought-provoking questions that require students to explain their thinking and processes. the majority of classroom discourse is teacher-to-student discourse. the majority of discourse is student-to-student discourse. the teacher encourages students to memorize rules, procedures, and formulas. the teacher encourages students to construct knowledge. prior knowledge is assessed as new concepts emerge. a1t1_fm_v1.qxd 4/11/08 10:30 am page xv 2008 text sampler page 15vi collaborative classroom 2008 carnegie learning, inc. collaborative classroom lesson: creating collaborative classroom guidelines preparing for the lesson arrange the class seats in groups of three or four such that students face each other . position the desks in such a way that students need only do a half turn of their heads if you call their attention to the front of the room. give poster boards to each group. give colorful markers to each group. expected student growth students will gain experience in working cooperatively, listening and respecting the ideas of others, and coming to a consensus regarding the final product. students will learn how to share power with the teacher . initiating the activity ask students if they have ever worked in groups. ask students to think about good and not-so-good group experiences. have students make lists of things that happen in groups or things that they think should happen in groups to have a group work more productively to complete a task. direct students to develop social guidelines for group work in class. the guidelines should be phrased positively and refer to observable behavior . lists of guidelines should not be too long. facilitating the activity monitor student behavior . offer assistance only if necessary. for instance, students may be making their lists too long. interrupt the process to reinforce cooperative skills or to provide directions. student presentations have all groups present their guidelines. have students determine which guidelines are similar and record those. have students look at the remainder of the guidelines and determine which should be included in the list of guidelines. students should be able to justify their choices. as all students will be using these guidelines, there should be consensus on the final list. lesson closure indicate that the final list will be generated and every member in the class must agree by signing off on the list. by doing so, students have agreed to honor the list of guidelines and will be held accountable. indicate that groups not adhering to the guidelines may have their group grades reduced. a1t1_fm_v1.qxd 4/11/08 10:30 am page xvi 2008 text sampler page 16ollaborative classroom 1 2008 carnegie learning, inc. collaborative classroom presentation rubric the rubric below can be used to help you score group presentations to the entire class. the presentation scores, which range from 1 to 5, are detailed in the rubric. it is a good idea to copy the presentation rubric and distribute it to the class so that students understand how they are scored. 5 you earn a 5 for your presentation if your presentation is nearly perfect. your mathematics must be correct with only a very minor flaw (not having to do with the main idea of the problem). your public speaking skills must also be perfect or quite close to perfect. you must look at your audience. your must present yourself well and not make distracting gestures or hand motions during the presentation. your rate of speech must be neither too fast nor too slow. 4 you earn a 4 for your presentation if you miss one thing within the mathematical content of your presentation. or you earn a 4 for your presentation if there is one thing that you do not do very well within the public speaking part of the presentation. 3 you earn a 3 for your presentation if you can complete the problem, but your public speaking skills are poor . this score means that you do not make eye contact, you speak inaudibly, your mumble your words, etc. or you earn a 3 for your presentation if you have some content knowledge and make one major error , as well as omit one of the important aspects of good public speaking. 2 you earn a 2 for your presentation if you stand up for your presentation but really have very little content knowledge. this score means that you are unable to complete the problem and your speaking skills are poor . 1 you earn a 1 for your presentation for being willing to stand up and try to present. 0 you earn a 0 for your presentation if you refuse to stand up and try to present. a1t1_fm_v1.qxd 4/11/08 10:30 am page 1 2008 text sampler page 17ourse description – bridge to algebra carnegie learning tm bridge to algebra is designed as the course taken immediately prior to an algebra i course. it can be implemented with students who lack the prerequisites necessary for success in algebra i as well as advanced middle school students. the first part of bridge to algebra focuses heavily on numeracy. students work with multiple representations such as models and number lines to develop a strong conceptual understanding of fractions, decimals, and percents. students use that conceptual knowledge to develop an understanding of algorithms used to operate on and convert between various numbers. students are also introduced to ratios and proportions, signed numbers, exponents, roots, and absolute value. the second part of bridge to algebra focuses on algebra. students use their intuitive understanding of linear relationships to detect and describe linear patterns using graphs, tables, and equations. students solve simple one- and two-step linear equations and begin to develop an understanding of slope as a rate of change. the third part of bridge to algebra focuses on select topics in geometry, probability, and statistics. students are introduced to geometric topics including angle relationships, similarity, area and perimeter, volume and surface area, and the pythagorean theorem. students find simple and compound probabilities. students explore measures of central tendency and ways of representing data visually. 2008 text sampler page 18ridge to algebra text setv contents 2008 carnegie learning, inc. contents contents 1 number sense and algebraic thinking • p. 2 1.1 money, money, who gets the money? introduction to picture algebra • p. 5 1.2 collection connection factors and multiples • p. 11 1.3 dogs and buns least common multiple • p. 15 1.4 kings and mathematicians prime and composite numbers • p. 19 1.5 i scream for ice cream prime factorization • p. 23 1.6 powers that be powers and exponents • p. 27 1.7 beads and baubles greatest common factor • p. 29 fractions • p. 36 bas1fm00.qxd 8/31/07 3:11 pm page iv 2008 text sampler page 20ontents v 2008 carnegie learning, inc. operations with fractions and mixed numbers • p. 70 3.1 who gets what? adding and subtracting fractions with like denominators • p. 73 3.2 old-fashioned goodies adding and subtracting fractions with unlike denominators • p. 77 3.3 fun and games improper fractions and mixed numbers • p. 81 3.4 parts of parts multiplying fractions • p. 85 3.5 parts in a part dividing fractions • p. 89 3.6 all that glitters adding and subtracting mixed numbers • p. 93 3.7 project display multiplying and dividing mixed numbers • p. 97 3.8 carpenter , baker , mechanic, chef working with customary units • p. 101 decimals • p. 108 4.1 cents sense decimals as special fractions • p. 111 4.2 what's in a place? place value and expanded form • p. 115 4.3 my dog is bigger than your dog decimals as fractions: comparing and rounding decimals • p. 119 4.4 making change and changing hours adding and subtracting decimals • p. 123 4.5 rules make the world go round multiplying decimals • p. 127 4.6 the better buy dividing decimals • p. 129 4.7 bonjour! working with metric units • p. 133 3 4 contents bas1fm00.qxd 8/31/07 3:11 pm page v 2008 text sampler page 21i contents 2008 carnegie learning, inc. contents ratio and proportion • p. 142 5.1 heard it and read it ratios and fractions • p. 145 5.2 equal or not, that is the question writing and solving proportions • p. 149 5.3 the survey says using ratios and rates • p. 155 5.4 who's got game? using proportions to solve problems • p. 159 percents • p. 166 6.1 one in a hundred percents • p. 169 6.2 brain waves making sense of percents • p. 173 6.3 commissions, taxes, and tips finding the percent of a number • p. 177 6.4 find it on the fifth floor finding one whole, or 100% • p. 181 6.5 it's your money finding percents given two numbers • p. 185 6.6 so you want to buy a car percent increase and percent decrease • p. 189 integers • p. 196 7.1 i love new york negative numbers in the real world • p. 199 7.2 going up? adding integers • p. 203 7.3 test scores, grades, and more subtracting integers • p. 207 7.4 checks and balances multiplying and dividing integers • p. 211 7.5 weight of a penny absolute value and additive inverse • p. 215 7.6 exploring the moon powers of 10 • p. 219 7.7 expanding our perspective scientific notation • p. 223 5 6 7 bas1fm00.qxd 8/31/07 3:11 pm page vi 2008 text sampler page 22ontents vii 2008 carnegie learning, inc. contents algebraic problem solving • p. 228 8.1 life in a small town picture algebra • p. 231 8.2 computer games, cds, and dvds writing, evaluating, and simplifying expressions • p. 237 8.3 selling cars solving one-step equations • p. 241 8.4 a park ranger's work is never done solving two-step equations • p. 245 8.5 where's the point? plotting points in the coordinate plane • p. 251 8.6 get growing! using tables and graphs • p. 255 8.7 saving energy solving problems using multiple representations • p. 261 geometric figures and their properties • p. 270 9.1 figuring all of the angles angles and angle pairs • p. 273 9.2 a collection of triangles classifying triangles • p. 279 9.3 the signs are everywhere quadrilaterals and other polygons • p. 283 9.4 how does your garden grow? similar polygons • p. 287 9.5 shadows and mirrors indirect measurement • p. 291 9.6 a geometry game congruent polygons • p. 295 8 9 bas1fm00.qxd 8/31/07 3:11 pm page vii 2008 text sampler page 23iii contents 2008 carnegie learning, inc. contents area and the pythagorean theorem • p. 302 10.1 all skate! perimeter and area • p. 305 10.2 round food around the world circumference and area of a circle • p. 311 10.3 city planning areas of parallelograms, triangles, trapezoids, and composite figures • p. 315 10.4 sports fair and square squares and square roots • p. 321 10.5 are you sure it's square? the pythagorean theorem • p. 325 10.6 a week at summer camp using the pythagorean theorem • p. 329 probability and statistics • p. 338 11.1 sometimes you're just rained out finding simple probabilities • p. 341 11.2 socks and marbles finding probabilities of compound events • p. 345 11.3 what do you want to be? mean, median, mode, and range • p. 351 11.4 get the message? histograms • p. 357 11.5 go for the gold! stem-and-leaf plots • p. 363 11.6 all about roller coasters box-and-whisker plots • p. 367 11.7 what's your favorite flavor? circle graphs • p. 371 10 11 bas1fm00.qxd 8/31/07 3:11 pm page viii 2008 text sampler page 24ontents ix 2008 carnegie learning, inc. contents volume and surface area • p. 378 12.1 your friendly neighborhood grocer three-dimensional figures • p. 381 12.2 carnegie candy company volumes and surface areas of prisms • p. 385 12.3 the playground olympics volumes and surface areas of cylinders • p. 391 12.4 the rainforest pyramid volumes of pyramids and cones • p. 395 12.5 what on earth? volumes and surface areas of spheres • p. 401 12.6 engineers and architects nets and views • p. 405 12.7 double take similar solids • p. 409 linear functions • p. 416 13.1 running a tree farm relations and functions • p. 419 13.2 scaling a cliff linear functions • p. 423 13.3 biking along slope and rates of change • p. 427 13.4 let's have a pool party! finding slope and y-intercepts • p. 435 13.5 what's for lunch? using slope and intercepts to graph lines • p. 441 13.6 healthy relationships finding lines of best fit • p. 449 12 13 bas1fm00.qxd 8/31/07 3:11 pm page ix 2008 text sampler page 25contents 200 carnegie learning, inc. contents number systems • p. 458 14.1 is it a bird or a plane? rational numbers • p. 461 14.2 how many times? powers of rational numbers • p. 467 14.3 sew what? irrational numbers • p. 471 14.4 worth 1000 words real numbers and their properties • p. 475 14.5 the house that math built the distributive property • p. 481 transformations • p. 488 15.1 worms and ants graphing in four quadrants • p. 491 15.2 maps and models scale drawings and scale models • p. 497 15.3 designer mathematics sliding and spinning • p. 503 15.4 secret codes flipping, stretching, and shrinking • p. 509 15.5 a stitch in time multiple transformations • p. 515 glossary • p. g1 index • p. i1 14 15 bas1fm00.qxd 8/31/07 3:11 pm page x 8 2008 text sampler page 262008 carnegie learning, inc. bridge to algebra student text bas1fm00.qxd 8/31/07 3:11 pm page i 2008 text sampler page 276 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 36 2008 text sampler page 28hapter 2 • fractions 37 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 37 2008 text sampler page 298 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 38 2008 text sampler page 302008 carnegie learning, inc. 2 lesson 2.1 • dividing a whole into fractional parts 39 comic strips dividing a whole into fractional parts objectives in this lesson, you will: • use fractions to represent parts of a whole. key terms • fraction • numerator • denominator will exactly the same size? use complete sentences to explain. 2.1 bas10200.qxd 8/31/07 3:37 pm page 39 2008 text sampler page 310 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 40 2008 text sampler page 32esson 2.1 • dividing a whole into fractional parts 41 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 41 2008 text sampler page 332 chapter 2 • fractions 2008 carnegie learning, inc. 2 with a fraction. 2. arrange your strips bas10200.qxd 8/31/07 3:37 pm page 42 2008 text sampler page 342008 carnegie learning, inc. 2 lesson 2.1 • dividing a whole into fractional parts 43 bas10200.qxd 8/31/07 3:37 pm page 43 2008 text sampler page 354 chapter 2 • fractions 2008 carnegie learning, inc. 2 bas10200.qxd 8/31/07 3:37 pm page 44 2008 text sampler page 36ridge to algebra teacher's implementation guide volume 1 cl_te_fm_vol1_i_vi_5.qxd 8/31/07 12:18 pm page i 2008 carnegie learning, inc. 2008 text sampler page 3736 chapter 2 • fractions 105 4 3 30 45 72 3 16 6 30 quarters 10 quarters cl_te_ch02.qxd 8/31/07 12:50 pm page 36 2008 carnegie learning, inc. 2008 text sampler page 38fractions 37 cl_te_ch02.qxd 8/31/07 12:50 pm page 37 2008 carnegie learning, inc. 2008 text sampler page 3938 chapter 2 • fractions cl_te_ch02.qxd 8/31/07 12:50 pm page 38 2008 carnegie learning, inc. 2008 text sampler page 40esson 2.1 • comic strips: dividing a whole into fractional parts 39a 2 comic strips dividing a whole into fractional parts 2.1 objectives in this lesson, students will • use fractions to represent parts of a whole. key terms • fraction • numerator • denominator materials • strips of paper to represent a newspaper comic strip, 9 per student (note: prepare strips before class.) nctm content standards number and operations standards grades 6-8 expectations • work flexibly with fractions, decimals, and percents to solve problems • understand the meaning and effects of arith-metic operations with fractions, decimals, and integers lesson overview within the context of this lesson, students will be asked to • use fractions to represent parts of a whole. • label parts of a whole with fractions. • use diagrams to represent fractions. students will physically create accurate representations of fractions. the focus will be on types of fractions that may be easier to represent in this form. students review definition of and parts of a frac-tional number . students will also begin to explore equivalent fractions, which will be covered more in-depth later in this unit. also later in the unit, students will work with fractions in physical as well as abstract representations. students will need to keep the fraction strips they create during this lesson for future lessons, or you may wish to store the strips for the students. essential questions the following key questions are addressed in this section: 1. how can you use fractions to represent parts of a whole? 2. what does each digit in a fraction represent? learning by doing lesson map get ready cl_te_ch02.qxd 8/31/07 12:50 pm page 39 2008 carnegie learning, inc. 2008 text sampler page 419b chapter 2 • fractions warm-up place the following questions or an applicable subset of these questions on the board or project on an overhead projector before students enter class. students should begin working as soon as they are seated. while students are working on the warm-up exercises, you can attend to clerical tasks like tak-ing attendance or returning student work. 1. write the fraction that is represented by the shaded part in each figure. motivator start the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson concerns dividing things evenly to represent fractions. the motivating questions ask about students' experience relating physical objects using the vocabulary of fractions. students briefly dis-cuss the vocabulary and definition of fractions. we all have eaten half of a sandwich before. how did you know that you had eaten "one half"? how else could it have been cut to equal one half? what other things that you've seen can you quickly rec-ognize as being one half? what about one quarter or one third? when introducing this lesson, discuss different ways to write fractions: using words (one half), numbers ( ), pictures, etc. have students identify the different parts of the representations, such as "one" in "one half" being just part of the whole. 1 2 2 2 or 1 6 9 or 2 3 6 7 2 4 or 1 2 3 10 1 3 2 show the way cl_te_ch02.qxd 8/31/07 12:50 pm page 40 2008 carnegie learning, inc. 2008 text sampler page 42roblem 1 grouping have students work in pairs. each stu-dent should work together , but fold and label his or her own strips. ask for a volunteer to read problem 1 aloud. after a student has read the problem, ask students guiding ques-tions to make sure they understand the main task as presented in part a. have a student restate the problem in his or her own words. guiding questions • how can you divide something into equal parts without measuring it? • what do you think is the most accurate way of dividing something? • why should the parts be equal in size? • how would you represent parts that are not equal in size? common student errors parts b or c, tell students that they may not have to use normal math vocabulary in their sentences, but that their answer for b may still include number words. if they are still having trouble expressing this simplistic task, help them focus on the normal word to describe two equal parts (half). explore together will the same size? use complete sentences to explain. 1 2 i put the ends together , then folded the paper in half to get two equal parts. the halves match. when i look at either side of the folded paper , i can't see any part of the other half behind the front half. i could measure each half with a ruler , but that will prove what i already can tell by just looking at the paper . 1 2 lesson 2.1 • comic strips: dividing a whole into fractional parts 39 2 cl_te_ch02.qxd 1/14/08 1:34 pm page 39 if students are having trouble with 2008 carnegie learning, inc. 2008 text sampler page 43nvestigate problem 1 math path ask a volunteer to read math path and the information about fractions below it. the information is a continuation of the warm-up, which is a review of basic fraction facts. this activity gets students started on creating their own fractions strips, which they will use throughout the unit. have students work in pairs to complete questions 1-4. problem 2 as in the math path, students continue to create fraction strips. common student errors students may label strips consecutively instead of individually. for example, stu-dents may write , , , instead of labeling each section . help them focus on each section on its own. 1 4 4 4 3 4 2 4 1 4 explore together 1 2 the denominator is two to represent two equal parts. the numerator is one to represent a single frame. there are 4 equal parts. each part should be labeled . 1 4 there are 8 equal parts. each part should be labeled . 1 8 there are 16 equal parts. each part should be labeled . answers will vary. 1 16 see diagram in part (a). 40 chapter 2 • fractions 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 40 2008 carnegie learning, inc. 2008 text sampler page 44nvestigate problem 2 key formative assessments • which fraction strips are easiest to compare? • what do you notice about the sec-tions of equal size? • how can you predict how many sections of one strip will equal another? problem 3 have a volunteer read problem 3. you may wish to have students suggest ways to divide strips evenly into thirds before allowing students to work in pairs to do so. common student errors students may try to fold the strips into thirds by folding each outward edge to meet in the middle. this only creates a strip with the outer sections equaling one quarter each and the inner section equaling one half. explore together it got harder with each fold. you could find fractions with the denominators: 2, 4, 8, 16, 32, 64, 128, etc. if you had a large enough piece of paper to fold into that many smaller sections. 2 4 6 it takes 6 of the parts labeled as to make up three of the parts labeled as . it takes 1 of the parts labeled as to make up eight of the parts labeled as . 1 2 1 8 it is more difficult to fold a strip into three parts than to fold it in two parts. 1 6 1 12 1 16 1 16 lesson 2.1 • comic strips: dividing a whole into fractional parts 41 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 41 2008 carnegie learning, inc. 2008 text sampler page 45nvestigate problem 3 grouping depending on how well the students are doing with the concepts and how well they are working together , allow pairs or individuals to complete questions 1-4. while students are working, circulate around the classroom to observe their progress. if students are having trouble with question 2, have them compare only two strips at a time. this should make it easier for them to focus on finding the largest parts. more than two strips may be too much input, making it easy to forget what they should be looking for on each. for question 3, you may want to model ways to divide each figure, such as a crosshatch over the circle for . you may also wish to have students find a template for each shape, such as a penny for the circle. tell students that the divisions do not need to be exact, but as close as they can get. don't let students spend too much time agoniz-ing over making exact divisions. 3 4 explore together with a fraction. 2. arrange your strips 1 20 1 10 1 5 it takes 2 of the parts labeled as to make up 1 of the parts labeled as . it takes 2 of the parts labeled as to make up four of the parts labeled as . 1 10 1 10 1 5 1 5 42 chapter 2 • fractions 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 42 2008 carnegie learning, inc. 2008 text sampler page 46nvestigate problem 3 after pairs or individuals finish, you may wish to have students compare their answers in pairs, groups, or as a class. ask students why their answers in question 3 look alike or unlike. students will note that they may or may not have divided certain shapes the same way. ask students to express whether there's a right way or a wrong way to divide shapes. students should decide that as long as the parts are all equal, it is fine to divide shapes up different ways (such as horizontally, vertically, etc.). explore together shapes with odd-numbered divisions were generally the hardest because it was harder to make each part equal. lesson 2.1 • comic strips: dividing a whole into fractional parts 43 2 cl_te_ch02.qxd 8/31/07 12:50 pm page 43 2008 carnegie learning, inc. 2008 text sampler page 473a chapter 2 • fractions 2 wrap-up close to close your lesson, ask students to answer the following questions. their answers should help summarize the lesson and the learning for the day. you may want to review the essential questions from the get ready and/or the key formative and guiding questions found throughout the lesson map. how can you divide something into equal parts? why should the parts be equal in size? what does the numerator in the fraction represent? what does the denominator in the fraction represent? how do you write a fraction after you've folded a strip of paper? how do you write a fraction after drawing a diagram? another alternative for closing your lesson is using the open-ended writing task provided on the next page. t ies to the cognitive tutor software students will be working in a variety of software units when they encounter this les-son. ties between text and software can be made because both software and text include modeling with algebraic expressions, graphing, and examining the relation-ship between written, numeric, graphic, and algebraic representations. for more specific information on correlations between the software and text, con-sult the software implementation guide. cl_te_ch02.qxd 8/31/07 12:50 pm page 44 2008 carnegie learning, inc. 2008 text sampler page 48esson 2.1 • comic strips: dividing a whole into fractional parts 43b 2 follow-up assignment use assignment 2.1 in the student assignments book. see the teacher's resources and assessments book for answers. assessment see the assessments provided in the teacher's resources and assessments book for this chapter . open-ended writing task your next comic strip in the newspaper is two lines or two strips long. how will you divide each line or strip? how much of your strip will each frame be? which would you use to have the largest frames to draw in? which would you use to have the greatest number of frames? reflections insert your reflections on the lesson as it played out in class today. what went well? what did not go as well as you would have liked? how would you like to change the lesson in order to improve the things that did not go well? how would you like to change the lesson in order to capitalize on the things that did go well? are there ways in which you feel the lesson could have been enriched for students? cl_te_ch02.qxd 8/31/07 12:50 pm page 45 2008 carnegie learning, inc. 2008 text sampler page 492008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 50 2008 text sampler page 502008 carnegie learning, inc. bridge to algebra teacher' s resources and assessments 1_bam1_fm.qxd 3/24/08 9:01 am page i 2008 text sampler page 51ontents iii 2008 carnegie learning, inc. contents contents section 1 assessments with answers section 2 assignments with answers section 3 skills practice with answers 1_bam1_fm.qxd 3/24/08 9:01 am page iii 2008 text sampler page 522008 carnegie learning, inc. 2 chapter 2 assignments 15 assignment name ___________________________________________________ date _____________________ assignment name ___________________________________________________ date _____________________ assignment for lesson 2.1 comic strips dividing a whole into fractional parts write the fraction that is represented by the fraction model. 1. 2. 3. divide and shade each rectangle to represent the fraction. 4. 5. 6. 7. 8. divide and shade the circle to represent 3 8 . 7 12 1 8 3 5 2 3 assignment name ___________________________________________________ date _____________________ assignment name ___________________________________________________ date _____________________ assignment for lesson 2.1 1 4 2 5 5 9 bag1_te_v1.qxd 1/7/08 10:34 am page 8 2008 text sampler page 532008 carnegie learning, inc. 2 chapter 2 skills practice 203 skills practice name ___________________________________________________ date _____________________ lesson 2.1 reflect & review 1. you and your friends have 18 for lunch. you buy three sandwiches for 4 each and three drinks for 1 each. assuming that the tax has already been included, do you have enough to pay for your lunch? show all your work. 2. this summer , you earned 1920 in 12 weeks. how much money did you earn each week? 3. what are the factors of 64? 4. use mental math to find the quotient practice 5. identify the numerator of the fraction 6. identify the denominator of the fraction 7. write the fraction that has a numerator of 10 and a denominator of 19. 8. write the fraction that has a denominator of 11 and a numerator of 5. complete each statement. 9. is the same as ___ s. 10. is the same as ___ s. 11. is the same as ___ s. represent each fraction by drawing the specified figure. 12. of a circle 13. of a rectangle 14. of a square 7 9 1 4 4 5 1 25 1 5 1 18 4 9 1 12 1 6 15 44 . 3 8 . 1200 60. 10 19 1, 2, 4, 8, 16, 32, 64 amount earned each week 1920 12 160 total cost yes. 3(4) 3(1) 15 18; 3 44 5 11 2 8 20 bak1_te_v1.qxd 1/9/08 4:25 pm page 8 5 2008 text sampler page 54hapter 2 assessments 15a 2 teacher notes pre- and post-tests this pre-assessment is designed to be completed by students before they start work on chapter 2. students should be given about 20–30 minutes to complete this pre-assessment. the teacher can use this assessment formatively, to understand what material students have mastered going into the start of chapter 2. this post-assessment is designed to be completed at the culmination of chapter 2. this test is a parallel form to the pre-assessment. in accordance, students should be allowed 20–30 minutes to complete this post-assessment. student performance on the post- and pre-assessment can be compared in order to measure gains in student learning from chapter 2. mid-chapter test this mid-chapter assessment is designed to be completed by students when they have completed the first four lessons in chapter 2. students should be given about 20–30 minutes to complete this mid-chapter assessment. students should be encouraged to spend about 5–10 minutes on each question in this mid-chapter test. test item 1 this test item aims to get students thinking about what happens when you are asked to compare two fractions without knowing what the whole is that the fractions refer to. if students have a hard time with this problem, it may be helpful to suggest they use different amounts for the magician and the ringleader , to see what the effect is on the comparison. test item 2 this test item asks students to sort fractions into two piles. if students have a hard time with these problems, it may mean that they do not have a firm grasp on the syntax associated with fractions. for instance, you may find that students want to group statements such as and together , when in fact, these two statements are not the same. 6 4 6 4 bam102.qxd 9/5/07 12:57 pm page 13 2008 carnegie learning, inc. 2008 text sampler page 555b chapter 2 assessments 2 teacher notes page 2 teacher notes page 2 end of chapter test this is an end of chapter assessment. give students one class period to complete this assessment. test item 1 for this test item students must determine which fractions are equivalent. students must be able to reduce fractions successfully. this problem also requires students to be organized in their approach, and this should be noted to students. test item 2 students must be able to compare fractions. students may complete this task by dividing the numerator by the denominator of each fraction and then ordering the decimals on a number line. or , students may write all of the fractions with a common denominator , in which case 32 is the least common denominator . test item 3 this test item requires students to estimate the height of each bar in the graph. student answers may vary slightly because of this estimation. test item 5 for this item, students must understand that there are values between and one way to show this is to write equivalent fractions. for instance, and you can see that is between and standardized test practice this test will provide practice for standardized tests that students may take during the school year . content from all previous chapters will be included on the practice exams. to prepare students for standardized testing, allow students 15 to 20 minutes of time to complete the exam. emphasize that students should work quickly but carefully to perform well. 5 6 . 4 6 9 12 5 6 2 2 10 12 . 4 6 2 2 8 12 5 6 . 4 6 teacher notes page 2 teacher notes page 2 bam102.qxd 9/5/07 12:57 pm page 14 2008 carnegie learning, inc. 2008 text sampler page 56hapter 2 assessments 15 2 pre-test name ___________________________________________________ date _____________________ 1. what fraction of the figure is shaded? 2. write the fraction that is represented by the point on each number line. 3. use a number line to represent . 4. use a rectangle to represent 5 8 . 2 3 0 1 2 3 4 0 1 2 3 4 the shaded region represents of the figure. both points are at on the number lines. 0 1 2 sample answer: bam102.qxd 9/5/07 12:57 pm page 15 2008 carnegie learning, inc. 2008 text sampler page 576 chapter 2 assessments 2 pre-test page 2 5. you share 5 kenya and jada started with the same amount of colored paper . kenya used of her colored paper . jada used of hers. who has more paper now, kenya or jada? explain your reasoning. 9. maya's book has 168 pages. she reads 24 pages each night before she goes to bed. what fraction of her book does she read each night? write your answer in simplest form. how long will it take maya to finish the book? 1 3 1 3 , 2 3 , and 3 5 3 4 , 6 8 , and 9 12 1 4 kenya used more paper than jada because one third is more than one fourth. so, jada has more paper left than kenya. maya reads of her book nightly. it will take her 7 days, or 1 week, to finish the book. 168 1 7 each person receives or pizzas. 1 1 4 5 4 16 2008 carnegie learning, inc. 2008 text sampler page 58hapter 2 assessments 17 2 10. find the missing numbers in the equivalent fractions below. 11. order the fractions from least to greatest. 1 2 , 2 3 , and 3 5 2 5 24 ? 3 8 15 ? pre-test page 3 name ___________________________________________________ date _____________________ in order from least to greatest, the fractions are 5 and the missing value is 40. the missing value is 60. 5 6 ; 8 5 5 5 ; bam102.qxd 9/5/07 12:57 pm page 17 2008 carnegie learning, inc. 2008 text sampler page 598 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 18 2008 carnegie learning, inc. 2008 text sampler page 60hapter 2 assessments 19 2 post-test name ___________________________________________________ date _____________________ 1. what fraction of the figure is shaded? 2. write the fraction that is represented by the point on the number line. 3. use a number line to represent . 4. use a rectangle to represent 2 5 . 4 5 0 1 2 the shaded region represents of the figure. 7 the point on the number line is . 5 sample answer: sample answer: 0 1 2 4 5 bam102.qxd 9/5/07 12:57 pm page 19 2008 carnegie learning, inc. 2008 text sampler page 610 chapter 2 assessments 2 5. you share 3 on thursday afternoon of the library computers were being used by students. what fraction of the computers were not being used? 9. sumi collects mugs. she has 5 yellow, 8 blue, and some red mugs. the 5 yellow mugs represent of her total mugs. how many mugs does sumi have? how many red mugs does she have? 1 5 6 9 2 5 , 3 7 , and 3 8 2 3 , 6 9 , and 8 12 post-test page 2 if of the computers were being used, then were not being used. 6 9 9 6 9 if 5 yellow mugs represent of the total mugs, then sumi has 25 mugs in all. because 5 are yellow and 8 are blue, then of the mugs are red. 5 5 8 5 each person receives pizza. 20 2008 carnegie learning, inc. 2008 text sampler page 62hapter 2 assessments 21 2 10. find the missing numbers in the equivalent fractions below. 11. order the fractions from least to greatest. 5 8 , 1 4 , 1 3 , 1 2 , 3 8 , 7 16 , and 3 4 4 ? 16 24 2 3 ? 18 post-test page 3 name ___________________________________________________ date _____________________ to solve this problem, students can place each fraction on a number line. or , they can write the fractions with the common denominator of 48 and then place them in order . in order from least to greatest, the fractions are 4 , 1 3 , 3 8 , 7 16 , 1 2 , 5 8 , and 3 4 . the missing value is 12. the missing value is 6. 4 6 ( 4 4 ) 16 24 ; 2 3 ( 6 6 ) 12 18 ; bam102.qxd 9/5/07 12:57 pm page 21 2008 carnegie learning, inc. 2008 text sampler page 632 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 22 2008 carnegie learning, inc. 2008 text sampler page 64hapter 2 assessments 23 2 mid-chapter test name ___________________________________________________ date _____________________ 1. what is the wrong with the reasoning below? 2. sort these statements into two piles. each pile must contain matching statements. you can win half of my money at my booth. you can win more at my booth because i give away one quarter of my money to winners. 6 4 4 6 6 4 4 6 4 6 6 4 if i share 6 pizzas equally among 4 people, how much pizza will each person get? if i share 4 pizzas equally among 6 people, how much pizza will each person get? pile 1: "if i share 6 pizzas equally among 4 people ..." pile 2: "if i share 4 pizzas equally among 6 people ..." 6 6 6 6 6 6 the reasoning is wrong because it is impossible to compare two fractions without knowing the total amounts that they refer to. for example, suppose that the magician has 100. then players in his game can win 50. then suppose that the ringleader has 400. then players in his game can win 100. so, the ringleader's statement is true. however , if both the magician and the ringleader have the same amount of money, then the players at the magician's booth will win more money. bam102.qxd 9/5/07 12:57 pm page 23 2008 carnegie learning, inc. 2008 text sampler page 654 chapter 2 assessments 2 mid-chapter test page 2 3. the bar graph shows the heights of students in a class. use the graph to answer the following questions. what fraction of the students are 150 centimeters tall? what fraction of the students are taller than 150 centimeters? what fraction of students are in the tallest group? 148 10 8 6 4 2 0 height (centimeters) number of students 149 150 151 152 you can find the total number of students in the class by adding the number of students from each column. so, there are students in the class. there are 8 students who are 150 centimeters tall, so of the students are 150 centimeters tall. 8 5 8 9 7 there are 9 students who are 151 centimeters tall and 7 students who are 152 centimeters tall, so of the students are taller than 150 centimeters. 9 7 6 there are 7 students in the tallest group, so of the students are in the tallest group. 7 bam102.qxd 9/5/07 12:57 pm page 24 2008 carnegie learning, inc. 2008 text sampler page 66hapter 2 assessments 25 2 end of chapter test name ___________________________________________________ date _____________________ 1. nine fractions are shown below. which fractions are equivalent? 2. in the united states, hardwood flooring comes in different thicknesses. maple, beech, or birch is inch thick. both oak and pecan flooring can be inch, inch, or inch thick. put these thicknesses in order , thickest first. 1 2 25 32 3 8 25 32 16 20 4 5 8 10 15 18 20 24 7 8 10 12 12 15 5 6 5 inch inch 8 inch all of the fractions reduce to one of three values in simplest form. students should group the fractions as follows: are equivalent fractions. are equivalent fractions. is not equivalent to any of the given fractions. 7 8 5 6 5 8 and 5 8 5 and 6 bam102.qxd 9/5/07 12:57 pm page 25 2008 carnegie learning, inc. 2008 text sampler page 676 chapter 2 assessments 2 end of chapter test page 2 3. the bar graph shows the number of people who visited a musum each day during one week. about how many people visited each day? about how many people visited in total during the week? about what fraction of people visited on wednesday? about what fraction of people visited before wednesday? 4. explain why the fractions are equivalent fractions. 5. find a number that lies between 6. the fractions are in simplest form. explain what is meant by simplest form. 3 5 , 3 10 , 2 15 , and 7 20 4 6 and 5 6 . 3 5 , 6 10 , 9 15 , and 12 20 monday tuesday wednesday thursday friday 500 450 400 350 300 250 200 150 100 50 0 students must estimate for this problem. the graph shows that about 250 people visited on monday, 310 people visited on tuesday, 150 people visited on wednesday, 250 people visited on thursday, and 440 people visited on friday. about people visited during the week. 5 5 5 about of the visitors came to the museum on wednesday. 5 8 about of the visitors came to the museum before wednesday. 5 56 5 the fractions all reduce to 5 is between and 5 6 6 9 fractions are in simplest form when the numerator and denominator have no common factors other than 1. bam102.qxd 9/5/07 12:57 pm page 26 2008 carnegie learning, inc. 2008 text sampler page 68hapter 2 assessments 27 2 1. what is the value of the expression , when a. 10 b. 13 c. 15 d. 21 2. what is the prime factorization of 96? a. b. c. d. 3. which number is exactly divisible by 7? a. 27 b. 34 c. 57 d. 91 4. what is the greatest common factor (gcf) of 18 and 54? a. 3 b. 6 c. 9 d. 18 12 8 9 6 3 3 2 3 2 5 3 3? 4 3 standardized test practice name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 27 2008 carnegie learning, inc. 2008 text sampler page 698 chapter 2 assessments 2 5. simplify the expression a. b. c. 1 d. 6. the figure below represents one whole divided into equal parts. what fractional part of the figure is shaded? a. b. c. d. 7. three circles are divided into 4 equal pieces. how many pieces are there? a. 4 b. 7 c. 10 d. 12 4 7 1 4 3 4 1 3 1 8 10 9 10 6 10 4 2 3 5 2 . standardized test practice page 2 bam102.qxd 9/5/07 12:57 pm page 28 2008 carnegie learning, inc. 2008 text sampler page 70hapter 2 assessments 29 2 8. you want to divide 2 circles so that you have a total of 12 equal pieces. into how many pieces should each circle be divided? a. 6 b. 10 c. 14 d. 24 9. which fraction is equivalent to ? a. b. c. d. 10. which fraction is the simplest form of ? a. b. c. d. 7 18 8 12 4 9 2 3 72 108 12 20 6 8 6 7 2 3 3 4 standardized test practice page 3 name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 29 2008 carnegie learning, inc. 2008 text sampler page 710 chapter 2 assessments 2 11. which fraction lies between and ? a. b. c. d. 12. a strip is divided into 5 equal parts. what fraction represents 2 parts of the strip? a. b. c. d. 13. which fraction is equivalent to ? a. b. c. d. 3 7 3 5 2 3 1 2 15 35 2 10 2 7 2 5 2 3 2 8 4 5 1 4 1 2 5 8 3 8 standardized test practice page 4 bam102.qxd 9/5/07 12:57 pm page 30 2008 carnegie learning, inc. 2008 text sampler page 72hapter 2 assessments 31 2 14. which fraction is written in simplest form? a. b. c. d. 15. which set of fractions is written in order from least to greatest? a. b. c. d. 1 2 , 1 3 , 1 4 5 16 , 2 8 , 2 4 1 3 , 2 9 , 5 12 1 3 , 2 4 , 4 6 9 18 4 16 3 9 2 5 standardized test practice page 5 name ___________________________________________________ date _____________________ bam102.qxd 9/5/07 12:57 pm page 31 2008 carnegie learning, inc. 2008 text sampler page 732 chapter 2 assessments 2 bam102.qxd 9/5/07 12:57 pm page 32 2008 carnegie learning, inc. 2008 text sampler page 74ridge to algebra homework helper 2008 carnegie learning, inc. bah1fm00.qxd 1/11/08 11:24 am page i 2008 text sampler page 752008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 76 2008 text sampler page 76chapter 2 • fractions 2008 carnegie learning, inc. 2 comic strips dividing a whole into fractional parts 2.1 students should be able to answer these questions after lesson 2.1: • how can you use fractions to represent parts of a whole? • what does each digit in a fraction represent? • how can you use pictures to represent a fraction? directions read question 1 and its solution. then, for questions 2 and 3 draw a picture to represent the fraction. finally, write the fraction to answer the question. 1. the school newspaper will print 6 comic strips on a page. you have created 5 comic strips. how much of the page is going to have your work? 2. the newspaper will print 3 different stories on each page. you wrote 1 story. how much of the page will contain your work? 3. the newspaper will have 8 ads on a page. you put in 3 ads—one ad for your tutoring business, one ad to sell candy for the spanish club, and one ad to sell your bicycle. how much of the page shows your ads? draw a rectangle to represent a page. divide the rectangle into 6 equal parts. step 2 step 1 shade the amount of the page that will contain your work. write the fraction to answer the question. five out of the 6 parts, or , of the page will be your work. 5 6 step 4 step 3 bah10200.qxd 12/17/07 3:21 pm page 8 2008 text sampler page 77ourse description – algebra i carnegie learning tm algebra i is designed as a first-year algebra course. it can be implemented with students at a variety of ability and grade levels. algebra i focuses heavily on linear functions. students use their intuitive understanding of linear relationships to detect and describe linear patterns. students are introduced to multiple representations of functions including verbal, numeric, graphical, and algebraic. students develop an understanding of the equivalence of relationships and the ability to convert between representations. students explore the graphs of linear functions and develop an understanding of slope as a rate of change. students model data with a linear function and use the regression equation to make predictions. students solve linear equations and inequalities, including absolute value equations and inequalities. students solve systems of linear equations and inequalities graphically and algebraically. algebra i also includes select topics in non-linear algebra, probability, and statistics. students are introduced to quadratic, polynomial, and exponential functions. these functions are covered in more depth in algebra ii. students find simple and compound probabilities. students explore measures of central tendency and ways of representing data visually. 2008 text sampler page 78lgebra i text setv contents 2008 carnegie learning, inc. contents contents patterns and multiple representations p. 2 proportional reasoning, percents, and direct variation p. 48 2.1 left-handed learners using samples, ratios, and proportions to make predictions p. 51 2.2 making punch ratios, rates, and mixture problems p. 57 2.3 shadows and proportions proportions and indirect measurement p. 63 2.4 tv news ratings ratios and part-to-whole relationships p. 69 2.5 women at a university ratios, part-to-part relationships, and direct variation p. 73 2.6 tipping in a restaurant using percents p. 81 2.7 taxes deducted from your paycheck percents and taxes p. 87 1 2 a1s1_fm.qxd 4/11/08 7:48 am page iv 2008 text sampler page 80ontents v 2008 carnegie learning, inc. contents solving linear equations p. 92 3.1 collecting road tolls solving one-step equations p. 95 3.2 decorating the math lab solving two-step equations p. 101 3.3 earning sales commissions using the percent equation p. 107 3.4 rent a car from go-go car rentals, wreckem rentals, and good rents rentals using two-step equations, part 1 p. 113 3.5 plastic containers using two-step equations, part 2 p. 125 3.6 brrr! it's cold out there! integers and integer operations p. 131 3.7 shipwreck at the bottom of the sea the coordinate plane p. 139 3.8 engineering a highway using a graph of an equation p. 143 linear functions and inequalities p. 150 4.1 up, up, and away! solving and graphing inequalities in one variable p. 153 4.2 moving a sand pile relations and functions p. 159 4.3 let's bowl! evaluating functions, function notation, domain, and range p. 165 4.4 math magic the distributive property p. 169 4.5 numbers in your everyday life real numbers and their properties p. 175 4.6 technology reporter solving more complicated equations p. 183 4.7 rules of sports solving absolute value equations and inequalities p. 187 3 4 a1s1_fm.qxd 4/11/08 7:48 am page v 2008 text sampler page 81i contents 2008 carnegie learning, inc. contents writing and graphing linear equations p. 194 5.1 widgets, dumbbells, and dumpsters multiple representations of linear functions p. 197 5.2 selling balloons finding intercepts of a graph p. 205 5.3 recycling and saving finding the slope of a line p. 211 5.4 running in a marathon slope-intercept form p. 219 5.5 saving money writing equations of lines p. 227 5.6 spending money linear and piecewise functions p. 233 5.7 the school play standard form of a linear equation p. 239 5.8 earning interest solving literal equations p. 245 lines of best fit p. 248 6.1 mia's growing like a weed drawing the line of best fit p. 251 6.2 where do you buy your music? using lines of best fit p. 259 6.3 stroop test performing an experiment p. 267 6.4 jumping correlation p. 273 6.5 human chain: wrist experiment using technology to find a linear regression equation, part 1 p. 279 6.6 human chain: shoulder experiment using technology to find a linear regression equation, part 2 p. 285 6.7 making a quilt scatter plots and non-linear data p. 291 5 6 a1s1_fm.qxd 4/11/08 7:48 am page vi 2008 text sampler page 82ontents vii 2008 carnegie learning, inc. contents systems of equations and inequalities p. 296 7.1 making and selling markers and t -shirts using a graph to solve a linear system p. 299 7.2 time study graphs and solutions of linear systems p. 307 7.3 hiking trip using substitution to solve a linear system p. 315 7.4 basketball tournament using linear combinations to solve a linear system p. 323 7.5 finding the better paying job using the best method to solve a linear system, part 1 p. 329 7.6 world oil: supply and demand using the best method to solve a linear system, part 2 p. 333 7.7 picking the better option solving linear systems p. 339 7.8 video arcade writing and graphing an inequality in two variables p. 345 7.9 making a mosaic solving systems of linear inequalities p. 351 quadratic functions p. 356 8.1 website design introduction to quadratic functions p. 359 8.2 satellite dish parabolas p. 369 8.3 dog run comparing linear and quadratic functions p. 377 8.4 guitar strings and other things square roots and radicals p. 385 8.5 tent designing competition solving by factoring and extracting square roots p. 389 8.6 kicking a soccer ball using the quadratic formula to solve quadratic equations p. 397 8.7 pumpkin catapult using a vertical motion model p. 403 8.8 viewing the night sky using quadratic functions p. 411 7 8 a1s1_fm.qxd 4/11/08 7:48 am page vii 2008 text sampler page 83iii contents 2008 carnegie learning, inc. contents properties of exponents p. 418 9.1 the museum of natural history powers and prime factorization p. 421 9.2 bits and bytes multiplying and dividing powers p. 425 9.3 as time goes by zero and negative exponents p. 429 9.4 large and small measurements scientific notation p. 433 9.5 the beat goes on properties of powers p. 437 9.6 sailing away radicals and rational exponents p. 443 polynomial functions and rational expressions p. 450 10.1 water balloons polynomials and polynomial functions p. 453 10.2 play ball! adding and subtracting polynomials p. 459 10.3 se habla español multiplying and dividing polynomials p. 463 10.4 making stained glass multiplying binomials p. 471 10.5 suspension bridges factoring polynomials p. 477 10.6 swimming pools rational expressions p. 483 probability p. 490 11.1 your best guess introduction to probability p. 493 11.2 what's in the bag theoretical and experimental probabilities p. 499 11.3 a brand new bag using probabilities to make predictions p. 503 11.4 fun with number cubes graphing frequencies of outcomes p. 507 11.5 going to the movies counting and permutations p. 513 9 1 0 1 1 a1s1_fm.qxd 4/11/08 7:48 am page viii 2008 text sampler page 84ontents 1 2008 carnegie learning, inc. contents 11.6 going out for pizza permutations and combinations p. 521 11.7 picking out socks independent and dependent events p. 527 11.8 probability on the shuffleboard court geometric probabilities p. 533 11.9 game design geometric probabilities and fair games p. 537 statistical analysis p. 544 12.1 taking the psat measures of central tendency p. 547 12.2 compact discs collecting and analyzing data p. 553 12.3 breakfast cereals quartiles and box-and-whisker plots p. 557 12.4 home team advantage? sample variance and standard deviation p. 563 quadratic and exponential functions and logic p. 568 13.1 solid carpentry the pythagorean theorem and its converse p. 571 13.2 location, location, location the distance and midpoint formulas p. 575 13.3 "old" mathematics completing the square and deriving the quadratic formula p. 583 13.4 learning to be a teacher vertex form of a quadratic equation p. 587 13.5 screen saver graphing by using parent functions p. 593 13.6 science fair introduction to exponential functions p. 601 13.7 money comes and money goes exponential growth and decay p. 607 13.8 camping special topic: logic p. 615 glossary p. g-1 index p. i-1 1 2 1 3 a1s1_fm.qxd 4/11/08 7:48 am page 1 2008 text sampler page 852008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 86 2008 text sampler page 862008 carnegie learning, inc. algebra i student t ext a1s1_fm.qxd 4/11/08 7:48 am page i 2008 text sampler page 87ooking 2 chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 a1s10100.qxd 4/11/08 7:55 am page 2 2008 text sampler page 88s10100.qxd 4/11/08 7:55 am page 3 2008 text sampler page 89 90esson 1.8 using tables, graphs, and equations, part 1 37 1 .8 a1s10108.qxd 4/11/08 8:02 am page 37 2008 text sampler page 918 a1s10108.qxd 4/11/08 8:02 am page 38 2008 text sampler page 92nvestigate problem 1 lesson 1.8 using tables, graphs, and equations, part 1 39 2008 carnegie learning, inc. 1 variable quantity lower bound upper bound interval number of shirts total cost t ake note remember to label your graph clearly and add a title to the graph. (label) (units) (label) (units) a1s10108.qxd 4/11/08 8:02 am page 39 2008 text sampler page 9s10108.qxd 4/11/08 8:02 am page 40 2008 text sampler page 942008 carnegie learning, inc. algebra i t eacher's implementation guide volume 1 a1t1_fm_v1.qxd 4/11/08 10:30 am page i 2008 text sampler page 95chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 looking 217 142 25 57 512 6.83 33.9 161.85 126.65 the amount of fencing needed is 32 feet and the area is 63 square feet. the amount of fencing needed is 70 meters, and the area is 294 square meters. the amount of fencing needed is 410 inches, and the area is 9100 square inches. a1t10100.qxd 4/1/08 9:41 am page 2 2008 text sampler page 96t10100.qxd 4/1/08 9:41 am page 3 2008 text sampler page 97t10100.qxd 4/1/08 9:41 am page 4 2008 text sampler page 98esson 1.8 using tables, graphs, and equations, part 1 37a nctm content standards grades 9–12 expectations algebra standards generalize patterns using explicitly defined and recursively defined functions. interpret representations of functions of two variables. use symbolic algebra to represent and explain mathematical relationships. draw reasonable conclusions about a situation being modeled. approximate and interpret rates of change from graphical and numerical data. measurement standards make decisions about units and scales that are appropriate for problem situations involving measurement. 1 .8 lesson overview within the context of this lesson, students will be asked to: represent a problem situation in a sentence, using a table of values, using an equation in two variables, and using a graph. determine an initial value for the number of shirts ordered if given the final result of the total cost of the order . write an equation with two variables. classify variables as dependent or independent. identify the advantages and disadvantages of each form of representing the problem situation: a sentence, an equation, a table of values, and a graph. essential questions the following key questions are addressed in this lesson: 1. what is a set-up fee? 2. how can you determine the initial number of shirts ordered if given the total cost of an order of shirts? 3. what is an independent variable? 4. what is a dependent variable? 5. what different representations can you use to represent a problem situation? get ready learning by doing lesson map a1t10108.qxd 4/1/08 9:49 am page 35 2008 text sampler page 997b chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1 warm up place the following questions or an applicable subset of these questions on the board before students enter class. students should begin working as soon as they are seated. evaluate each expression. 1. 32 2. 27 3. 4 4. 9 5. 39 6. 364 7. 3 8. 32 9. 0 motivator begin the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson is about making custom t -shirts. the motivating questions are about working for a custom t -shirt shop. ask the students the following questions to get them interested in the lesson. suppose your club wanted to have t -shirts printed with a design that you drew. what work would have to be done by the t -shirt shop to set up your design to put onto t -shirts? would the cost for the work by the t -shirt shop to scan the design into their computer system be different if the shop was going to print 1 shirt, 10 shirts, or 100 shirts? why might a t -shirt shop charge for that service as a fee that is not based on the number of shirts made? how much money do you think a t -shirt shop should charge for the set-up service? imagine the following situation. a club decides to have t -shirts made and goes to the t -shirt shop to set up the design and to get a price for printing the shirts. before they have any shirts printed, they advertise the shirts at school. the principal and teachers feel that the statement on the shirts is offensive, so the group is not allowed to use that design. in that situation, why would the group still have to pay the set-up fee? (8 4 2)45 5(7) 3 9 6 ( 12 6 ) 3.6(100) 4 54 15 6(14) 21 7 64 (2 8) 16(3 2) 11 3(12) 4 show the way a1t10108.qxd 4/1/08 9:49 am page 36 2008 text sampler page 100esson 1.8 using tables, graphs, and equations, part 1 37 2008 carnegie learning, inc. 1 problem 1 students will calculate the cost for orders of t -shirts for various given values. this problem is intentionally mathematically related to the 8 an hour problem in lesson 1.6. the rate of change in both lessons is a constant 8, but the y-intercept was zero in lesson 1.6 and the y-intercept is 15 in this lesson. grouping ask for a student volunteer to read the scenario and problem 1 aloud. have a student restate the problem. pose the guiding questions below to verify student understanding. have students work together in small groups to complete parts (a) through (d) of problem 1. guiding questions what information is given in this problem? what is a custom t -shirt? what does the number 8 represent in this situation? what does the number 15 represent in this situation? what is a set-up fee? why would the t -shirt shop charge for the set-up? common student errors students often will incorrectly add the cost per shirt of 8 to the set-up cost of 15 to get 23 and then multiply the number of shirts by 23. if this happens, refocus the students by asking them what the set-up fee is for , and how many times they would have to pay the set-up fee. grouping call the class back together to have the students discuss and present their work for parts (a) through (d) of problem 1. sample answer: i will calculate the total cost of orders. the total cost of an order is the cost of each shirt ordered plus a set-up fee. the cost of one shirt is 8 and the set-up fee is 15. sample answer: there is a set-up fee in this problem situation. total cost in dollars: 100(8) 15 815 an order of 100 shirts will cost 815. the total costs were found by first multiplying the number of shirts by the cost of one shirt and then adding the set-up fee to the result. total cost in dollars: 3(8) 15 39 an order of 3 shirts will cost 39. total cost in dollars: 10(8) 15 95 an order of 10 shirts will cost 95. explore together a1t10108.qxd 4/1/08 9:49 am page 37 2008 text sampler page 1018 the numbers of shirts were found by first subtracting the set-up fee from the amount of money available and then dividing the result by the cost for one shirt. the decimal portion of each result is dropped, because a customer cannot receive a partial shirt. number of shirts: because the customer cannot receive a partial shirt, the customer receives 4 shirts. 50 15 8 4.375 39 47 95 215 815 975 1215 number of shirts: because the customer cannot receive a partial shirt, the customer receives 5 shirts. 60 15 8 5.625 number of shirts: because the customer cannot receive a partial shirt, the customer receives 25 shirts. 220 15 8 25.625 3 4 10 25 100 120 150 the table shows sample answers. explore together investigate problem1 students will calculate the number of shirts that can be purchased for various given amounts of money. students will create a table of values for the problem situation. grouping ask for a student volunteer to read question 1 aloud. have a student restate the problem. pose the guiding questions below to verify student understanding. have students work together in small groups to complete questions 1 through 6. guiding questions how did you find the cost for an order of t -shirts in parts (a) through (d) of problem 1? what are you asked to find in question 1? how is question 1 different from parts (a) through (d)? how can you find the number of shirts that can be purchased for a given amount of money? what is the smallest possible number of shirts that could be ordered? why might a group pay the set-up fee for a shirt design, but not buy any shirts? do you think that it is very common for a group to pay the set-up fee, but order no shirts? what is the largest reasonable number of shirts of one design that you think may be ordered? when might a person or group order such a large number of shirts of the same design? notes in question 2, students are asked to create a table of values for this problem situation. it is important that they be required to create a table of values that is reasonable for the situation and that also models examples of different-sized orders. many students will use the values of 1 through 5 for the number of t -shirts, if allowed to do so. such a limited table makes graphing and fully understanding the situation very difficult. always require that students' tables and graphs encompass the entire situation. a1t10108.qxd 4/1/08 9:49 am page 38 2008 text sampler page 102esson 1.8 using tables, graphs, and equations, part 1 39 2008 carnegie learning, inc. 1 investigate problem 1 variable quantity lower bound upper bound interval number of shirts total cost the variable quantities are the number of shirts ordered s and the total cost c in dollars. the constant quantities are the cost per shirt in dollars and the set-up fee in dollars. 0 150 10 0 1500 100 c s 10 20 30 40 50 0 60 70 80 90 100 110 150 120 130 140 100 200 300 400 0 500 600 700 800 900 1000 1100 1200 1300 1400 1500 u.s. shirts number of shirts total cost (dollars) c 8s 15 the total cost depends on the number of shirts ordered. the variable s is the independent variable and the variable c is the dependent variable. explore together investigate problem 1 students will classify the variables in this situation as dependent or independent and create a graph. grouping call the class back together to have the students discuss and present their work for questions 1 through 6. notes when discussing question 3, talk about the benefits of using meaningful variables such as s or n for the number of shirts and c or d for the cost of the shirts in dollars. some students are exposed so frequently to the variables x and y that it is more difficult for them to solve equations in geometry and physics courses with meaningful variables. later in this course, the students will have considerable practice writing, solving, and graphing equations in terms of the variables x and y. common student errors students may still struggle to classify each variable as a dependent or an independent variable. be ready to refocus students when debriefing questions 4 through 6. grouping have students work in small groups to complete questions 7 through 9. pose the guiding questions below to verify student understanding. guiding questions what bounds would be appropriate for the number of shirts? explain. what bounds would be appropriate for the total cost of the order? explain. what interval is appropriate for the number of shirts? explain. what interval is appropriate for the total cost of the order? explain. t ake note remember to label your graph clearly and to add a title to the graph. a1t10108.qxd 4/1/08 9:49 am page 39 2008 text sampler page 10 the equation is c 8s 15, where c represents the total cost in dollars and s represents the number of shirts ordered. sample answer: sentences allow you to understand what information you need to find. however , sentences don't give a visual representation of a problem situation. a table gives you specific values for the problem situation. however , it does not show values between those given. a graph allows you to find different values for the problem situation and to visually see how the data in the problem situation are related. however , the values may not be exact. an equation allows you to generalize the problem situation and find any value exactly. however , an equation doesn't give a visual representation of the problem situation. explore together investigate problem 1 students will analyze the various representations that they used to model the problem situation. grouping call the class back together to have the students discuss and present their work for questions 6 through 9. common student errors some students may not have noticed the various representations of the problem situation. you may need to call the students together and discuss question 9 if several of the groups are unable to complete the question on their own. guiding questions where in this lesson was the problem situation represented in a sentence? were there any other parts of this lesson in which the problem situation was represented in a sentence? where was this problem situation represented using a table in this lesson? where was this problem modeled using a graph in this lesson? where was this problem modeled using an equation? key formative assessments compare your graph in lesson 1.6 for the 8 an hour problem with your graph for this situation. how are the graphs similar? how are they different? what was different in the scenario of the 8 an hour problem and the u.s. shirts problem? how do the differences in the scenarios and graphs relate to each other? summarize the advantages and disadvantages for each representation used in this lessont10108.qxd 4/1/08 9:49 am page 40 2008 text sampler page 104esson 1.8 using tables, graphs, and equations, part 1 40a 2008 carnegie learning, inc. 1 close review all key terms and their definitions. include the terms independent variable and dependent variable. you may also want to review any other vocabulary terms that were discussed during the lesson, which may include representations and models, and for more advanced students possibly slope and y-intercept. remind the students to write the key terms and their definitions in the notes section of their notebooks. you may also want the students to include examples. ask the students to summarize the mathematics from this lesson. ask the students to construct another scenario that could be modeled by the equation of . then ask how the graph of such an equation would compare to the graph of the problem situation for the problem they just finished. finally, verify their thoughts by graphing this equation as a whole class. discuss with the students how the values for c will compare from the equation in the u.s. shirts problem and the new equation . ties to the cognitive tutor software in some cognitive tutor software units, students graph the relationship between quantities by plotting points. this helps students understand the correspondence between rows in a table and points on a graph. students should reflect on the fact that the graph of the relationship is a straight line. this is no coincidence. in these problems, the dependent quantity changes at a constant rate, and the graph of a relationship with a constant rate of change will always be a line. c 8s 12 c 8s 15 c 8s 12 wrap up assignment use the assignment for lesson 1.8 in the student assignments book. see the teacher's resources and assessments book for answers. assessment see the assessments provided in the teacher's resources and assessments book for chapter 1. open-ended writing t ask ask the students to compare the problem situation in the consultant's problem in the previous lesson with the problem situation in the u.s. shirts problem. how are they similar and how are they different? how did the similarities and differences affect the different representations of the problem situations? follow up a1t10108.qxd 4/1/08 9:49 am page 41 2008 text sampler page 1050b chapter 1 patterns and multiple representations 2008 carnegie learning, inc. 1_______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ notes a1t10108.qxd 4/1/08 9:49 am page 42 2008 text sampler page 1062008 carnegie learning, inc. algebra i t eacher's resources and assessments a1tra_fm.qxd 5/6/08 9:42 am page i 2008 text sampler page 1072008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 108 2008 text sampler page 108 a1tra_fm.qxd 5/6/08 9:42 am page iii 2008 text sampler page 1092008 carnegie learning, inc. 1 chapter 1 assignments 15 u.s. shirts using tables, graphs, and equations, part 1 define each term in your own words. 1. variable quantity 2. constant quantity evaluate each algebraic expression for the value given. show your work. 3. when 4. when 5. when you want to save money for college. you have already saved 500, and you are able to save 75 each week. 6. if you continue to save money at this rate, what will your total savings be in 3 weeks? what will your total savings be in 10 weeks? what will your total savings be in 6 months? (hint: there are four weeks in one month.) 7. use a complete sentence to explain how you found the total savings in question 6. 8. if you continue to save money at this rate, how long will it take you to save 2000? how long will it take you to save 8000? how long will it take you to save 11,750? 9. use a complete sentence to explain how you found the answers to the number of weeks in question 8. r 10 1 2 r 30 m 4 10 2m s 20 8s 15 assignment name ___________________________________________________ date _____________________ assignment for lesson 1.8 sample answer: a variable quantity is a quantity that does not have a fixed value. sample answer: a constant quantity is a quantity that has a fixed value. 8(20) 15 175 10 – 2(4) 2 (10) 30 35 1 2 725; 1250; 2300 sample answer: the total savings were found by first multiplying the weekly savings rate by the number of weeks and then adding the amount already saved. 20 weeks; 100 weeks; 150 weeks sample answer: the numbers of weeks were found by first subtracting the amount already saved from the amount saved and then dividing the result by the weekly savings rate. a1g1_1_te.qxd 5/6/08 6:56 am page 10 2008 text sampler page 1102008 carnegie learning, inc. 1 16 chapter 1 assignments variable quantity lower bound upper bound interval time total savings 10. complete the table using the data from questions 6 and 8. be sure to fill in your labels and units. quantity name unit 11. use the grid below to create a line graph of the data from the table in question 10. first, choose your bounds and intervals. be sure to label your graph clearly. time total savings weeks dollars 3 725 10 1250 20 2000 24 2300 100 8000 150 11,750 0 150 10 0 15,000 1000 s t 10 20 30 40 50 0 60 70 80 90 100 110 150 120 130 140 1000 2000 3000 4000 0 5000 6000 7000 8000 9000 10,000 11,000 12,000 13,000 14,000 15,000 college savings time (weeks) total savings (dollars) a1g1_1_te.qxd 5/6/08 6:56 am page 11 2008 text sampler page 1112008 carnegie learning, inc. 1 chapter 1 assignments 17 name ___________________________________________________ date _____________________ 12. write an algebraic equation for the problem situation. use a complete sentence in your answer . sample answer: the equation s 75t 500, where s represents the total savings in dollars and t represents the time in weeks. a1g1_1_te.qxd 5/6/08 6:56 am page 12 2008 text sampler page 112hapter 1 assessments 1 2008 carnegie learning, inc. 1 pre-t est name ___________________________________________________ date _____________________ find the next two terms in each sequence. 1. 2. 2, 4, 8, 16, 32, ______ , ______ 3. use the nth term to list the first five terms of the sequence. show your work. use the sequence below to answer questions 4 through 6. 4. complete the table by filling in the number of hexagons in each term of the sequence. 5. write an expression showing the relationship between the term and the number of hexagons in that term. let n represent the term. 6. use the expression from question 5 to find the 10th term of the sequence. show your work. 7. write the power as a product. 3 4 a 5 ____________ a 4 ____________ a 3 ____________ a 2 ____________ a 1 ____________ a n 2n 4 term (n) 1 2 3 4 5 number of hexagons 64 128 2(1) 4 6 2(2) 4 8 2(3) 4 10 2(4) 4 12 2(5) 4 14 3 6 9 12 15 3n 3(10) 30 (3)(3)(3)(3) a1m101.qxd 5/2/08 3:03 pm page 1 2008 text sampler page 113chapter 1 assessments 2008 carnegie learning, inc. 1 pre-t est page 2 8. write the product as a power . (7)(7)(7)(7)(7)(7)(7)(7) 9. perform the indicated operations. show your work. 10. you and your classmates have set up a phone chain to call each other if school is can-celled due to bad weather . you call two classmates, then each of them calls two class-mates, and so on until everyone in your class has been notified. there are 31 students in your class. draw a diagram to show how each student would be reached to be notified of school cancellations. 11. find the sum of the numbers from 1 to 10. show your work. 12. write an expression for the sum of the numbers from 1 to n. 4 2 (8 3)6 7 8 46 16 (5)6 16 30 1 2 3 4 5 6 7 8 9 10 55 or 55 10(11) 2 n(n 1) 2 a1m101.qxd 5/2/08 3:03 pm page 2 2008 text sampler page 114hapter 1 assessments 3 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 13 through 16. the spanish club at your school is selling animal piñatas to raise money for a trip to mexico city. the club earns a profit of 3 on each piñata sold. the sale runs for 5 weeks. the number of piñatas sold each week are 15, 22, 8, 35, and 42. 13. make a table to show the number of piñatas sold and the profit made for each week of the sale. 14. create a bar graph to display the profit for each week of the sale. pre-t est page 3 name ___________________________________________________ date _____________________ 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 spanish club piñata sale profts proft (dollars) week 1 week 2 week 3 week 4 week 5 week number of piñatas profit piñatas dollars week 1 15 45 week 2 22 66 week 3 8 24 week 4 35 105 week 5 42 126 a1m101.qxd 5/2/08 3:03 pm page 3 2008 text sampler page 1155. create a graph to display the relationship between the number of piñatas sold and the profit. first, choose your bounds and intervals. be sure to label your graph clearly. 16. write an algebraic equation that you could use to show the profit for any number of piñatas sold. 4 chapter 1 assessments 2008 carnegie learning, inc. 1 pre-t est page 4 variable quantity lower bound upper bound interval number of piñatas profit sample answer: p 3n, where p is profit and n is the number of piñatas sold. p n 3 6 9 12 15 0 18 21 24 27 30 33 45 36 39 42 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 spanish club piñata sale profits number of piñatas profit (dollars) 0 45 3 0 150 10 a1m101.qxd 5/2/08 3:03 pm page 4 2008 text sampler page 116hapter 1 assessments 5 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 17 and 18. two airlines offer special group rates to your school's spanish club for the trip to mexico city. the mexican air airline offers a roundtrip airfare of 250 per person. the fiesta airline offers a roundtrip airfare of 150 per person if the club agrees to pay a one-time group rate processing fee of 1000. 17. which airline offers the better deal if only nine students from the spanish club are able to fly to mexico city? show all your work and use complete sentences in your answer . 18. which airline offers the better deal if 20 students are able to fly to mexico city? show all your work and use complete sentences in your answer . pre-t est page 5 name ___________________________________________________ date _____________________ mexican air: fiesta: total cost in dollars: 250(9) 2250 150(9) 1000 2350 the total cost for nine students on mexican air is 2250, and the total cost for nine students on fiesta is 2350. therefore, mexican air offers a better deal if only nine students are able to fly to mexico city. mexican air: fiesta: total cost in dollars: 250(20) 5000 150(20) 1000 4000 the total cost for 20 students on mexican air is 5000, and the total cost for 20 students on fiesta is 4000. therefore, fiesta offers a better deal if 20 students are able to fly to mexico city. a1m101.qxd 5/2/08 3:03 pm page 5 2008 text sampler page 117chapter 1 assessments 2008 carnegie learning, inc. 1 a1m101.qxd 5/2/08 3:03 pm page 6 2008 text sampler page 118hapter 1 assessments 7 2008 carnegie learning, inc. 1 find the next two terms in each sequence. 1. 2. 4, 7, 10, 13, ______ , ______ 3. use the nth term to list the first five terms of the sequence. show your work. use the sequence below to answer questions 4 through 6. 4. complete the table by filling in the number of triangles in each term of the sequence. 5. write an expression showing the relationship between the term and the number of triangles in that term. let n represent the term. 6. use the expression from question 5 to find the 10th term of the sequence. show your work. 7. write the power as a product. 5 6 a 5 _______________ a 4 _______________ a 3 _______________ a 2 _______________ a 1 _______________ a n 3n 5 post-t est name ___________________________________________________ date _____________________ term 1 2 3 4 5 number of triangles 2 3 4 5 6 16 19 3(1) 5 8 3(2) 5 11 3(3) 5 14 3(4) 5 17 3(5) 5 20 n 1 10 1 11 (5)(5)(5)(5)(5)(5) a1m101.qxd 5/2/08 3:03 pm page 7 2008 text sampler page 119chapter 1 assessments 2008 carnegie learning, inc. 1 post-t est page 2 8. write the product as a power . (3)(3)(3)(3)(3)(3)(3)(3)(3) 9. perform the indicated operations. show your work. 10. you and your classmates have set up an email chain to notify each other if school is cancelled due to bad weather . you email three classmates, then each of them emails three classmates, and so on until everyone in your class has been notified. there are 40 students in your class. draw a diagram to show how each student would be reached to be notified of school cancellations. 11. find the sum of the numbers from 1 to 200. show your work. 12. write an expression for the sum of the numbers from 1 to n. (2 1) 3 5(2) 3 9 17 3 3 10 27 10 200(201) 2 20,100 n(n 1) 2 a1m101.qxd 5/2/08 3:03 pm page 8 2008 text sampler page 120hapter 1 assessments 9 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 13 through 16. a local ballet company is selling tickets for their upcoming performances of swan lake. the company earns a profit of 8 on each ticket they sell. the first week that tickets are on sale, they sold 30 tickets on monday, 27 on tuesday, 18 on wednesday, 6 on thursday, and 41 tickets on friday. 13. make a table to show the number of tickets sold each day during the first week and the profit made on each of those days. 14. create a bar graph to display the profit for each day of ticket sales in the first week. post-t est page 3 name ___________________________________________________ date _____________________ 25 50 75 100 0 125 150 175 200 225 250 275 300 325 350 375 profts from ticket sales for swan lake proft (dollars) monday tuesday wednesday thursday friday day day number of tickets profit tickets dollars monday 30 240 tuesday 27 216 wednesday 18 144 thursday 6 48 friday 41 328 a1m101.qxd 5/2/08 3:03 pm page 9 2008 text sampler page 1210 chapter 1 assessments 2008 carnegie learning, inc. 1 post-t est page 4 variable quantity lower bound upper bound interval number of tickets profit (dollars) 15. create a graph to display the relationship between the number of tickets sold and the profit. first, choose your bounds and intervals. be sure to label your graph clearly. 16. write an algebraic equation that you could use to show the profit for any number of tickets sold. sample answer: p 8n, where p is profit and n is the number of tickets sold. p n 3 6 9 12 15 0 18 21 24 27 30 33 45 36 39 42 25 50 75 100 0 125 150 175 200 225 250 275 300 325 350 375 profits from ticket sales for swan lake number of tickets sold profit (dollars) 0 45 3 0 375 25 a1m101.qxd 5/2/08 3:03 pm page 10 2008 text sampler page 122hapter 1 assessments 11 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 17 and 18. the director of the local ballet company needs to print the programs for swan lake. janet's print shop charges .25 a program plus a 35 set-up fee. the printing press charges .18 a program plus a 50 set-up fee. 17. which printing company offers the better deal if 200 programs are printed? show your work and use complete sentences in your answer . 18. which printing company offers the better deal if 300 programs are printed? show your work and use complete sentences in your answer . post-t est page 5 name ___________________________________________________ date _____________________ janet's print shop: the printing press: total cost in dollars: 0.25(200) 35 85 0.18(200) 50 86 the total cost for printing 200 programs at janet's print shop is 85, and the total cost for printing 200 programs at the printing press is 86. therefore, janet's print shop offers a better deal if 200 programs are printed. janet's print shop: the printing press: total cost in dollars: 0.25(300) 35 110 0.18(300) 50 104 the total cost for printing 300 programs at janet's print shop is 110, and the total cost for printing 300 programs at the printing press is 104. therefore, the printing press offers a better deal if 300 programs are printed. a1m101.qxd 5/2/08 3:03 pm page 11 2008 text sampler page 1232 chapter 1 assessments 2008 carnegie learning, inc. 1 a1m101.qxd 5/2/08 3:03 pm page 12 2008 text sampler page 124hapter 1 assessments 13 2008 carnegie learning, inc. 1 mid-chapter t est name ___________________________________________________ date _____________________ read the scenario below. use the scenario to complete questions 1 through 8. a local concrete company has hired you for the summer . your first project is to help pour a driveway. before you can pour the driveway, you must put wood forms in place to hold the concrete until it is dry. the first form in a driveway is made by constructing a square out of four 2 4's that are each ten feet long. the second form is made by using one side of the first form and three other 2 4's to make a second square. this process is continued to the end of the driveway. the diagrams show the first two steps in completing the form. 1. draw a diagram that shows steps 3, 4, and 5 in the form construction. step 3 step 4 step 5 2. the completed driveway will be 80 feet long. the company would refer to this as an eight-form driveway. draw a diagram that shows what the completed form will look like. (hint: draw the eighth step in the form construction.) step 1 step 2 a1m101.qxd 5/2/08 3:03 pm page 13 2008 text sampler page 1254 chapter 1 assessments 2008 carnegie learning, inc. 1 3. the company needs to know the number of 2 4's needed to complete the driveway. they will also need to know the area of the driveway in order to properly bill their clients. complete the table to help organize this information. 4. write the sequence of numbers formed by the area of one form, two forms, three forms, and so on. use a complete sentence to describe the pattern produced by the area. 5. use a complete sentence to describe the relationship between the number of forms and the number of 2 4's required to complete the forms. 6. write an expression that gives the number of 2 4's needed to complete a driveway with n forms. 7. use your expression from question 6 to determine the number of 2 4's needed for a 12-form driveway. show your work. 8. a. write the sequence of numbers formed by the number of 2 4's. b. find the 10th term of this sequence. show your work. c. use a complete sentence to explain what the 10th term represents. mid-chapter t est page 2 number of forms 1 2 3 4 5 6 number of 2 4's area (square feet) 4 7 10 13 16 19 100 200 300 400 500 600 100, 200, 300, 400, 500, 600, … sample answer: add 100 to the previous term to get the next term. sample answer: multiply the number of forms by 3 and then add 1 to get the number of 2 4's needed. sample answer: 3n 1 3(12) 1 37 4, 7, 10, 13, 16, 19, … 3(10) 1 31 the 10th term represents the number of 2 x 4's it would take to make the forms for a 10-form driveway. a1m101.qxd 5/2/08 3:03 pm page 14 2008 text sampler page 126hapter 1 assessments 15 2008 carnegie learning, inc. 1 write each power as a product. 9. 7 4 10. 12 6 write each product as a power . 11. (15)(15)(15)(15)(15) 12. (6)(6)(6)(6)(6)(6)(6)(6)(6) perform the indicated operations. show your work. 13. 14. evaluate each expression for the given value of the variable. show your work. 15. evaluate when r is 12. 16. evaluate when t is 36. 17. use the nth term to list the first five terms of the sequence. show your work. a 5 _______________ a 4 _______________ a 3 _______________ a 2 _______________ a 1 _______________ a n 20 2n t 4 2r 8 25 (3 5) 2 4 (6 3) 3 2(1 4) mid-chapter t est page 3 name ___________________________________________________ date _____________________ 20 – 2(1) 18 20 – 2(2) 16 20 – 2(3) 14 20 – 2(4) 12 20 – 2(5) 10 (7)(7)(7)(7) (12)(12)(12)(12)(12)(12) 15 5 6 9 37 3 3 2(5) 27 10 33 25 8 16 17 16 32 2(12) 8 24 8 36 4 9 a1m101.qxd 5/2/08 3:03 pm page 15 2008 text sampler page 1276 chapter 1 assessments 2008 carnegie learning, inc. 1 read the scenario below. use the scenario to answer questions 18 through 20. a local college has decided to build new sidewalks to connect the main administration building to the other buildings on campus. they can only build two new sidewalks a month. it will take 6 months to connect the administration building to all of the other buildings on campus. the diagrams show the number of sidewalks that have been built after 1, 2, and 3 months. 18. draw diagrams that would represent the number of sidewalks after 4, 5, and 6 months. 19. complete the table below to show the number of sidewalks built after 1, 2, 3, 4, 5, and 6 months. 20. the college wants to put in sidewalks connecting the library to the other buildings on campus. there is already a sidewalk connecting the library to the main administration building. how many more sidewalks will need to be built in order to connect the library to the remaining buildings on campus? 21. write an algebraic expression to find the sum of the numbers from 1 to n. 22. use your answer to question 21 to find the sum of the numbers from 1 to 85. show your work. 2 months 3 months 1 month mid-chapter t est page 4 number of months 1 2 3 4 5 6 number of sidewalks 2 4 6 8 10 12 6 months 5 months 4 months 11 n(n 1) 2 3655 7310 2 85(85 1) 1 85(86) 2 a1m101.qxd 5/6/08 9:52 am page 16 2008 text sampler page 128hapter 1 assessments 17 2008 carnegie learning, inc. 1 term (n) 1 2 3 4 5 sequence 5 25 125 625 3125 term (n) 1 2 3 4 5 sequence 5 15 25 35 45 end of chapter t est name ___________________________________________________ date _____________________ for each sequence, find the next two terms and describe the pattern. 1. 3, 8, 13, 18, 23, ______ , ______ 2. 3, 9, 27, 81, ______ , ______ 3. 4. for each sequence, find the expression for the nth term and describe the pattern. 5. 6. 28 33 add 5 to the previous term to find the next term. 243 729 multiply the previous term by 3 to find the next term. start with the previous term. draw a concentric circle around the other circles to find the next term. start with the previous term. add a row of triangles below the figure to find the next term. the expression is 10n – 5, so multiply the term number (n) by 10 and then subtract 5. the expression is 5 n , so raise 5 to the power of the term number (n). a1m101.qxd 5/2/08 3:03 pm page 17 2008 text sampler page 1298 chapter 1 assessments 2008 carnegie learning, inc. 1 end of chapter t est page 2 7. write the power as a product. 8. write the product as a power . 5 4 (8)(8)(8) use the nth term to list the first five terms of each sequence. show your work. 9. 10. ________________ ________________ ________________ ________________ ________________ read the scenario below. use the scenario to answer questions 11 through 23. you are a volunteer for the school store. one of the most popular items is strawberry-banana-orange juice. there are two local vendors that will deliver the juice to the school store at the beginning of every month. healthy drinks, inc. charges .39 per bottle with a delivery fee of 25. the squeeze charges .18 per bottle with a delivery fee of 65. 11. you want to stock the store with juice at the beginning of the school year . how much will it cost to purchase 300 bottles of strawberry-banana-orange juice from healthy drinks, inc.? show your work and use a complete sentence in your answer . 12. how much will it cost to purchase 300 bottles of strawberry-banana-orange juice from the squeeze? show your work and use a complete sentence in your answer . a 5 a 4 a 3 a 2 a 1 a n (n 1) 2 3 a n 3(n 1) 2 ________________ ________________ ________________ ________________ ________________ a 5 a 4 a 3 a 2 a 1 (5)(5)(5)(5) 8 3 cost for 300 bottles of strawberry-banana-orange juice: 0.39(300) 25 142 it will cost 142 to purchase 300 bottles of strawberry-banana-orange juice from healthy drinks, inc. cost for 300 bottles of strawberry-banana-orange juice: 0.18(300) 65 119 it will cost 119 to purchase 300 bottles of strawberry-banana-orange juice from the squeeze. (1 – 1) 2 3 3 (2 – 1) 2 3 4 (3 – 1) 2 3 7 (4 – 1) 2 3 12 (5 – 1) 2 3 19 3(5 1) 2 9 3(4 1) 2 15 2 3(3 1) 2 6 3(2 1) 2 9 2 3(1 1) 2 3 a1m101.qxd 5/2/08 3:03 pm page 18 2008 text sampler page 130hapter 1 assessments 19 2008 carnegie learning, inc. 1 13. complete the table summarizing the cost of purchasing strawberry-banana-orange juice from each vendor based on last year's actual monthly sales. remember to label units. 14. let c represent the cost of purchasing bottles of juice from healthy drinks, inc. and b represent the number of bottles. write an equation that relates c and b for this problem situation. 15. let c represent the cost of purchasing bottles of juice from the squeeze and b represent the number of bottles. write an equation that relates c and b for this problem situation. 16. what was the average number of bottles of juice sold in a month last year? show your work and use a complete sentence in your answer . end of chapter t est page 3 name ___________________________________________________ date _____________________ month number of bottles of juice purchased cost of purchasing from healthy drinks, inc. cost of purchasing from the squeeze bottles dollars dollars september 187 october 229 november 162 december 137 january 171 february 201 march 192 april 258 may 214 june 79 97.93 98.66 114.31 106.22 88.18 94.16 78.43 89.66 91.69 95.78 103.39 101.18 99.88 99.56 125.62 111.44 108.46 103.52 55.81 79.22 average number of bottles of juice sold in a month: 1830 10 183 the average number of bottles of juice sold in a month was 183 bottles. c 0.39b 25 c 0.18b 65 a1m101.qxd 5/2/08 3:03 pm page 19 2008 text sampler page 1310 chapter 1 assessments 2008 carnegie learning, inc. 1 17. create a bar graph to display the costs of purchasing strawberry-banana-orange juice from healthy drinks, inc. each month. 18. create a graph displaying the cost of purchasing strawberry-banana-orange juice from both healthy drinks, inc. and the squeeze. first, choose your bounds and intervals. be sure to label your graph clearly. end of chapter t est page 4 variable quantity lower bound upper bound interval bottles cost 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 cost of purchase from healthy drinks, inc. cost (dollars) sep oct nov dec jan feb mar apr may jun 0 300 20 0 150 10 a1m101.qxd 5/2/08 3:03 pm page 20 2008 text sampler page 132hapter 1 assessments 21 2008 carnegie learning, inc. 1 19. estimate the number of bottles of juice for which the total costs for each company are the same and explain how you found your answer . use complete sentences in your answer . 20. for how many bottles of juice is healthy drinks, inc. more expensive to order from? use a complete sentence in your answer . end of chapter t est page 5 name ___________________________________________________ date _____________________ c b 20 40 60 80 100 0 120 140 160 180 200 220 300 240 260 280 10 20 30 40 0 50 60 70 80 90 100 110 120 130 140 150 cost of juice from healthy drinks, inc. and the squeeze number of bottles of juice cost (dollars) c 0.18b 65 c 0.39b 25 sample answer: the total costs are about the same when about 190 bottles of juice are purchased. the total costs are the same where the lines intersect each other on the graph. to find this number , start at the point of intersection and move straight down to the horizontal axis to read the number of bottles of juice for this total cost. healthy drinks, inc. is more expensive to order from when you order more than 190 bottles of juice. a1m101.qxd 5/2/08 3:03 pm page 21 2008 text sampler page 1332 chapter 1 assessments 2008 carnegie learning, inc. 1 21. for how many bottles of juice is the squeeze more expensive to order from? use a complete sentence in your answer . 22. the faculty sponsor who is responsible for the school store has asked you to write a report that compares the costs of ordering from each vendor . she would also like you to make a recommendation about which vendor you would choose if you had to order from the same vendor for the entire school year . use complete sentences in your answer . 23. if you were able to choose a different vendor each month, would you? use complete sentences in your answer . end of chapter t est page 6 the squeeze is more expensive to order from when you order 190 or fewer bottles of juice. sample answer: the total cost is the same when approximately 190 bottles of juice are purchased. if you purchase less than 190 bottles, healthy drinks, inc. is cheaper . if you purchase more than 190 bottles, the squeeze is cheaper . because the average number of bottles sold each month last year is 183, i would recommend healthy drinks, inc. as the vendor for the entire school year . answers will vary. a1m101.qxd 5/2/08 3:03 pm page 22 2008 text sampler page 134hapter 1 assessments 23 2008 carnegie learning, inc. 1 1. which choice shows the next two terms in the sequence? 1, 101, 2, 102, 3, 103, … a. 4 and 140 b. 4 and 401 c. 4 and 104 d. 4 and 114 2. which choice shows the next two items in the sequence? a. b. c. d. 3. which statement describes the pattern? –1, 10, –100, 1000, … a. start with the previous term, and multiply by 10 to get the next term. b. start with the previous term, and add a zero to get the next term. c. start with the previous term, and multiply by 100 to get the next term. d. start with the previous term, and multiply by –10 to get the next term. 4. simplify a. –1 b. 3 c. 14 d. 18 5 2 (3 6) 14 7 . standardized t est practice name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 23 2008 text sampler page 1354 chapter 1 assessments 2008 carnegie learning, inc. 1 5. which expression is equivalent to 4 6 ? a. (4)(6) b. (4)(4)(4)(4)(4)(4) c. (6)(6)(6)(6) d. 46 6. how many cups of flour are needed for 10 loaves of bread? a. 12 b. 16 c. 20 d. 24 7. evaluate when . a. 14 b. 18 c. 23 d. 41 8. which expression represents the nth term of the sequence? a. 7n 1 b. 7n c. 6n d. n 7 p 6 3p 5 standardized t est practice page 2 loaves of bread 1 2 3 4 5 cups of flour 2 4 6 8 10 term (n) 1 2 3 4 5 sequence 6 13 20 27 34 a1m101.qxd 5/2/08 3:03 pm page 24 2008 text sampler page 136hapter 1 assessments 25 2008 carnegie learning, inc. 1 9. which number represents a 4 ? a. 6 b. 8 c. 10 d. 12 10. a family of 8 has just signed a contract for a new cellular phone service so that they can call each other for free. to try it out, each person in the family calls every other person once unless that person has already called them. how many calls does the family make? a. 10 b. 15 c. 21 d. 28 11. what is the sum of the numbers from 1 to 300? a. 301 b. 4515 c. 45,150 d. 300,001 12. the average speed of an airplane is 325 miles per hour . which expression shows the distance in miles an airplane could travel in n hours? a. b. 325n c. d. n 325 325 n 325 n a n 2n 4 standardized t est practice page 3 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 25 2008 text sampler page 1376 chapter 1 assessments 2008 carnegie learning, inc. 1 13. james earns 6.25 an hour at work. the table shows his hours and earnings for each week in one month. which graph correctly displays the relationship between hours worked and earnings? a. b. c. d. 0 15 30 45 60 75 90 105 120 135 150 0 2 4 6 8 10 12 14 16 18 20 h e time worked (hours) earnings (dollars) james' earnings 0 15 30 45 60 75 90 105 120 135 150 0 2 4 6 8 10 12 14 16 18 20 e h earnings (dollars) time worked (hours) james' earnings 0 2 4 6 8 10 12 14 16 18 20 0 15 30 45 60 75 90 105 120 135 150 e h earnings (dollars) time worked (hours) james' earnings 0 2 4 6 8 10 12 14 16 18 20 0 15 30 45 60 75 90 105 120 135 150 h e time worked (hours) earnings (dollars) james' earnings standardized t est practice page 4 week time worked earnings hours dollars week 1 15 93.75 week 2 18 112.50 week 3 12 75 week 4 13 81.25 a1m101.qxd 5/2/08 3:03 pm page 26 2008 text sampler page 138hapter 1 assessments 27 2008 carnegie learning, inc. 1 14. james earns 6.25 an hour at work. the table in question 13 shows his hours and earnings for each week in one month. which bar graph correctly displays the relationship between the week and the time worked? a. b. c. d. 0 15 30 45 60 75 90 105 120 135 150 week time worked per week 1 2 3 4 time worked (hours) 0 15 30 45 60 75 90 105 120 135 150 week time worked per week 1 2 3 4 earnings (dollars) 0 2 4 6 8 10 12 14 16 18 20 week time worked per week 1 2 3 4 time worked (hours) 0 2 4 6 8 10 12 14 16 18 20 time worked (hours) time worked per week 1 2 3 4 week standardized t est practice page 5 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 27 2008 text sampler page 1398 chapter 1 assessments 2008 carnegie learning, inc. 1 15. james earns 6.25 an hour at work. which algebraic equation shows the amount of money e that james earns in n hours? a. b. c. d. 16. angelica earns 7.50 each hour she works. how many hours will she have to work to buy a bicycle that costs 90? a. 6 b. 9 c. 12 d. 15 17. angelica earns 7.50 each hour she works. how much money will angelica earn if she works for 6 hours and 12 minutes? a. 45 b. 46.50 c. 46.75 d. 135 18. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. the total cost of an order was 562. how many frisbees were printed? a. 47 b. 275 c. 281 d. 550 e 6.25n n 6.25e e 6.25 n e 6.25 n standardized t est practice page 6 a1m101.qxd 5/2/08 3:03 pm page 28 2008 text sampler page 140hapter 1 assessments 29 2008 carnegie learning, inc. 1 19. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. which algebraic equation shows the cost c of printing f frisbees? a. b. c. d. 20. t -shirts & more print shop will print any image on a frisbee for a cost of 2 per frisbee and a one-time charge of 12 to set up the frisbee design. you say it, we print it will print any image on a frisbee for a cost of 5 per frisbee and no set-up fee. which statement is true? a. you say it, we print it is a better buy if you purchase more than four frisbees. b. t -shirts & more print shop is always the better buy. c. you say it, we print it is always the better buy. d. t -shirts & more print shop is the better buy if you purchase more than four frisbees. c 2f 12 f 2c 12 c 12f 2 c 2f standardized t est practice page 7 name ___________________________________________________ date _____________________ a1m101.qxd 5/2/08 3:03 pm page 29 2008 text sampler page 1412008 carnegie learning, inc. algebra i homework helper a1h1_fm.qxd 5/7/08 1:34 pm page i 2008 text sampler page 1422 chapter 1 homework helper 2008 carnegie learning, inc. 1 students should be able to answer these questions after lesson 1.8: what are the four different ways you have learned to represent a problem situation? what are the advantages and disadvantages of each method? read question 1 and its solution. then, use similar steps to complete questions 2 and 3. 1. you have 183 in your savings account. you want to add 30 to your account each week. how much will you have in your account after 6 weeks? step 1 describe the problem situation. i will calculate the total amount in my savings. this amount is amount saved each week (30) plus the starting value (183). step 2 total in dollars: after 6 weeks, i will have 363 in my savings account. 2. a storage company charges 100 to store your furniture in their warehouse plus an additional 35 per month. how much will they charge to rent their space for 3 months? use a complete sentence in your answer . 3. a car-rental company charges 30 to rent a car plus .25 per mile. how much will the rental company charge a customer who drives 200 miles? use a complete sentence in your answer . use the scenario in question 3 to answer questions 4 through 8. 4. how much will the rental company charge a customer who drives 364 miles? 5. the rental company charges you 135.25 to rent a car . how many miles did you travel? 6. what are the two variable quantities in this problem situation? assign letters to represent these quantities and include the units that are used to measure these quantities. use a complete sentence in your answer . 7. which of the variables from question 6 is the independent variable and which is the dependent variable? use a complete sentence in your answer . 8. write an algebraic equation for the problem situation. use a complete sentence in your answer . 6(30) 183 363 u.s. shirts using t ables, graphs, and equations, part 1 1 .8 directions a1h101.qxd 5/7/08 1:34 pm page 12 2008 text sampler page 143ourse description – geometry carnegie learning tm geometry is designed to be taken after an algebra course. it can be implemented with students at a variety of ability and grade levels. the course assumes number fluency and basic algebra skills such as equation solving. geometry focuses heavily on developing spatial relationships, measurement, and reasoning. students develop properties of figures in two and three dimensions and use these properties to prove statements and calculate measurements. students are introduced to the basic building blocks of geometry: points, lines, and angles. students develop properties of angles and angle pairs, including angles formed by parallel lines. students explore triangles using the pythagorean theorem, special right triangles, the triangle inequality, and trigonometric ratios. students explore quadrilaterals and understand the relationship between squares, rectangles, parallelograms, trapezoids, and rhombi. students explore circles including angles, arcs, chords, tangents, and sectors. students explore polygons including area and perimeter, similarity, congruence, and angle sums. students use reflections, rotations, translations, dilations, and symmetry to transform shapes in the coordinate plane. students calculate slope, distance, and midpoint and use these measures to explore shapes in the coordinate plane. students calculate the volume and surface area of three dimensional figures. students explore ways to represent three dimensional figures including nets and cross sections. 2008 text sampler page 144eometry text setv contents 2008 carnegie learning, inc. contents contents perimeter and area p. 2 1.1 building a deck introduction to polygons, perimeter , and area p. 5 1.2 weaving a rug area and perimeter of a rectangle and area of a parallelogram p. 13 1.3 sailboat racing area of a triangle p. 21 1.4 the keystone effect area of a trapezoid p. 27 1.5 traffic signs area of a regular polygon p. 33 1.6 photography circumference and area of a circle p. 39 1.7 installing carpeting and tile composite figures p. 49 volume and surface area p. 56 2.1 backyard barbecue introduction to volume and surface area p. 59 2.2 turn up the volume volume of a prism p. 65 2.3 bending light beams surface area of a prism p. 73 2.4 modern day pyramids volume of a pyramid p. 79 2.5 soundproofing surface area of a pyramid p. 85 2.6 making concrete stronger volume and surface area of a cylinder p. 91 2.7 sand piles volume and surface area of a cone p. 97 2.8 ball bearings and motion volume and surface area of a sphere p. 103 1 2 ges1_fm.qxd 4/25/08 9:28 am page iv 2008 text sampler page 146ontents v 2008 carnegie learning, inc. contents introduction to angles and t riangles p. 110 3.1 constellations naming, measuring, and classifying angles p. 113 3.2 cable-stayed bridges special angles p. 121 3.3 designing a kitchen angles of a triangle p. 129 3.4 origami classifying triangles p. 137 3.5 building a shed the triangle inequality p. 145 right t riangle geometry p. 150 4.1 tiling a bathroom wall simplifying square root expressions p. 153 4.2 installing a satellite dish the pythagorean theorem p. 157 4.3 drafting equipment properties of 45º–45º–90º triangles p. 163 4.4 finishing concrete properties of 30º–60º–90º triangles p. 167 4.5 meeting friends the distance formula p. 175 4.6 treasure hunt the midpoint formula p. 183 parallel and perpendicular lines p. 188 warehouse space points of concurrency in triangles p. 235 3 4 5 ges1_fm.qxd 4/25/08 9:28 am page v 2008 text sampler page 147i contents 2008 carnegie learning, inc. contents simple t ransformations p. 240 6.1 paper snowflakes reflections p. 243 6.2 good lighting rotations p. 253 6.3 web page design translations p. 261 6.4 shadow puppets dilations p. 269 6.5 cookie cutters symmetry p. 277 similarity p. 280 7.1 ace reporter ratio and proportion p. 283 7.2 framing a picture similar and congruent polygons p. 289 7.3 using an art projector proving triangles similar: aa, sss, and sas p. 297 7.4 modeling a park indirect measurement p. 305 7.5 making plastic containers similar solids p. 311 congruence p. 316 8.1 glass lanterns introduction to congruence p. 319 8.2 computer graphics proving triangles congruent by using sss and sas p. 323 8.3 wind triangles proving triangles congruent by using asa and aas p. 329 8.4 planting graph vines proving triangles congruent by using hl p. 337 8.5 koch snowflake fractals p. 341 6 7 8 ges1_fm.qxd 4/25/08 9:28 am page vi 2008 text sampler page 148ontents vii 2008 carnegie learning, inc. contents quadrilaterals p. 348 9.1 quilting and tessellations introduction to quadrilaterals p. 351 9.2 when trapezoids are kites kites and trapezoids p. 357 9.3 binocular stand design parallelograms and rhombi p. 363 9.4 positive reinforcement rectangles and squares p. 369 9.5 stained glass sum of the interior angle measures in a polygon p. 373 9.6 pinwheels sum of the exterior angle measures in a polygon p. 377 9.7 planning a subdivision rectangles and parallelograms in the coordinate plane p. 383 circles p. 388 10.1 riding a ferris wheel introduction to circles p. 391 10.2 holding the wheel central angles, inscribed angles, and intercepted arcs p. 397 10.3 manhole covers measuring angles inside and outside of circles p. 401 10.4 color theory chords and circles p. 407 10.5 solar eclipses tangents and circles p. 413 10.6 gears arc length p. 417 10.7 playing darts areas of parts of circles p. 421 9 1 0 ges1_fm.qxd 4/25/08 9:28 am page vii 2008 text sampler page 149iii contents 2008 carnegie learning, inc. contents right t riangle t rigonometry p. 426 11.1 wheelchair ramps the tangent ratio p. 429 11.2 golf club design the sine ratio p. 435 11.3 attaching a guy wire the cosine ratio p. 439 11.4 using a clinometer angles of elevation and depression p. 443 extensions in area and volume p. 446 12.1 replacement for a carpenter's square inscribed polygons p. 449 12.2 box it up nets p. 455 12.3 tree rings cross sections p. 459 12.4 minerals and crystals polyhedra and euler's formula p. 463 12.5 isometric drawings compositions p. 469 glossary p. g-1 index p. i-1 1 2 1 1 ges1_fm.qxd 4/25/08 9:28 am page viii 2008 text sampler page 1502008 carnegie learning, inc. geometry student t ext ges1_fm.qxd 4/25/08 9:28 am page i 2008 text sampler page 15188 ges10500.qxd 4/24/08 9:10 am page 188 2008 text sampler page 152 ges10500.qxd 4/24/08 9:10 am page 189 2008 text sampler page 15390 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 ges10501.qxd 4/24/08 9:11 am page 190 2008 text sampler page 154esson 5.4 parallel and perpendicular lines in the coordinate plane 213 y-intercept point-slope form slope-intercept form parallel lines perpendicular lines reciprocal negative reciprocal horizontal line vertical line 5.4 ges10504.qxd 4/24/08 9:12 am page 213 2008 text sampler page 15514 ges10504.qxd 4/24/08 9:12 am page 214 2008 text sampler page 156esson 5.4 parallel and perpendicular lines in the coordinate plane 215 2008 carnegie learning, inc. 5 ges10504.qxd 4/24/08 9:12 am page 215 2008 text sampler page 157 ges10504.qxd 4/24/08 9:12 am page 216 2008 text sampler page 158esson 5.4 parallel and perpendicular lines in the coordinate plane 217 2008 carnegie learning, inc. 5 ges10504.qxd 4/24/08 9:12 am page 217 2008 text sampler page 159 ges10504.qxd 4/24/08 9:12 am page 218 2008 text sampler page 160esson 5.4 parallel and perpendicular lines in the coordinate plane 219 2008 carnegie learning, inc. 5 ges10504.qxd 4/24/08 9:12 am page 219 2008 text sampler page 16120 ges10504.qxd 4/24/08 9:12 am page 220 2008 text sampler page 162esson 5.4 parallel and perpendicular lines in the coordinate plane 221 2008 carnegie learning, inc. 5 ges10504.qxd 4/24/08 9:12 am page 221 2008 text sampler page 163 perpendicular ges10504.qxd 4/24/08 9:12 am page 222 2008 text sampler page 1642008 carnegie learning, inc. geometry t eacher's implementation guide volume 1 get1_fm_v1.qxd 4/29/08 9:05 am page i 2008 text sampler page 16588 2 5 4 2 3 2.5 miles ( 3, 3 1 2 ) 5 miles library get10500.qxd 4/18/08 12:26 pm page 188 2008 text sampler page 166 get10500.qxd 4/18/08 12:26 pm page 189 2008 text sampler page 16790 chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 get10500.qxd 4/18/08 12:26 pm page 190 2008 text sampler page 168esson 5.4 parallel and perpendicular lines in the coordinate plane 213a perpendicular lines point-slope form reciprocal slope-intercept form negative reciprocal y-intercept horizontal line parallel lines vertical line materials graph paper rulers protractors nctm content standards grades 9–12 expectations algebra standards analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, and local and global behavior . use symbolic algebra to represent and explain mathematical relationships. draw reasonable conclusions about a situation being modeled. approximate and interpret rates of change from graphical and numerical data. 5.4 geometry standards use cartesian coordinates and other coordinate systems, such as navigational, polar , or spherical systems, to analyze geometric situations. lesson overview within the context of this lesson, students will be asked to: determine whether lines are parallel. find the equations of lines parallel to given lines. determine whether lines are perpendicular . find the equations of lines perpendicular to given lines. determine equations of horizontal and vertical lines. essential questions the following key questions are addressed in this lesson: 1. how can you determine whether lines are parallel given the equations? 2. how can you determine whether lines are parallel given the graphs of the lines? 3. what formula is used to find the equation of a line given two points? 4. how can you determine whether lines are perpendicular given the equations? 5. how can you determine whether lines are perpendicular given the graph? 6. how do you determine equations of horizontal and vertical lines? 7. what is the slope of a horizontal line? 8. what is the slope of a vertical line? get ready learning by doing lesson map get10504.qxd 4/23/08 12:31 pm page 211 2008 text sampler page 16913b chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5 warm up place the following questions or an applicable subset of these questions on the board before students enter class. students should begin working as soon as they are seated. graph the following equations on your graph paper . 1. 2. 3. 4. motivator begin the lesson with the motivator to get students thinking about the topic of the upcoming problem. this lesson is about designing a parking lot. the motivating questions are about parking lots. ask the students the following questions to get them interested in the lesson. who has his or her driver's license? are you good at parallel parking? do you prefer parking in a parking lot? at what angles are cars normally parked in a parking lot? is there a reason why lines in a parking lot are on a diagonal? y 1 1 3 4 5 2 3 4 5 1 2 3 4 5 5 4 3 2 1 x y 1 1 2 3 4 5 2 3 4 5 1 2 4 5 5 4 3 2 1 x 9x 4y 12 3x 2y 8 y 1 1 2 3 4 5 3 4 5 1 2 3 4 5 5 4 3 2 1 x y 1 1 2 2 3 5 6 7 8 1 2 3 4 5 5 4 2 1 x y 2x 1 y 2x 5 show the way get10504.qxd 4/18/08 1:02 pm page 212 2008 text sampler page 170esson 5.4 parallel and perpendicular lines in the coordinate plane 213 2008 carnegie learning, inc. 5 problem 1 students will determine that parallel lines have the same slopes and different y-intercepts. grouping ask for a student volunteer to read the scenario and problem 1 aloud. pose the guiding questions below to verify student understanding. have students work together as a whole class to complete part (a) through part (f) of problem 1. guiding questions what information is given in this problem? how much does each square on the coordinate plane represent? why do the lines need to be slanted in some parking lots instead of horizontal or vertical lines? common student errors some students will have forgotten how to read a graph. when the students complete the warm up questions discuss graphing by using slope-intercept form and by using a table of values (or a graphing calculator). r q p sample answer: the segments are all the same distance apart, which means they are parallel. the vertical distance between and and between and is 6 meters. ef cd cd ab explore together get10504.qxd 4/23/08 12:39 pm page 213 2008 text sampler page 17114 sample answer: line p: line q: line r: y 12 3 2 (x 0); y 3 2 x 12 y 6 3 2 (x 0); y 3 2 x 6 y 0 3 2 (x 0); y 3 2 x sample answer: the y-intercepts are all multiples of 6. ; sample answer: the slope would be the same as the other lines, and the y-intercept would be 6 units above the y-intercept for line r. y 3 2 x 18 the slope is the same for all three lines. this slope is . 3 2 sample answer: the y-intercepts tell you how many units one line is above (or below) the other . explore together problem 1 grouping students will be working on part (a) through part (f) as a whole class. after completing part (d), have a student read the take note boxes aloud. guiding questions what does slope mean? how do we look at the graph and determine the slope? what happens to the value of rise if we count down and then over? what happens to the value of the run when we count left instead of right? what do x 1 and y 1 represent? what does m represent? what does b represent? what does y-intercept mean? get10504.qxd 4/23/08 12:40 pm page 214 2008 text sampler page 172esson 5.4 parallel and perpendicular lines in the coordinate plane 215 2008 carnegie learning, inc. 5 investigate problem 1 students will make conclusions about the slopes of parallel lines. students will examine equations to determine whether the lines are parallel. grouping ask a student volunteer to read question 1 aloud. have students work together in small groups to complete questions 1 through 4. common student errors students might need direction in question 3 that they need to use point-slope form. in question 4, students who are weak equation solvers will struggle with isolating the variable y. some students will not recognize that they need to change the equations to slope-intercept form in order to compare the slopes. guiding questions will the slopes of parallel lines always be the same? explain. will the y-intercepts of parallel lines ever be the same? explain. is it easier to pick out the value of slope from an equation in standard form or an equation in slope-intercept form? how do you determine whether two lines are parallel without graphing the lines? call the class back together to have the students discuss and present their work for questions 1 through 4. parallel lines in the coordinate plane have the same slope. parallel lines in the plane have different y-intercepts. sample answer: any line that is parallel to the line given by will have a slope of and a y-intercept that is not 4. 2 y 2x 4 y 2x; y 2x 5, y 2x 3 a line parallel to the line given by must have a slope of 5. because you know that the line must pass through the point (4, 0), you can use the point-slope form to write the equation. y 0 5(x 4); y 5x 20 y 5x 3 the slopes are the same and the y-intercepts are different, so the lines are parallel. y 2x 4 y 2x 4 y 2x 5 2x y 4 y 2x 5 explore together get10504.qxd 4/23/08 12:41 pm page 215 2008 text sampler page 173 q r s p all of the angles are right angles. sample answer: lines q, r, and s are parallel, lines p and q are perpendicular , lines p and r are perpendicular , and lines p and s are perpendicular . explore together problem 2 students will discover that perpendi-cular lines have negative reciprocal slopes. grouping ask for a student volunteer to read the scenario for problem 2 aloud. pose the guiding questions below to verify student understanding. have students work together in pairs to complete part (a) through part (h). guiding questions what information is given in this problem? what does each square represent? what would you guess about the relationship of the lines? which lines look to be parallel? what do you know about the slopes of each line? which slopes are positive? which slope is negative? common student errors remind students that they should not assume that lines are parallel or perpendi-cular based on a diagram. there should be marks on the diagram indicating parallel or perpendicular lines, or students should measure the angles. get10504.qxd 4/23/08 12:41 pm page 216 2008 text sampler page 174esson 5.4 parallel and perpendicular lines in the coordinate plane 217 2008 carnegie learning, inc. 5 problem 2 students will continue to investigate the slope of perpendicular lines. grouping students will be working together in pairs to complete part (d) through part (f). notes students may be unfamiliar with the symbols , which indicates that lines are perpendicular , and , which indicates that lines are parallel. call the class back together to discuss and present their work for part (a) through part (h). investigate problem 2 students will be formally introduced to the terms reciprocal and negative reciprocal in question 1. grouping ask a student volunteer to read question 1 aloud. have the students complete question 1 individually. note that question 1 is continued on the next page. sample answer: the slopes of perpendicular lines have a product of –1. sample answer: the slopes must have opposite signs. line p: ; lines q, r, and s: 2 1 1 2 sample answer: the slopes have opposite signs. the absolute value of the slope of line p is the reciprocal of the slopes of lines q, r, and s. sample answer: the slopes of lines q, r, and s will be the same because the lines are parallel, and the slopes of lines q, r, and s will be different from the slope of line p because the lines are not parallel. the product of the slopes is –1. explore together get10504.qxd 4/23/08 12:42 pm page 217 2008 text sampler page 175 3 1 2 1 5 sample answer: yes, taken together with the slopes, the y-intercepts can give you a general idea of where the lines intersect in the plane. sample answer: any line that is perpendicular to the line given by will have a slope of . 1 2 y 2x 4 y 1 2 x; y 1 2 x 5; y 1 2 x 3 a line perpendicular to the line given by must have a slope of . because you know that the line must pass through the point (4, 0), you can use the point-slope form to write the equation. y 0 1 5 (x 4); y 1 5 x 4 5 1 5 y 5x 3 explore together investigate problem 2 students will make conclusions about the slopes of perpendicular lines. students will examine equations to determine whether the lines are perpendicular . grouping question 1 is continued from the previous page. call the class back together to discuss and present their answer for question 1. ask a student volunteer to read the paragraph at the top of the page aloud. pose the guiding questions below to verify student understanding. guiding questions what does reciprocal mean? what does negative reciprocal mean? what is the product of negative reciprocals? grouping have the students work in small groups to complete questions 2 through 6. pose the guiding questions to verify student understanding. common student errors students might need direction in question 5 that they need to use point-slope form. in questions 6, students who are weak equation solvers will struggle with isolating the variable y. some students will not recognize that they need to change the equations to slope-intercept form in order to compare the slopes. guiding questions will the slopes of perpendicular lines ever be the same? explain. will the y-intercepts of perpendicular lines ever be the same? explain. is it easier to pick out the value of slope from an equation in standard form or an equation in slope-intercept form? how do you determine whether two lines are parallel without graphing the lines? get10504.qxd 4/23/08 12:43 pm page 218 2008 text sampler page 176esson 5.4 parallel and perpendicular lines in the coordinate plane 219 2008 carnegie learning, inc. 5 investigate problem 2 students will summarize their findings from investigate problem 2. students will be working in small groups to complete question 6. call the class back together to discuss and present their work for questions 2 through 6. grouping ask a student volunteer to read question 7 aloud. complete question 7 as a whole class. then have students work in groups to complete questions 8 and 9. call the class back together to discuss and present their answers for questions 8 and 9. the same negative reciprocals of each other you can draw one line through the given point that is perpendicular to the given line. you can draw one line through the given point that is perpendicular to the given line. you can draw one line through the given point that is parallel to the given line. the slopes are not negative reciprocals because . so, the lines are not perpendicular . 2(2) 4 y 2x 4 y 2x 4 y 2x 5 2x y 4 y 2x 5 explore together get10504.qxd 4/23/08 12:47 pm page 219 2008 text sampler page 17720 p s r q sample answer: the angles formed are right angles because they meet at the intersection of grid lines. sample answer: (0, 4), (1, 4), (2, 4) sample answer: (0, 8), (1, 8), (2, 8) sample answer: (0, 12), (1, 12), (2, 12) sample answer: for each line, the x-coordinates are different, but the y-coordinates are the same. explore together problem 3 students will investigate vertical and horizontal lines that are also parallel and perpendicular lines. grouping ask for a student volunteer to read the scenario of problem 3 aloud. pose guiding questions to verify student understanding. complete part (a) through part (d) as a whole class. guiding questions what information is given in this problem? what does each square represent? what would you guess about the relationship of the lines? which lines look parallel? what do you know about the slopes of each line? get10504.qxd 4/18/08 1:02 pm page 220 2008 text sampler page 178esson 5.4 parallel and perpendicular lines in the coordinate plane 221 2008 carnegie learning, inc. 5 problem 3 students will continue to investigate horizontal and vertical lines. students will be working as a whole class to complete part (c) and part (d). investigate problem 3 students will formalize their findings about horizontal and vertical lines. solve and discuss question 1 together as a whole class. just the math the terminology for horizontal and vertical lines is formally presented in question 1. grouping ask a student volunteer to read question 1 aloud. have a student restate the problem. then, have students work in pairs to complete question 1. when the students are finished, call the class back together to discuss and present their work for question 1. the equations should be and because no matter what the x-coordinates are, the y-coordinates are constant. y 12 y 8, y 4, sample answer: (4, 0), (4, 1), (4, 2) the x-coordinates are the same and y-coordinates are different. the equation should be because no matter what the y-coordinates are, the x-coordinates are constant. x 4 the value of y does not change at all. the slope of any horizontal line is zero because the rise is always zero and zero divided by any number is zero. the value of x does not change at all. explore together get10504.qxd 4/23/08 12:47 pm page 221 2008 text sampler page 179 perpendicular ; no, because you cannot divide by zero. 1 0 sample answer: the statement about parallel lines is true for horizontal lines because you can compare the slopes, but this statement is not true for vertical lines because you cannot compare their slopes. the statement about perpendicular lines is not true for horizontal and vertical lines because vertical lines have no slope. sample answer: any vertical line and any horizontal line are perpendicular . all all y 1; x 2 y 0 x 5 explore together investigate problem 3 students will write equations of horizontal and vertical lines. grouping ask for a student volunteer to read question 2 aloud. then, have students work with a partner to complete questions 2 through 4. call the class back together to discuss and present their answers for questions 2 through 4. key formative assessments how can you determine whether lines are parallel given the equations of the lines? how can you determine whether lines are parallel given the graph of the lines? what formula is used to find the equation of a line given two points? how can you determine whether lines are perpendicular given the equations of the lines? how can you determine whether lines are perpendicular given the graph of the lines? how do you determine equations of horizontal and vertical lines? what is the slope of a horizontal line? what is the slope of a vertical line? get10504.qxd 4/24/08 8:39 am page 222 2008 text sampler page 180esson 5.4 parallel and perpendicular lines in the coordinate plane 222a 2008 carnegie learning, inc. 5 close review all key terms and their definitions. include the terms slope, y-intercept, point-slope form, slope-intercept form, parallel lines, perpendicular lines, reciprocal, negative reciprocal, horizontal line, vertical line, zero slope, and undefined slope. remind the students to write the key terms and their definitions in the notes section of their notebooks. you may also want the students to include examples. ask the students to explain the meaning of the term slope. write several equations on the board in slope-intercept form. have the students determine which lines are parallel, perpendicular , or intersecting. ask the students to compare the similarities and contrast the differences between parallel and perpendicular lines. wrap up assignment use the assignment for lesson 5.4 in the student assignments book. see the teacher's resources and assessments book for answers. assessment see the assessments provided in the teacher's resources and assessments book for chapter 5. open-ended writing t ask ask the students to design a parking lot that is 150 feet by 400 feet. they must maximize the number of cars in the parking lot. how will they paint the lines in the parking lot? explain and show a diagram. follow up get10504.qxd 4/18/08 1:02 pm page 223 2008 text sampler page 18122b chapter 5 parallel and perpendicular lines 2008 carnegie learning, inc. 5_______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ _______________________________________________________________________________________________ notes get10504.qxd 4/18/08 1:02 pm page 224 2008 text sampler page 1822008 carnegie learning, inc. geometry t eacher's resources and assessments 1_gem1_fm_v1.qxd 4/25/08 8:03 am page i 2008 text sampler page 183 1_gem1_fm.qxd 5/1/08 8:44 am page iii 2008 text sampler page 184hapter 5 assignments 67 2008 carnegie learning, inc. 5 assignment name ___________________________________________________ date _____________________ assignment for lesson 5.4 parking lot design parallel and perpendicular lines in the coordinate plane state whether each pair of lines is parallel, perpendicular , or neither . explain your answer using a complete sentence. 1. 2. 3. 4. 5. 6. y x y x y 7x 7x 2 y y 1 2 x 2y 2x 10 y 1 6 x 5 y 6x y 1 2 x 9 y 2x 6 y 4x y 4x 18 the lines are neither parallel nor perpendicular because their slopes are not equal and the product of their slopes is not –1. the lines are perpendicular because the product of their slopes is –1. the lines are neither parallel nor perpendicular because their slopes are not equal and the product of their slopes is not –1. the lines are parallel because their slopes are equal. the lines are parallel because their slopes are equal. the lines are perpendicular because the product of their slopes is –1. geg105.qxd 4/22/08 10:30 am page 67 2008 text sampler page 1858 chapter 5 assignments 2008 carnegie learning, inc. 5 7. 8. write the equations of 3 lines that are parallel to 9. write the equations of 3 lines that are perpendicular to 10. write the equation of a line that is perpendicular to 11. write the equation of a line that is perpendicular to the line in your answer to question 10. 12. is the line from your answer in question 11 parallel, perpendicular or neither to the original line in number 10? explain. y 1 3 x 2. y 4x 2. y 2 3 x 7. y 2 x 5 the line is parallel to the line given in question 10 because it has the same slope. sample answer: y 2 3 x 3, y 2 3 x 25, y 2 3 x 1.75 the lines are perpendicular because x 5 is a vertical line and y 2 is a horizontal line. sample answer: y 1 4 x, y 1 4 x 5, y 1 4 x 2 3 sample answer: y 3x 2 sample answer: y 1 3 x 7 geg105.qxd 4/22/08 10:30 am page 68 2008 text sampler page 186hapter 5 assessments 77 2008 carnegie learning, inc. 5 pre-t est name ___________________________________________________ date _____________________ is each pair of lines parallel or skew? use a complete sentence to explain your reasoning. 1. 2. use the figure shown below to answer questions 3 through 6. 3. which line given in the figure is a transversal? use a complete sentence to explain your reasoning. 4. name a pair of alternate interior angles in the figure. use a complete sentence in your answer . a b c 1 2 3 4 5 6 7 8 a b p m sample answer: the lines are skew, because they are not coplanar and they do not intersect. angle 4 and or are alternate interior angles. 3 and 6 5 sample answer: line c is a transversal, because it intersects two lines at different points. sample answer: the lines are parallel, because they are coplanar and they do not intersect. gem105.qxd 4/22/08 12:43 pm page 77 2008 text sampler page 1878 chapter 5 assessments 2008 carnegie learning, inc. 5 5. name a pair of alternate exterior angles in the figure. use a complete sentence in your answer . 6. name a pair of corresponding angles in the figure. use a complete sentence in your answer . 7. in the figure shown below, line x is parallel to line y and the measure of 1 is 64º. find the missing angle measures without using a protractor . use complete sentences to explain how you found your answers. z x y 1 2 3 4 5 6 7 8 pre-t est page 2 angle 1 and 5 or 2 and 6 or 3 and 7 or 4 and 8 are corresponding angles. angle 2 and 7 or 1 and 8 are alternate exterior angles. m 2 is 116 , because 1 and 2 are supplementary angles. m 3 is 116 , because 2 and 3 are vertical angles. m 4 is 64 , because 1 and 4 are vertical angles. m 5 is 64 , because 4 and 5 are alternate interior angles. m 6 is 116 , because 2 and 6 are corresponding angles. m 7 is 116 , because 2 and 7 are alternate exterior angles. m 8 is 64 , because 4 and 8 are corresponding angles. º º º º º º º gem105.qxd 4/22/08 12:43 pm page 78 2008 text sampler page 188hapter 5 assessments 79 2008 carnegie learning, inc. 5 pre-t est page 3 name ___________________________________________________ date _____________________ pre-t est page 3 8. write the equation of a line that is parallel to the line and passes through the point (–3, 1). show all your work and use complete sentences to explain how you found your answer . 9. write the equation of a line that is perpendicular to the line and passes through the point (–3, 1). show your work and use complete sentences to explain how you found your answer . 10. write the equations for a horizontal line and a vertical line that pass through the point (–6, 4). horizontal line: ________ vertical line: ________ y 3x 4 y 3x 4 sample answer: a line parallel to the line must have a slope of 3. because you know the line must pass through the point (–3, 1), use the point-slope form to write the equation. y 1 3(x (3) ); y 3x 10 y 3x 4 sample answer: a line perpendicular to the line must have a slope of . because you know that the line must pass through (–3, 1), use the point-slope form to write the equation. y 1 1 3 (x (3) ); y 1 3 x 1 3 y 3x 4 y 4 x –6 gem105.qxd 4/22/08 12:43 pm page 79 2008 text sampler page 1890 chapter 5 assessments 2008 carnegie learning, inc. 5 11. what is the length of ? use a complete sentence to explain your reasoning. 12. in the figure shown below, bisects a, which has a measure of 84 . what is the value of x? use a complete sentence in your answer . a b c x d º ad y 1 1 2 3 4 5 6 7 8 9 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 9 8 7 6 5 4 3 2 1 x a b c x y xy pre-t est page 4 sample answer: is a midsegment. so it is parallel to and half the length of the side that it does not intersect, . therefore, the length of is 7 units. xy ab xy the value of x is 42 . º gem105.qxd 4/22/08 12:43 pm page 80 2008 text sampler page 190hapter 5 assessments 81 2008 carnegie learning, inc. 5 13. sketch the perpendicular bisector of segment . match each point of concurrency with its definition. 14. incenter a. the point at which the altitudes of the triangle intersect. 15. circumcenter b. the point at which the angle bisectors of the triangle intersect. 16. centroid c. the point at which the medians of the triangle intersect. 17. orthocenter d. the point at which the perpendicular bisectors of the triangle intersect. y x xy pre-t est page 5 name ___________________________________________________ date _____________________ a c d b gem105.qxd 4/22/08 12:43 pm page 81 2008 text sampler page 1912 chapter 5 assessments 2008 carnegie learning, inc. 5 gem105.qxd 4/22/08 12:43 pm page 82 2008 text sampler page 1922008 carnegie learning, inc. geometry homework helper geh1_fm.qxd 4/28/08 10:13 am page i 2008 text sampler page 1936 chapter 5 homework helper 2008 carnegie learning, inc. 5 students should be able to answer these questions after lesson 5.4: how are the slopes of parallel and perpendicular lines determined? how are the equations of parallel and perpendicular lines determined? how are the equations of horizontal and vertical lines determined? read question 1 and its solution. then, write the equations of one line parallel to and one line perpendicular to the given line in questions 2 and 3. 1. write the equation of one line that is parallel and one that is perpendicular to the line represented by the equation . step 1 identify the slope of the equation. the slope is the coefficient of the x-term. in this example, the slope is 2. step 2 determine the slopes of the parallel and perpendicular lines. parallel lines have the same slope. lines parallel to have a slope of 2. perpendicular lines have slopes that are negative reciprocals of each other . lines perpendicular to have a slope of . step 3 write the equations of the lines. the value of the constant in the equation does not affect whether the line is parallel or perpendicular . parallel line: perpendicular line: 2. 3. parallel line: ____________________ parallel line: ____________________ perpendicular line: ____________________ perpendicular line: ____________________ read question 4 and its solution. then complete question 5. 4. write the equation of a horizontal line and a vertical line that passes through the point (–1, 2). step 1 a horizontal line will pass through the y-coordinate. the horizontal line has the equation step 2 a vertical line will pass through the x-coordinate. the vertical line has the equation 5. write the equation of a horizontal line and a vertical line that passes through (3, –6). x 1. y 2. y 2 3 x 1 y 3x 5 y 1 2 x 4 y 2x 7 1 2 y 2x 1 y 2x 1 y 2x 1 parking lot design parallel and perpendicular lines in the coordinate plane 5.4 directions geh105.qxd 4/28/08 10:17 am page 36 2008 text sampler page 1942008 carnegie learning, inc. 2008 carnegie learning, inc. 2008 text sampler page 195 2008 text sampler page 195ourse description – algebra ii carnegie learning tm algebra ii is designed as a second-year algebra course. it can be implemented with students at a variety of ability and grade levels. algebra ii focuses heavily on functions. throughout the course, students understand and compare the properties of classes of functions including quadratic, polynomial, exponential, logarithmic, rational, radical, and circular. students extend their understanding of linear functions to include a wide range of function types. for each family of function, students explore graphical behavior and characteristics including general shape, x- and y-intercepts, rate of change, extrema, intervals of increase and decrease, domain, and range. students simplify expressions using techniques including factoring and properties of exponents and radicals. students develop ability to solve equations for each function and understand the relationship between solutions algebraically and graphically. students develop an understanding of arithmetic and geometric sequences as linear and exponential functions with whole number domains. students determine arithmetic and geometric series, including infinite series. students explore conic sections algebraically and graphically. algebra ii also includes select topics in probability and statistics. students find simple and compound probabilities, including experimental probabilities, and are introduced to combinations and permutations. students explore variation, standard deviation, and variance. 2008 text sampler page 196lgebra ii text set2008 carnegie learning, inc. contents linear functions, equations, and inequalities p. 1 in equations and inequalities in one and two variables p. 55 1.8 inverses and pieces functional notation, inverses, and piecewise functions p. 67 systems of linear equations and inequalities p. 75 2.1 finding a job introduction to systems of linear equations p. 77 2.2 pens-r-us solving systems of linear equations: graphing and substitution p. 83 2.3 tickets solving systems of linear equations: linear combinations p. 91 2.4 cramer's rule solving systems of linear equations: cramer's rule p. 99 2.5 consistent and independent systems of linear equations: consistent and independent p. 105 2.6 inequalities–infinite solutions solving linear inequalities and systems of linear inequalities in two variables p. 113 2.7 three in three or more solving systems of three or more linear equations in three or more p. 121 contents 1 2 iv contents a2s10100.qxd 7/23/08 1:01 pm page iv 2008 text sampler page 1982008 carnegie learning, inc. quadratic functions p. 129 3.1 lots and projectiles introduction to quadratic functions p. 131 3.2 intercepts, vertices, and roots quadratic equations and functions p. 137 3.3 quadratic expressions multiplying and factoring p. 143 3.4 more factoring special products and completing the square p. 155 3.5 quadratic formula solving quadratic equations using the quadratic formula p. 165 3.6 graphing quadratic functions properties of parabolas p. 173 3.7 graphing quadratic functions basic functions and transformations p. 181 3.8 three points determine a parabola deriving quadratic functions p. 193 3.9 the discriminant the discriminant and the nature of roots/vertex form p. 199 the real number system p. 209 4.1 thinking about numbers counting numbers, whole numbers, integers, rational and irrational numbers p. 211 4.2 real numbers properties of the real number system p. 217 4.3 man-made numbers imaginary numbers and complex numbers p. 223 4.4 the complete number system operations with complex p. 229 polynomial functions p. 235 5.1 many terms introduction to polynomial expressions, equations, and functions p. 237 5.2 roots and zeros solving polynomial equations and inequalities: factoring p. 245 3 contents 4 5 contents v a2s10100.qxd 7/23/08 1:01 pm page v 2008 text sampler page 1992008 carnegie learning, inc. 5.3 successive approximations, tabling, zooming/tracing, and calculating solving polynomial equations: approximations and graphing p. 253 5.4 it's fundamental the fundamental theorem of algebra p. 259 5.5 when division is synthetic polynomial and synthetic division p. 263 5.6 remains of a polynomial the remainder and factor theorems p. 273 5.7 out there and in between extrema and end behavior p. 279 exponential and logarithmic functions p. 289 6.1 the wizard and the king introduction to exponential functions p. 291 6.2 a review properties of whole number exponents p. 297 6.3 exponents, reciprocals, and roots integral and rational exponents p. 307 6.4 the hockey stick graph applications of exponential functions p. 313 6.5 log a what? inverses of exponential functions: logarithmic functions p. 325 6.6 properties of logarithms the remainder and factor theorems p. 333 6.7 continuous growth, decay, and interest solving exponential and logarithmic equations p. 339 rational equations and functions p. 351 7.1 cars and growing old introduction to rational functions p. 353 7.2 rational expressions, part i simplifying, adding, and subtracting rational expressions p. 361 7.3 rational expressions, part ii multiplying and dividing rational expressions p. 365 7.4 solutions solving rational equations and inequalities p. 369 7.5 holes and breaks graphing rational functions and discontinuities p. 383 contents 7 vi contents 6 a2s10100.qxd 7/23/08 1:01 pm page vi 2008 text sampler page 2002008 carnegie learning, inc. 7.6 work, mixture, and more applications of rational equations and functions p. 399 radical equations and functions p. 411 8.1 inverses of inverses introduction to radical functions and expressions p. 413 8.2 radical expressions simplifying, adding, and subtracting radical expressions p. 423 8.3 solutions solving radical equations p. 429 8.4 graphs graphing radical functions p. 437 conic sections p. 449 9.1 conics? conic sections p. 451 9.2 round and round circles p. 457 9.3 it's a stretch ellipses p. 465 9.4 more asymptotes hyperbolas p. 477 9.5 ultimate focus parabolas p. 493 9.6 antennas, whispering rooms, and more applications of conics p. 507 t rigonometric ratios and circular functions p. 527 10.1 the unit circle angle measures p. 529 10.2 circular functions, part i sine and cosine functions p. 537 10.3 circular functions, part ii tangent function p. 543 10.4 you mean there are more? other circular functions p. 551 10.5 arc functions inverses of circular functions p. 557 8 9 1 0 contents contents vii a2s10100.qxd 7/23/08 1:01 pm page vii 2008 text sampler page 2012008 carnegie learning, inc. t rigonometric graphs, identities, and equations p. 565 11.1 ups and downs graphs of circular functions p. 567 11.2 transformations amplitude, period, phase shift p. 575 11.3 identical? trigonometric identities p. 587 11.4 solutions solving trigonometric equations p. 595 11.5 rabbits and seasonal affective disorder applications of circular functions p. 601 11.6 angle-angle-side and angle-side-angle law of sines p. 613 11.7 side-angle-side and side-side-side law of cosines p. 621 sequences and series p. 627 12.1 college tutoring introduction to arithmetic sequences p. 629 12.2 too much homework! introduction to geometric sequences p. 637 12.3 sums a lot arithmetic and geometric series p. 641 12.4 summing forever sum of infinite geometric series p. 647 counting methods and probability p. 655 13.1 rolling, flipping, and pulling probability and sample spaces p. 657 13.2 multiple trials compound probability p. 665 13.3 counting permutations and combinations p. 673 13.4 pascal and independent events pascal's triangle and the binomial theorem p. 681 13.5 the theoretical and the actual experimental versus theoretical probability p. 689 contents 1 3 viii contents 1 1 1 2 a2s10100.qxd 7/23/08 1:01 pm page viii 2008 text sampler page 2022008 carnegie learning, inc. statistics p. 695 14.1 averages measures of central tendency, quartiles, and percentiles p. 697 14.2 spread variation and standard deviation p. 703 14.3 normal? distribution p. 709 14.4 line of best fit linear regressions p. 719 14.5 not all data are linear other regressions p. 727 matrices p. 735 15.1 arrays, arrays! introduction to matrices and matrix operations p. 737 15.2 rows times columns matrix multiplication p. 743 15.3 solving systems of linear equations matrices p. 747 15.4 multiplicative inverses solving matrix equations p. 753 15.5 calories and lunch applications of matrices p. 759 glossary g-1 index i-1 1 4 1 5 contents ix contents a2s10100.qxd 7/23/08 1:01 pm page ix 2008 text sampler page 2032008 carnegie learning, inc. algebra ii student t ext a2s10100.qxd 7/23/08 1:01 pm page i 2008 text sampler page 2042008 carnegie learning, inc. equations and inequalities p. 55 1.8 inverses and pieces functional notation, inverses, and piecewise functions p. 67 inventory is the list of items that businesses stock in stores and warehouses to supply customers. businesses in the united states keep about 1.5 trillion dollars worth of goods in inventory. you will use linear functions to manage the inventory levels of a business. 1 chapt er linear functions, equations, and inequalities chapter 1 linear functions, equations, and inequalities 1 1 a2s10101.qxd 7/10/08 12:46 pm page 1 2008 text sampler page 205chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. a diagram, symbols, or some other representation? work with your partner • how did you do 1 a2s10101.qxd 7/10/08 12:46 pm page 2 2008 text sampler page 206se the table to graph the functions, and indicate the transformations, both in terms of transforming the equation and the graph, which were performed on the basic function to arrive at the transformed function. 2008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 37 1 algebraic graphical transformations transformations add a constant shift up subtract a constant shift down multiply or divide by a positive constant dilation multiply by 1 reflection 1 a2s10101.qxd 7/22/08 4:46 pm page 37 2008 text sampler page 207lgebraic transformation: graphical transformation: 38 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1. basic function y x algebraic transformation: graphical transformation: 2. y x 3 1 a2s10101.qxd 7/10/08 1:14 pm page 38 2008 text sampler page 2082008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 39 3. y x 4 algebraic transformation: graphical transformation: 4. y 2x algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 39 2008 text sampler page 2090 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 5. y 2x 1 algebraic transformation: graphical transformation: 6. y 3x algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 40 2008 text sampler page 2102008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 41 7. y 4x 1 algebraic transformation: graphical transformation: 8. y 3x 5 algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 41 2008 text sampler page 2112 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 9. y 2 3 x 1 algebraic transformation: graphical transformation: 10. y 1 2 x 3 algebraic transformation: graphical transformation: 1 a2s10101.qxd 7/10/08 1:14 pm page 42 2008 text sampler page 2122008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 43 1 a2s10101.qxd 7/10/08 1:14 pm page 43 2008 text sampler page 2134 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 4 2 6 8 –4 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x – 1 2 a2s10101.qxd 7/10/08 1:14 pm page 44 2008 text sampler page 2142008 carnegie learning, inc. lesson 1.5 basic functions and linear transformations 45 1 4 2 6 8 6 8 4 –6 –4 –8 –2 y x –8 –6 y 3x y x – 1 3 a2s10101.qxd 7/10/08 1:14 pm page 45 2008 text sampler page 215algebraii teacher's implementationguide 2008 text sampler page 216 2008 text sampler page 216esson 1.5 basic functions and linear transformations 37a 2008 carnegie learning, inc. 1 1.5 shifts and flips basic functions and linear t ransformations objectives define basic functions. use translations, dilations and reflections to transform linear functions. graph parallel lines. graph perpendicular lines. key t erms basic function dilation reflection line of reflection nctm content standards grades 9–12 expectations algebra standards understand relations and functions and select, convert flexibly among, and use various representations for them. use symbolic algebra to represent and explain mathematical relationships. geometry standards understand and represent translations, reflections, rotations, and dilations of objects in the plane by using sketches, coordinates, vectors, function notation, and matrices. use various representations to help understand the effects of simple transformations and their compositions. essential ideas algebraic and geometric transformations essential questions 1. describe the similarities and differences between the graph of y x and the graph of y x 5. 2. how can you tell the graph of a linear function has been shifted down when you look at its equation? 3. how can you tell the graph of a linear function has been dilated when you look at its equation? 4. describe the similarities and differences between the graph of y x and the graph of y 3x. 5. describe the similarities and differences between the graph of y 5x and the graph of y 5x. 6. can you look at two equations and tell if the lines are parallel? explain. 7. can you look at two equations and tell if the lines are perpendicular? explain.7b chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 show the way warm up provide students with a graph of y 2x and ask them to write a different equations that would be parallel to the equation y 2x the graph should be of any equation with a slope of 2. perpendicular to the equation y 2x the graph should be of any equation with a slope of . the same as the equation y 2x the original graph. motivator 1. describe the difference between the graph of y 2x and the graph of y 2x 8 without graphing the linear functions. the second graph is the first shifted down 8 units. 2. describe the difference between the graph of y 2x and the graph of y 2x 8 without graphing the linear functions. the second graph is the first shifted up 8 units. 3. identify the x- and y-intercept in the equation y 2x 8. the x-intercept is 4 and the y-intercept is 8. 4. identify the slope in the equation y 2x 8. the slope is 2. 1 2esson 1.5 basic functions and linear transformations 37 2008 carnegie learning, inc. 1 problem 1 this activity is both a review of geometric transformations and an introduction to transformations of linear functions from a basic function and through algebraic transformations. students will also find how the slopes of parallel and perpendicular lines are related. grouping have a student read the first paragraph of problem 1. you may want to ask some or all of the following questions: have we worked with other families of functions? if so, what were they? what is functional notation? use the table to graph the functions and indicate the transformations, both in terms of transforming the equation and the graph, which were performed on the basic function to arrive at the transformed function. algebraic graphical transformations transformations add a constant shift up subtract a constant shift down multiply or divide by a positive constant dilation multiply by 1 reflection8 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 guiding questions have another student read the rest of problem 1. you may want to ask some or all of the following questions: where have you used these algebraic transformations? where have you used these geometric transformations? how do the algebraic transformations relate to the geometric transformations? note make sure that students know how to graph the basic function and the transformed function or image of the transformation. grouping have students work in pairs or groups to complete problem 1. this should take about 15 minutes. be sure to instruct the students to graph the basic function on each graph first so the transformation is visual. algebraic transformation: graphical transformation: 1. basic function y x algebraic transformation: graphical transformation: 2. y x 3 there were no transformations. there were no transformations. add 3. the line shifts up 3 units. 2 4 6 8 –2 –4 2 o 6 8 4 –6 –8 –4 –2 y x –8 –6 y x 4 6 8 –2 –4 2 o 6 8 4 –6 –8 –2 y x –8 –6 y x y x 3esson 1.5 basic functions and linear transformations 39 2008 carnegie learning, inc. 1 note make sure to circulate throughout the class to monitor student progress and facilitate student learning. most students will have little difficulty with vertical shifts. 3. y x 4 algebraic transformation: graphical transformation: 4. y 2x algebraic transformation: graphical transformation: 4 2 6 8 –2 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y x – 4 subtract 4. the line shifts down 4 units. double the slope. there is a dilation of 2. 4 2 6 8 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 2x0 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 common student errors students may struggle with the change in slope being a dilation. unlike in transformations of geometric figures where a dilation enlarges or shrinks the figure, with linear functions, a dilation just changes the slope—the unit rate of change. students should be encouraged to apply multiple transformations in different orders to determine if order matters. as you circulate, if you see "different" student solutions, make sure that these are reported out when the groups share their answers. in the event of different visuals for the same problem, usually due to scaling, ask students to explain their thinking, and ask the class to accept or deny their explanations with a supporting counter example where appropriate. note the change in sign of the slope produces a reflection. 5. y 2x 1 algebraic transformation: graphical transformation: 6. y 3x algebraic transformation: graphical transformation: multiply by 3. there is a dilation of 3 followed by a reflection around the x-axis. double slope and then subtract 1. there is a dilation of 2 followed by a shift down of 1 unit. 4 2 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x y 2x – 1 4 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 3x y –3xesson 1.5 basic functions and linear transformations 41 2008 carnegie learning, inc. 1 7. y 4x 1 algebraic transformation: graphical transformation: 8. y 3x 5 algebraic transformation: graphical transformation: 6 8 2 6 8 4 –6 –4 –8 –2 y x –8 –6 y x y 4x y –4x y –4x 1 multiply by 4 and then add 1. there is a dilation of 4, followed by reflection around x-axis, and then shift up 1 unit. 4 2 6 8 2 o 6 8 4 –6 –4 –8 –2 y x –6 y x y 3x y 3x – 5 multiply by 3 and then subtract 5. there is a dilation by a factor of 3 and then shift down 5 units.2 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 9. y 2 3 x 1 algebraic transformation: graphical transformation: 10. y 1 2 x 3 algebraic transformation: graphical transformation: 4 2 6 8 2 o 6 8 4 –6 –4 –8 y x –6 –8 –2 –4 y x y 3x y x 2 3 y x 1 2 3 multiply by and then subtract 3. 1 2 there is a dilation by a factor of , reflection around the x-axis and shift down 3 units. 1 2 multiply by and then add 1. 2 3 there is a dilation by a factor of and then shift up 1 unit. 2 3 4 2 6 8 6 8 4 –6 –8 y x –6 –8 –2 –4 y x y x – 3 – 1 2 y x – 1 2 y x 1 2esson 1.5 basic functions and linear transformations 43 2008 carnegie learning, inc. 1 grouping after an appropriate amount of time, pull the class back together to share their solutions. problem 2 problem 2 asks students to determine how the slopes of parallel and perpendicular lines are related. the relationship for parallel lines is more obvious in that vertical shifts produce parallel lines. grouping have students work though problem 2 in pairs or groups for about 15 minutes. perform a dilation by a factor of 4. perform a shift down 7 units. perform a dilation by a factor of 2, a reflection around the x-axis, and a shift up 7 units. perform a dilation by a factor of 7, a reflection around the x-axis, and a shift down 11 units. 2 6 8 –4 2 o 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y 2x 5 the lines are parallel.4 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 4 2 6 8 –4 6 8 4 –6 –4 –8 –2 y x –8 –6 y 2x y x – 1 2 the lines are parallel. equations with the same slope have graphs that are parallel. 4 6 8 –4 2 o 6 8 4 –6 –4 –8 y x –8 –6 y –3x y –3x – 5esson 1.5 basic functions and linear transformations 45 2008 carnegie learning, inc. 1 math note the relationship for slopes of perpendicular lines may be shown by drawing the appropriate right triangles and proving the triangles congruent. the angle formed by the lines is the sum of the two acute complimentary angles and therefore forms a right angle, thereby making the lines perpendicular . although this is not a "proof" this is a fairly straightforward illustration of this relationship. grouping after an appropriate amount of time, pull the class back together to share their solutions. essential ideas algebraic and geometric transfor-mations 4 2 6 8 6 8 4 –6 –4 –8 –2 y x –8 –6 y 3x y x – 1 3 the right triangles are congruent because their angles are congruent. the angles formed by the lines are right angles and the lines are perpendicular . the right triangles are congruent because their angles are congruent. the angles formed by the lines are right angles and the lines are perpendicular . linear functions with slopes that are negative reciprocals have graphs that are perpendicular . 4 2 6 8 4 –6 –4 –8 –2 y 6 y y x – 1 36 chapter 1 linear functions, equations, and inequalities 2008 carnegie learning, inc. 1 follow up assignment use the assignment for lesson 1.5 in the student assignments book. see the teacher's resources and assessments book for answers. assessment see the assessments provided in the teacher's resources and assessments book for chapter 1. check students' understanding using y 3x, ask the students to write equations and sketch graphs that would model the following trans-formations: a vertical shift of 7 a horizontal shift of 3 a dilation of 4 a reflection and dilation of .5 a reflection and vertical shift of 3 notesalgebraii teacher's resourcesandassessments 2008 text sampler page 229 2008 text sampler page 2292008 carnegie learning, inc. contents 2008 text sampler page 2302008 carnegie learning, inc. 1 shifts and flips basic functions and linear transformations graph the basic function y x on each grid. then graph the given function and describe the transformation that was performed on the basic function to result in the given function. describe the transformation both algebraically and graphically. 1. 2. y 3x y x 5 assignment name _____________________________________________ date _____________________ assignment for lesson 1.5 y x 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 –5 –4 –3 –2 –1 5 6 4 3 2 1 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 2 1 –1 –2 –3 –4 –5 –6 –7 –8 0 algebraically: subtract 5. algebraically: multiply by 3. graphically: shift down 5 units. graphically: dilate by 3. chapter 1 assignments 13 2008 text sampler page 2312008 carnegie learning, inc. 1 3. 4. graph the basic function y x on each grid. then graph the function that results after performing the given transformation on the parent function. then write an equation for the new function. 1. shift up 2 units. 2. reflect in the x -axis. y 2x 3 y 1 2 x 4 algebraically: multiply by then add 4. algebraically: multiply by 2, then subtract 3. graphically: dilate by and shift up graphically: dilate by 2, reflect in 4 units. x-axis, and shift down 3 units. 1 2 1 2 , y x 12 10 8 6 4 2 –2 –4 –6 –8 –10 –10 –8 –6 –4 –2 10 12 8 6 4 2 0 y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 y x –5 –4 –3 –2 –1 5 6 4 3 2 1 3 4 5 6 2 1 –1 –2 –3 –4 –5 0 y x y x 2 14 chapter 1 assignments 2008 text sampler page 2322008 carnegie learning, inc. 1 name______________________________________________ date _____________________ 3. dilate by 4. then shift down 3 units. 4. dilate by then reflect in x -axis. 1 3 . y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y x –5 –6 –4 –3 –2 –1 5 4 3 2 1 3 4 5 2 1 –1 –2 –3 –4 –5 –6 0 y 1 3 x y 4x 3 chapter 1 assignments 15 2008 text sampler page 23316 chapter 1 assignments 2008 carnegie learning, inc. 2008 text sampler page 234hapter 1 assessments 1 1. tommy's farm has a water tank to supply all the cattle with fresh water . the tank contains 400 gallons of water . water is pumped out to the animals at a rate of 15 gallons per minute. a. write a linear equation that represents the volume of water in the tank at any given time since it was full. b. to the nearest gallon, how much water will be left in the tank after 10 minutes? c. to the nearest minute, how long will it take the entire tank to empty? d. for what time period will the tank contain more than 200 gallons? show your answer as an inequality and to the nearest minute. pre-t est name _________________________________________________________ date _________________________ 2008 carnegie learning, inc. 1 400 15t x gallons after 10 minutes, 250 gallons will be left in the tank. x 250 400 150 x 400 15(10) x 400 15t x minutes it will take the tank about 27 minutes to empty. t 27 t 400 15 400 15t 400 15t 0 minutes t 13 15t 200 400 15t 200 2008 text sampler page 235chapter 1 assessments 2. a. graph the equation . b. what are the x-intercept, y-intercept, and slope of the line? c. if the graph is shifted up 4 units, write the equation of the new graph. 3. determine the equation of the line in slope-intercept form that passes through the points (4, 6) and (1, 3). y 2x 2 pre-t est page 2 2008 carnegie learning, inc. 1 x-intercept 1 y-intercept 2 m 2 (4, 6) and (1, 3) substitute (4, 6). y x 2 b 2 6 (4) b y x b m ( y 2 y 1 ) (x 2 x 1 ) (3 6) (1 4) 3 3 1 y 1 6, y 2 3 x 1 4, x 2 1 y mx b y 2x 6 y x –8 –6 –4 –2 8 6 4 2 6 8 4 2 –2 –4 –6 –8 0 2008 text sampler page 236hapter 1 assessments 3 4. determine the equation of the line that is perpendicular to the line y 2x 5 and passes through the point (8, 4). 5. determine the inverse of the function . f(x) 5x pre-t est page 3 name _________________________________________________________ date _________________________ 2008 carnegie learning, inc. 1 f 1 (x) x 5 f(x) 5x perpendicular line: substitute (8, 4). y 1 2 x 8 b 8 4 1 2 (8) b y 1 2 x b y mx b m 1 2 m 2 y 2x 5 2008 text sampler page 237chapter 1 assessments 2008 carnegie learning, inc. 1 2008 text sampler page 238
ICME 8 - Topic Group 19 Presentation Summary How does a Computer Algebra System actually work to Teach and Learn Mathematics? It has been conjectured that the use of a CAS to teach and learn secondary mathematics could give pupils a wider access to an experimental practice of mathematics and a deeper understanding of concepts in algebra and calculus. A French project including 25 teachers and 460 pupils has been achieved to study those new possibilities. Using questionnaires as well as classroom observations, we examined how those possibilities worked. We found actual benefits of the CAS, but also unexpected phenomena produced by the computational transformation of mathematical concepts when instanced inside a CAS. First, we observed that pupils' understanding of the results obtained in a CAS were often different from the expectations of the teachers. The reason was that the teachers understood those results using their mathematical knowledge when the pupils' interpretation was closer to the explicit display. As another example, the means that a CAS brings to express functions differ subtly from the usual mathematical way. For instance, defining a piece wise continuous functions inside a CAS appeared to be a real problem to pupils. Our conclusion is that one cannot just think of a CAS as an easier equivalent to paper and pencil. On the contrary, much can be expected from the computational insight that a CAS may bring on mathematical concepts.
This guide is designed as a teaching tool, aimed at both teachers and student teachers. Its main purpose is to enhance the value of "The Mathematical Experience" as a textbook. It includes a sample syllabus, outlines for group work, sample exams and hints for grading essays. [via] These provocative essays take a modern look at the 17th-century thinker's dream, examining the influences of mathematics on society, particularly in light of technological advances. They survey the conditions that elicit the application of mathematic principles; the applications' effectiveness; and how applied mathematics transform perceptions of reality. 1987 edition.The New Mexico Mathematics Contest for high-school students has been held annually since 1966. Each November, thousands of middle- and high-school students from all over New Mexico converge to battle with elementary but tricky math problems. The 200 highest-scoring students meet for the second round the following February at the University of New Mexico in Albuquerque where they listen to a prominent mathematician give a keynote lecture, have lunch, and then get down to round two, an even more challenging set of mathematical mind-twisters. Liong-shin Hahn was charged with the task of creating a new set of problems each year for the New Mexico Mathematics Contest, 1990-1999. In this volume, Hahn has collected the 138 best problems to appear in these contests over the last decades. They range from the simple to the highly challenging--none are trivial. The solutions contain many clever analyses and often display uncommon ingenuity. His questions are always interesting and relevant to teenage contestants. Young people training for competitions will not only learn a great deal of useful mathematics from this book but, and this is much more important, they will take a step toward learning to love mathematics.
Huntsville, TX SAT the integration of material from other programs such as Microsoft Word into PowerPoint. Pre-algebra begins the student's entry into higher math. In many ways it is more important than the upper level math courses I am e...
GCSE Foundation: This is a modular course and exams will be taken in November, March and June. GCSE Higher: This is a linear course and exams will be taken in June EXAM BOARD: Edexcel WHERE DOES IT LEAD? Whatever job you go for, a maths qualification very clearly demonstrates that you are willing to face a difficult challenge and stick with it. It shows employers and universities that you have powers of analysis, reasoning and problem solving that other candidates may not be able to offer. Mathematics qualifications improve your chances of employment considerably – most jobs include some degree of measurement, money handling and mental arithmetic and a large number of jobs will require some use of skills such as percentages.
You are here Request Information Step 1 Desired Program Step 2 Contact Information First Name * Desired Program * MATH101: Introduction to College Math Course Credits: 3 Course Hours Per Week: 9 Course Overview This course presents college math concepts combined with the real-world experience of developing a personal financial plan. Basic Microsoft Excel spreadsheet skills are introduced as a way to develop mathematical and financial planning skills. The course also introduces basic algebraic concepts such as solving for unknowns in an equation. In the course project, Personal Financial Plan: Using Math for Decision-Making, students use Microsoft Excel to create a financial plan and analyze data represented in the plan. (Students must have access to Microsoft Excel to complete the project.) Students will also complete sets of problems using an artificial intelligence-based math software program called ALEKS. (Students are required to purchase an access code for the software program through the JIU bookstore.) Course Learning Objectives Apply mathematical formulas to to analyze personal financial data. Use basic Microsoft Excel skills to create a financial plan. Use basic algebraic operations for decision-making. Special Requirements Students must purchase an access code to a web-based, artificial intelligence-based math software program that assesses knowledge through adaptive questioning – known as ALEKS. Students must purchase the access code through the online JIU bookstore. Students will need to have access to or purchase Microsoft Excel to complete this course.
Elementary And Intermediate Algebra with Infotrac A Combined Approach 9780534490249 ISBN: 0534490247 Pub Date: 2005 Publisher: Thomson Learning Summary: Kaufmann and Schwitters have built this text's reputation on clear and concise exposition, numerous examples, and plentiful problem sets. This traditional text consistently reinforces the following common thread: learn a skill; use the skill to help solve equations; and then apply what you have learned to solve application problems. This simple, straightforward approach has helped many students grasp and apply fundam...ental problem solving skills necessary for future mathematics courses in an easy-to-read format
Specification Aims To prove some basic results in number theory. Brief Course Description The distribution of the prime numbers appears to be rather irregular, although they certainly thin out as x increases. How can we describe the distribution? How many primes are there less than x? The key to all this is the Riemann Zeta function, the Riemann zeros, and the famous Explicit Formulas in Number Theory. The Riemann Formula counts up precisely the number of primes less than x. This formula contains oscillatory terms corresponding to the Riemann zeros, and gives rise to the "music of the primes". Syllabus The Euler product. The Hadamard product. The functional equation. The trivial zeros of the zeta function. [6] The von Mangoldt explicit formula. The oscillatory terms. [6] The Riemann explicit formula. The oscillatory terms. The Riemann approximation to π(x). The prime number theorem. The largest known prime. The distribution of prime numbers. The music of the primes.[6]
Trigonometry - Technical Tutoring Site provides an introduction to trigonometry, includes illustrative examples and exercises in basic trigonometry as well as a summary of the important basic trig identities and formulaeVagn Lundsgaard Hansen A personal home page with links to: a mathematical story "I am the greatest," solving and proving/explaining the isoperimetric problem for quadrilaterals; Mathematics and the Public - some experiences with the popularization of mathematics; and Mathematics ...more>> Virtual Polyhedra - George W. Hart A growing collection of over 1000 virtual reality polyhedra to explore, complementing Hart's Pavilion of Polyhedreality. Includes instructions for building paper models of polyhedra including modular origami, with ideas for classroom use. Each of the ...more>> Visualizing An Infinite Series - Cynthia Lanius This lesson uses a trapezoid successively divided in 4ths as a visual representation of an infinite geometric series, a mathematical concept that is often only treated symbolically. Explorations are outlined for investigating a number of series, and links ...more>> The Web Wizard's Math Challenge - J. Mooser An ongoing Internet contest; register (free) to submit answers and score points. Problems are designed to be readily understood but not readily solved. Speed counts, because points are awarded based on the order in which contestants submit correct answers. ...more>> wrotniak.net - J. Andrzej Wrotniak Includes shareware and freeware programs for Windows written by Wrotniak: scientific and regular calculators, a spherical geometry calculator, a logic and strategy game, a statistics graphing program, and a simple program to compute the area of a polygon. ...more>>
Three years of high school mathematics at the level of algebra and above, and a satisfactory score on the SAT/ACT or Math Placement Test. NOTE: If you find a student has not satisfied the prerequisites for this course, contact the front office IMMEDIATELY for their removal. We do not waive prerequisites for ANY students. Learning Outcomes: The following learning outcomes are the MINIMUM set of objectives for the course. These objectives will be measured. While there may be more information covered in the course, make sure these outcomes are listed on your syllabus along with the topical outline. Set Theory Determine what constitutes a set, and identify elements of that set. Determine the relationships between sets and elements of sets using the concepts of union and intersection. Consumer Math and Financial Management Calculate the cost of an item that has been marked up or down. Calculate the percent increase/decrease of a change in quantity. Use financial formulas such as compound interest or the monthly payment formula to answer financial investment and payment questions. Geometry Find missing angles using properties such as supplementary and complementary angles. Determine triangle dimensions and angles by relating similar triangles and applying the Pythagorean theorem. Determine the perimeter and area of various shapes. Use trigonometry to determine missing sides and angles of triangles. Statistics Construct a group frequency distribution and histogram using a given data set. Determine the mean, median, mode and standard deviation of a given data set. Each instructor will make up their own final exam. It must contain a minimum set of questions. You must contact the assessment coordinator for information on these questions. Please submit a copy of your final exam to the assessment coordinator for approval by the second-to-last week of classes.
Specification Aims To introduce students to noncommutative algebra. Brief Description of the unit Nature is inherently noncommutative---just try putting on your shoes and socks in the wrong order---and noncommutative structures are increasingly important throughout mathematics and physics. In this course, we will examine in detail some of the most important noncommutative algebras that appear "in nature" and prove some of the basic structure theorems about noncommutative rings. One of the most fundamental such algebras is the quaternions---if you ignore the fact that it is not commutative, then this is a field that is 4-dimensional as a real vector space! Famously when Hamilton discovered it in the 19-th century he carved its formulae on a bridge lest he forget them. Another is the Weyl algebra---sometimes called the "algebra of quantum mechanics" since its structure encodes the Uncertainty Principle, and sometimes called a ring of differential operators since it encodes the algebraic aspects of differential equations. Learning Outcomes On successful completion of this course unit students will have a deepened understanding of algebra; understand fundamental noncommutative algebras like the quaternions and the Weyl algebra (or algebra of quantum mechanics); understand some of the basic structure theorems in noncommutative algebra, like the Wedderburn density theorem, the classification of simple Artinian rings and the structure of the Jacobson radical.
Introductory Algebrais typically a 1-semester course that provides a solid foundation in algebraic skills and reasoning for students who have little or no previous experience with the topic. The goal is to effectively prepare students to transition into Intermediate Algebra. Table of Contents Tools to Help Students Succeed ix Additional Resources to Help You Succeed xi Preface xiii Applications Index xxiii Prealgebra Review 1 (1) Factors and the Least Common Multiple 2 (7) Fractions 9 (10) Decimals and Percents 19 Group Activity: Interpreting Survey Results 28 (1) Vocabulary Check 29 (1) Highlights 29 (3) Review 32 (2) Test 34 Real Numbers and Introduction to Algebra 1 (88) Tips for Success in Mathematics 2 (6) Symbols and Sets of Numbers 8 (11) Exponents, Order of Operations, and Variable Expressions 19 (10) Adding Real Numbers 29 (9) Subtracting Real Numbers 38 (10) Integrated Review--Operations on Real Numbers 46 (2) Multiplying and Dividing Real Numbers 48 (12) Properties of Real Numbers 60 (8) Simplifying Expressions 68 (21) Group Activity: Magic Squares 77 (1) Vocabulary Check 78 (1) Highlights 78 (5) Review 83 (4) Test 87 (2) Equations, Inequalities, and Problem Solving 89 (89) The Addition Property of Equality 90 (9) The Multiplication Property of Equality 99 (9) Further Solving Linear Equations 108 (10) Integrated Review--Solving Linear Equations 116 (2) An Introduction to Problem Solving 118 (12) Formulas and Problem Solving 130 (12) Percent and Mixture Problem Solving 142 (12) Solving Linear Inequalities 154 (24) Group Activity: Investigating Averages 164 (1) Vocabulary Check 165 (1) Highlights 165 (3) Review 168 (5) Test 173 (2) Cumulative Review 175 (3) Exponents and Polynomials 178 (76) Exponents 179 (12) Negative Exponents and Scientific Notation 191 (9) Introduction to Polynomials 200 (10) Adding and Subtracting Polynomials 210 (7) Multiplying Polynomials 217 (7) Special Products 224 (9) Integrated Review--Exponents and Operations on Polynomials 231 (2) Dividing Polynomials 233 (21) Group Activity: Modeling with Polynomials 240 (1) Vocabulary Check 241 (1) Highlights 241 (3) Review 244 (5) Test 249 (2) Cumulative Review 251 (3) Factoring Polynomials 254 (70) The Greatest Common Factor 255 (10) Factoring Trinomials of the Form x2 + bx + c 265 (7) Factoring Trinomials of the Form ax2 + bx + c 272 (7) Factoring Trinomials of the Form ax2 + bx + c by Grouping 279 (4) Factoring Perfect Square Trinomials and the Difference of Two Squares 283 (10) Integrated Review--Choosing a Factoring Strategy 291 (2) Solving Quadratic Equations by Factoring 293 (9) Quadratic Equations and Problem Solving 302 (22) Group Activity 311 (1) Vocabulary Check 312 (1) Highlights 312 (3) Review 315 (4) Test 319 (2) Cumulative Review 321 (3) Rational Expressions 324 (82) Simplifying Rational Expressions 325 (10) Multiplying and Dividing Rational Expressions 335 (9) Adding and Subtracting Rational Expressions with the Same Denominator and Least Common Denominators 344 (9) Adding and Subtracting Rational Expressions with Different Denominators
LESSON PLANS Understanding Quadratics Introduction Quadratic functions are explored in various forms from Algebra 1 through college level math. These functions can be used to model the motion of objects falling to earth. If you know an object's initial velocity and height, you can find out how long it will take for it to hit the ground if it is dropped. In this activity, students will: • Solidify their understanding of standard form and vertex form for the equation of a quadratic. • Gain a firm understanding of the real roots of a quadratic function. • Learn how the discriminant is calculated. • See the relationship between the equations and graphs of quadratic functions. • Use quadratic functions to solve free-fall problems Materials This is a software only lab, so you just need to have LabVIEW installed
Math Java Applets (Popularity: ): About 15 applets covering a number of math problems and principles. Manipula Math with Java (Popularity: ): Over 200 applets for middle school students, high school students, college students, and all who are interested in mathematics. Interactive programs and a lot of animation that helps with understanding ... Java Demos for Probability and Statistics (Popularity: ): College professor's applets. Chaos and Fractals Applets (Popularity: ): Several java applets for use in exploring the topics of chaos and fractals. Experimental Math Applets (Popularity: ): Some applets covering Besicovitch sets, conformal compactifaction, honeycombs, exponent calculator, the complex plane, elementary complex maps, Möbius transforms, multi-valued functions, the complex derivative, the complex integral, Taylor and Laurent expansions. Spirograph Applet (Popularity: ): Makes a spirograph, just like the kid toy. TenBlocks and IntegerZone (Popularity: ): TenBlocks turns the times tables into a series of puzzles. IntegerZone lets users explore aspects of arithmetic and number theory using the integers themselves as the interface. Graph Explorer (Popularity: ): A Java applet for graphing functions, with smooth zooming and panning across graphs, and variable parameters which can be used for animation. Java Applets for Visualization of Statistical Concepts (Popularity: ): These applets are designed for the purpose of computer-aided education in statistic courses. The intent of these applets is to help students learn some abstract statistics concepts easier than before. ... xFunctions (Popularity: ): The xFunctions applet covers several aspects of calculus and pre-calculus mathematics, including graphs, parametric curves, derivatives, Riemann sums, and integral curves.
Algebra 2 DVD with Books Algebra 2 for Distance Learning The first semester of Algebra 2 reviews and expands concepts learned for graphing and solving linear and quadratic equations. Second semester the focus shifts to a more advanced look at radical, exponential, rational, and logarithmic equations and functions. The course introduces trigonometry, matrix algebra, probability, statistics, and analytic geometry to expose the students to higher mathematical studies. The TI-83 Plus graphing calculator is used throughout the year to build concepts and expand understanding of the material. Mrs. Carrie Finney teaches this course. Recommended Viewing Schedule: five 45-minute lessons a week; 176
MATH 110: College Algebra CATALOG COURSE DESCRIPTION:algebraic models and then applying algebraic techniques to find a solution. D. Exploring exponential and logarithmic functions, including application problems, and the efficient and appropriate use of logarithms and their properties. E. Learning the techniques of solving systems of equations and appropriately applying these processes to word problems. 2.Communication Skills: A. Producing both written and oral communication throughout the course; particular attention is paid to the accurate and appropriate use of the language of algebra. B. Using technology - calculators, in some cases graphing calculators - to solve problems and to be able to communication solutions and explore options. 3.Life Value Skills: A. Developing an appreciation for the intellectual honesty of deductive reasoning; a mathematician's work mustAesthetic Skills: A. Developing an appreciation for the austere intellectual beauty of deductive reasoning. B. Developing an appreciation for mathematical elegance. ADDITIONAL COMMENTS: Most simply put, this course is designed to give students the algebraic tools required in subsequent courses, specifically MATH 180 (Elementary Functions), MATH 230 (Elements of Statistics), or MATH 270 (Managerial Mathematics). By "tools" here I mean etc.)I have come to think I will try to stress this repeatedly during the course. In general, you: Homework: There will be a daily homework assignment. While these generally I believe that various people have various learning styles; some of us learn best by working alone in silence, others learn best by talking about the material with colleagues. I will try to build into the course some amount of group work, not because I think it is the best way for all of you to learn but because I know that some of you learn best by talking about things; what I am trying to do is provide an environment in which you can all latch onto some activities. Use of Technology: It is important to make use of technology, specifically computers and calculators, in doing mathematics. While we do not require the purchase of a specific calculator, you will find it very helpful to acquire and learn to use a graphing calculator of some sort, probably a TI-83. I will use an overhead display frequently in class and it will be to your advantage if you can "play along". I will also give you an introduction to DERIVE, a computer algebra system package created by Texas Instrument. Grading Procedure: In general I use the following scale: 90% for an "A", 80% for a "B", 70% for a "C", and 60% for a "D". There will be nearly 1,000 points during the semester: 6 Labs (about 20 points each), 3 Exams (100 points each), 3 Quizzes (50 points each), 7 Chapter Reviews (about 300 points in all), and a Final Exam (150 points).I am trying to create a situation in which you have adequate opportunity
The three applets on this page are: One-to one or Not?, Graphs of Inverse Functions, and Inverse Functions. The first one allows students to decide if a function is one-to-one, and if not, it shows a... More: lessons, discussions, ratings, reviews,... This JavaSketchpad sketch has a direct varation line with a slider for k. There are seven questions that guide the student through a discovery process that relates k to the slope and to the rate of c... More: lessons, discussions, ratings, reviews,... When the sun moves around in the sky, the shadow cast by a vertical stick moves on the ground. The tip of the shadow traces some kind of curve. The curve depends on the observation point, and it also ... More: lessons, discussions, ratings, reviews,... This shows users the mechanics of matrix muliplication using color coding and how each entry in the solution matrix is derived. The user can scroll down and practice multiplying matrices. After deterThe NA_WorkSheet Demo (beta version) is a collective aggregation of algorithms coded in Java that implements various Numerical Analysis solutions/techniques in one easy to use open source tool. The to... More: lessons, discussions, ratings, reviews,... On this online calculator calculate mathematical expressions and complex numbers. You can do matrix algebra and solve linear systems of equations and graph all 2D graph types. You can also calculate z... More: lessons, discussions, ratings, reviews,... Students can look at graphs of degrees 0 through 5 and see the effects of changing the coefficients a, b, c, d and e on each graph by moving sliders. Allows students to see how the different cubic fu... More: lessons, discussions, ratings, reviews,... This applet solves only the real roots of polynomial equations up to a maximum of order five. Complex number solutions are not available, however it can be included in the future if users request s... More: lessons, discussions, ratings, reviews,... Choose values of p and q in the equation x^2 + px + q = 0, and see determinations of the real roots by completing the square and use of the quadratic formula as well as a graphical visualization of th... More: lessons, discussions, ratings, reviews,... Students graph a quadratic function in vertex form by using a, h and k sliders. The applet makes the translations, reflections and stretches very visual and clear. There are no exercises or supportin... More: lessons, discussions, ratings, reviews,... This applet allows the user to explore the graphs of horizontal of vertical parabolas in vertex form by using sliders to adjust a, h and k. The equation changes with the change in a, h and k on the s
inking Mathematically Blitzer continues to raise the bar with his engaging applications developed to motivate readers from diverse majors and backgrounds. Thinking ...Show synopsisBlitzer continues to raise the bar with his engaging applications developed to motivate readers from diverse majors and backgrounds. Thinking Mathematically, Fifth Edition, draws from the author's unique background in art, psychology, and math to present math in the context of real-world applications. The author understands the needs of nervous readers and provides helpful tools in every chapter to help them master the material. Voice balloons are strategically placed throughout the book, showing what an instructor would say when leading a student through a problem. Study tips, chapter review grids, Chapter Tests, and abundant exercises provide ample review and practice Thinking Mathematically Blitzer starts off with Inductive and Deductive reasoning and builds from there. Other chapters include Logic, Number Representation, Number Theory, Measurement , Geometry, Counting Methods, Probability Theory, Statistics, and ending with Mathematical Systems. Well written, easy to understand, using
Math Studio A mathematics learning space, located in North 330A&B "The learning of mathematics is enhanced when students feel they belong to a community of learners; when they have the opportunity to interact with each other, tutors, and instructors, both in and out of the classroom." — from the Greenfield Community College Math Department Philosophy The Math Studio is the space that helps to create a "community of learners" outside of the formal classroom. In the Math Studio, students from beginning level to advanced courses work with and help each other, or receive assistance from math faculty. Math texts and references, computers, calculators, and math manipulatives are available for student use to enhance and support their learning. Here are some of the things students have to say about the Studio: It's a place to study and do your homework and realize you're not the only one having difficulty. It is a place where you realize your own strengths when you help someone else and it's a place with a feeling of camaraderie and teamwork. You are not learning in a vacuum all by yourself. I am a very busy single mom, taking three classes while working part-time. I normally do most of my mathematics homework in the Studio and have received much needed support and help there. I have many doubts about my math abilities and the Studio has been a place to relax as well as get help. If you would like an individual one-on-one peer tutoring appointment, contact the Peer Tutoring Program at 775-1330.
Written to support OCR's Mathematics GCSE Specification A at Higher tier, featuring comprehensive lesson plans, photocopiable assessments and revision activities, and covering the content of Foundation tier Units A, B and C, and the updated Assessment Objectives, including AO3 - Problem Solving Top page Complete description OCR Mathematics for GCSE Specification A - Higher Teacher and Assessment Pack has been written and edited by experienced examiners and authors, combining their teaching and examining expertise to deliver relevant and meaningful coverage of the course at Foundation tier. The content also supports delivery of the revised Assessment Objectives - including Problem Solving and Quality of Written Communication. - Full teaching guidance covering all the topics required by the Foundation tier of the course - Answers to all of the questions and revision activities in the student's and homework books, and additional photocopiable assessments for student practice The series comprises dedicated student books, teacher's resources, homework books, and online digital assessment and resources Top page
Introduction to Algebra 1 Hotify Introduction to Algebra In the magnitude. In the introduction of algebra, a set is defined as the pair of binary operations. Usually, set and operations includes the identity element. Operations given are commutative or associat News/India/Introduction-to-Algebra Read this article hotklixed More than a Year Ago! In

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