text
stringlengths 8
1.01M
|
|---|
Algebra Combat, players alternate turns to see who can solve sets of single-variable linear equations with the best time and accuracy. Equations are randomly generated in one of four difficulty levels. All solutions are integers less than 40 and presented as multiple choice.
Beginners may need to jot down a step or two on a piece of paper. More advanced players will be able to solve these problems on the fly. With continued gameplay, students develop increased confidence with entry level algebraic expressions, negative integers, and quick mental calculations.
Both practice and competition modes are included. All rounds contain 5 equations to solve. Scores and times are provided following each round. A "Current Rankings" score tally also follows each competition round. Best rounds of all time are entered into the "Top Fighters" hall of fame along with the player's name, time, score, and difficulty level played.
In keeping with the cage fight theme, players are encouraged to give their fighters a cool name and type in a little "trash talk" before each round. All trash talk messages are then delivered to the opposing player at the beginning of his or her turn… ensuring matches are both spirited and entertaining.
|
Calculating Drug Dosages-workbook And Cd - 2nd edition
Summary: Learning drug calculation is one of a nursing student's most time-consuming and important tasks. Calculating Drug Dosages, 2nd edition, makes learning dosage calculation--even reviewing basic math principles--fun and easy! All modules have been updated to address medication errors and safety and the National Patient Safety Goals set forth by JCAHO. Plus, a new software interface presents a bold improvement to an already successful learning tool.
Why is this so...show moreftware better than the book or software your students are currently using?
It teaches all major methods -- linear ratio and proportion, fractional ratio and proportion, dimensional analysis, and the formula method. Every method can be used for every problem.
Students can work at their own pace through the entire program or focus on areas of weakness
Section quizzes allow your students to check their progress and see where they need further review
Allows students to print test scores so you can determine where your students need extra help
Workbook allows your students to practice their mathematical skills when they are away from a computer38 +$3.99 s/h
New
Extremely_Reliable Richmond, TX
Buy with confidence. Excellent Customer Service & Return policy.
$4.79 +$3.99 s/h
New
sherbiebooks NY Seaford, NY
2007 Paperback New
$5.20 +$3.99 s/h
New
BookCellar-NH Nashua, NH
0803615329 BRAND NEW. Still in plastic wrap
|
Lecture 1: Introduction to matrices
Embed
Lecture Details :
What a matrix is. How to add and subtract them.
Course Description :
Matrices, vectors, vector spaces, transformations. Covers all topics in a first year college linear algebra course. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn't a prereq) so don't confuse this with regular high school algebra.
|
Book Description: Lie groups, Lie algebras, and representation theory are the main focus of this text. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.
Featured Bookstore
New
$6.46
Used
$6.46
|
Math Review Tips
Buy the textbook. Too many people think that they can just borrow their book from other students, use the one in the Learning Center, or just do the class without it. There is no review tool which equals the textbook.
Ask specific questions. While doing homework, identify specific areas in which you're having difficulties. This can be crucial when trying to figure out what to study. It also can be extremely helpful when getting help from a tutor or professor.
Utilize the Learning Center. The Learning Center can be a great way to review or get help on the problems that you're having. With tutors and staff that are available to help you, you have many people that are willing to help you accomplish your goal.
Attend all classes. If you're not spending time in classes, you're most likely not learning very much. Math textbooks are hard to just sit down and study from scratch, and class periods can give you a great feel for the material as well as possibly answer questions you had.
Read the appropriate section before class. With a bit of preparation, class time can make much more sense with a bit of background in the topic before the professor talks about it.
Online Math Review Websites
Khan Academy - Thousands of lessons on topics ranging from arithmetic to calculus and beyond.
Spanish
Audio
Librivox - A website dedicated to creating and compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the spanish language that you can download and listen to at your leisure. Be sure to check out "Las Fábulas de Esopo" – Aesop's Fables.
Grammar
Dictionaries
Word Reference - A very useful dictionary with definitions in English and Spanish, a verb conjugator, and great forums that address finer language points that no textbook could ever attempt. Diccionario de la Lengua Española - A terrific reference with definitions in Spanish.
French
Audio
Librivox - A website dedicated to compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the french language that you can download and listen to at your leisure. Indo-European Languages - A terse but comprehensive review of French grammar and vocabulary with extensive audio examples.
Grammar
Indo-European Languages - A terse but comprehensive review of French grammar and vocabulary with extensive audio examples.
Dictionaries
Word Reference - A very useful dictionary with definitions in English and French, a verb conjugator, and great forums that address finer language points that no textbook could ever attempt.
Russian
Audio
Librivox - A website dedicated to compiling audiobooks in the public domain. Here you will find hundreds of hours of recordings in the russian language that you can download and listen to at your leisure.
|
The Number System (Dover Books on Mathematics)
Book Description: This book explores arithmetic's underlying concepts and their logical development. It offers an informal and intuitive understanding of the rigorous logical approach, in addition to a detailed, systematic construction of the number systems of rational, real, and complex numbers. Numerous exercises help students test their progress and practice concepts. 1956 edition.
Buyback (Sell directly to one of these merchants and get cash immediately)
Currently there are no buyers interested in purchasing this book. While the book has no cash or trade value, you may consider donating it
|
Mount Wilson CalculusIt is essential to understand the students? thought processes because each student is uniquely suited to a particular method. In order to lead students in a right direction, I ask them what problems they are really having difficulty with and take the necessary time to explain the concepts to them. However, understanding the theories is fundamental but is not enough
|
paul77 (Offline)
New Member
Write a Message
Sorry I have all the materials listed in Cambridge Econ intranet but I don't have the materials for maths and stats as it seems like there is nowhere to get them Actually, my friends told me that he was given some handouts for the maths and stats and no digital materials.
The materials are very big. I don't know how to post them. They cannot be sent by emails.
|
With Algebra, you get to work puzzles by playing with letters, numbers, and symbols. Algebra problems are pretty abstract, and learning the proper algebraic procedures gives you tools to actually solve these abstract problems instead of guessing.
|
ChiliMATH is here! This site contains free online math tutorials created to supplement class lectures and to guide students in solving math problems in a straightforward way. My goal is for students to build confidence as they develop their own mathematical skills and knowledge in the process. One secret to succeed in Math is doing a lot of practice. ChiliMATH offers many worked examples which can be printed easily for offline use. I hope that you find these resources helpful in your studies.
WebMath (Popularity: ): A set of tutorials on various topics in introductory mathematics, as well as free software. Algebra Review in Ten Lessons (Popularity: ): Review of (high school) algebra. Acrobat Reader required. Stroh MathPage (Popularity: ): Organized, easy to understand, help with math, specifically algebra Purplemath (Popularity: ): Includes illustrated tutorials, categorized links, homework guidelines, and a study skills survey. Algebra - Mathematical Abstraction from Concrete Experience (Popularity: ): An explanation of how the Montessori student learns algebra while interacting with manipulatives, physical objects, represented by expressions incorporating the numerals and variables of mathematics. Algebra Wizard (Popularity: ): Free e-zine, Algebra Times, helps students, teachers and homeschoolers with algebra. MathDork - Math, Algebra interactive turtorials (Popularity: ): Unique learning experience using animation to make math more intuitive and fun. Animated images, sounds create a visually appealing experience that draws the student into the material. Algebra Online Learning Sites (Popularity: ): Algebra and math site links in a table to help you find various math subjects from algebra to calculus, algebra games to quizzes. Algebra Homework Help, Online Solvers (Popularity: ): Interactive homework problems. Topics include Pre-Algebra, linear Algebra and other college Algebra. IB Math Studies: Functions 1 (Popularity: ): Site on Functions and Relations from the IB Mathematical Studies textbook. Pet-Abuse.Com (Popularity: ): If you had asked us before October 16 of 2001 if we thought for one second that our cats were in danger in the peaceful town of Del Mar, California, ... The Animal Welfare Information Center (Popularity: ): Provides information for improved animal care and use in research, teaching, and testing. The Animal Welfare Information Center (AWIC) is mandated by the Animal Welfare Act (AWA) to provide information ...
|
This collection is included inLens:Connexions Featured Content By: Connexions
Comments:
"This is the "main" book in Kenny Felder's "Advanced Algebra II" series. This text was created with a focus on 'doing' and 'understanding' algebra concepts rather than simply hearing about them inHomework: Graphing
Summary: This module provides practice problems designed to develop some concepts related to graphing.
The following graph shows the temperature throughout the month of March. Actually, I just made this graph up—the numbers do not actually reflect the temperature throughout the month of March. We're just pretending, OK?
Figure 1
Exercise 1
Give a weather report for the month of March, in words.
Exercise 2
On what days was the temperature exactly 0°C?
Exercise 3
On what days was the temperature below freezing?
Exercise 4
On what days was the temperature above freezing?
Exercise 5
What is the domain of this graph?
Exercise 6
During what time periods was the temperature going up?
Exercise 7
During what time periods was the temperature going down?
Exercise 8
Mary started a company selling French Fries over the Internet. For the first 3 days, while she worked on the technology, she lost $100 per day. Then she opened for business. People went wild over her French Fries! She made $200 in one day, $300 the day after that, and $400 the day after that. The following day she was sued by an angry customer who discovered that Mary had been using genetically engineered potatoes. She lost $500 in the lawsuit that day, and closed up her business. Draw a graph showing Mary's profits as a function of days
|
Secondary Mathematics II [2011]
The focus of Mathematics II is on quadratic expressions, equations, and functions; comparing their characteristics
and behavior to those of linear and exponential relationships from Mathematics I as organized into 6 critical areas,
or units. The need for extending the set of rational numbers arises and real and complex numbers are introduced so
that all quadratic equations can be solved. The link between probability and data is explored through conditional
probability and counting methods, including their use in making and evaluating decisions. The study of similarity
leads to an understanding of right triangle trigonometry and connects to quadratics through Pythagorean
relationships. Circles, with their quadratic algebraic representations, round out the course. The Mathematical
Practice Standards apply throughout each course and, together with the content standards, prescribe that students
experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of
problem situations.
Critical Area 1: Students extend the laws of exponents to rational exponents and explore distinctions between rational
and irrational numbers by considering their decimal representations. In Unit 3, students learn that when quadratic
equations do not have real solutions the number system must be extended so that solutions exist, analogous to the
way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution. Students
explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and
complex numbers. The guiding principle is that equations with no solutions in one number system may have solutions
in a larger number system.
Critical Area 2: Students consider quadratic functions, comparing the key characteristics of quadratic functions to
those of linear and exponential functions. They select from among these functions to model phenomena. Students
learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In
particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function.
When quadratic equations do not have real solutions, students learn that that the graph of the related quadratic
function does not cross the horizontal axis. They expand their experience with functions to include more specialized
functions—absolute value, step, and those that are piecewise-defined.
Critical Area 3: Students begin this unit by focusing on the structure of expressions, rewriting expressions to clarify
and reveal aspects of the relationship they represent. They create and solve equations, inequalities, and systems of
equations involving exponential and quadratic expressions.
Critical Area 4: Building on probability concepts that began in the middle grades, students use the languages of set
theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound
events, attending to mutually exclusive events, independent events, and conditional probability. Students should make
use of geometric probability models wherever possible. They use probability to make informed decisions.
Critical Area 5: Students apply their earlier experience with dilations and proportional reasoning to build a formal
understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and
apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right
triangles and the Pythagorean Theorem. It is in this unit that students develop facility with geometric proof. They
use what they know about congruence and similarity to prove theorems involving lines, angles, triangles, and other
polygons. They explore a variety of formats for writing proofs.
Critical Area 6: In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a
radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths
and angle measures. In the Cartesian coordinate system, students use the distance formula to write the equation of a
circle when given the radius and the coordinates of its center, and the equation of a parabola with vertical axis when
given an equation of its directrix and the coordinates of its focus. Given an equation of a circle, they draw the graph in
the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines
and circles or a parabola and between two circles. Students develop informal arguments justifying common formulas
for circumference, area, and volume of geometric objects, especially those related to circles.
*In some cases clusters appear in more than one unit within a course or in more than one course. Instructional notes will indicate how these standards grow over time. In some cases only certain standards within a cluster are included in a unit.
Core Standards of the Course
Unit 1: Extending the Number System Students extend the laws of exponents to rational exponents and explore distinctions between rational and irrational numbers by considering their decimal representations. In Unit 2, students learn that when quadratic equations do not have real solutions the number system must be extended so that solutions exist, analogous to the way in which extending the whole numbers to the negative numbers allows x+1 = 0 to have a solution. Students explore relationships between number systems: whole numbers, integers, rational numbers, real numbers, and complex numbers. The guiding principle is that equations with no solutions in one number system may have solutions in a larger number system.
Extend the properties of exponents to rational exponents.
N.RN.1N.RN.2
Rewrite expressions involving radicals and rational exponents using the properties of exponents.
Use properties of rational and irrational numbers.
Connect N.RN.3 to physical situations, e.g., finding the perimeter of a square of area 2.
N.RN.3
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
Perform arithmetic operations with complex numbers.
Limit to multiplications that involve i2 as the highest power of i.
N.CN.1
Know there is a complex number i such that i2 = –1, and every complex number has the form a + bi with a and b real.
N.CN.2
Use the relation i2 = –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
Perform arithmetic operations on polynomials.
Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x.
A.APR.1
Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.
Unit 2: Quadratic Functions and Modeling Students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. When quadratic equations do not have real solutions, students learn that that the graph of the related quadratic function does not cross the horizontal axis. They expand their experience with functions to include more specialized functions - absolute value, step, and those that are piecewise - defined.
Interpret functions that arise in applications in terms of the context.
Focus on quadratic functions; compare with linear and exponential functions studied in Mathematics I.
F.IF.4★.
F.IF.5
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★
F.IF.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★
Analyze functions using different representations.
For F.IF.7b, compare and contrast absolute value, step and piecewisedefined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range and usefulness when examining piecewise-defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Mathematics I on exponential functions with integer exponents. For F.IF.9, focus on expanding the types of functions considered to include, linear, exponential, and quadratic.
Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored.
F.IF.7
Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★
Graph linear and quadratic functions and show intercepts, maxima, and minima.
F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.
Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.
Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.
F.IF.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.
Build a function that models a relationship between two quantities.
Focus on situations that exhibit a quadratic or exponential relationship.
F.BF.1
Write a function that describes a relationship between two quantities.★
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
Build new functions from existing functions.
For F.BF.3, focus on quadratic functions and consider including absolute value functions.. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0.
F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, kf Include recognizing even and odd functions from their graphs and algebraic expressions for them.
F.BF.4
Find inverse functions.
Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1.
Construct and compare linear, quadratic, and exponential models and solve problems.
Compare linear and exponential growth studied in Mathematics I to quadratic growth.
F.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.
Unit 3: Expressions and Equations Students begin this unit by focusing on the structure of expressions, rewriting expressions to clarify and reveal aspects of the relationship they represent. They create and solve equations, inequalities, and systems of equations involving exponential and quadratic expressions.
Interpret the structure of expressions.
Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from the integer exponents found in Mathematics I to rational exponents focusing on those that represent square or cube roots.
A.SSE.1
Interpret expressions that represent a quantity in terms of its context.★A.SSE.2Write expressions in equivalent forms to solve problems.
It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal.
A.SSE.3
Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.★
Factor a quadratic expression to reveal the zeros of the function it defines.
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.
Create equations that describe numbers or relationships.
Extend work on linear and exponential equations in Mathematics I to quadratic equations. Extend A.CED.4 to formulas involving squared variables.
A.CED.1
Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
A.CED.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.
Solve equations and inequalities in one variable.
Extend to solving any quadratic equation with real coefficients, including those with complex solutions.
A.REI.4
Extend to solving any quadratic equation with real coefficients, including those with complex solutions.
Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form.
Solve quadratic equations by inspection (e.g., for x2Use complex numbers in polynomial identities and equations.
Limit to quadratics with real coefficients.
N.CN.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
Solve systems of equations.
Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x2 + y2 = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple 32 + 42 = 52.
A.REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
Unit 4: Applications of Probability Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.
Understand independence and conditional probability and use them to interpret data.
Build on work with two-way tables from Mathematics I Unit 4 (S.ID.5) to develop understanding of conditional probability and independence.
S.CP.1S.CP.2
Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.
S.CP.3S.CP.4
Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.
S.CP.5.
S.CP.6
Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.
S.CP.7
Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.
S.CP.8S.CP.9
(+) Use permutations and combinations to compute probabilities of compound events and solve problems.
Use probability to evaluate outcomes of decisions.
This unit sets the stage for work in Mathematics III, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e. incomplete information) requires an understanding of probability concepts.
S.MD.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).
S.MD.7
(+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game).
Unit 5: Similarity, Right Triangle Trigonometry, and Proof Students apply their earlier experience with dilations and proportional reasoning to build a formal understanding of similarity. They identify criteria for similarity of triangles, use similarity to solve problems, and apply similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean theorem.
It is in this unit that students develop facility with geometric proof. They use what they know about congruence and similarity to prove theorems involving lines, angles, triangles, and other polygons. They explore a variety of formats for writing proofs.
Understand similarity in terms of similarity transformations
G.SRT.1
Verify experimentally the properties of dilations given by a center and a scale factor:
A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.
The dilation of a line segment is longer or shorter in the ratio given by the scale factor.
G.SRT.2G.SRT.3
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
Prove geometric theorems.
Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of perpendicular bisectors and angle bisectors as preparation for G.C.3 in Unit 6.
G.CO.9
Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
G.CO.10
Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.
G.CO.11
Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Prove theorems involving similarity.
G.SRT.4
Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.
G.SRT.5
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.
Use coordinates to prove simple geometric theorems algebraically.
G.GPE.6
Find the point on a directed line segment between two given points that partitions the segment in a given ratio.
G.SRT.6
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
G.SRT.7
Explain and use the relationship between the sine and cosine of complementary angles.
G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★
Prove and apply trigonometric identities.
In this course, limit θ to angles between 0 and 90 degrees. Connect with the Pythagorean theorem and the distance formula. A course with a greater focus on trigonometry could include the (+) standard F.TF.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could continue to be limited to acute angles in Mathematics II.
Extension of trigonometric functions to other angles through the unit circle is included in Mathematics III.
F.TF.8
Pro.
Unit 6: Circles With and Without Coordinates In this unit students prove basic theorems about circles, such as a tangent line is perpendicular to a radius, inscribed angle theorem, and theorems about chords, secants, and tangents dealing with segment lengths and angle measures. They study relationships among segments on chords, secants, and tangents as an application of similarity. In the Cartesian coordinate system, students use the distance formula to write the equation of a circle when given the radius and the coordinates of its center, and the equation of a parabola with vertical axis when given an equation of its directrix and the coordinates of its focus. Given an equation of a circle, they draw the graph in the coordinate plane, and apply techniques for solving quadratic equations to determine intersections between lines and circles or a parabola and between two circles. Students develop informal arguments justifying common formulas for circumference, area, and volume of geometric objects, especially those related to circles.
Understand and apply theorems about circles.
G.C.1
Prove that all circles are similar.
G.C.2G.C.3
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.
G.C.4
(+) Construct a tangent line from a point outside a given circle to the circle.
Find arc lengths and areas of sectors of circles
Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course.
G.C.5Translate between the geometric description and the equation for a conic section.
Connect the equations of circles and parabolas to prior work with quadratic equations. The directrix should be parallel to a coordinate axis.
G.GPE.1
Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
G.GPE.2
Derive the equation of a parabola given a focus and directrix.
G.GPE.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).
Explain volume formulas and use them to solve problems.
Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k2 times the area of the first. Similarly, volumes of solid figures scale by k3 under a similarity transformation with scale factor k.
G.GMD.1
Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.
G.GMD.3
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★
★ Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol
|
Background and Goals: The sequence Math 115-116-215 is the standard
complete introduction to the concepts and methods of calculus. It is taken by
the majority of students intending to major in mathematics, science, or
engineering as well as students heading for many other fields. The emphasis is
on concepts and solving problems rather than theory and proof. All sections
are given two uniform midterms and final exam.
Content:
The course presents the concepts of calculus from four points of
view: geometric (graphs), numeric (tables), symbolic (formulas), and
verbal descriptions. Students will develop their reading, writing and
questioning skills, as well as their ability to work cooperatively. Topics
include functions and graphs, derivatives and their applications to
real-life problems in various fields, and an introduction to integration.
Alternatives:
Math
185 (Honors Anal. Geom. and Calc. I) is a more theoretical
course which covers much of the same material.
Math
175 (Combinatorics and Calculus) includes some of the material
of Math 115 together with some combinatorial mathematics.
A student whose preparation is insufficient for Math 115
should take
Math105 (Data, Functions and Graphs).
Subsequent Courses:
Math116
(Calculus II) is the natural successor. A student who has
done well in this course could, after consulting with a math honors advisor, enter the honors sequence
at this point by taking
Math186 (Honors Anal. Geom. and Calc.II).
|
A. Beck, M. Bleicher and D. Crowe,Excursion into Mathematics,
Worth Publishers, 1969 (ISBN 0-87901-004-5). The first chapter (about 80 pages)
introduces graph theory and many of its most interesting topics. This
book is written for those with two or three years of high school mathematics.
K. H. Rosen, Discrete Mathematics and its Applications, Random
House, NY, 1988. (ISBN 0-394-36768-5, QA39.2 R654) This college text,
written for students that have completed college algebra, present graph
theory in chapters seven and eight, and does so from an algorithmic viewpoint.
L. Steen editor, For All Practical Purposes : Introduction to
Contemporary Mathematics 3ed. W. H. Freeman and Company, New York 1994.
(ISBN 0-7167-2378-6, QA7.F68 1994) This excellent text by the Consortium
for Mathematics and Its Applications aims to show what mathematics is
good for and what mathematicians do. This freshman college level book
starts with two very practical chapters on graph theory.
|
MATHEMATICS
Mathematics is a powerful tool for solving practical problems, as well as a highly creative field of
study, combining logic and precision with intuition and imagination. A major goal of mathematics
is to reveal and explain patterns, whether the patterns appear in the arrangement of swirls on a
pinecone, fluctuations in the value of currency, or as the detail in an abstract geometric figure.
As Reformed Christians, we believe that God has created, redeemed, and still sustains every
aspect of the world around us. It is our task to appreciate the beauty and well-orderedness of
his creation and to use our God-given abilities to subdue it and to use it for his purposes. Thus,
the aim of the mathematics department is to use analytical thinking to help prepare students to
be Christians who are qualified and professional in their chosen vocations. This is accomplished
in three ways: we work to develop students who are proficient in mathematics, to educate
students for a life of Christian service, and to develop in the students good work habits. The
proficiency in mathematics requires the background, skills, and analytical thinking necessary for
these students to succeed professionally in their chosen work environment: graduate school,
industry, or the elementary or secondary school classroom. The secondary mathematics education
major is designed for teachers in grades 6 through 12. This program meets the major graduation
requirements only for students completing the secondary education certification program.
Trinity Christian College's proximity to metropolitan Chicago offers its mathematics majors
unique education and employment advantages. The wide diversity of industry, business, and
institutions near Trinity allow for a broad range of internship opportunities. Trinity's math
department is an active participant in the mathematics division of the Association of Colleges
in the Chicago Area (ACCA) and the Illinois Section of the Mathematical Association of
America (ISMAA). Activities include area-wide competitions, lectures and conferences, and
annual opportunities for presentations of student research. The department sponsors annual
Mathematics Triathlon competitions for students in grades 3 through 8 from area Christian
schools.
|
Tips for simplifying tricky operations Get the skills you need to solve problems and equations and be ready for algebra class Whether you're a student preparing to take algebra or a parent who wants to brush up on basic math, this fun, friendly guide has the tools you need to get in gear. From positive, negative, and whole numbers to fractions, decimals,... more...,... more...
Improve your score on the Analytical Reasoning portion of the LSAT If you're like most test-takers, you find the infamous Analytical Reasoning or "Logic Games" section of the LSAT to be the most elusive and troublesome. Now there's help! LSAT Logic Games For Dummies takes the puzzlement out of the Analytical Reasoning section of the exam and shows... more...... more...
Manage your time and ace the mathematics section of the SAT Scoring well on the mathematics section of the SAT exam isn't guaranteed by getting good grades in Algebra and Geometry. Turn to SAT Math For Dummies for expert advice on translating your classroom success into top scores. Loaded with test-taking strategies, two practice tests, and hundreds... more...
Multiply your chances of success on the ACT Math Test The ACT Mathematics Test is a 60-question, 60-minute subtest designed to measure the mathematical skills students have typically acquired in courses taken by the end of 11th grade, and is generally considered to be the most challenging section of the ACT. ACT Math For Dummies is an approachable,... more...... more...
The fun and easy way® to understand the basic concepts and problems of pre-algebra Whether you're a student preparing to take algebra or a parent who needs a handy reference to help kids study, this easy-to-understand guide has the tools you need to get in gear. From exponents, square roots, and absolute value to fractions, decimals, and percents,... more...
|
Master of Science:
Master of Science:
Mathematics for Elementary Education
The MATH 140-141 sequence is designed for preservice elementary school teachers. These courses
are required for admission to the Elementary Education Program in the College of Education.
The courses emphasize a problem-solving, calculator-based, activity-oriented approach to the study
of mathematics. Arithmetic, algebraic, geometric, and statistical interpretations of topics are
integrated. The classes are offered in a laboratory setting to encourage interaction between
students in a cooperative learning atmosphere. Course work includes not only tests and homework
but also group projects and independent investigations.
Success in these courses requires a mastery of precollegiate mathematics, including algebra.
Students who do not demonstrate sufficient mathematical strength are placed into algebra courses.
Transfer students who have taken math for teachers courses may be able to receive credit for Math 140
or Math 141. We recommend looking at the sample proficiency exams in order to gauge your preparedness.
To confirm your readiness to take one of the courses, we encourage you to contact
Janice Nekola (312-413-3750) in the Office of Mathematics Education. You can arrange
to take a practice exam that we can grade and then counsel you appropriately.
|
Basic elements offinite-dimension vector spaces as such as bases and generator systems,
linear maps and dual vector space. Study of alternate multilinear forms and
concept of determinant. Basic properties of determinants and their calculation.
Matrix diagonallization. Basic concepts of affine spaces with special mention
to dimensions 2 and 3. Definition of projective space and projectivization of an
affine space.
Professor: María Luz PUERTAS
GONZÁLEZ.
Teaching Method: Theoretical and practical lectures. Solution of
problems by the student in the lecture hall.
An introductory subject which briefly tackles an
introduction to Descriptive Statistics, followed by some basic elements of
probability from the Kolgomorov Axiomacy. Following, the concept of random
variable is treated, basically the study of the Distribution Function, types of
variables, changes of variable and their characteristics. The subject ends with
an overview of the different types of distribution models.
Professor: Francisco HERRERA CUADRA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Numerical Method I
Part One;
Second term subject; 4 hours per week. 4.5 Cred. E.C.T.S.
In many scientific disciplines the presence of
mathematical problems requiring the solution of large linear equation systems,
mostly difficult to be explicitally solved, is frequently found.
This subject develops direct and indirect solution
methods for linear equation systems, analyzing the method´s convergence order. A
basic study of MatrixAlgebra for its
use in some practical problems like population matrixes, constant coefficients
differential equation systems a.o. are also introduced.
Professor: Manuel GÁMEZ CÁMARA;
Antonio ANDÚJAR RODRÍGUEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Task works.
Second Year
First term subjects
Matrix Algebra: Canonic Forms.
Part One; First term subject; 6 hours per week. 6
Cred. E.C.T.S.
Rings and Modules: Factorization in a domain of
principal ideals.Basic concepts of
the theory of modules. Special classes of modules. Theorem of structures. Submodules
of free modules. Theorems of descomposition. Applications of the Structure
Theoreme: Finitelly generated abelian groups. Canonic forms of matrixes. Effective
calculus of canonic forms. Multilinear Algebra: Tensor product of modules and
algebras: tensor algebra of one module. External algebra of one module:
determinants.
Professor: María Jesús ASENSIO DEL
ÁGUILA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written/oral examination.
Real
Analysis II
Part One; First term subject; 7 hours per week. 7
Cred.E.C.T.S.
Analysis of some real variables: Lebesque´s Integral.
Vector functions and sets: continuity,
differentiability.
Extremi. Inverse functions and Implicits.
Conditioned extremi.
Professor: Antonio JIMÉNEZ VARGAS.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Calculus of Probabilities and
Mathematical Statistics.
Part One; First term subject; 7 hours per week. 7
Cred.E.C.T.S.
This subject is a continuation of the subject
"Introduction to the Calculus of Probabilities". It starts with some
basic elements of the Measurement Theory in order to harshen the approach to
Probability. Following, the study of bidimensional random variables is tackled,
specifying the importance in the obtention of the relationship among
unidimensional variables. This is generalized to n-dimensional variables. The
subject ends with the study of Succesions of random variables and their boundary
theorems. An introduction to Sampling Statistics and their distributions is
given.
Professor: Francisco HERRERA CUADRA.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Geometry II
Part One; First term subject; 5 hours per week. 5
Cred. E.C.T.S.
General vision of vector spaces, particularly
euclidean spaces. Study of projectivities among projective spaces and
classification ofP1, P2,
P3 ´s. Study of the double ratio and armonic quaternin the projective line and of the dual
projective space, the principle of duality and dual projectivities. Study of
projective hyperquadrics, classified for conics and quadrics, tackling
polarity, tangency, cone of tangents and tangential hyperquadric. Study of the
hyperquadrics sheafs and classification of the conic and quadric sheafs, and
their reduced equations. At last, the study of euclidean hyperquadrics, metric
elements and unvariants for conics and quadratics.
Professor: Rosendo RUIZ SÁNCHEZ.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Requirement: Theoretical/practical written examination.
Numerical Methods II
Part One; First term subject;5 hours per week. 5 Cred. E.C.T.S.
The approximation to functions of one real variable is
one of the pillars of numerical methods. The first chapter considers Hermite´s
interpolation, together with the convergence and minimization of the
interpolation error. Splines interpolation is also studied. The second chapter
consists of an approximation in euclidean spaces and in the uniform norm
(theorems of existence and unicity, characterization of the best approximation,
constructive methods, ortogonal polynomials). Rational approximation is briefly
introduced.These topics support others
like: numerical derivation methods (based upon Taylor expansion, the
interpolation polynomial, together with the convergence acceleration by means
of the Richardson´s extrapolation) and numerical integration (simple and
compound Newton-Cotes quadratures, Romberg method and Gaussian quadratures).
Professor: Juan José MORENO BALCÁZAR; Andrei MARTÍNEZ
FINKELSHTEIN.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written
examination. Task works.
Second term
subjects
Advanced Mathematical Statistics.
Part One; Second term subject; 5 hours per week: 5
Cred. E.C.T.S.
The major techniques and methods of the Statistical
inference are developed, specially in parametrical models, studying in depth
the sufficiency, completion, invariance, unbiasness, efficiency and asintotic
properties of statistics. The Theory of Point Estimation, the Hypothesis Test
and the Regions of Confidence are studied in depth. The basic techniques of
non-parametrical Inference and linear models are presented, together with an
introduction to the Theory of Decision.
Three differentiated parts. The first one completes
some of the topological concepts not studied in previous subjects (separation
axiomes and numberability, included in the Tietze´s extension theorem and the
Urysolin´s theorem. The second part studies the differential curves in plane
and space by means of the Frenet-Serre equations (thus, locally). It also uses
topology introduced in the first part to demonstrate the Jordan´s curve theorem
(by means of the Brouwer´s fixed pointtheorem)
and the Isopermetric theorem (the most important global theorem for plane
curves). In the third part, the followig concepts are introduced: smoothing of
maps among them, Gauss mapping, fundamental quadratic forms, the various types
of curvature and methods for their calculation, and the Gauss´s Egrgium
Theorem. This forms the starting point of the most classical topics of the
Intrinsec Geometry.
In many scientific disciplines the presence of
mathematical problems that require the solution of equations, in most of the
cases difficult or impossible to be explicitally solved, is frequently found. This
subject introduces numerical methods developed to obtain approximate solutions
for non-linear equations. The search of the approximate solution does not pose
any ambiguity because error bounding is also studied. An approximate solution
with a pre-determined margin of error can be equally found.
Re-activation of the language knowledge by means of
grammar exercises and syntax structurization. Reading and comprehension of
scientific texts. Study of the retorical functions of the scientific speech
pursuying cohesion, organization and coherence of the information to be
transmited. Specific lexicon and word formation. Arithmetical operations in
spoken english. Mathematical definitions. The use of mathematical simbols
taking notes in English-spoken Congresses. Listening
exercises.
The linear finite differences equation with constant
coefficients. The space of possible strings. The obtaining of the Riemann
integrals. The factorial function (Gamma) as a solution for a equation in
linear difference, of first degree and variable coefficients. Linear SDO and
complete linears. Improper integrals. Convergence criteria in improper integral
of first and second specie. The Cauchy Criteria. Circular elemental functions.
Solutions of the EDO. The Flee´s transform. Transform of EDO and linear EDP of
constant coefficients. Series of Fourier integrals. The theorem of convolution.
Fourier inverse transform.
The objective of this optional subject is to complete
the geometry background of the student in aspects not tackled in the core
subjects of the programme.
The name of the subject comes from the study of the
vector metric Geometry, that is to say quadratic forms (symmetrical or not) and
the type of Geometry they determine. The student is also introduced in the
study of convex bodies in Rn, the simplest objects after vector
spaces. Other classical topics of elementary geometry like tessellates,
simplicees and polyhedrons are also treated.
This subject tackles the classification and solution
of problems. Different classifications of mathematical problems are analyzed,
classroom problems, classification of arithmetical problems. Phases in the
problem-solving process. Teaching models of the problem-solving process. Heuristic
techniques and strategies. Factors and requirements of the problem-solving
process. Obstacles and blockings in the problem-solving process. Elaboration
and analysis of protocols. Practicals about the theoretical contents by means
ofindividual and small-groups
problem-solving.
Holomorphic functions: Basic Theory: the concept of
derivative, Cauchy-Riemann´s equations. Holomorphic functions. Series of
powers. Analytical functions. Exponential function, trigonometric functions, multiform
functions. The Local Cauchy´s theory: curvilinear integral. Cauchy´s theorem
for star-shaped domains. Cauchy´s integral formula. Taylor´s expansion in
serie. Equivalence between analyzicity and holomorphy: Riemann´s theory of
avoidable singularities. The Principle of identity. The principle of maximal
module. Theorems of the open map and the inverse function. Singularities:
Laurent´s expansion in serie. Classification of singularities. The Theory of
residues. Mappings. The Principle of the argument. The Theorems of Rouché and
Hurwitz.
Requirement: Every 1st and 2nd year
Mathematic subjects should have been passed.
Basics in Physics
Part Two; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
In this subject some of the most fundamental Nature
processes are studied. The subject has been structured in three blocks tackling
different aspects of the Physical sciences: Mechanics, Thermodynamics and
Electricity and Magnetism. The first block tackles Classical mechanics
(Kinematics and Dynamics of the particle; the gravitational Field; Dynamics of
the Rigid Solid; Elasticity) with an introduction to the Quantum Mechanics. The
second block tackles concepts like Heat and his propagation, ideal and real
gases, and the Principles of Thermodynamics. The last block shows and studies
the electric and magnetic fields in the vacuum.
Professor: Francisco LUZÓN
MARTÍNEZ.
Teaching Method: Theoretical and practical lectures.
Mét. Exámen: Written examination.
Second Term
Functional Analysis I
Part Two; Second term subject; 5 hours per week. 5.5
Cred. E.C.T.S.
Basic theory of normed spaces: linear and continuous
mappings among normed spaces, finite dimension normed spaces, topological dual.
Hilbert´s spaces: The theorems of the optimal approximation, of the orthogonal
projection and Riesz-Fréchet´s. Orthonormal bases. Operators in Hilbert´s
spaces, spectral theorem for a compact normal operator. Fundamental Principles
of the Functional Analysis and Duality in Banach´s Spaces: the Hahn-Banach´s
Theorems (the extension and separation theorems). Banach´s reflexive spaces,
weak topologies, the Helly´s, Goldstine and Milman-Pettis Theorems. The
Banach-Alaoglú Theoreme. Consequences of the Baire Theoreme: Theorems of the
open map, of the Banach´s isomorphisms and of the closed graphic. The
Steinhaus-Banach Theoreme.
Professor: Juan Carlos NAVARRO
PASCUAL.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination. Practical reports.
Numerical Calculus I
Part Two; Second term subject; 6 hours per week. 6.5
Cred. E.C.T.S.
The mathematical modelization of the real phenomena
around us requires the solution of ordinary differential equations or equations
in partial derivatives. In most of the cases these equations do not show
analytically explicit solutions, so the use of numeric methods is necessary in
order to obtain a solution. The mathematical fundaments of numeric methods for
the solution ofproblems of initial
values are studied: 1-step methods (special attention paid to the Runge-Kutta
methods), multi-step methods (the Adams methods), and methods for special
problems (stiff problems, and so on). Numeric methods to solve contour problems
are also studied: shots, resolution in differences, variational methods, and so
on. Finally solving methods for Fredholm and Volterra´s integral equations are
introduced.
This subject consists of two parts: 1st.- Advanced
Geometry of regular surfaces. Intrinsic Geometry of surfaces. Global Theorems
of the Theory of surfaces. Need of abstraction and generalization of the
surface concept. 2 nd .- Theory of the differentable varieties.
Assessment Method: Written/oral examination. Work presentation. Work in
groups. Presentations.
Mathematic Didactics in
Baccalaureate
Second Part; First term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
Curriculum of Mathematics. Teaching. Learning.
Elements of curriculum design. The process in class. Mathematics programmes in
second-grade education. Design and development of the curriculum. The community
of mathematic lecturers. Comparative vision of the Mathematics Curriculum.
The objective of this subject is to introduce and
analyze the major models of design of statistical experiments. After tackling
the Analysis of Variance technique we develope the completely randomized,
randomized, factorial, nestedand mixed
models. The analysis of models is carried out by using statistical packages
such as Statgraphics. The methodology of response surfaces is also introduced.
Second Part; First term subjects; 4 hours per week. 4.5
Cred. E.C.T.S.
First order partial derivative equations. The general
Cauchy´s problem. The Cauchy-Kowalewsky Theorem. The Unicity Theorem. Quadratic
equations. Classification. The Divergence Theorem. The potential equation. The
waves equation. An introduction to the Theory of Partial Derivative Equations
modern expansion.
Professor: Bernardo LAFUERZA GUILLÉN.
Teaching Method:
Assessment Method:
Statistical Inference II
Part Two; First term subject;3 hours per week. 3.5 Cred. E.C.T.S.
The purpose of this subject is to tackle statistics
from different points of view: Bayes model and theory of decision. The subject
is divided in three different parts: 1,. Interpretation of probability:
Classical method, frequentist model and subjective model. 2.- Approach to
statistical problems under the bayes perspective. 3.- Introduction and study of
statistical problems by means of Theory of Decission´s tools.
The objective of this subject is to instruct the
students in the analysis of efficient algorithms, the different techniques in
the conception of algorithms, and let them know the basic tools for the
development of their own algorithms applied to mathematics. The following
topics will be developed: 1- Analysis of the algorithm´s efficiency. 2.-
Algorithms "divide and rule". 4.- Voracious algorithms. 4.- Dynamic
programmation. 5.- Graph exploration. 6.- Elements ofcalculation complexity.
Definition of the first group of homotopy in a
topological space. Calculation of the first group of homotopy of the unit
circunference.The Seifert-van Kampen
theorem and his application to the calculation of the first homotopy group in
different vector spaces. Definition of the singular homotopy groups of a
topological space. Contruction of the Mayer-Vietoris string of a topological
pair. The Scission theorem and construction of the Mayer-Vietoris string and
its use for the calculation of the sphere´s singular homology groups. Singular
homology techniques used for the demonstration of classical theorems in
Topology: the theorem of dimension invariance. The Brouwer´s theorem of the
fixed point and the Jordan-Brouwer´sseparation theoreme.
Global description of the Universe; a description
surging from interpretation of observational aspects through fundamental
physical theories: the theory of radiation, classical and relativity dynamics
and nuclear physics.
The two-bodies problem. Motion of the Solar System
bodies. Earth motion. Double-stars.
Brief introduction to spheric trigonometry. Introduction
to co-ordinates systems and co-ordinate change systems. Problems associated to
the daily movement (rising and setting of stars, maximal disgressions and first
vertical). Correction in the astronomic co-ordinates: refraction and light
aberration, equinox precession, parallax. The problem of time: Sidereal, true,
mean, civil, and official. The solar system: the sun, the moon, the planets. Eclipses.
Professor: David LLENA CARRASCO.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Numerical Calculus II
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
The theoretical fundaments and mostly used algorithms
in interpolation and approximation of functions of some real variables are
studied. The general analysis of the problems of existence and unisolvency, is
followed by the interpolation in regular grids. The interpolation of scattered
data is discussed: unisolvency, error, algorithms, together with different
approximation methods (uniform, square minimums, a.o.) The local methods are
analyzed; a central place occupied by multivariated splines and their use for
approximation and interpolation. In this sense, triangulation and surface
partitioning methods are studied. As a Bezier´s application, the Coon patches
and the method of finite elements.
This subject tackles the bonds between Statistics and
Computer Science. We start studying the generation of randomized numbers and
variables (in general), followed by the Monte Carlo´s simulation and its use
for integrals estimation. The use of statistical techniques in the construction
of expert systems is also studied. Different statistical packages are used
during practicals.
The objective of Computational Geometry is to tackle
geometry problems with computational methods. The focus of the subject lies on
the discovery of effective algorithms (necessary to introduce first the
concepts of algorithm and efficiency) for rather simple problems (due to the
impossibility for the student to solve complex problems which are in some cases
still object of research).An example
of the treated topics would be: Voronoi´s diagrams, "guarded
vigilance" (the Chevall´s art gallery theoreme), uses in visibility and
robotics.
Professor: Mª Luz PUERTAS GONZÁLEZ y
M. A. SÁNCHEZ GRANERO.
Teaching Method: Theoretical and practical lectures.
Assessment Method: Written examination.
Differential Geometry
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
In the solving-process of many mathematical problems
lies the idea of reasonably disturbing the given hypothesis in order to
simplify the situation; this is the idea of "general position" in
Geometry, or "non-degenerated case" in Analysis. The most fruitful
expression of this argument is the transversality notion, which, created by
René Thorn, introduced the Differential Topology.In order to tackle this notion, it was necessary to introduce the
fundaments of the study of varieties with boundary (fibrated, inmersions and
summersions, orientations) and from there, reach the Sand-Brown theorem and the
Whitney´s theorem. On the other hand the construction of tubular environments
in the normal fibrate gives us some approximation theorems that, if combined
with transversality, constitute a powerful tool to classify curves, demonstrate
the Brouwer´s fixed point theorem and introduce the concept of degree.
Teaching Method: Theoretical and practical lectures. Practicals.
Seminars. Discussion on specific topics which in relation with the subject,
have been collected in recent journals. Literature retrieval.
Assessment Method: Written examination about basic theoretics and
problems. Presentation of complementary exercises. Work in groups (max. two students).
Presentation of complementary advanced topics.
Requirement: In order to follow this subject it is necessary to be
proficient in Mathematics Didactics.
Numerical Solution of Partial
Derivative Equations
Part Two; Second term subject; 4 hours per week. 4.5
Cred. E.C.T.S.
The subject starts introducing physical situations
modelled with P.D.E. The classical equations of Mathematical Physics are
studied, together with the maximum-minimum concepts for armoric and parabolic
functions. By means of the variable separation method mixed problems like the
heat and waves equations are solved, together with problems of the Dirichlet
type in rectangle and disc. The core part of the programme is dedicated to the
study of explicit and implicit methods in finite differences, together with the
finite elements method and the Galerkin semi-discrete methods. Mathematica and
Ansys will be used as software packages.
Automata
Theory and Formal Languages
Part Two; Second term subject; 4 hours per week. 4.5 Cred. E.C.T.S.
Finite
automata. Regular expressions. Context
free grammar. Battery automata. Turing´s machines. Computability. Chomsky´s
hierarchy. This subject deals with some of the principles or fundaments of
Computer Science supporting the global theoretical and practical frame of this
science, for example, the automata theory, the computation theory and the
formal languages theory.
Professor: Manuel CANTÓN GARBÍN.
Teaching Method: Theoretical/practical (problem-solving) lectures.
Assessment Method: Two examinations during this term.
Requirement: Basic knowledge of Mathematics.
[1] Subjects that been of a different
degree, the student can choose between the ones that have been offered by the
University in order to complete the number of credits needed.
[2] All subjects in this term are Optional or FreeConfiguration subjects.
* Every subject of this term period are Optionals or Free Configuration.
|
This site is designed to give students access to lessons that they may have missed in class as well as homework reminders! Extra credit is also available! If you need access to your Course 2 textbook, go to Math Connects to get online and complete assignments. For Algebra students go to Algebra 1 Online. If you need further information about your school or the school system, please click the links below:
Go to Hines Middle School
Go to NNPS
Core 1 - Algebra
Algebra is the first step to success in all future math courses and in logical thinking and problem solving life skills.
|
Calculus: A gateway to mathematical analysis
Calculus is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations. Calculus is a gateway to advanced courses in mathematics devoted to the study of functions and limits, broadly called mathematical analysis. Calculus has widespread applications in science, economics, and engineering and can solve many problems for which algebra alone is insufficient. Calculus has two major branches, differential calculus and integral calculus.
Isaac Newton (1642-1727) and German philosopher and mathematician Gottfried Wilhelm Leibniz (1646-1716) invented calculus independently. Historically calculus is known as infinitesimal calculus. It constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus.
In differential calculus, the term' derivative' is used to denote the 'rate of change'. In simple term it describes how quickly a variable changes with respect to another variable. For example, the velocity of a car can be calculated using the concept of derivative which is the rate of change of distance with respect to time ( ). Historically, it helped dealing with many problems in physics.
Conceptually, the integral calculus is the opposite to differential calculus and hence it is also called ánti-derivative. In integral calculus, the term 'integral' is used to denote the summation of values. This is represented by an elongated 'S' symbol ò.
For example, the concept of integration can be used to calculate the area of irregular shapes.
The area of DEF cannot be solved similar to the area of ABC. Calculus helps solving such problems. Area under DEF can be broken into an infinite number of recognisable areas (infinitesimals) and then summing up all those small areas—this is the concept of integral calculus. Historically this is known as method of exhaustion. Greek mathematician Archimedes developed this idea in the 3rd century BC while calculating the area of a parabolic segment.
|
Screenshots
Description
* You can use functions: all trigonometry, all hyperbolic, logarithm, arithmetic and statistics. And logically the operators "+,-,/,*,^,!,_".
* You can enter 4 different numerical systems in the same expression: binary, decimal, octal, hexadecimal.
* You may have the answer in 4 different number systems (binary, decimal, octal, hexadecimal)
* You can create variables of type "variable name=expression" and use them in the following expressions.
* You can standardize the form of calculations (degrees, radians, gradians) or/and if you want you can force a function to be calculated in radians, degrees, and gradians, using "rad", "deg", "gon". Example "radsin(pi/6)" ="degsin(30)".
* You can use three types of parentheses "{}" "[,]" "(,)"
* You can solve expressions with exponents like "2^_2^3^_4" and using parentheses you can calculate with even more complexity.
* You can get the answer in the form of SI prefixes (milli, micro, everyone!) Example "1E-9" = "1n".
* And more...
|
The purpose of this class is to offer students a refresher math class. Topics will
include operations with real numbers (fractions, integers, decimals), solving linear
equations, introduction to graphing, and multiplying polynomials. This is a great
course for students who are struggling in math 092 or apprehensive about taking math
courses as they advance towards their degree requirements.
|
The study of mathematics helps students to develop thinking skills, order their thoughts, develop logical arguments, and make valid inferences. The goal of all mathematics courses is to enable students to discern mathematical relationships, reason logically, and use mathematical techniques effectively.
Course List, Textbooks and text coverage
The Mathematics Department's Course List provides a listing and description of the classes at Lowell offered by the Mathematics Department along with prerequisites for prospective students. Here (updated Sept. 2009) is a list of textbooks currently used with the sections covered by course semester.
The Mathematics Program
All entering ninth-grade students are tested for math placement. Placement is determined based on a combination of the mathematics placement test and/or the Algebra I California Standards Test that should be given in middle school. If students are unhappy with their initial placement, they may attempt to audit the higher course along with the course they were placed into and move into the more advanced course sequence should be successful in the audited course. Entering the honors path after Accelerated Math 2H is difficult because honors math courses move at a much more rapid pace than regular math courses.
|
Covers
all topics to date including and not limited to: steps of the modeling
process, plotting data, fitting curves to data, linear regression,
correlation coefficient, extrapolation, interpolation, how a
mathematical model can be good.
|
Book Description: To succeed in the lab, it is crucial to be comfortable with the math calculations that are part of everyday work. This accessible introduction to common laboratory techniques focuses on the basics, helping even readers with good math skills to practice the most frequently encountered types of problems. Discusses very common laboratory problems, all applied to real situations. Explores multiple strategies for solving problems for a better understanding of the underlying math. Includes hundreds of practice problems, all with solutions and many with boxed, complete explanations; plus hundreds of "story problems" relating to real situations in the lab. MARKET: A useful review for biotechnology laboratory professionals.
|
Students use a graphing calculator, online calculators, and other tools to explore the profitability of raising corn. The lesson is most suitable for the end of the first year of algebra when students have knowledge of using in/out tables, identifying types of lines, and developing lines of best fit, equations, and interpreting graphs. Aligned to the California State Standards. From the Schools of California Online Resources for Educators SCORE Mathematics Lessons.
|
Homogeneous Systems
In this lesson our instructor talks about homogeneous systems. He discusses the definition of homogeneous systems and procedure for finding a basis for the null space. He ends the lesson with two example problems and relationship between homogeneous and non-homogeneous systems.
This content requires Javascript to be available and enabled in your browser.
Homogeneous Systems introduces concepts early in a familiar, concrete real number setting, develops them gradually, and returns to them again and again throughout the text so that when discussed in the abstract, these concepts are more accessible.
|
Elementary Linear Algebra with sophomore-level courses in Linear Algebra or Matrix Theory.This text presents the basic ideas of linear algebra in a manner that offers students a fine balance between abstraction/theory and computational skills. The emphasis is on not just teaching how to read a proof but also on how to write a proof. This book presents the basic ideas of linear algebra in a manner that users will find understandable. It offers a fine balance between abstraction/theory and computational skills, and gives readers a... MOREn excellent opportunity to learn how to handle abstract concepts.Included in this comprehensive and easy-to-follow manual are these topics: linear equations and matrices; solving linear systems; real vector spaces; inner product spaces; linear transformations and matrices; determinants; eigenvalues and eigenvectors; differential equations; and MATLAB for linear algebra.Because this book gives real applications for linear algebraic basic ideas and computational techniques, it is useful as a reference work for mathematicians and those in field of computer science.
|
I have recently completed my second semester of calculus, and this textbook was used for each class. I was quite impressed by it. It starts out simply enough with a unit on functions and gradually leads the student deeper into more advanced concepts, such as differentiation, integration, integration techniques, numerical series, and vector analysis. The book also includes a chapter giving an introduction to differential equations, and it contains several appendices dealing with trigonometry, logarithms, complex numbers, and more; making this book excellent as a math reference as well as a class textbook. That's why I have no intention of selling it back after the class!
Each new concept is illustrated with several examples, and numerous exercises accompany each section. The author strikes a good balance between being overly abstract and overly concrete, so you can make this book work for you no matter what your style of learning math may be. It contains interesting side notes on the history of mathematics, and the pages are laid out in a way that's pleasing to the eye. All in all, a very well-constructed book.
I'm NOT saying that this book makes learning calculus easy-such a book does not exist, unless you're a math prodigy! Stewart's Calculus will, however, give you thorough guidance as you learn this difficult yet fascinating subject. Calculus is very hard to learn with any approach, but with confidence and plenty of effort, it can be mastered and I've found it a very fulfilling area of study. Hopefully you will also.
The bookseller (dreamboat books) was great to deal with, would gladly work with them again. The book is o.k., for a government book.
|
This course emphasizes the extension of mathematical knowledge and skills to prepare students for success in their everyday lives, in the workplace, strengthening and extending key foundational mathematical concepts and skills by solving authentic, everyday problems. Students have opportunities to extend their mathematical literacy and problem-solving skills and to continue developing their skills in reading, writing, and oral language through relevant and practical math activities.
MFM2P - Mathematics: Foundations (Applied):
This course enables students to consolidate their understanding of relationships relationships. Students will investigate similar triangles, the trigonometry of right-angled triangles, and the measurement of three-dimensional objects. Students will consolidate their mathematical skills as they solve problems and communicate their thinking.
MPM2D - Mathematics: Principles (Academic): multistep problems and communicate their thinking.
|
Mathrealm
Algebra 1 Explorer software will turn struggling students into successful math learners and average students into accelerated math learners!
Spark your students' interest in algebra with state-of-the-art graphing tools and interactive lessons that encourage students to explore and learn algebraic concepts.
The Algebra 1 Explorer features a comprehensive algebra 1 curriculum that is packed with resources for your class, including student self-paced tutorials, interactive class presentation material, as well as student learning activities and worksheets.
Algebra World software will turn struggling students into successful math learners and average students into accelerated math learners!
Algebra World teaches and reinforces introductory algebra concepts and meets NCTM standards. Mathematics topics have series of lessons and real world examples that accompany them. Equations and their relationship to word problems are emphasized throughout the program.
Pre-Algebra World creatively develops key concepts with images that students will remember for a lifetime. Learning activities in this highly visual and interactive program actively engage students and reflect NCTM standards. This entire imaginary world is designed to motivate students to learn and use math.
Thousands of students, teachers and schools are using Pre-Algebra World to get great results. It provides a proven and dynamic learning environment that motivates students to succeed.
This program provides comprehensive overview of trigonometry fundamentals and helps students make connections from abstract concepts to the world around them. This engaging and easy-to-follow program builds many bridges from math to science, history, and everyday occurrences.
The Trigonometry Explorer offers something for every student of Trigonometry. For those who need to review the basic building blocks of Trigonometry, the creative animated and engaging lessons on degrees, angles and triangles provide a great starting place. The functions, plotting and inverse functions lessons are for those who have a grasp of the basics. For the student who is bored and uninspired, with Trigonometry, the applications on radio waves, sound, surveying and many more show how Trigonometry is used to explain everyday phenomena and solve real word problems.
|
The Elements of Cantor Sets thorough introduction to the Cantor (Ternary) Set and its applications and brings together many of the topics (advanced calculus, probability, topology, and algebra) that mathematics students are required to study, but unfortunately are treated as separate ideas. This book successfully bridges the gap between how several mathematical fields interact using Cantor Sets as the common theme. While the book is mathematically self-contained, readers should be comfortable with mathematical formalism and have some experience in reading and writing mathematical proofs. Chapter coverage includes: a biography of Cantor; an introduction to the Cantor (Ternary) Set; Self-Similar Sets and Fractal Dimensions; sums of Cantor Sets; the role of Cantor Sets to create pathological functions; and additional topics such as continued fractions, Ana Sets, and p-adic numbers.
|
130 Elementary Calculus
Credits: 3 This course includes the following topics: differentiation and integration of polynomials; rational, logarithmic, and exponential functions; and interpretation and application of these processes. 3.0-0.0-3.0
Prerequisite(s)........... MAT-110 with a grade of "C" or higher, or acceptable placement score
|
10 Units 1000 Level Course
Available in 2013
Continues the development of linear algebra and calculus from MATH1210. In algebra, students learn how matrices may be simplified by an appropriate choice of coordinates. Known as the eigenvalue technique, this is a very powerful mathematical tool and is used throughout Science and Engineering. In calculus, students learn the mathematics behind algorithms used in calculators for computing exponential, trigonometric and other functions. They are also introduced to functions of several variables and to the notion of a differential equation, which are key concepts in mathematical modelling.
This course is a sequel to MATH1210 and is likewise intended for prospective mathematics majors and those who have a strong background in mathematics. There is substantial overlap with MATH1120; students' performance on this common material is compared and used to scale the marks to ensure that comparable students achieve comparable grades.
Not to be counted for credit with MATH1120.
Objectives
1. To provide students with a firm foundation for later studies in algebra and analysis, the two main branches of mathematics.
2. To reinforce students' understanding of high school mathematics by developing a more rigorous approach.
3. To develop students' capacity for effective reasoning, and their ability to use their mathematical skills elsewhere.
|
SL Random Number Generator is a program designed togenerate random numbers.Program can exclude specific digits from random numbers.Output can be decimal, binary, hexadecimal, octal.Numbers can be sorted.Program can print or save all generated numbers.
Popular Downloads of Education \ Mathmatics...
If you are a math student or teacher or just a person who is interested in high-school algebra or college calculus, I would recommend you this program.Graphmatica presents an interactive algebraic equation grapher that can be used as an aide to plotting mathematical curves. Graphmatica remembers...
Equation Maker is fast and versatile math editor. This program designed for edit rich-text content with almost any math formulas and graphics. It works like TeX or HTML but with simple terms.It search in ASCII text for math formulas, hyperlinks and pictures but without any tags. It can be used as...
Simple Solver is a free Windows application that can simplify Boolean equations, truth tables and digital logic. The application includes six different tools: Boolean, Logic Design Draw, Logic Design Auto, Logic Simulation, Permutation and Random Number. These tools are built on years of...
Logic Minimizer is an innovative, versatile application for simplifying karnaugh maps and logical expressions step by step. It is geared for those involved in engineering fields, more precisely digital and formal logic scholars and academics, digital devices constructors or anybody involved with...
Random Number Generator PPC is a Windows Mobile based software that produces sequences of random numbers.Program can exclude digits and numbers from generated random sequences.User can save results to a text file.Generated random numbers can be positive or negative.
|
Reviews
Top 3
College Freshman Austin Community College Cypress Creek, Cedar Park TX
There was a time earlier in my life that the subject Math was my favorite. Now Iím dreading learning a new formula or how to learn one thing, then a week later, forget it, but donít forget. In this particular selection there are study skills that help you remember, at least till you take the exam!
The three I favor the most are: Do math homework everyday. This insures that you donít forget. Secondly, write down questions for the instructor. Definitely want to find out what I donít know. Itís hard trying to figure out a math problem, get an answer and apply it to the rest of the problems, only to find out that Iím using the wrong process! Last, make time to study. I work two jobs, one part time, and one full time. I canít seem to find enough time to think, so making time to study, or planning time to study should I say is somewhat difficult. However, if I just sit down with my schedules, there is always 2-3 hrs every other day.
When I was in middle school, Math was one of my favorite subjects, but once I got into harder math, it slowly became my least favorite subject. I didnt know how to study it or even where to start. Well, the way this article broke down the problem solving and study skills, it made me feel like I have hope in math. I think anyone who struggles in math should read this article. It may really help you.
MATH
KCOOPER, College Freshman ACC, Austin
This review is very helpful because it is short and sweet. Straight to the point. I am already studying a ton for my math class and I do not want to have to study how to study for long periods of time. It is helpful because the main ideas are in bold so you can just skim through it real quick to refresh your memory.
Math is one of the subjects that most students have difficulties with it. Each individual might have different method for learning math.
I believe that one way to improve math study skill is to practice different types of problems and try not just to memorize the steps but to learn how to solve a problem. Also, those students who want to be successful have to practice as many problems as they can. The more a student practices, the more likely he/she will learn faster and will understand each problem better.
Besides practicing, I believe that organization is another important factor to improve math study skills. A student needs to be organized in writing math. This means that a problem should be written neatly and the steps should be organized. Writing neatly can help the reader to understand the solution and it helps the student when he/she is referring back to it for tests.
Another way to improve math study skills is to draw pictures and diagrams that would help the student to understand the concept. With the use of diagrams and picture, the student can have a visual sense of the problem too.
This is really helpful! I agree with numbers 19 and 22. Whenever I'm taking notes, I always try to write step-by-step instructions on how to solve different types of equations. I also try to write down different examples on how to solve the various equations.
I have always been intrigued by math and its methods. It is simply amazing how much the mind is exercised and expanded by constantly doing math problems and exercises. But one thing that I have always come across with fellow students and some of my tutees, is that they will know the method needed to approach a problem, but have no idea why it is that way, and what is the reason for that method. This not only invalidates the point of the instruction or lesson, but also might hinder the student's comprehension which might come in handy when the difficulty of problems goes up. So it's always very important to understand not only how the method is applied but also why.
A good, straight-forward handout. Many of the tips I already use (such as taking extra notes on the difficult steps), but there are a few that I don't often think about. In particular, interviewing potential instructors, getting help early on, and scheduling a study session right after class would help a lot with creating an effective learning environment.
Puzzled with Math
Rachel, College Junior University of Alaska Fairbanks, Fairbanks, AK
This is a great article that addresses a few key issues that many people have with math. In particular these are some very good tips:
"While doing homework, write down questions for the instructor/ tutor." – Many times it is difficult to speak up in class and get help, however, in my experience it is even more challenging to try and dig your way out of a pit of bad scores and frustration. Actively participating in lectures is perhaps the most important math study skill – and probably the one that people forget about the most!
"Schedule a study period after your math class." – Math is definitely a subject where practice makes perfect. This means that no practice makes…well, it isn't pretty. Taking the time to do your math homework is crucial to your performance in the class! Even if HW isn't worth much, without the practice your exam scores will likely suffer as well.
I have taken several math courses. Some of them I have thrived in and others I have not fared as well. I think that printing and going through all of the steps in this selection would be super helpful to students. Bring it to your tutor or study group if you have them (which in my opinion you should)!
Finally, a point that isn't included is that math does not have to be torture. As children we are constantly exposed to communication, language, writing, and history which help to make these subjects a little more comfortable to study. Advanced math? We do not always have the opportunity to be so associated with the subject, and consequently tend to shiver at the thought of it. Math can truly be fun though, if you look at the problems as puzzles or games rather than impossible tasks, you can usually work through the problem in your own way and enhance your learning by doing so.
About
Spotlight
How to Study Model
Write a Review
Be the first from your college to be published here. All you have to do is to write a comment about how a study skills handout was helpful. Or you could add some handy tips of your own. Click on write review.
Find a Tutor
Looking for a free or paid private tutor?
Just for Fun
"Im hiding." Look for this image on the study skills pages for more cartoons!
|
Search
Academics at Champion
MATHEMATICS - Precalculus/Trigonometry
ADVANCED MATH: This is a 11th/12th grade preparatory course for students intending to take college level Calculus.
This year long course is based primarily on the California State Standards for Trigonometry and Mathematical Analyses. This class includes further study of trigonometry, logarithms, and graphs of rational functions, matrices, probability, and statistics. It also includes an introduction to basic topics in calculus such as sequences, limits, and derivatives.
|
AP Calculus A/B/C
AP Calculus A
AP Calculus A is a mathematics course designed to provide students with the prerequisite skills necessary for AP Calculus B and the Advanced Placement exam. This online calculus course addresses various learning styles through projects, discussions, online interactivities, as well as traditional coursework. AP Calculus A students are serious and enthusiastic about learning and are expected to be proactive throughout the entire course. A strong background in Algebra II, Trigonometry, and Pre-Calculus is necessary in order to be successful in this course. The major topics explored in this calculus course include limits and continuity, derivatives and differentiation techniques, related rates, optimization problems, and numerous applications.
AP Calculus B
AP Calculus B is a mathematics course designed to provide students with the prerequisite skills necessary for AP Calculus C and the Advanced Placement exam (AP Calculus AB). This calculus course addresses various learning styles through projects, discussions, online interactivities, as well as traditional coursework. AP Calculus B students are serious and enthusiastic about learning and are expected to be proactive throughout the entire course. A strong background in AP Calculus A is necessary in order to be successful in this course. The major topics explored in this online calculus course include integration and differentiation techniques, differential equations, areas and volumes of bounded regions, arc length, and surfaces of revolution, and numerous applications.
AP Calculus C
AP Calculus C is a continuation of AP Calculus B and is designed to provide students with the prerequisite skills necessary for the Advanced Placement (AP Calculus BC) exam. This online calculus course addresses various learning styles through projects, discussions, online interactivities, as well as traditional coursework. AP Calculus C students are serious and enthusiastic about learning and are expected to be proactive throughout the entire course. A strong background in AP Calculus B is necessary in order to be successful in this course. The major topics explored in this calculus course include derivatives of vector-valued and parametrically defined functions, integration by partial fractions, improper integrals, series convergence (Taylor and Maclaurin), L'Hopitals Rule, and numerous applications.
All courses are written to California Department of Education standards and to national standards, as applicable.
|
Find a Lionville provide that. A continuation of Algebra 1 (see course description). Use of irrational numbers, imaginary numbers, quadratic equations, graphing, systems of linear equations, absolute values, and various other topics. May be combined with some basic geometryI am certified to teach secondary English and social studies in PA. My
|
Websites that are usefull
by
interactmath.org This is a tutorial website for many high school topics. When you get to the website, just click "enter". The new screen will ask you for a textbook. The Pearson 2011 edition is very similar to the one we use. Select it and choose a level to start. It checks your answer for correctness and will walk you through a problem if you wish.
phschool.com/atschool/txtbk_res_math.html This is the website of the textbook manufacturer. Choose the Alg II 2004 edition. You will be able to pick a particular section from any chapter for a tutorial on the major topics from the section.
Geogebra.org This is an interactive graphing program that is rated very high in quality. You can download it for free. It is used anywhere from simple geometry to high level calculus. The tutorial walks you through many examples. I highly recommend this for every high school student.
Purplemath.com This is a web site that has excellent concept development for students to understand an area of mathematics. I use many of their explanations in the Daily Lessons section.
KhanAcademy.com This is a web site contain short instructional videos developed by a MIT professor. It coverts the basic procedures on a very wide range of topics covered in the high school curriculum. This is highly recommended for the student to view before they even cover the material in the class.
|
Find a CartersFrequently used Microsoft PowerPoint during MBA coursework. Can help you use templates, create appealing transitions, and link or embed various items. Tutored Prealgebra topics during high school and college.
|
College Algebra Placement Test
You don't have the latest version of Adobe Flash Player.
College Algebra Placement Test Study Guide The student may use any calculator he or she wishes but may not use a computer. (The student must furnish the calculator.) The time limit for the test is three hours. The test consists of 26 questions.
About half the points come from 17 multiple-choice problems, while the other half come from 9 open-ended and graphing problems. The problems on the test are of average difficulty. They are at the level of difficulty generally found in the regular exercises in any College Algebra textbook.
The current course textbook is College Algebra: A Graphing Approach , 3 rd Edition; Larson, Hostetler, and Edwards; however, any College Algebra textbook should provide sufficient information. To pass this placement test, a student should have a working knowledge of the following topical areas: Functions: The student should be able to: " Apply the definition of a function. " Perform operations on a function (add, subtract, multiply, divide).
" Evaluate a composite function. " Determine the domain and the range of a function. " Find the inverse of a function or indicate there is no inverse.
odd and even functions.<br><br>
" Translate a function on a graph and then determine the corresponding change in the equation of the function. " Recognize the graphs of elementary families of functions (linear, square root, greatest integer, absolute value, quadratic, cubic, rational exponential, logarithmic). Logarithms: The student should be able to: " Translate back and forth between a logarithmic equation and the corresponding exponential equation.<br><br>
" Define and use common and natural logarithms. " Graph exponential and logarithmic functions. " Use the properties of logarithms to simplify expressions.<br><br>
" Use the properties of logarithms to solve logarithmic equations and basic exponential equations. 7/01 Graphs: The student should be able to: " Write the equation of a parabola, circle, hyperbola, or ellipse based on its graph. " Graph a parabola, circle, hyperbola, or ellipse based on its equation.<br><br>
" Graph absolute value, exponential, logarithmic, or rational functions. " Graph the solution of a system of inequalities (linear and non-linear), identifying the coordinates of the vertices. General: Besides the topics specifically listed above, the placement test assumes that the student has acquired knowledge of the fundamental concepts of algebra.<br><br>
The student should be able to: " Solve equations and inequalities. " Apply the properties of exponents. " Simplify radicals.<br><br>
" Simplify algebraic expressions. " Work with complex numbers. 7/01<br><br>
|
Homework exercises
have been liberally chosen from the following texts.:
Principles of Modern Algebra by J. Eldon Whitesitt
AModern
Introduction to Basic Mathematics by Mervin L. Keedy
A First Course in Abstract Algebra by John B. Fraleigh
Our Current Math 103 Textbook
The topics selected for coverage are based on the curriculum
in ALGEBRA I, published by GLENCOE and are specifically related to the SOL's
for the State of Virginia.
There are six laboratories scheduled during the course.Each laboratory will involve hands on tools
and demonstration of how the topics of the course arise in the middle school
curriculum and how they are taught.A
seventh laboratory may be scheduled if time permits.
qIs the operation of
subtraction on the set of all integers (a) commutative? (b) associative? In
each case explain your answer.
qShow that division by
non-zero elements is right distributive, but not left distributive over
addition in the set of rational numbers.
qFor the set of
integers, define the operation * as follows:a*b = a + b – ab where and b are any integers and + and -are the usual addition and subtraction and
ab is the usual product of integers.Find the identity element in the set relative to this operation.Let n be an arbitrary integer and find the
inverse of n relative to the operation *.
·Rational Numbers
HW:
qFor the set of rational
numbers, we define two binary operations $ and # (in terms of ordinary
arithmetic operations) as follows: a $ b = 2ab and a#b = a + 2b for any two
rational numbers a and b.
qFind a nonzero divisor
of zero in M the set of all 2x2 matrices with integer entries.
qLet E be the set of all
English words and let S be the set of all "letter strings" [finite lists of
letters, possibly repeated] of our alphabet.Notice that E is a subset of S.In each instance below a process *that makes sense oneach of
these two sets is described.[The Greek
letters a and b represent arbitrary
elements of these sets.]Answer each of
the following questions for each exercise:
1.Give two more examples to show how * works.
2.Is * a binary operation on S? on E?Justify each answer with a reason or
counterexample.
3.If * is a binary operation on the set, say whether *
is associative and/or commutative, supporting your answers with reasons or
counterexamples.
a)a*b is formed by putting a and b next to each other,
forming a single string.For example,
moon*glow = moonglow.
[This
is like the password process used by CompuServe and other electronic services.]
b)a*b is the word or string
that in alphabetical order comes first.For example, trip*trap = trap.
[This is a basic part of any
word-sorting algorithm.]
c)a*b is the string of
letters, including repeated letters, that are common to a and b, arranged in
alphabetical order.For example,
beaker*knee = eek.
d)a*b is the number of
letters in the longer word or string, or if they have the same number of
letters, it's that number.For example,
movie*theater = 7.
Connection to lessons in ALGEBRA 1 (Glencoe): 1-6, 1-8,
5-1.Here will be an opportunity to
identify the whole numbers, integers, rational numbers and matrices as distinct
algebraic structures by virtue of the properties of the binary operations in
each structure.
qWrite out each element
of S3.Label the identity as
r1 and each of the other elements as r2,r2, r3, r4, r5, r6.Fill out the complete "Multiplication Table"
for S3 using your elements as labeled in the previous problem
qConsider the Subset A3
of S3 as follows: .Show that A3
is closed for the operation of multiplication and that each element in A3
has a multiplicative inverse in A3.
qProve Theorems 2,4 and
6
qThe converse of theorem
1 is: For any elements a,x and y in a group G, if x = y then a¨x = a¨y.
a)Does this require proof as a result in group theory?
Why or why not? (Hint:consider the
definition of abinary operation and
use this in your discussion.)
b)Euclid stated this same idea by saying "If equals are
multiplied by equals, the results are equal."Discuss the appropriateness of saying "multiply both sides of an
equation by the same thing" or "multiply one equation by another" in the
context of Group theory.
qDiscuss what each of
theorems 1-4 says about elements of (a) the integers and (b) the rational
numbers in the context of the kinds of equations with (a) integer and (b)
rational number coefficients that are guaranteed to have unique solutions for
x.
·Theorems
7.In any group G, the
identity is its own inverse.
8.For any elements x and
y in a group G, (x¨y) -1 = y -1¨a –1.
9.For any element x in a
group G, (x –1) –1 = x.
10.Let a and b be elements of an arbitrary group G, and
let m be a natural number.Then, for any natural number n,
a)en = e where e is the identity of G.
b)For any a and b in G,
(ab)n = anbn
if an only if ab=ba.
c)am+n = am an.
d)(am)n = amn = (an)m.
·Definitions: order of a
group, order of an element
·Cayley tables
HW:
qFactor 3 in two
different ways in Z6.Can 5
be factored into factors in Z6 other than by using 1 as a factor?
qFind two equations of
the form ax = b, with a and b in Z6 and with b ¹ 1, which cannot be
solved in Z6.
qLet the following table
define a binary operation on the set {2,4,6,8}
*
2
4
6
8
2
4
8
2
6
4
8
6
4
2
6
2
4
6
8
8
6
2
8
4
Answer each exercise below in reference to*
a)Determine 4*8
b)Which element, if any,
is the identity?
c)Which element, if any,
is the inverse of 8?
qEach table below defines
a binary operation on the indicated set..
ª
p
q
r
s
t
à
0
1
2
3
4
p
s
r
t
p
q
0
0
1
2
3
4
q
t
s
p
q
r
1
1
2
3
4
0
r
q
t
s
r
p
2
2
3
4
0
1
s
p
q
r
s
t
3
3
4
0
1
2
t
r
p
q
t
s
4
4
0
1
2
3
Answer each exercise below with
reference to the appropriate table.
a)Calculate (qªr)ªp and 1à(3à2).
b)Is there an identity
for à?If so what is it?
c)Does q have an
inverse?If so, what is t?If not, why not?
d)Does 1 have an
inverse?If so, what is t?If not, why not?
e)Solve for x: 2àx = 3.
qFill in the table below
so that you create a binary operation on the set {a,b,c,d} with the following
three properties:
i.c is the identity
element,
ii.b is the inverse of d,
and
iii.the operation is
commutative.
*
a
b
c
d
a
b
c
d
qProve Theorems 7,9 and
10
·Subgroups
·Theorems
11.If H is any subgroup of a group G, then the identity
element of G is in H and is the identity element of H.
12.A subset H of a Group G is a subgroup if and only if:
a)H is not the empty set.
b)For every pair of
elements a and b in H, the product ab –1 is an element of H.
13.If a is an
element of a group G, the set H = {ak| k is any integer} is a
subgroup of G.
14.If a cyclic
group G with generator a has order n, then an = e and the distinct
elements of G are the elements {a, a2,a3, …, an
= e}.
HW:
qProve that the set of
even integers is a subgroup of the additive group of the integers.
qFind all subgroups of
the symmetric group S3.
qFind the order of each
element in each group S3 and Z7 under addition.
qIf H and K are
subgroups of G, prove that HÇK is also a subgroup of G.
·Definitions:set multiplication, one to one
correspondence, partition
·cosets
·Theorems
15.If H is a subgroupof the group G and if a and b are elements of G, the following
conditions are equivalent.
a)aÎ Hbb) Ha = Hb.c) ab -1ÎH
16.If H is a subgroup of G, the right cosets of H in G
form a partition of G.
17.(LaGrange) If H is a subgroup of a finite group G,
then the order of H is a divisor of the order of G.
HW:
qProve theorems 15 and 16 for left cosets.
qLet G be the additive group Z24.Let H be the subgroup H = { 0,6,12,18}.Determine whether each of the following
statements id true or false.
a)7º11 mod Hb) 10º 5 mod Hc)
13º19 mod Hd)3º15 mod H
qLet G and H be as
above.For each of the following
congruences, find three distinct values of x for which the congruence is true.
a)7ºx mod Hb)xº17 mod Hc)xº12 mod Hd)10ºx mod H
qConsider the following
information concerning groups W, X, Y, and Z.
Group W contains 22 elements.
Group X contains 23 elements.
Group Y contains 24 elemetns.
Group Z contains 25 elements.
Use
LaGrange's theorem to answer the following questions.
a)Which group has the
most different sizes of possible subgroups, and which has the fewest?
b)What sizes of subgroups
are possible in group W ? in Group Z ?
qSuppose G is a 50 element
group.
a)What are the possible
sizes for subgroups of G?
b)For each possible size,
assume there is a subgroup of that size, and specify how many distinct cosets
that subgroup has.
c)Answer a) and b) if the
group has 55 elements.
Connection to lessons in ALGEBRA
1 (Glencoe): 3-1,3-5,5-6
Laboratory 2:
ØHands on Equations
(single operation).
ØCups and counters
Laboratory 3:
ØModular Arithmetic.
IIIRings (4 weeks)
·Definitions and
examples including Z6, Z7
·Theorems
1.For any a,x,y in a ring R, if a+x = a+ y then x = y
and if x+a = y+a then x = y.
2.There is only one additive identity in any ring.Each element of a ring has a unique additive
inverse.
3.For any a, b in a ring R, the equation a + x = b has
one and only one solution in R.
qSolve the following
equations in the ring of 2x2 matrices over the set of integers and check your
solution.
qb)
q
qExplain why theorems
1-3 hold in the context of what you know about groups and the definitions of a
ring.
·integral domains
·fields
·Theorems
5.There is only one multiplicative identity in any field.Each non-zero element of a field has a
unique multiplicative inverse.
6.For any nonzero a in a field F and any elements x and
y in F, if ax = ay then x = y and if xa = ya then y = y.
7.For any nonzero a in a
field F and any element b in F, the equation ax = b has one and only one
solution in F.
8.In any field, if x and
y are non-zero then (xy) –1 = x –1 y –1.
9.For any element a in a
ring R, a0=0.
10.For any element in a ring R, 0a = 0.
11.In any field F, 0 ¹ 1.
12.In any field F, 0 has no multiplicative inverse.
13.For any elements a,b in a ring R, - ab = -
(ab).
14.For any elements a,b in a ring R, - a
– b = ab.
15.For any nonzero a in a field F and any elements b and
c in F, the equation ax + b = c has one an only one solution in F.
16.For all a,b,c,d,e,f in a field F, if ae + - (bd)
¹ 0 then the system of
equations given by ax + by = c AND dx + ey = f has one and only one solution .
17.For any a, c in a field F and for any nonzero b,d in
F, (ab –1)(cd –1) = (ac)(bd)–1.
18.Let a and b be elements of an arbitrary ring R, and
let m be a natural number.Then, for any natural number n,
a)n0 = 0 (0ÎR is the zero of R) and 1n = 1 (the latter assuming R has a unity 1).
b)For any a and b in R, n(a + b) = na + nb and (ab)n = anbn (the latter if an only if ab=ba).
c)(m + n)a = ma + na and am+n = aman.
d)nm(a) = n(ma) = m(na) and (am)n = amn = (an)m.
19.Let F be any field, then F is an integral domain.
HW:
qUse arithmetic mod 6 to
show that theorem 7 DOES NOT HOLD in every ring.
qShow that 3 and 4 have
no multiplicative inverses in the set Z6.
qExplain why theorems
5-7 hold in the context of what you know about groups and the definitions of a
ring.
qProve Theorems 10, 15,
16 and 17.
qFrom arithmetic mod 7,
find three examples to illustrate each of theorems 13, 14 and 17.
qUse arithmetic mod 6 to
show that theorem 15 DOES NOT HOLD in every ring.
qAssume that all of the
coefficients in the following are from Z7 and all operations are
modulo 7.Perform the indicated
operations and rewrite your answers leaving no minus signs in them.
q(2x – 3)2
q(x – 2)(x + 5) g)a-2h)a-5
qAssume that R is the
ring of 2x2 matrices over Q with the usual matrix operations.Compute each of the following for the
element .
q3ab)10ac)a3d)a4e)(-3)af)(-10)ag)
a-2h)a-3
q
·polynomial rings
·Theorems
20.Let R be any ring, then R[x] is a ring.
21.Let R be any ring with nonzero zero divisors, then
R[x] also has nonzero zero divisors.
22.If R is an integral domain then R[x] is an integral
domain.
23.Let R be an integral domain. If p(x) and q(x) are any
polynomials in R[x] then
a)Either p(x) + q(x) = 0
or Deg( p(x)+q(x)) £ Max( Deg(p(x),Deg(q(x))
b)Either p(x)q(x) = 0 or
Deg(p(x)q(x)) = Deg(p(x)) + Deg(q(x))
24.Let R be an integral domain.If p(x) and q(x) are any polynomials in R[x]
with degree greater than or equal to one then Deg(p(x)) < Deg(p(x)q(x)) and
Deg(q(x) < Deg(p(x)q(x)).
HW:
qProve that the
polynomials over each of the rings, the integers, the rational numbers and the
real numbers form an integral domain.
qFind the sum and
product of each of the polynomials in the given polynomial ring.
qf(x) = 2x2 +
3x + 4, g(x) = 3x2 + 2x + 3 in Z6[x].
qf(x) = 2x3+
4x2 + 3x + 2, g(x) = 3x4 + 2x + 4 in Z7[x].
·The division algorithm
·Theorems
25.The Division Algorithm.Let F be a field.For any
f(x) in F[x] and for any nonzero g(x) in F[x], there exist elements q(x) and
r(x) in F[x] such that f(x) = q(x)g(x) + r(x) where either r(x) is identically
zero or else 0£Deg(r(x))<Deg(g(x))
HW:
qFind q(x) and r(x) as described in the division
algorithm so that f(x) = q(x) g(x) + r(x) with r(x) either equal to zero or
else Deg(r(x)) < Deg(g(x)) in the given polynomial ring.
1.A polynomial equation f(x) = 0 with f(x) in F[x] has
a solution x =a in F if an only if (x-a) is a factor of f(x).
HW:
qCompute the values in Z7
for each of the following evaluation mappings.
qf2(x2+
3)b)f3((x4 + 2x)(x3 – 3x2
+ 3))c)f3(x4 + 2x) f3(x3
– 3x2 + 3)
qFor each ofthe following equations in Z7, either solve the equation or show the
equation has no solution in Z7 using arithmetic modulo 7.
q2 – x = 5 b)
3x + 2 = 5c) x2 = 1d) 3x2 = 6e)x3 = 6
qf)x2 + 2x + 2 = 0g) (x – 4)2 = 4
qFor each of the
following equations in Z5, either solve the equation or show the
equation has no solution in Z5 using arithmetic modulo 5.
qx – 3 = 2b) 2 – x = 4c)
3x = 4d) 2x = 3e) 4x = 2
qf)x2 = 4g) x2 = 3h)
x2 + 4x + 2 = 0.
qFind all zeros in the
indicated field of the given polynomial with coefficients in that field.
qx2 + 1 in Z2b)x3
+ 2x + 2 in Z7
qThe polynomial x4
+ 4 can be factored into a product of linear factors in Z5[x].Find this factorization.
qThe polynomial x3
+ 2x2 + 2x + 1 can be factored into a product of linear factors in Z7[x].Find this factorization.
qFind a polynomial f(x)
in Z7 which has no linear factors.What does this say about the solvability of the equation f(x) = 0?
qShow that x2
– 2 = 0 has no solutions in Q.
·LCM and GCD
·Euclid's Algorithm
·Theorems
2.Let F be a field.The Euclidean Algorithm holds in F[x].
3.Let F be a field.The integral domain F[x] can be extended to
a field by constructing, for each nonzero f(x), a multiplicative inverse
denoted by and extending the
operations of addition and multiplication to these new elements so that the
properties of a field will hold.This
is called the field of quotients for F[x].
HW:
qFind the g.c.d. of each
of the following pairs of polynomials over the field Q of rational numbers.
(a)2x3 – 4x2
+ x – 2 and x3 – x2 – x – 2
(b)x4 + x3
+ x2 + x + 1 and x3 – 1
(c)x5 + x4
+ 2x3 – x2 – x - 2 and x4 + 2x3 +
5x2 + 4x + 4
(d)x3 – 2x2
+ x+ 4 and x2 + x + 1
qLet f(x) = x2 – x - 2 and g(x) = 1/(3x2
– 9) and h(x) = 3x3 –6x2 +3x + 9 be in the field of
quotients for Q[x].Calculate each of
the following.
(a)f(x) + g(x)
(b)f(x)g(x)
(c)f(x) + g(x) + f(x)g(x)
– h(x)
Connection to lessons in
ALGEBRA 1 (Glencoe): 10-4.
Laboratory 6:
ØAlgeblocks and
factoring
ØDeveloping the sum and
product method for factoring a general trinomial
VGraph theory (3 weeks)
·Cayley graphs
·connected graphs
·edge paths
·vertex degree
·terminal vertices and
terminal edges
·vertex paths
·polygonal graphs
·crossing curves
·dual graphs
·Hamiltonian graph
·critical vertex
·H-edge
·simple graph
·nth order H-edges
·regular solid
·Theorems
1.If a graph has more
than two odd vertices, then it has no edge path.
2.A graph that has an
edge path is nearly connected.
3.If a graph is nearly
connected and has at most two odd vertices then it has an edge path.
4.If a graph has a vertex
path then it is connected.
5.If a graph has a vertex
path then it has at most two terminal vertices.
6.If a graph has a closed
vertex path then is has no terminal vertices.
7.A polygonal graph has a
crossing curve if and only if its dual graph has an edge path.
8.If a polygonal graph
has a crossing curve, the either all faces have even order, or there are
exactly two faces of odd order and the curve begins in one odd face and ends in
the other.
9.If a polygonal graph
has more than two odd faces, then it has no crossing curve.
10.A Hamilton graph contains no critical vertices.
11.In a Hamilton graph no vertex is incident with more
than 2 H-edges.
12.If the H-edges of a graph G include a closed path
among some, but not all, vertices then G is not a Hamilton graph.
13.An edge incident to a vertex of degree two is an
H-edge.
14.Starting with a single vertex and then expanding by
successively performing one of two types of construction one may construct any
polygonal graph.Type 1.Add a new vertex and join it by an edge to
an existing vertex. Type 2. Add a new edge [possibly a loop] by joining two
[not necessarily distinct] vertices already present.
16.In any regular solid, S,where D is the degree
of each vertex, R is the order of each face and e is the number of edges in the flat graph that
represents S.
HW:
·Consider the following
drawings.
D
I
J
(iii)
(i)
(ii)
(v)
(iv)
(vi)
Z
(vi)
qWhich
of the drawings as labeled above represent graphs?
qWhich
of the graphs above are connected?
qWhich
of the graphs above are flat?
qWhich
of the graphs above contain a loop?
qReferring to figure (vii) above:
a)Identify a path from X to X that uses no edge more
than once.
b)Identify a path from W to X that uses every edge at
least once.
c)Identify a path from W to Z that is an edge path.
d)How many different edge paths are there from W to X?
e)How many different edge paths are there from X to Y?
f)How many different
paths are there from X to Y?
qIf we define the distance between two vertices on a graph as the number of
edges along a path connecting the two vertices which has the smallest number of
edges:
a)In figure (vi), what is the distance from P to T?
b)In figure (vi), what is the distance from P to S?
c)In figure (vii), What is the distance from W to Y?
d)In figure (vii), what is the distance from W to Z?
qFor each of the following sets of conditions, give an
example of a graph with four vertices that satisfies all of the given
conditions.If this is not possible,
explain why not.
a)Disconnected and nonflat.
b)Disconnected, with an edge path.
c)Disconnected, with a vertex path.
d)Connected, with no vertex path.
e)With and edge path, but no vertex path.
f)With a vertex path, but
no edge path.
g)With an edge path and a
vertex path.
qConsider the figures below:
1.Find the degree of each vertex in figure (c).
2.Which of the six graphs have no edge path?
3.Which have an open edge path?
4.Which have a closed edge path?
5.Which are Euler graphs?
6.Identify the faces and the orders of the faces of
graphs (a), (b), (e) and (f).
7.Draw the duals of graphs (a), (b), (e) and (f).
8.Which of the graphs (a), (b), (e) and (f) have
crossing curves?
qConsider the figures
below:
Which
graphs have terminal vertices?
Which
graphs have critical vertices?
Which
graphs are not simple?
Identify
the H-edges of graphs (c), (d), (e) and (f).
Identify
the secondary H-edges of graphs (c), (d), (e) and (f).
Which
graphs are not Hamilton Graphs?
Which
graphs are Hamilton graphs?
Identify
the number of vertices, the number of edges and the number of faces for graphs
(c), (e) and (f).
qWhich
of the graphs of the tetrahedron, hexahedron, octahedron and icosahedron are
Euler graphs?Which are Hamilton
graphs?
qA
certain construction job has several phases to it and each phase requires a
certain amount of time to complete.The
table below summarizes the time required for each phase of the job.
Task
Days
A. Frame the walls
3
B. Panel walls
4
C. Hang acoustic
ceiling
3
D. Install wiring
2
E. Install electrical
fixtures
1
F. Do Plumbing
3
G. Install wet bar
2
H. Lay carpet
2
Total
20
There are always at least
two workers assigned to each job and so it will take 10 days to complete the
job with a two-person crew.A further
examination of the project suggests that certain phases must be completed
before other phases begin.The table
below summarizes the precedence requirements of the job.
Before
Starting Task
We Must
Complete Task
B
A,D,F
D
A
E
A,B,C,D
G
F
H
A,B,C,D,E,F,G
Use this information to
construct a directed graph to summarize the information and then study that
digraph to determine if a two-person crew can complete the job in less than 10
days.What about a three-person crew or
two two-person crews?Assume that a
crew will have to be paid for a whole day whether it works the whole day or
not.What option would be
cheapest?For example, if a
three-person crew can do the job in 5 days that is only 15 person days of labor
compared to 20 person days of labor if a two-person crew could do it in 10
days.
|
Review : Matrices and
Vectors
This section is intended to be a catch all for many of the
basic concepts that are used occasionally in working with systems of
differential equations. There will not
be a lot of details in this section, nor will we be working large numbers of
examples. Also, in many cases we will
not be looking at the general case since we won't need the general cases in our
differential equations work.
Let's start with some of the basic notation for
matrices. An n x m (this is often
called the size or dimension of the matrix) matrix is a
matrix with n rows and m columns and the entry in the ith row and jth column is denoted by aij. A short hand method of writing a general n x m
matrix is the following.
The size or dimension of a matrix is subscripted as shown if
required. If it's not required or clear
from the problem the subscripted size is often dropped from the matrix.
Special Matrices
There are a few "special" matrices out there that we may use
on occasion. The first special matrix is
the square matrix. A square matrix is any matrix whose size (or
dimension) is n x n.
In other words it has the same number of rows as columns. In a square matrix the diagonal that starts
in the upper left and ends in the lower right is often called the main diagonal.
The next two special matrices that we want to look at are
the zero matrix and the identity matrix.
The zero matrix, denoted 0n x m , is a matrix all of whose
entries are zeroes. The identity matrix is a square n x n
matrix, denoted In, whose
main diagonals are all 1's and all the other elements are zero. Here are the general zero and identity
matrices.
In matrix arithmetic these two matrices will act in matrix
work like zero and one act in the real number system.
The last two special matrices that we'll look at here are
the column matrix and the row matrix. These are matrices that consist of a single
column or a single row. In general they
are,
We will often refer to these as vectors.
Arithmetic
We next need to take a look at arithmetic involving
matrices. We'll start with addition and subtraction of two matrices.
So, suppose that we have two n
x m matrices, A and B. The sum (or difference) of these two matrices
is then,
The sum or difference of two matrices of the same size is a
new matrix of identical size whose entries are the sum or difference of the
corresponding entries from the original two matrices. Note that we can't add or subtract entries
with different sizes.
Next, let's look at scalar
multiplication. In scalar
multiplication we are going to multiply a matrix A by a constant (sometimes called a scalar) α. In
this case we get a new matrix whose entries have all been multiplied by the
constant, α.
Example 1 Given
the following two matrices,
compute A-5B.
Solution
There isn't much to do here other than the work.
We first multiplied all the entries of B by 5 then subtracted corresponding
entries to get the entries in the new matrix.
The final matrix operation that we'll take a look at is matrix multiplication. Here we will start with two matrices, An x p and Bp x m
. Note that A must have the same number of columns as B has rows. If this isn't
true then we can't perform the multiplication.
If it is true then we can perform the following multiplication.
The new matrix will have size n x m and the entry in
the ith row and jth column, cij, is found by multiplying row i of matrix A by column j of matrix B.
This doesn't always make sense in words so let's look at an example.
Example 2 Given
compute AB.
Solution
The new matrix will have size 2 x 4. The entry in row 1 and column 1 of the new
matrix will be found by multiplying row 1 of A by column 1 of B. This means that we multiply corresponding
entries from the row of A and the
column of B and then add the results
up. Here are a couple of the entries
computed all the way out.
Here's the complete solution.
In this last example notice that we could not have done the
product BA since the number of
columns of B does not match the
number of row of A. It is important to note that just because we
can compute AB doesn't mean that we
can compute BA. Likewise, even if we can compute both AB and BA they may or may not be the same matrix.
Determinant
The next topic that we need to take a look at is the determinant of a matrix. The determinant is actually a function that
takes a square matrix and converts it into a number. The actual formula for the function is
somewhat complex and definitely beyond the scope of this review.
The main method for computing determinants of any square
matrix is called the method of
cofactors. Since we are going to be
dealing almost exclusively with 2 x 2 matrices and the occasional 3 x 3 matrix
we won't go into the method here. We can
give simple formulas for each of these cases.
The standard notation for the determinant of the matrix A is.
Here are the formulas for the determinant of 2 x 2 and 3 x 3
matrices.
Example 3 Find
the determinant of each of the following matrices.
Solution
For the 2 x 2 there isn't much to do other than to plug it
into the formula.
For the 3 x 3 we could plug it into the formula, however
unlike the 2 x 2 case this is not an easy formula to remember. There is an easier way to get the same
result. A quicker way of getting the
same result is to do the following.
First write down the matrix and tack a copy of the first two columns
onto the end as follows.
Now, notice that there are three diagonals that run from
left to right and three diagonals that run from right to left. What we do is multiply the entries on each
diagonal up and the if the diagonal runs from left to right we add them up
and if the diagonal runs from right to left we subtract them.
Here is the work for this matrix.
You can either use the formula or the short cut to get the
determinant of a 3 x 3.
If the determinant of a matrix is zero we call that matrix singular and if the determinant of a
matrix isn't zero we call the matrix nonsingular. The 2 x 2 matrix in the above example was
singular while the 3 x 3 matrix is nonsingular.
Matrix Inverse
Next we need to take a look at the inverse of a matrix. Given a
square matrix, A, of size n x n
if we can find another matrix of the same size, B such that,
then we call B the
inverse of A and denote it by B=A-1.
Computing the inverse of a matrix, A, is fairly simple. First
we form a new matrix,
and then use the row operations from the previous section and try to convert this matrix into the form,
If we can then B
is the inverse of A. If we can't then there is no inverse of the
matrix A.
Example 4 Find
the inverse of the following matrix, if it exists.
Solution
We first form the new matrix by tacking on the 3 x 3
identity matrix to this matrix. This
is
We will now use row operations to try and convert the
first three columns to the 3 x 3 identity.
In other words we want a 1 on the diagonal that starts at the upper
left corner and zeroes in all the other entries in the first three columns.
If you think about it, this process is very similar to the
process we used in the last section to solve
systems, it just goes a little farther.
Here is the work for this problem.
So, we were able to convert the first three columns into
the 3 x 3 identity matrix therefore the inverse exists and it is,
So, there was an example in which the inverse did
exist. Let's take a look at an example
in which the inverse doesn't exist.
Example 5 Find
the inverse of the following matrix, provided it exists.
Solution
In this case we will tack on the 2 x 2 identity to get the
new matrix and then try to convert the first two columns to the 2 x 2
identity matrix.
And we don't need to go any farther. In order for the 2 x 2 identity to be in
the first two columns we must have a 1 in the second entry of the second
column and a 0 in the second entry of the first column. However, there is no way to get a 1 in the
second entry of the second column that will keep a 0 in the second entry in
the first column. Therefore, we can't
get the 2 x 2 identity in the first two columns and hence the inverse of B doesn't exist.
We will leave off this discussion of inverses with the
following fact.
Fact
Given a square matrix A.
1.If
A is nonsingular then A-1 will exist.
2.If
A is singular then A-1 will NOT exist.
I'll leave it to you to verify this fact for the previous
two examples.
Systems of Equations
Revisited
We need to do a quick revisit of systems of equations. Let's start with a general system of
equations.
where, is a vector whose components are the unknowns
in the original system of equations. We
call (2)
the matrix form of the system of equations (1) and
solving (2)
is equivalent to solving (1). The solving process is identical. The augmented matrix for (2)
is
Once we have the augmented matrix we proceed as we did with
a system that hasn't been wrote in matrix form.
Given the system of equation (2)
we have one of the following three possibilities for solutions.
1.There
will be no solutions.
2.There
will be exactly one solution.
3.There
will be infinitely many solutions.
In fact we can go a little farther now. Since we are assuming that we've got the same
number of equations as unknowns the matrix A
in (2)
is a square matrix and so we can compute its determinant. This gives the following fact.
2.If
A is singular then there will be
infinitely many nonzero solutions to the system.
Linear
Independence/Linear Dependence
This is not the first time that we've seen this topic. We also saw linear independence and linear
dependence back when we were looking at second
order differential equations. In that
section we were dealing with functions, but the concept is essentially the same
here. If we start with n vectors,
If we can find constants, c1,c2,…,cn with at least two
nonzero such that
then we call the vectors linearly dependent. If the only constants that work in (4)
are c1=0, c2=0, …,
cn=0 then we call the vectors linearly independent.
If we further make the assumption that each of the n vectors has n components, i.e. each
of the vectors look like,
we can get a very simple test for linear independence and
linear dependence. Note that this does
not have to be the case, but in all of our work we will be working with n vectors each of which has n components.
Fact
Given the n
vectors each with n components,
form the matrix,
So, the matrix X
is a matrix whose ith
column is the ith
vector, . Then,
If X is nonsingular (i.e. det(X) is not zero) then the n
vectors are linearly independent, and
if X is singular (i.e. det(X) = 0) then the n
vectors are linearly dependent and the constants that make (4)
true can be found by solving the system
Example 6 Determine
if the following set of vectors are linearly independent or linearly
dependent. If they are linearly
dependent find the relationship between them.
Solution
So, the first thing to do is to form X and compute its determinant.
This matrix is non singular and so the vectors are
linearly independent.
Example 7 Determine
if the following set of vectors are linearly independent or linearly
dependent. If they are linearly
dependent find the relationship between them.
Solution
As with the last example first form X and compute its determinant.
So, these vectors are linearly dependent. We now need to find the relationship
between the vectors. This means that
we need to find constants that will make (4)
true.
So we need to solve the system
Here is the augmented matrix and the solution work for
this system.
Now, we would like actual values for the constants so, if
use we get the following solution ,
,
and . The relationship is then.
Calculus with
Matrices
There really isn't a whole lot to this other than to just
make sure that we can deal with calculus with matrices.
First, to this point we've only looked at matrices with
numbers as entries, but the entries in a matrix can be functions as well. So we can look at matrices in the following
form,
Now we can talk about differentiating and integrating a
matrix of this form. To differentiate or
integrate a matrix of this form all we do is differentiate or integrate the
individual entries.
So when we run across this kind of thing don't get excited
about it. Just differentiate or
integrate as we normally would.
In this section we saw a very condensed set of topics from
linear algebra. When we get back to
differential equations many of these topics will show up occasionally and you
will at least need to know what the words mean.
The main topic from linear algebra that you must know
however if you are going to be able to solve systems of differential equations
is the topic of the next section.
|
Title
Resource Type
Views
Grade
RatingStudents read an article to explain the reasoning behind theorems. In this calculus lesson, students understand the underlying principles of theorems and how it helps them make sense of the problems. They know why they do what they do in AP Calculus.
Greg Kelly puts together another great slide presentation to demonstrate ways to combine derivative rules to evaluate more complicated functions. This pattern is called the chain rule. He shows step by step ways to solve these complicated problems.
Twelfth graders explore differential equations. In this calculus lesson, 12th graders explore Euler's Methods of solving differential equations. Students use the symbolic capacity of the TI-89 to compare Euler's Method of numeric solutions to a graphical solution.
Students analyze implicit differentiation using technology. For this calculus lesson, students solve functions dealing with implicit differentiation on the TI using specific keys. They explore the correct form to solve these equations.
In this circuits worksheet, students answer 25 questions about passive integrator circuits and passive differentiator circuits given schematics showing voltage. Students use calculus to solve the problems.
|
Your Curriculum
Mathematics
The Mathematics program at Mt. Hood is a curriculum focused on real applications, problem solving, appropriate technology use, conceptual understanding, mathematical skills, and a discovery/experiential approach to math. We enthusiastically welcome mathematics majors entering at all mathematical levels.
The Math department is pleased to honor exemplary mathematics students at all levels with recognition awards, which may include scholarship funds. Details are available from your current math instructor around the fifth week of the term.
There are many careers available for students majoring in math, including actuarial work, education and positions as the math experts in industry and computer science. For more information, please contact a math instructor, the Career Advising Center or visit the website of the Mathematical Association of America at
Curricular Outcomes At the completion of this curriculum, the student should be able to:
Apply mathematical concepts, skills, reasoning and modeling to solve problems arising from the real world
Model problem situations using mathematics visually, numerically, graphically, and/or algebraically and make connections among various models
Demonstrate a command of functions from multiple perspectives
Determine if a solution is reasonable, verify results, and compare solutions from different approaches
Use appropriate technology to analyze and solve mathematical problems
Describe and interpret, from multiple perspectives, the purpose and usefulness of the derivative concept
Describe and interpret, from multiple perspectives, the purpose and usefulness of the integral concept .
The following plan of classes is a general guide to prepare students to pursue a mathematicsSee an adviser to personalize this plan and/or to create a plan that starts with the math sequence before calculus. It is possible to start the calculus sequence as late as spring of the first year, take summer classes and finish by spring of the following year.
1 This plan aligns with the Associate of Science; refer to degree requirements, page 14 of the printed catalog. 2 Recommended electives: MTH243/244 (some schools, including PSU, require a statistics sequence for math majors); CS161; German, French or Russian (recommended for those pursuing graduate work in math); MTH211/212/213 (recommended for those interested in teaching math at any level, sequence starts fall/winter); PH211/212/213 (sequence starts fall). Other areas of study that would support continuing education and/or employment in mathematics: engineering, PHL191 Language and the Layout of Argument, economics, computer science, science. 3 Lab science is required by most universities for a Bachelor of Science degree; it is not required for MHCC graduation. 4Students hoping to teach at any level are strongly encouraged to apply for work as a tutor in the Learning Success Center for hands-on experience.
|
MATH 0910 Developmental Mathematics
Lecture/Lab/Credit Hours 5 - 0 - 5
This course presents basic computational skills for either review or initial mastery by the students. Topics include fractions; decimals; the solutions of ratio, proportion, and percent problems; operations with integers; and basic study skills for mathematics problem-solving and estimation. Topics may also include geometry, measurement, and basic algebraic concepts00 with a grade of P, or MCC placement test
|
Understandable Statistics (Hardcover)
9780618949922
ISBN:
0618949925
Edition: 9 Publisher: Houghton Mifflin Company
Summary: This algebra based text is a thorough yet approachable statistics guide for students. The new edition addresses the growing importance of developing students' critical thinking and statistical literacy skills with the introduction of new features and exercises.
Ships From:Alpharetta, GAShipping:Standard, Expedited, Second Day, Next DayComments: 0618949925 MULTIPLE COPIES AVAILABLE-Very Good Condition-May have writing or highlighting-May ha... [more] 06189499
|
A FREE, Powerful Algebra
End-of-Course Prep Tool
To help teachers and students succeed on the Algebra 1 End-of-Course exam
(EOC), the University of Florida and Study Edge have created Algebra Nation – a FREE, online, easy-to-use,
EOC preparation resource aligned with the latest state standards.
Read more about Algebra Nation here or click on the "Enter Algebra Nation" button below
to get started!
You can also watch Algebra Nation videos and ask questions on the go! Algebra
Nation is accessible through your iPhone, iPod Touch, iPad, or Android device.
Just search for the free 'Study Edge' app in the app store on your mobile device.
Join Us
To sign up for access to teacher resources and/or to receive periodic newsletters, click continue. If you have any questions, please e-mail help@algebranation.com.
How Does Algebra Nation Work?
24/7, Free Online Resources
Algebra Nation is a highly effective, intensive, interactive online
learning mechanism. This is a 24/7, free resource to help Florida's high school students prepare for
the Algebra End-of-Course exam.
Social and Collaborative Learning
Based on the latest research and featuring some of Florida's top math teachers and Study Experts,
Algebra Nation utilizes social learning and technological breakthroughs to construct and stage a
vibrant online learning system for students.
Better EOC Exam Results
Algebra Nation fosters a dynamic, social process – non-linear, hands-on, effective and fun that
helps students across Florida conquer the Algebra End-of-Course (EOC) exam, a computer based
exam all Algebra students need to pass for graduation credit.
|
MathMol (Mathematics and Molecules) is designed
to serve as an introductory starting point for K-12 students and teachers interested
in the field of molecular modeling and its application to mathematics.
MathMol Quick Tour:
What is molecular modeling? Why is it so important? What is the relationship between
molecules and mathematics?
Hypermedia
Textbook for Grades 6-12 Visit an experimental textbook of the future
that makes full use of the latest in Netscape capabilities, including: frames,
VRML, interactive Javascipt files, animated GIF's and MPEG files.
Library of
3-D Geometric Structures The MathMol Library of Geometric Structures
contains GIF and and 3-D (VRML) files of geometric structures that are found in
most intermediate school textbooks.
Review
of Mass, Density and Volume Review mass, density
and volume. Contains interactive javascripts that test your knowledge as you move
through the activity. Don't forget to try the Challenge questions at the end.
|
College textbooks may be in the top-ten of the worst things for sale, ever. It's not bad enough that the universe makes you feel worthless if you don't get a degree, and then laughs at you when you want a job. No, along the four- or five-year journey to your worthless diploma, they make you buy dozens of textbooks.
The future has brought slight reprieve to the textbook problem - you can buy them online for cheap, get free shipping, and resell them for more than the snotty guy at the campus bookstore wants to give you when the class is over. But the fundamental issue remains that introductory calculus, or chemistry, or whatever, has not changed in at least twenty years. The only difference is the word problems have changed.
Skrillex buys an ice cream cone whose height is h and radius r, topped with a sphere of ice cream with radius 1.1r. His friend Deadmau5 texts him on an iPhone 4, and while he texts back, the ice cream melts and runs into the cone. The cone has a leak which allows the melted ice cream to run out the bottom at rate 0.031r3 per minute (t). Express the surface area of the cone filled with melted ice cream as a function of time. Do not use rage faces in your solution.
|
This book covers *The use of MATLAB examples to provide motivation for the theory to come. *The incorporation of MATLAB code to allow students to understand how the theory is applied in practice. *Numerous computer exercises to familiarize the student with MATLAB and how it is used to solve real problems. *The incorporation of "real-world" problems from various disciplines in each chapter to illlustrate the application of the chapter concepts. *Discussion of discrete random variables first, followed by continuous random variables to minimize confusion
|
Mathematics
Head of Department: Mr Andrew Arratoon
Year 10
Mathematical education today is based on the ability to use Mathematical skills and techniques in life related situations and the ability to communicate solutions in a Mathematical manner. Students further develop the ability to apply Mathematical problem solving techniques to life related situations and are given the opportunity to develop further skills with calculators, computer software and other appropriate tools.
The Year 10 Mathematics courses are designed to give students a positive approach to Mathematics and to develop the skills needed in Senior Mathematical Studies.
Mathematics
The Mathematics course continues to develop skills and techniques necessary for Senior Mathematics. Students further develop the ability to apply Mathematical problem solving techniques to life related situations and are given the opportunity to develop further skills with calculators, computer software and other appropriate tools.
Career Options A good result in Year 10 Mathematics is desirable for students progressing to Mathematics B or Mathematics C in Years 11 and 12. Mathematics B leads to many University courses and this must be considered when choosing subjects.
Extreme Mathematics
Extreme Mathematics is modelled largely around the Year 10 Mathematics course, however it offers many additional learning opportunities in order to target students seeking a greater challenge or with a passion for Mathematics. The course is ideally suited to those who are considering doing Mathematics C in Year 11.
Some of the additional learning opportunities could include: • Introduction to the use of Graphics calculators as a learning tool (normally occurs during Year 11) • Participation in the Interschool Science and Engineering challenge • Deeper exploration of many concepts • Enhanced use of technology to support learning eg Graphmatica, Excel spreadsheets
Note that final placement into the Extreme Mathematics group will be a school decision should there be more students requesting this course than can be accommodated (usually one class). Assessment criteria will be the same as described above in Mathematics.
Supported Class
One Mathematics class in Year 10 will provide additional support for those students who are experiencing some difficulties with Mathematics.
The teaching in these classes will focus around achieving competency at 'C' level objectives only with a particular emphasis on life applications of Mathematics. Student placement into these classes will be based on previous results in Mathematics, and involve parent consultation.
Year 11 and 12
As a result of the rapid changes in technology and the consequential changes in Mathematics, the face of Mathematics education is changing, from an emphasis on mechanical calculations out of context, to one of life related problem solving. Skills and techniques continue to be important, but they are now put in context.
Mathematics B and C will be offered in Year 11 and 12. These subjects are intended for students who wish to pursue tertiary studies that require competency in formal Mathematics procedures and structures. Mathematics A will also be offered to Years 11 and 12. Mathematics A is intended to raise the level of competence in Mathematics required for intelligent citizenship and to provide students wishing to pursue general tertiary studies with a competence in Mathematics.
Prevocational Mathematics is best suited to those students looking at a 'trade' pathway after year 12.
Mathematics A
Background The study of Mathematics A will emphasise the development of positive attitudes towards the student's involvement in mathematics. This development is encouraged by an approach involving problem solving and applications, working systematically and logically, and communicating with and about mathematics. This often involves the use of computer software, calculators and other appropriate instruments.
Having completed the course of study, students of the subject should: • be able to recognise when problems in their everyday life are suitable for mathematical analysis and solution, and be able to attempt such analysis or solution with confidence • be aware of the uncertain nature of their world and be able to use mathematics to assist in making informed decisions in real life situations • be able to manage their financial affairs in an informed way • be able to visualise and represent spatial relationships in both two and three dimensions • be aware of the diverse applications of Mathematics • comprehend mathematical information which is presented in a variety of forms • communicate mathematical information in a variety of forms • be able to benefit from the availability of a wide range of mathematical instruments
Content The core of Mathematics A focuses on three strands of Mathematics: • Financial Mathematics • Applied Geometry • Statistics and Probability
Operations Research and further work on Probability and Statistics are offered through the extensions
Prerequisites A sound level of achievement in Year 10 B
Background The aim of Mathematics B is to encourage students to develop positive attitudes towards Mathematics by an approach involving problem solving and applications. Students will also be encouraged to work systematically and logically, and to communicate with and about Mathematics. This often involves the use of computer software, graphics calculators and other appropriate instruments. The School uses the TI 84 plus graphics calculator.
Having completed the course of study,
Throughout the course fundamental knowledge and skills are identified and these aspects are revised. The course is arranged in a spiral structure which allows for the revisiting of topics throughout the two years.
Prerequisite A 'B' level of achievement in Year 10 Mathematics or Extreme the C
Background The study of Mathematics C will give students the opportunity to extend their mathematical knowledge into new areas, and hence will provide an excellent preparation for the further study of mathematics in a wide variety of fields. The additional rigour and structure of the mathematics required in this subject will equip students with valuable thinking skills. Students will use the TI 84 plus graphics calculator.
Having completed the course of study • be aware of the wide range of mathematics vocations • have an appreciation of the power and beauty of mathematics
Content The course contains both core and option topics. All core topics must be studied, and two options topics are to be completed. The core topics are: • Introduction to groups • Real and Complex Number Systems • Matrices and Applications • Vectors and Applications • Calculus • Structures and Patterns
Prerequisite A 'B' level of achievement in Year 10 Mathematics or Extreme Mathematics is desirable. Students studying Mathematics C must also study Mathematics BCareer Options Tertiary pathways including Engineering, Aviation and Mathematics.
Prevocational Mathematics
Prevocational Mathematics is ideally suited to those students who do not wish to pursue tertiary studies after Year 12, but are likely to be looking at some sort of trade or vocational career pathway. The content addressed is chosen to align with the specific needs of the student's career pathways, as well as focusing on developing a range of numeracy skills. Content includes but is not limited to:- • Financial Literacy • Measurement • Navigation (maps, scales etc) • Domestic Maths (budgeting, home renovations etc) • Trade Maths – construction, plumbing, electrical, mechanical
While the subject is not one that contributes to OP scores, it does contribute to the QCE and fulfils the Numeracy requirement of the QCE.
Prerequisite Generally suited to students who have achieved a C or less in Year 10 Mathematics.
|
A course surveying topics utilized in computer science. Topics include problem-solving, logic, computer arithmetic, Boolean algebra and linear mathematics. Required of Math teaching majors. Prerequisites: acceptable score on placement exam, a grade of C or higher in one year of high school algebra, or a grade of C or higher in 001. Recommended for general education requirements-B.S. degree. Offered as needed.
(d) . . . study some basic concepts of Graph Theory. For example, consider the most efficient way for a mailman to deliver the mail in a certain part of a city.
(e) . . . study matching problems. For example, assigning bus operators to routes, or a basketball coach must assign a player to guard each player on the opposing team in such a way as to minimize the opponent's total score.
(f) . . . study network flows problems. For example, a long-distance telephone company must move messages from one city to another.
The number of of telephone calls that the company can handle at a given time is limited by the capacity of its cable and its switching equipment.
Content:: This course is aimed at the students who major/minor in Mathematics Education.
Course Philosophy and Procedure: Two key components of a success in the course are regular attendance and a fair amount of constant, every-day study. You should try to make sure that your total study time per week at least triples the time spent in class.
Grading will be based on three in-class exams (100 points each), a cumulative final exam (200 points), class participation, take-home problems, projects, group practice exams and portfolios.
Out of all these assignments, I attach a special importance to the Mathematical Reasoning project. It is a semester long project which consists of doing the following problems:
• Do also exercises 21 − 24, this time stating some of the basic principles/ formulas hidden in those that are true.
• The last group, still Appendix A.3, Supplementary exercises: Do 37 − 44.
Rules for the project:
• Each problem is worth 2 points, no partial credit.
• You can submit a solution to any problem, or any number of them, any time during the semester. The solutions are to be submitted electronically.
Please use the e-mail address given at the top of this syllabus.
The last day to submit is the last day of class.
• Please try to write your solutions so that they are correct and complete.
Justify your argument, state your reasoning.
• I will look up each of your solutions and return it to you with some comments. I either accept a solution of a problem, and mark down 2 points for you, or will return the problem back to you with some suggestions for correction and improvement.
I will keep returning a problem to you until it is done right. You don't lose any points for repeated attempts.
My grading scale is
A=90%, AB=87%, B=80%, BC=77%, C=70%, CD=67%, D=60%.
Americans with Disability Act:: If you are a person with a disability and require any auxiliary aids, services or other accommodations for this class, please see me and Wayne Wojciechowski in Murphy Center Room 320 (796- 3085) within ten days to discuss your accommodation needs.
|
Ideas for Sixth Form Mathematics: Further Pure Mathematics and Mechanics
This book offers a wealth of innovative lesson ideas for important areas of post-16 mathematics teaching, bringing variety to the Further Mathematics and Mechanics curriculum areas.
The material is arranged by topic and enables teachers to move away from an over-emphasis on routine textbook based work. Hints and suggestions are included to make it clear where the ideas may lead as well as additional material for extension.
|
Heart Of Mathematics - 4th edition
ISBN13:978-1118156599 ISBN10: 1118156595 This edition has also been released as: ISBN13: 978-1118371046 ISBN10: 1118371046
Summary: Burger's 4thedition of Heart of Mathematics builds on previous editions based on math appreciation and an emphasis on critical thinking. The text is noted for itsreadable writing style, broad range of topics, and presentatio...show moren of the classic mathematical ideas in a fun and interesting way. Topic coverage of the text is more traditional ''skill-drill topics'' such as graph theory and algebra with an entirely new graph theory section and additional computational exercises to the end of each section.Furthermore, this edition offers an engaging and mind-opening experience for even your most math-phobic users. It's written for non-math, non-science-oriented majors and encouraging them to discover the mathematics inherent in the world around them. Infused throughout with the authors' humor and enthusiasm, The Heart of Mathematics introduces students to the most important and interesting ideas in mathematics while inspiring them to actively engage in mathematical thinking....141.75 +$3.99 s/h
New
Textbookcenter.com Columbia, MO
Ships same day or next business day! UPS(AK/HI Priority Mail)/ NEW book
$146.66 +$3.99 s/h
New
TextbooksRus_com Grandview Heights, OH
Ships SAME or NEXT business day. We ship to APO/FPO addr. Choose EXPEDITED shipping and receive in 2-5 business days.
$146.80 +$3.99 s/h
New
BookCellar-NH Nashua, NH
1118156595161.50 +$3.99 s/h
New
nb-books Dobbs Ferry, NY
Hardcover New 1118156595 Brand New. In a hurry? Select Expedited Shipping at checkout to get your books in 2-5
|
MATH 021: Concise Calculus
Course Description:[Previous Course Code: MATH 001]
This course teaches fundamental concepts in calculus and provides mathematical
preparation for students who are going to take further courses in mathematics.
Key topics include: logic and sets, functions, limits and continuity,
differentiation and graphing, integration; improper integrals, sequence and
series, power and Taylor series.
|
Pointblank Prealgebra elementary math sets the stage for other math courses. I have worked with elementary math extensively and understand what we need to do to master it
|
Document Actions
Mathematica @ Hunter College
What is Mathematica? universities and colleges, and is commonly used in the following types of departments --- Mathematical Sciences, Physical Sciences, Business and Finance, Life Sciences, Engineering, Computer Science.
|
Presenting algebra exercises in which FUN is the "unknown quantity!" Each page features 12 to 24 skill-building algebra problems. After students have simplified expressions or solved equations, the answers provide clues for drawing lines to reveal a secret picture in the coordinate grid. A unique and fresh approach to algebra practice; great for students who enjoy visual challenges and direct feedbackThis was purchased as a Christmas gift. My son did not put down the first 2 books in this series that I purchased for weeks. Therefore, I foresee him enjoying this book for weeks after Christmas. The school is encouraging him to take algebra in 7th grade. Therefore, this will help him attain that goal.
Would you like this item placed in an attractive gift bag for an additional $1.95 per item? YES!
If you like Algebra Antics, we think you'll LOVE EquiLogic. Buy them together right now and we'll knock 5% off the price of that item!
Price for both: $35.65
|
Advanced functions and Modeling provides students an in-depth study of modeling and applying functions. Home, work, recreation, consumer issues, public policy, and scientific investigations are just a few of the areas from which applications will originate. Appropriate technology, from manipulatives to calculators and application software will be used regularly in instruction and assessments. This course meets the fourth year math graduation requirement for University Prep students.
Projects:
Statistics labs, simulations, and data collecting
Internet investigations
Trigonometry projects
Various hands-on activities and mini-projects
Absences/Tardies:
Three tardies are equivalent to 1 absence. The school tardy policy will be followed. More than 6 absences in one semester may result in NO CREDIT for the class.
Grading Policy:
Daily grades, activities, quizzes etc. count one time, homework average counts twice, tests count twice, projects count three times, and the 9 weeks test counts three times. The final exam is 25% of the semester grade. This is exam is now a NC State Exam. (Measure of Student Learning) If you miss an assignment or test it is YOUR responsibility to see me and make arrangements to make it up at a mutually agreed upon time. This must be done the week following your absence. Failure to do so will result in a zero for the assignment or test.
|
Pre-Algebra. Nice choice! During our long and celebrated (OK, so maybe we're exaggerating
a little) years in various math classes, we've found that a solid foundation
is extremely important. So we're glad you came here, and we hope
it helps you out!
In this section
of the site, we'll try to clear up some common problems encountered in
pre-algebra. We'll cover everything from the basics of equations
and graphing to everyone's favorite - fractions.
After each section,
there is an optional (though highly recommended) quiz that you can take
to see if you've fully mastered the concepts. Don't forget to visit the
message board
and the formula database.
|
2004 Paperback New Book New and in stock. 7/4IT ALL ADDS UP: A GREAT NEW METHOD FOR LEARNING REAL-WORLD MATH
Now anyone with an interest in the math of daily life can gain a deeper understanding. Everyday Math Demystified provides an effective, fun, and totally painless way to improve your understanding and mastery of the math you find in newspapers, on TV, at the bank or store, on vacation, in school — and just about everywhere.
With Everyday Math Demystified, you master the subject one simple step at a time — at your own speed. This unique self-teaching guide helps you decipher such topics as numbers and arithmetic, measurements, and fractions and graphs, and puts them into the context of real-life situations you're sure to encounter.
If you want to build or refresh your everyday math skills, here's a fast and entertaining self-teaching course that's specially designed to reduce anxiety. Get ready to:
* Conquer scientific and engineering math, such as logarithms, exponents, angles, magnitude and direction, and rates of change
* Take a "final exam" and grade it yourself!
Simple enough for real beginners but challenging enough for math-savvy readers, Everyday Math Demystified is your direct route to learning or brushing up on the mathematical aspects of daily life.Related Subjects
Meet the AuthorensChan
Posted June 26, 2012
Good reference to many squares in the place of math symbols in examples...
Great book!!!!
Though I find that the publisher did not take the time to
create a lot of the mathematical symbols in formulas; there are a lot of squares (example: Aset [] Bset)--this is the symbol used to indicate the symbol is unrecognized by, in this case an ipad 1st generation--that requires the reader to go back to the math symbols chart many, many times to figure out what symbol should be visible. Out side of those squares in math formulas this is a great reference.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
Anonymous
Posted June 6, 2004
What Every Adult Should Know
As an educator (retired), I got an advance copy of this book. My first reaction was, 'This is everyday math?!' Then in the third chapter it hit me: This is not necessarily what everyone knows (if that was the case, there would be no need for the book). It's what every American adult should know by the time they graduate from high school. Sadly, given the state of math education in this country, this book probably should have the subtitle 'in an Ideal World.' I recommend that anyone who wants to really understand math, and not just rush through it as some sort of evil necessity, study this book thoroughly -- after, or in addition to, their high school courses. And don't fret the abstract stuff. Math is abstract by its very nature.
Was this review helpful? YesNoThank you for your feedback.Report this reviewThank you, this review has been flagged.
|
9780073205991
Buy New Textbook
This is a hard-to-find title. We are making every effort to obtain this item, but do not guarantee stock.
$78.6432.76
Questions About This Book?
What version or edition is this?
This is the 3rd edition with a publication date of 1/3Connect Math Access Card for Elementary : With SMART CD-ROM for Windows and OLC Card
Elementary and Intermediate Algebra with Mathzone
Elementary and Intermediate Algebra with SMART CD
SSM/Elem and Interm Algebra
Student Solutions Manual for Elementary and Intermediate Algebra
Summary
Student Solutions Manual:The Student's Solutions Manual provides comprehensive, worked-out solutions to all of the odd-numbered exercises. The steps shown in the solutions match the style of solved examples in the textbook.
|
More About
This Textbook
Overview
An Introduction to the Theory of Numbers by G. H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D. R. Heath-Brown, this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory.
Updates include a chapter by J. H. Silverman on one of the most important developments in number theory - modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader.
The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.
Related Subjects
Meet the Author
Roger Heath-Brown F.R.S. was born in 1952, and is currently Professor of Pure Mathematics at Oxford University. He works in analytic number theory, and in particular on its applications to prime numbers and to Diophantine equations.
Table of Contents
Preface to the sixth editionAndrew Wiles Preface to the fifth edition
1. The Series of Primes (1)
2. The Series of Primes (2)
3. Farey Series and a Theorem of Minkowski
4. Irrational Numbers
5. Congruences and Residues
6. Fermat's Theorem and its Consequences
7. General Properties of Congruences
8. Congruences to Composite Moduli
9. The Representation of Numbers by Decimals
10. Continued Fractions
11. Approximation of Irrationals by Rationals
12. The Fundamental Theorem of Arithmetic in k(l), k(i), and k(p)
13. Some Diophantine Equations
14. Quadratic Fields (1)
15. Quadratic Fields (2)
16. The Arithmetical Functions ø(n), µ(n), *d(n), σ(n), r(n)
17. Generating Functions of Arithmetical Functions
18. The Order of Magnitude of Arithmetical Functions
19. Partitions
20. The Representation of a Number by Two or Four Squares
21. Representation by Cubes and Higher Powers
22. The Series of Primes (3)
23. Kronecker's Theorem
24. Geometry of Numbers
25. Elliptic Curves, Joseph H. Silverman Appendix List of Books Index of Special Symbols and Words Index of Names General
|
Starting from an elementary level Professor Jones discusses generalised functions and their applications. He aims to supply the simplest introduction for those who wish to learn to use generalised functions and there is liberal provision of exercises with which to gain experience. The study of more advanced topics such as partial differential equations, Laplace transforms and ultra-distributions should also make it a valuable source for researchers. The demands placed upon the reader's analytical background are the minimum required to approach this topic. Therefore, by selecting chapters it is possible to construct a short introductory course for students, a final-year option for honours undergraduates or a comprehensive postgraduate course.
|
Reviews
I believe that the math study skills link will help a lot of people learn to study better the math studylink not only tells you how you can learn to be a better student, but it also gives you a little encouragement to ask questions when you need the help. So don't be afraid to ask.
Good Advice!
Spartanburg Community College, Spartanburg, SC
"Math Study Skills" provides all the suggestions I give to my math students, but in a much more concise form. Great reading for any student who wishes to succeed in college-level math.
This article talks about the skills of studying math. As it mentioned "Math is learned by doing problems." The more you practice, the better you learn. Moreover, never be shy or afraid to ask your classmates and instructors for help. Everyone is nice. They would like to help you out.
Also, it's a good way to make you absorb the knowledge you learn in class by reading the textbook before the classes begin. Learning math is a wonderful trip, if you put your heart into it. It will give you a fantastic experience.
This resources is talking about math skills. To be active, students should take responsibility to themselves, they should attendclass and participate in the course. Besides, college math is different than high school. Class always is less time per week than in high school but you have more information. Thus if students missed the class, they have to take more time to catch up.
Moreover, asking questions is another important skill. Math is different than other courses. It focuses on "doing" rather than writing. If students don't know how to do, they can't get the right answer, but in the writing course, students can get some grades if they written down something that is related. Therefore, math skill is very important to learn. The best way to do it,is to just spend time and do problems over and over again until you totally understand
|
Honors Algebra I is the first of five sequential courses in the academic math track and is designed for the math students who possess a strong background in their math skills. Honors Algebra I is offered to students in grades 8. Major units of study include; solving first degree equations and inequalities; solving second degree equations, using four basic operations; learning monomials, polynomials, and algebraic fractions using factoring - all varieties, graphing linear equations; solving work problems with application of the above skills; and applying systems (3 ways). Math vocabulary and spelling skills are developed. Use of the graphing calculator will be applied to various concepts throughout the course. Students are exposed to related careers and emphasis is placed on the need for math skills in life work. Major assignments may include graphing projects. The depth of course coverage and the complexity of the algebra problems offered in the program serve to differentiate the different program levels of honors and college prep.
Program Purpose:
Students who complete the Honors Algebra One course will be exposed to problem solving, applications of algebra, reasoning skills, making geometric models and using the latest technology to develop a clear understanding of mathematical concepts as outlined in the New Jersey core curriculum standards.
Students will be given extensive opportunities to develop higher level thinking skills necessary for success in future advanced mathematics courses.
|
Stock Status:In Stock Availability: Usually Ships in 5 to 10 Business Days
Product Code:9781741251357
About this book
Author Information
This is a new maths dictionary that gives students a full understanding of the language of Mathematics, as well as mathematical content.
The Excel Junior High School Maths Study Dictionary (Years 7-10): - is extremely comprehensive with over 300 pages of definitions. - has over 1000 Maths terms defined. - is something you can use for 4 secondary years. - is based on Australian state syllabuses. - has been specifically colour-coded for each letter of the alphabet. - has over 1000 diagrams to help explain new concepts clearly. - has over 150 colour photographs to engage the student. - has actual Question and Answer examples in definitions.
Author: Michael J. Brown
A new maths dictionary that gives students a full understanding of the language of Mathematics, as well as mathematical content.
|
Lecture 38: Quadratic Inequalities
Embed
Lecture Details :
Solving quadratic inequalities using factoring
Course Description :
This is the original Algebra course on the Khan Academy and is where Sal continues to add videos that are not done for some other organization. It starts from very basic algebra and works its way through algebra II.
|
The Nuffield Mathematics Teaching Project produced a series of twenty modules aimed at supporting the teaching of topics in mathematics which in the project team's view students found difficult. The modules were aimed at students aged 11 and 12 in the first two years of secondary education. Some of the ideas and content would…
…
The Continuing Mathematics Project (CMP), published by Longman in the 1970s, was conceived as a response to the problem of insufficiently numerate students who were studying beyond '0' level.
The Dainton Report recommended that " ... normally all pupils should study mathematics until they leave school ...".…
…
Network is a series of books from Leapfrogs which were intended for middle and secondary schools. In content they are less restricting than a simple workcard and may be seen as a source of additional interesting material for class use.
There are three groups of books available:
Action books are each based on a theme and encourage…
This selection of books presented here and published by Longman is part of the prolific output of mathematics books written by K A Hesse to provide carefully graded examples for primary school students to practise computation and work through problems.
In each of the series the books follow a similar pattern, starting with simple…
Alpha Mathematics and Beta Mathematics, first published by Schofield & Sims in the 1970s, were developed to provide a cohesive, progressively planned course which, when completed, would give students of junior and middle school age a broadly-based foundation in mathematics, deemed essential for future progress.
Alpha Mathematics…
|
Symmetry and Pattern in Projective Geometry is a self-contained study of projective geometry which compares and contrasts the analytic and axiomatic methods. The analytic approach is based on homogeneous coordinates, and brief introductions to Plücker coordinates and Grassmann coordinates are presented. This book looks carefully at linear, quadratic,... more...
An important new perspective on AFFINE AND PROJECTIVE GEOMETRY This innovative book treats math majors and math education students to a fresh look at affine and projective geometry from algebraic, synthetic, and lattice theoretic points of view. Affine and Projective Geometry comes complete with ninety illustrations, and numerous examples and exercises,... of Logarithmic Identities invaluable. All the information included is practical, rarely used results are excluded. Great care has been taken to present all results concisely and clearly. Excellent to keep as a handy reference! If you don't
|
Introduction to junior/senior-level courses in Advanced Calculus, Analysis I, Real Analysis taken by math majors. The first semester is usually a general requirement; the second semester is often an option for the more motivated students. Designed to challenge advanced students while bringing weaker students up to speed, this text is an advantageous alternative to most other analysis texts which either tend to be too easy (designed for an Intermediate Analysis course) or too difficult (designed for students headed for a Ph.D. in Pure Math... MOREematics)both of which usually tend to slight multidimensional material. Hailed for its readability, practicality, and flexibility, this text presents the Fundamental Theorems from a very practical point of view. Introduction to Analysis starts slowly and carefully, with a focused presentation of the material; saves extreme abstraction for the second semester; provides optional enrichment sections; includes many routine exercises and examples; and liberally supports (with examples and hints) what little theory is developed in the exercises. Offering readability, practicality and flexibility, Wade presents Fundamental Theorems from a practical viewpoint. Introduces central ideas of analysis in a one-dimensional setting, then covers multidimensional theory. Offers separate coverage of topology and analysis. Numbers theorems, definitions and remarks consecutively. Uniform writing style and notation. Practical focus on analysis. For those interested in learning more about analysis.
|
Pencils (for homework, quizzes, and tests) and a pen (for correcting and notes)
Folder for organization of papers
Ruler, protractor, and compass (for home use - do have some for classroom use)
A good attitude!
Grade Makeup:
Homework 40%
Tests 40%
Quizzes 20%
Algebra I (2nd Semester)Graph paper
A good attitude!
Grade Makeup:
Homework 40%
Quizzes/ Tests 60%
College Alg. & Trig.A good attitude!
Grade Makeup:
Homework 25%
Quizzes/ Tests 75%
Applied Math I
Materials Needed:
A scientific calculator
3-ring binder
Pencils (for homework, quizzes, and tests) and a pen (for correcting and notes)
Standard/metric ruler
A good attitude!
Grade Makeup:
Homework 20%
Quizzes/Tests 50%
Labs 30%
Classroom Rules and Expectations:
In order to make these classes the best they can be I expect students to follow these simple guidelines:
¸ Respect yourself, classmates, and the teacher.
¸ Come to class prepared to learn.
¸ Listen to and follow all verbal and written instructions.
¸ Ask questions!
Grading Procedure:
Progress reports will be sent home every 6 weeks. The semester grade or final grade for my classes is the result of running totals from the 3 6-week grading periods and a final exam (10%), if given; It is not an average. This cumulative grade can be found on the gradesheets sent home and is designated as S1. (G1, G2, and G3 are the individual 6-week grades).
|
Featuring hundreds of exercises, this book offers plenty of opportunities for practice on the math found in sixth, seventh, eighth, and ninth grade curriculums. It gives your child the tools to master: integers rational numbers; patterns equations; graphing functions and more.
|
already helped many students in this area. MatLab is a computer software program using the general syntaxing of both C and Java. MatLab allows one the ability to either write the code for the problem at hand or to use built-in programs that can assist to get the calculation complete.
|
Reasoning: [1.1] Students will demonstrate an understanding of axiomatic-deductive systems in the linear algebra context. [1.2] Students will read and understand proofs given in the text and in class. [1.3 and 1.4] Students will be able to make conjectures and prove or disprove them, and will deduce how to apply theory to solving assigned problems.
Problem Solving: [2.1 and 2.2] Students will solve numerous assigned problems using routine application of basic theory, and in some cases non-routine application of a variety of results, sometimes from other areas of mathematics.
Technology: [3.1 and 3.2] Students will appropriately use calculators and a computer statistical package (Matlab) to assist them in solving linear algebra problems.
Communication: [ 4.1, 4.2 and 4.3] Students will accurately and appropriately use the language of mathematics for oral in-class presentations of solutions to problems and in written solutions to problems on assignments and exams
|
This access kit will provide you with a code to get into MyMathLab, a personalized interactive learning environment, where you can learn mathematics and statistics at your own pace and measure your
|
Mathway - put in a math problem and not just receive the answer, but receive an explanation of how the problem was solved, step by step. The site covers basic math, algebra, trigonometry, and calculus.
|
Quadratic Equations Introducing various
techniques by which quadratic equations can be solved - factorization,
direct formula. Relationship between roots of a quadratic equation.
Cubic and higher order equations - relationship between roots and
coefficients for these. Graphs and plots of quadratic equations.
Series and Progressions Arithmetic,
Geometric, Harmonic and mixed progressions. Notes, formulas and solved
problems. Sum of the first N terms. Arithmetic, Geometric and Harmonic
means and the relationship between them.
Introduction to Matrices - Part IIProblems
and solved examples based on the sub-topics mentioned above. Some of
the problems in this part demonstrate finding the rank, inverse or
characteristic equations of matrices. Representing real life problems in
matrix form.
Determinants
Introduction to determinants. Second and third order determinants,
minors and co-factors. Properties of determinants and how it remains
altered or unaltered based on simple transformations is matrices.
Expanding the determinant. Solved problems related to determinants.
Simultaneous linear equations in multiple variables
Representing a system of linear equations in multiple variables in
matrix form. Using determinants to solve these systems of equations.
Meaning of consistent, homogeneous and non-homogeneous systems of
equations. Theorems relating to consistency of systems of equations.
Application of Cramer rule. Solved problems demonstrating how to solve
linear equations using matrix and determinant related methods.
Introductory problems related to Vector Spaces -
Problems demonstrating the concepts introduced in the previous
tutorial. Checking or proving something to be a sub-space, demonstrating
that something is not a sub-space of something else, verifying linear
independence; problems relating to dimension and basis; inverting
matrices and echelon matrices.
More concepts related to Vector Spaces
Defining and explaining the norm of a vector, inner product,
Graham-Schmidt process, co-ordinate vectors, linear transformation and
its kernel. Introductory problems related to these.
More
advanced cases of evaluating limits, conditions for continuity of
functions, common approximations used while evaluating limits for ln ( 1
+ x ), sin (x); continuity related problems for more advanced functions
than the ones in the first group of problems (in the last tutorial).
|
Mathway - put in a math problem and not just receive the answer, but receive an explanation of how the problem was solved, step by step. The site covers basic math, algebra, trigonometry, and calculus.
|
Formulas for functions of one variable This is a chart of functions with one variableLicense information
Related content
No related items provided in this feed
Basic Math - Number Patterns Studying number patterns is important for two reasons. First, they help one better understand the concepts of arithmetic and provide a basis for understanding the concepts of more complex mathematics (algebra, trigonometry, calculus). Second, pattern recognition is a useful problem-solving skill, both in mathematics and in real-world situations. Patterns involving odd and even numbers are investigated. Patterns in multiples of certain numbers lead to an understanding of divisibility rules. SequeStatistical Reasoning II Statistical Reasoning in Public Health II provides an introduction to selected important topics in biostatistical concepts and reasoning through lectures, exercises, and bulletin board discussions. Author(s): John McGreadyMethods in Biostatistics II Presents fundamental concepts in applied probability, exploratory data analysis, and statistical inference, focusing on probability and analysis of one and two samples. Author(s): Brian Caffo
License information
Related content
Content within individual OCW courses is (c) by the Johns Hopkins University and individual authors unless otherwise noted. JHSPH OpenCourseWare materials are licensed under a Creative Commons License
Analyzing Statistics S.S. Europe and Russia Students will gather statistical information on countries in Europe and Russia from almanacs. The information will be recorded in a chart. Students will then take the information and make line or bar graphs. Students will analyze the information by answering higher level thinking questions.Supporting Teachers Intervention in Collaborative Knowledge Building In the context of distributed collaborative learning, the teacher's role is different from traditional teacher-centered environments, they are coordinators/facilitators, guides, and co-learners. They monitor the collaboration activities within a group, detect problems and intervene in the collaboration to give advice and learn alongside students at the same time. We have designed an Assistant to support teachers intervention in collaborative knowledge building. The Assistant monitors the collabo Author(s): Chen Weiqin
License information
Related content
Rights not set
No related items provided in this feed
The Effective Provision of Pre-School Education (EPPE) Project: Technical Paper 8a - Measuring the IA changing climate for educational research? The role of research capability-building As part of the Teaching and Learning Research Programme, the ESRC have funded a totally new kind of project, which is likely to be watched with interest by others in social science more generally. This Research Capacity-Building (RCB) project (grant number L139251106) is an innovative attempt to invigorate an entire research field. Among its aims are to support and encourage: the management of complex projects, a widening of methodological approaches, the further combination of different approac Author(s): Creator not set
Artificial Intelligence: Natural Language Processing This course is designed to introduce students to the fundamental concepts and ideas in natural language processing (NLP), and to get them up to speed with current research in the area. It develops an in-depth understanding of both the algorithms available for the processing of linguistic information and the underlying computational properties of natural languages. Wordlevel, syntactic, and semantic processing from both a linguistic and an algorithmic perspective are considered. The focus is on m Author(s): No creator set
License information
Related content
No related items provided in this feed
Examining the Burdens of Gendered Racism: Implications for Pregnancy Outcomes Among College-Educated Objectives: As investigators increasingly identify racism as a risk factor for poor health outcomes (with implications for adverse birth outcomes), research efforts must explore individual experiences with and responses to racism. In this study, our aim was to determine how African American college-educated women experience racism that is linked to their identities and roles as African American women (gendered racism).
Methods: Four hundred seventy-four (474) African American women collaborate Author(s): Jackson, Fleda Mask,Phillips, Mona Taylor,Hogue, C
License information
Related content
Rights not set
No related items provided in this feed
Formulas for functions of two variables This website features a chart of functions with two variables
|
Essentials of GeometryA
This course will explore the basic concepts of Geometry. Throughout the course we will use a variety of tools including the graphing calculator, Geometer's SketchPad, Geogebra, and Aleks
|
Sums of series Teacher Resources
Title
Resource Type
Views
Grade
Rating
Students find common ratios of geometric sequences on a spreadsheet. Using this data they create scatter plots of the sequences to determine how each curve is related to the value of the common ratio. Students then generate geometric sequences on a spreadsheet and develop a formula to find any term of the sequence.
For this sequences and series worksheet, students solve 18 multiple choice problems. Students find the next terms in a sequence, the sum of series, determine if a series is convergent, expand binomials, etc.
For this sequences and series worksheet, students complete seven activities covering arithmetic sequences, geometric sequences, and series. Fully worked out examples, formulas, and explanations are included.
In this college level Calculus worksheet, students examine sequences to determine if they are increasing, decreasing, or non monotonic. Students find the sum of series and determine if they diverge. The one page worksheet contains twenty-one problems. Answers are not provided.
In this sequences and series worksheet, students solve 16 multiple choice problems. Problem sets change each time the worksheet is accessed. They cover a variety of sequences such as arithmetic and geometric sequences. The worksheet addresses sums of series (with sigma notation). Each problem includes a hint button. The worksheet is self-scoring.
Students analyze geometric series in detail. They determine convergence and sum of geometric series, identify a series that satisfies the alternating series test and utilize a graphing handheld to approximate the sum of a series.
|
umi-umd-5639
Course: TOMOS 1903, Fall 1920 School: Maryland Rating:
Word Count: 22997
Document Preview. However, simply refocusing attention on these ignored aspects of algebra will not alone ensure that students avoid common pitfalls. After examining evidence that students are very prone to overgeneralize, IBEYOND PEMDAS: TEACHING STUDENTS TO PERCEIVE ALGEBRAIC STRUCTURE
by Ethan Michael Merlin
Thesis submitted to the faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Master of Arts 2008
Advisory Committee: Professor Lawrence M. Clark, Chair Professor Daniel Chazan Professor James T. Fey
Copyright by Ethan Michael Merlin 2008
ii TABLE OF CONTENTS
Introduction What is expression transformation? Evidence that students have difficulty with expression transformation Simplifying numeric expressions and evaluating algebraic expressions
1 3 5 11
The non-notational precedence conventions: The hierarchy of operations 14 Notational precedence conventions: Grouping symbols Parsing: Implicit activity versus explicit activity Transforming expressions Generalization, and the two faces of algebra The rules of algebra Transforming an expression: Matching given expressions to rule expressions Subexpressions and structural templates Transforming an expression: Using subexpressions Using reference as support for expression transformation Diagnosing student difficulties with expression transformation Reification: Mathematical objects as compressed processes Subexpressions and structural templates as mathematical objects PEMDAS: Structure (or lack thereof) in the traditional curriculum Diagnoses that support or overlap with mine The role of overgeneralization 25 27 30 33 35 36 42 46 50 53 17 19 20 20 23
iii Matz: Overgeneralizing rule-revision strategies Kirshner & Awtry: Overgeneralizing memorable visual sequences The vicious circle of reification The pragmatic value of reification The vicious circle and algebra: A window for overgeneralization Competing impulses: The connectionist view of mind Kirshner on the role of spacing in parsing decisions Kirshner on connectionism Landy & Goldstones research on formally irrelevant distractions Other examples of competition in algebra performance Instructional strategies Instructional strategies for helping students achieve the processperception of parsing Instructional strategies for inducing reification of structure vertically Instructional strategies for inducing reification of structure horizontally Some supporting and overlapping curricular recommendations Conclusion and implications References 82 85 88 92 98 103 55 59 63 63 65 67 67 71 74 79 81
1 Introduction Scholars of algebra education have long lamented that few students become competent in the subject despite years of exposure in school. For instance, Kieran (1983) states that algebra is known to be a school subject which presents many cognitive obstacles to the student encountering it for the first time (p. 162). Similarly, Booth (1984), after conducting a major study of student algebra proficiency, reports that items representing the highest level of understanding were answered successfully by only a small percentage of children, and that in many cases the level of understanding improved relatively little as the child progressed (p. 2). Herscovics (1989) surveys data about student achievement in algebra and concludes that only a minority of pupils completing an introductory course achieve a reasonable grasp of the course content (p. 60). Lee & Wheeler (1989) reach a similar conclusion, which they report with alarm: It is tempting to describe high school algebra as it is unveiled in our research as a disaster area (p. 53). In this paper, I will propose some instructional strategies that I believe can modestly improve student performance on some algebra tasks. These proposals stem jointly from my experience as a classroom teacher and from empirical findings and theoretical frameworks in the scholarly literature. I have deliberately chosen the word modestly to describe the hoped-for improvement in student algebra performance. Much evidence both scientific and anecdotal suggests that symbolic algebra will always be somewhat difficult for many students. There is no all-encompassing solution to the problems of algebra education.
2 Nonetheless, I am optimistic about the possibility of real, measurable improvements in student ability to master certain aspects of symbolic algebra. We are only just beginning to understand the cognitive tasks involved in learning algebra and doing algebra. Insofar as our understanding of human cognition is still evolving, it is perhaps not only possible but indeed likely that there are distinctly better ways to teach algebra that are yet to be discovered and implemented. In this paper, I offer some instructional strategies to help students avoid common errors while performing the algebra task called transforming expressions. Evidence shows that transforming expressions is a major stumbling block for many algebra students. Using Sfards (1991) theory of reification, I will or, more generally, to structure. We will see, however, that simply refocusing attention on these ignored aspects of algebra will not alone ensure that students avoid the common pitfalls. After examining evidence that students are very prone to overgeneralize, I will3 What is expression transformation? In algebra, an expression is a single number, a single variable, or multiple numbers and/or variables linked by one or more arithmetic operations. If the expression contains at least one variable, then it is an algebraic expression. If it contains only numbers, then it is a numeric expression. These definitions are consistent with those used by many student texts in recent decades (e.g. Stein, 1956; Payne et al., 1972; Dolciani et al., 1983; Foerster, 1994). Note that expressions are not statements: they cannot be true or false. In particular, they do not contain relation symbols like the equal sign. Expressions are simply numbers, unknowns, and operations strung together. In this study, we will only consider expressions in which the operations are limited to the basic arithmetic operations of addition, subtraction, multiplication, division, exponentiation, and root-taking; we will not consider logarithms, trigonometric functions, or other more advanced functional operations. Transforming an expression is a relatively complex task to describe precisely. It will be necessary first to describe two simpler tasks of algebra. Two of the most basic activities of elementary algebra are simplifying numeric expressions and evaluating algebraic expressions. Simplifying a numeric expression consists of performing the indicated operations in a proper sequence to obtain a single number. Evaluating an algebraic expression consists of replacing each of the variables in the expression with given numbers and then simplifying the resulting numeric expression. Examples A and B below show typical instances of the sorts of exercises one encounters early in many algebra textbooks, as well as the sort of simplification work that a competent student might perform:
4 Example A Simplify 2 + 5(4 6) 2 .
Example B Evaluate 5 x 2 2 xy using x = 2 and y = 3 .
5 Two algebraic expressions are said to be equivalent if they yield identical numerical results upon being evaluated for any allowable value of the variable. For instance, the expressions 3 x + 4 y + 2 x and y + 5 x + 3 y are equivalent: both expressions yield the number 22 when evaluated with x = 2 and y = 3 ; both expressions yield the number 58 when evaluated with x = 6 and y = 7 ; and so on. This characterization of expression equivalence is the one in use in many textbooks (e.g. Payne et al., 1972; Dolciani et al., 1983). Transforming an algebraic expression consists of replacing that expression with an equivalent expression. For instance, one might transform 3 x + 4 y + 2 x into the equivalent y + 5 x + 3 y , or perhaps into the equivalent 5 x + 4 y , which requires fewer symbols to write. Context or instruction will dictate what sort of equivalent expression is appropriate. Simplifying an algebraic expression and factoring an algebraic expression are two common types of transforming, but it is not necessary to elaborate here precisely what those terms mean. Note that thus far I have not described how to transform an expression but only what expression transformation is.
Evidence that students have difficulty with expression transformation While teachers and researchers have identified algebra in general as an area of difficulty for many students, they have reported expression transformation as an area of particular difficulty. Because of the frequency and rule-like regularity with which students are prone to make certain errors, Sleeman (1984) and others refer to these classic errors as mal-rules. The central aim of this paper is to contribute research-
6 based instructional strategies to help students master the skill of expression transformation and avoid producing these well-known errors. Marquis (1988) is a good example of someone who has collected some of these common errors together into one article. She describes the universality of a certain set of errors made by students who are attempting to transform algebraic expressions. She provides a list of twenty-two such errors:
a ax 12. x = b bx
1. 3 = 3
2. 32 33 = 95 3. a 2 b5 = (ab)7
13. 14.
xa + xb a + b = x + xd d
x y =
xy
4. x + y 3( z + w) = x + y 3 z + w 15. If 2(2 z ) < 12 then z < 4 . 5.
r (6 s ) r 12 2 s = 4 2 4
16.
1 1 x y
=
y 1 x
6. 3a + 4b = 7 ab 7. 3 x 1 =
1 3x
17. a 2 a 5 = a10 18. (3a) 4 = 3a 4
a b ab = ba ab
8.
x2 + y2 = x + y x+ y y = x+ z z 1 1 = x y x+ y x r x+r += y s y+s
19.
9.
20. ( x + 4) 2 = x 2 + 16
r 6s r 6s = 4 4 4
10.
21.
11.
22. (a 2 )5 = a 7
7 Marquis emphasizes that the regular occurrence of such mistakes in algebra is strikingly independent of context: After a few years of teaching mathematics courses in high school, teachers know which concepts and manipulations will cause difficulty for students. From year to year, class to class, students often make the same algebraic mistakes over and over (p. 204). She also emphasizes the stubborn persistence of these errors. Student use of these mal-rules, according to Marquis, does not seem to be merely an appropriate developmental step on the path to eventual mastery; on the contrary, she explains that in upper-level mathematics courses, students indication of mastery of the new concepts may be obscured by common algebraic errors (p. 204). Many teachers corroborate Marquis observations about the universality and persistence of the sorts of errors on Marquis list. Grossman (1924), for instance, provides the following examples of cancellation errors:
3x + 7 3y + 8
3x + 7 3+ 7
8( x + 7) 9(5 x + 7)
3( x + 7) + 11 3( x + 7) + 2
Grossman describes such mistakes as commonplace: Every teacher of experience knows that a great many of his algebra pupils all the way from the first year in high school up to college continue with almost comical regularity to make strange mistakes in the subject of cancellation in fractionsmistakes that show clearly that the essence of the matter has escaped them. (p. 104)
8 Schwartzman (1977) provides the following examples of distribution errors: ( a + b) 2 = a 2 + b 2
x2 y2 = x2 y 2 = x y a ( xy ) = ax ay
He confirms that these are three mistakes that my students frequently make (p. 594), echoing others in noting the persistence and universality of such mal-rules. Martinez (1988), too, describes common algebra errors of this sort. He provides a list of students common errors which he takes to be representative of the kinds of difficulties that seem to be correlated with students misunderstanding of factors and terms (p. 747). Nearly all of the errors on his list are very similar to the errors Marquis included:
1. x + x 2 = x 3 3. 3[2 + ( x 1) + 1] = 6 + ( x 1) + 1 5. x( x 1) + ( x 1) = ( x 1) 2 x
x+2 = 10 2 x + 2 = 12
2. ( x)( x) = 2 x 4. (a + b) 2 = a 2 + b 2 6.
a2 + b2 = a + b
a+b = 1 + b, a 0 a
7.
8.
9.
2( x + y ) =2 xy
10.
4( x + y ) =1 4x + y
Algebra beginners are not the only students who fall victim to these mal-rules; Parish & Ludwig (1994) provide a list of twenty typical mathematical errors made by high school and lower division college students (p. 235). They write that the
9 mathematics and science teaching professions are well aware of the fact that certain types of mathematical errors are continually repeated by students. Large numbers of typical errors are documented in the literature (p. 235). This sampling of teacher observations typifies a broader awareness in the profession that algebra students are universally prone to persistently make mal-rule errors. The authors of some major textbooks also seem to show an awareness that students are prone to making these sorts of errors when transforming expressions. Steins 1956 text is typical in this regard. In this text, each lesson beings with a Procedure that provides precise step-by-step instructions for performing the skill under consideration. For the lesson on Reduction of Fractions (p. 146), however, Stein departs from the usual format by including a step in the procedure for what not to do (emphasis mine):
I. Aim: To reduce algebraic fractions to lowest terms. II. Procedure
1. Find the largest common factor of both numerator and denominator. If the numerator, or denominator, or both are polynomials, factor them if possible. 2. Divide both numerator and denominator by the largest common factor. 3. Do not cancel term with term. See sample solutions 6, 7, and 8. 4. Check by going over the work again or by numerical substitution. (p. 146)
10 Stein repeats the warning against canceling terms in his commentary (p. 146) to Sample Solution 6 (emphasis mine):
6.
3x x = 3( x + y ) x + y
Divide numerator and denominator by 3.
Do not cancel x in answer.
Remove parentheses from x + y in answer. Answer,
x x+ y
Foersters (1994) Algebra 1 text shows a similar awareness of student susceptibility to this and other common errors. In a lesson on Simplifying Rational Algebraic Expressions, Foerster includes the following warning: Do not read more into the definition of canceling than is there! For instance, in
x7 , you cannot cancel the x+2
xs (p. 461). To reinforce the point, Foerster includes a set of exercises with the
instruction Can canceling be done? If so, what can be canceled? (p. 462). Elsewhere, Foerster includes the following exercise designed to draw student attention to a mal-rule frequently followed when squaring a binomial: Explain the error in the work below: ( x + 4) 2 = x 2 + 16 (p. 199). The newer Core-Plus Mathematics Project texts also show awareness of common student algebra errors. For instance, Coxford et al. (2003) includes the following reflection question (p. 205):
11 Youve found that there are some common errors that occur when people operate on symbolic expressions in an effort to produce equivalent but more useful forms. What advice would you give to help someone understand why each of the following pairs of expressions is not equivalent?
a. a ( x + b) is not equivalent to ax + b . b. x b is not equivalent to b x . c. (ax)(bx) is not equivalent to abx . d. ( x b) is not equivalent to x b .
Elsewhere, the same authors include the following writing prompt: When beginning students are asked to expand expressions like ( x + a) 2 , one very common error is often made. What do you think that error is, and how could you help someone avoid making the error? (p. 214). The Teachers Guides suggested response begins as follows: Students often forget the middle term and expand ( x + a) 2 to x 2 + a 2 (p. T214). Thus, evidence from teachers and from textbooks makes clear that students are particularly prone to have difficulty with the task of expression transformation. In particular, they are universally prone to make certain predictable and persistent errors.
Simplifying numeric expressions and evaluating algebraic expressions
To help students overcome the transformation errors described above, we need to diagnose what goes wrong when students produce these common transformation errors. Before trying to diagnose the problem, however, I will
12 characterize competent performance of expression transformation. In order to give a precise account of competent performance of this skill, I must first give a more precise account of the two simpler algebraic skills mentioned earlier, namely simplifying numeric expressions and evaluating algebraic expressions. Simplifying numeric expressions and evaluating algebraic expressions are not trivial tasks for novices. Reexamining Examples A and B above, we can isolate three distinct competencies involved in successful performance of these skills. First, students need to correctly perform arithmetic operations. For instance, in Example A, the simplifier needed to know that five times four equals twenty. Second, students need to successfully interpret and use algebraic syntax, that is, the symbolic notation of algebra. For instance, in Example A, the simplifier needed to know that the juxtaposition of 5 and (4 6) 2 indicates multiplication. Similarly, in Example B, the evaluator needed to introduce an appropriate notation for multiplication other than juxtaposition once x and y were replaced with negative numbers. The third competency is determining an appropriate order of precedence for the operations in the expression. When multiple operations or even multiple instances of the same operation are indicated in an expression, different numerical results are sometimes obtained depending upon which operations are given precedence over which others. Consider, for instance, Examples C and D below:
13
EXAMPLE C
EXAMPLE D
Simplify 2 + 3(5) .
Simplify 2 + 3(5) .
In Example C, the simplifier treated the addition as more precedent than the multiplication. In Example D, the simplifier treated the multiplication as more precedent than the addition. Two different answers result. If expressions containing multiple operations are to have unambiguous meaning, then it is necessary for all users of algebra to make the same (or equivalent) decisions about operation precedence. Several researchers (e.g. Sleeman, 1984; Ernest, 1987; Thompson & Thompson, 1987; Kirshner & Awtry, 2004) have illustrated the notion of operation precedence using tree diagrams, which do not allow for the ambiguity present in standard algebraic notation. In a tree diagram, a more precedent operation appears lower on the tree than a less precedent one. Here are tree diagrams illustrating the precedence decisions of the simplifiers in Examples C and D above:
14
+
+
5
2
2
3
3
5
For virtually all multi-operation expressions written in standard (i.e. non-tree) notation, mathematical conventions dictate an agreed-upon order of operation precedence, thereby eliminating ambiguity and determining a unique interpretation. Some of the conventions that determine operation precedence involve the use of notations that appear on the page, while others do not involve visible notations.
The non-notational precedence conventions: The hierarchy of operations
Some teachers (e.g. Schwartzman, 1977; Rambhia, 2002) describe the nonnotational conventions for operation precedence by referring to the six basic
operations (addition, subtraction, multiplication, division, exponentiation, and roottaking) as occupying levels in a hierarchy of operations. Kirshner (1989) provides the following table of the hierarchy of operations:
In this hierarchy, exponentiation and its inverse (root-taking) occupy the level of highest precedence; multiplication and its inverse (division) occupy the middle level; and addition and its inverse (subtraction) occupy the level of lowest precedence.
15 Revisiting the expression 2 + 3(5) from Examples C and D above, we see that according to this hierarchical convention, it is standard to interpret the multiplication as more precedent than the addition in the expression. Thus, Example C above is incorrect while Example D is correct, according to convention. This hierarchical convention is non-notational in that knowledge external to the written form of an expression dictates the order of precedence. (We will see later, however, that Kirshner (1989) identifies hints present in standard algebraic notation regarding the hierarchical precedence order.) The core of the hierarchy that exponentiation is more precedent than multiplication, which is in turn more precedent than addition has deep mathematical underpinnings and is universally accepted. Miller (2006), in a study of mathematical notations, indicates that this basic convention was followed in the earliest books employing symbolic algebra in the 16th century. Similarly, Peterson (2000) affirms that the basic hierarchy of precedence appears to have arisen naturally and without much disagreement as algebraic notation was being developed in the 1600s and the need for such conventions arose. This history indicates that the order of precedence is not simply the sort of convention adopted willy-nilly that could have been otherwise; rather, it reflects something essential and deep about the operations themselves. Peterson posits that distributive relationships among the operations make this hierarchy the only natural one: exponents and radicals distribute over multiplication and division, while multiplication and division distribute over addition and subtraction. Both Peterson and Wu (2007) explain that since polynomials primary objects of study in symbolic algebra are sums of products, the easy
16 representation of polynomials in symbolic notation naturally motivated a convention in which multiplication is understood to be more precedent than addition. Thus, although teachers often present the hierarchy as a convention designed strictly to eliminate ambiguity (e.g. Rambhia, 2002), historical evidence belies this conclusion and points to something mathematically deep and essential about this convention. While the core notion of exponentiation-precedes-multiplication-precedesaddition is uniformly accepted, its application becomes slightly more controversial when dealing with two related situations: the presence of any of the three inverse operations, and the appearance of multiple operations from the same hierarchical level in the same expression. Typically, the convention is stated in a manner similar to Kirshner (1989, p. 276): In case of an equality of levels, the left-most operation has precedence. In other words, perform all exponents and/or roots from left to right, then all multiplications and/or divisions from left to right, then all additions and/or subtractions from left to right. Yet a case can be made that this manner of stating the convention over-specifies the order. For one thing, in an expression such as (a 3)( x + 2) , it makes no difference whether the subtraction or addition is performed first because the precedence of the intervening multiplication. In tree notation, it is easy to see the independence of these two operations in the separateness of the trees two main branches:
*
a
+ 3
x
2
17 Moreover, since addition and multiplication are associative operations, it truly does not matter in expressions like a + b + c (or abc ) which addition (or multiplication) is performed first, even without the presence of an intervening higher-precedence operation. And, while Miller (2006) affirms that most modern textbooks seem to agree that all multiplications and divisions should be performed in order from left to right, he acknowledges that as recently as 1929 there was no agreement as to whether this was so or whether all multiplications should precede all divisions. Rather than overspecify unnecessarily, Wu (2007) prefers to avoid the question of what to do about multiple occurrences of operations from the same level of the hierarchy. His simpler formulation of the convention is exponents first, then multiplications, then additions (p. 2), but the cost of this economical formulation is the elimination of the inverse operations as independent operations, requiring a more sophisticated understanding of subtraction as addition of the opposite, division as multiplication by the reciprocal, and root-taking as raising to a fractional power. Thus, in formulating the details of the hierarchy of operation precedence, there is a trade-off between precision and accessibility to beginners.
Notational precedence conventions: Grouping symbols Notational conventions for indicating operation precedence are syntactical
indications in the written form of an expression. These include use of parentheses, brackets, and braces (often called grouping symbols); for instance, compare 2( x + 3) , in which the addition understood as precedent, and 2 x + 3 , in which the multiplication is understood as precedent. Notational precedence conventions also
18 include use of the horizontal fraction bar and the radical symbol; for instance, compare
x+3 3 (addition more precedent) with x + (division more precedent), and 2 2
x + 3 (addition more precedent) with
x + 3 (root-taking more precedent).
Kirshner (1989) points out that raised notation also indicates precedence; for instance, compare 2 n +3 (addition more precedent) with 2 n + 3 (exponentiation more precedent). If there are multiple operations within a grouped or raised portion of an expression, the conventional hierarchy of operations determines precedence. Wu (2007) points out that there is a trade-off between utilizing notational and non-notational conventions to indicate operation precedence. On the one hand, a well-placed set of parentheses can help stave off parsing errors, and so grouping symbols are sometimes written even when formally unnecessary. Redundant parentheses, such as those in (3 102 ) + (4 10) + 5 , can serve to reinforce the usual precedence hierarchy rather than to override it. Taken to the extreme, the use of notational precedence indicators could obviate the need for learning a hierarchy of operations. For instance, the expression 2 + 5(4 6) 2 from Example A above could be written as the formally-equivalent 2 + (5((4 6) 2 )) . But, as Wu points out, our current algebraic conventions make for a notational simplicity (p. 2) in comparison to such parentheses-laden expressions, despite the fact that these conventions place a certain demand upon the user to memorize and apply the operation hierarchy.
19
Parsing: Implicit activity versus explicit activity
We are now ready to define parsing. The term parsing comes from computer science and linguistics, where it is used to describe the process of breaking down a sequence of symbols or words into component parts. Many researchers (e.g. Sleeman, 1984; Kirshner, 1989; Jansen, Marriott, & Yelland, 2007; Landy & Goldstone, 2007a) have written about parsing as an important component of algebraic ability. In the context of algebra, parsing an expression means breaking the expression into pieces using the precedence conventions. We can therefore say that, along with performing arithmetic operations and interpreting and using algebraic syntax, parsing is the third component of competent performance of simplifying numeric expressions, as in Example A, and evaluating algebraic expressions, as in Example B. Note, however, that in written performances like Examples A and B, the act of parsing is itself invisible. We can usually infer what parsing decisions were made from what is visible; at least one parsing decision is implicit in each step. But each step also involves performance of arithmetic and interpretation of syntax. Moreover, students sometimes perform more than one operation per step, in which case their parsing decisions are even less apparent. We can contrast the implicitness of the parsing in Examples A and B with more explicit forms of parsing. One explicit form of parsing would involve inserting the redundant parentheses. Drawing an expression tree is another explicit way to parse an expression. Both inserting redundant parentheses and drawing expression trees involve indicating visibly the order of operation precedence.
20
Transforming expressions
Now that we have considered competent performance of the two more basic skills of simplifying a numeric expression and evaluating an algebraic expression, we are ready to return to the primary skill under consideration in this paper, namely expression transformation. Thus far, I have only described the result of transforming an expression: when an expression is transformed correctly, an equivalent expression is produced. I have also shown examples of classic incorrect expression transformation. It is time to identify exactly what skills and concepts are involved in correctly transforming an expression. How does one go about finding an appropriate equivalent expression? The following account builds upon writings in the information processing tradition, such as Matz (1980), Kirshner (1985), and Ernest (1987). Note that the following explanation of expression transformation is an idealization of algebraic behavior. I am not, in fact, making any claims about what actually transpires in the brains of users of algebra.
Generalization, and the two faces of algebra
To explain how one goes about the relatively complex task of transforming an algebraic expression, I must first say a bit about how algebraic symbolism is used for the purpose of generalization. In the process of describing algebras capacity for generalization, I will elaborate upon two mutually reinforcing and codependent aspects of algebra, which I will call the referential and structural aspects of algebra.
21 In contrast to expression transformation, which certainly resides within the realm of algebra, the two more simple skills discussed fall basically within the realm of arithmetic. True, the two do involve algebraic syntax for indicating operations, and they do involve algebraic conventions for operation precedence. Evaluating even involves variables, albeit briefly. However, we can characterize these activities as arithmetic rather than as algebra because they do not involve generalization. More than anything else, it is generalization and not the presence of variables that demarcates the territory of algebra. Kaput (1995) describes two ways in which algebra functions as a tool for generalizing. First, algebra can generalize arithmetic facts. For instance, each of the arithmetic facts 5 + 0 = 5 , 12 + 0 = 12 , and 7 + 0 = 7 are instances of the algebraic generalization m + 0 = m . Second, algebra can generalize relationships among varying quantities. For instance, the relationship between the perimeter p of a rectangle and its base b and height h can be described by the generalization
p = 2b + 2h .
When algebra is used to generalize arithmetic facts or relationships among quantities, its variables and expressions refer to numbers or quantities. These referred to numbers or quantities have an existence that is independent of the algebra symbolism used to express the generalizations on paper. Yet algebraic symbolism is powerful not only because of its capacity for symbolically capturing generalizations. Algebraic symbolisms power resides also in its ability to be considered without referential context. For instance, while the expression 2b + 2h can be considered in the context of rectangle and perimeter, it
22 also can be considered divorced from this context as a string of syntactically-related symbols. The task of transforming an expression is naturally performed while operating in such a decontextualized framework, as I will explain below. We can idealize the importance of algebra in human activity as follows: first, a real situation about related quantities is generalized using algebraic symbolism; then, the symbolism is transformed into an equivalent form that is somehow different from the original form; and finally, the new form is usefully related back to the context that generated it. In the idealized description just provided of how people use algebra, the first and last phase of the process are alike in their focus on external referents, while the middle phase differs from the other two in its focus away from external referents and toward formal use of the symbolic language. Many researchers have therefore pointed out that it is possible to speak about two aspects (e.g. Kaput, 1995) or faces (e.g. Kirshner, 2006) of algebra, one focused on external referents and one focused away from them. Kirshner, for instance, describes the two faces of algebra as follows: The empirical face points outward toward domains of reference, toward modeling phenomena in the world, toward application, toward number, quantity, and shape. The structural face points inward to the logical infrastructure, to the grammar of rules and procedures abstracted from external realms of interpretation. (p. 13) Henceforth, I will describe these two aspects or faces of algebra as referential and
structural.
23 Referential and structural algebra are co-dependent. Both are essential functions of algebra in human activity. Kirshner (2006) affirms the complementarity and necessity of the referential and structural aspects of algebra: Internal structure and external reference are complementary and equally vital aspects of algebraic knowledge (p. 14). Kaput (1995) agrees that the two aspects are complementary, touting both the traditional power of algebra [that] arises from the internally consistent, referent-free operations that it affords (p. 76) and the importance of the semantic starting point where the formalisms are initially taken to represent something (p. 76). Indeed, both algebras referential function and its structural function are powerful and necessary for algebra to be important for people.
The rules of algebra
Expression transformation is one of the tasks for which the structural aspect of algebra evolved. The proces of transforming an expression can naturally be described by disregarding contextual referents and approaching algebraic symbolism formally. That said, in order for algebra to function in this decontextualized realm, one must have a supply of previously accepted generalizations of arithmetic facts. These previously accepted generalizations are often simply referred to as the rules of algebra. There is no official list of the rules of algebra. Nearly all algebra texts would include any commutative, associative, identity, and distributive properties of the basic arithmetic operations as rules of algebra. They might also include properties
24 of exponents and properties of fractions. The rules are usually stated as equations, as shown here:
Examples of rules of algebra
Commutative Property of Multiplication: Distributive Property of Multiplication over Addition: Cancellation Property of Fractions:
mn = nm
m(n + p ) = mn + mp mp m = np n
Note that each rule comprises two expressions, which we shall refer to as the rules
expressions.
A thoroughly formal and axiomatic approach to algebra would designate a minimal number of these rules as postulates and then regard the other rules as proven
theorems. Postulates are arithmetic generalizations accepted empirically and
inductively without proof. Theorems, on the other hand, are derived from known rules. In practice, however, competent users of algebra need not concern themselves with the distinction between postulate and theorem. In fact, even expert users of algebra are not necessarily aware of the distinction: the rule 0 m = 0 is typically regarded as a theorem in formal treatments of algebra, yet for most users of algebra this rule is simply an empirical generalization of known arithmetic facts. Since we are describing the use of algebra, we need not concern ourselves with any distinction between postulates and theorems, but can simply regard all previously and canonically accepted conclusions as rules.
25
Transforming an expression: Matching given expressions to rule expressions
The rules of algebra can be used to transform expressions in three distinct ways. I will describe the most basic way here; the other two will be described a bit later. The basic way to use a rule to transform an expression involves matching the given expression to one of the expressions in a rule. Suppose, for instance, that a person encounters the expression 2( x + 3) and wishes to transform it. That person would compare the expression to each known rule of algebra, searching for a rule expression of which 2( x + 3) is an instance. That person might select m(n + p ) , one of the expressions in the Distributive Property of Addition over Multiplication
m(n + p ) = mn + mp . In this case, 2 is playing the role of m, x is playing the role of
n, and 3 is playing the role of p. After making this match, the person would turn to
the rules second expression in this case mn + mp and substitute 2, x, and 3 for m,
n, and p respectively, obtaining 2 x + 2(3) . The result, 2 x + 2(3) , is equivalent to the
given expression 2( x + 3) . In the preceding example, 2( x + 3) was an instance of the rule expression
m(n + p ) in the most basic of ways: each variable in the rule expression m(n + p )
corresponded directly with either a number or a single variable in the given expression 2( x + 3) . Tree diagrams make the simplicity of the match even more apparent:
26 Tree diagram for 2( x + 3) : * 2 Tree diagram for m(n + p ) : *
+
m
+
x
3
n
p
The trees of the two expressions have identical branch patterns and identical operations at the nodes of the branches; the only difference between the two expressions is that ones tree terminates in 2, x, and 3 while the others tree terminates in a, b, and c. Were such one-to-one matching of given expression to rule expression the only method of expression transformation, then it would indeed be a limited activity, for it would only be usable if one were lucky enough to know a rule with an expression that constituted a one-to-one match to the given expression. Were one-toone matching the only method of expression transformation, then in order to be prepared for all the possible expressions one might need to transform, one would need to memorize many, many rules. Fortunately, such one-to-one matching is not the only way to transform expressions. The extraordinary power of expression transformation resides in the fact that it often can be carried out when the given expression and the rule expression do not match in this one-to-one fashion. There are two additional ways in which one can transform expressions. Both depend upon a careful characterization of subexpression and structural template.
27
Subexpressions and structural templates
Tree diagrams provide a useful medium for illustrating the notion of a subexpression. Consider, for instance, the following tree, which shows the expression 2 x + 3 y 2 : Tree diagram for 2 x + 3 y 2 : + * 2 *
x
3
^
y2
A subexpression of 2 x + 3 y 2 is an expression obtained by considering any of the trees numbers, variables, or operations and the portion of the tree lying below it. Thus, if we take the leftmost of the two multiplication signs and everything below it, we obtain the subexpression 2 x . If we take the rightmost of the two multiplication signs and everything below it, we obtain the subexpression 3y 2 . If we take the exponentiation sign and everything below it, we obtain the subexpression y 2 . If we take just the 3, we obtain the subexpression consisting only of the number 3. The complete list of subexpressions of 2 x + 3 y 2 is: 2; x; 3; y; y 2 ; 2 x ; 3y 2 ; 2 x + 3 y 2 . We come here to a crucial observation: to determine an expressions subexpressions, one must be able to parse that expression. Parsing, in other words, is a prerequisite skill for determining subexpressions. One cannot identify subexpressions unless one knows how to parse according to the conventions about operation precedence. y 2 is a subexpression of 2 x + 3 y 2 because when the
28 operations of 2 x + 3 y 2 are performed according to the order of precedence, 2 and x are multiplied. On the other hand, even though x + 3 and 3 y are perfectly good expressions in their own right, and even though these symbol strings occur in 2 x + 3 y 2 , neither is a subexpression of 2 x + 3 y 2 . We also need to define a set of notions that are closely related to the notion of subexpression. The dominant operation of an expression (or of a subexpression) is the least precedent operation of that expression (or subexpression). For example, addition is the dominant operation in the expression 2 x + 3 while multiplication is the dominant operation in the expression 2( x + 3) . When addition (or subtraction) is the dominant operation, then the subexpressions created by that addition (or subtraction) are called terms. So the terms of 2 x + 3 are the subexpressions 2 x and 3. When multiplication is the dominant operation, then the subexpressions created by that multiplication are called factors. So the factors of 2( x + 3) are 2 and x + 3 . When division is the dominant operation, then the first subexpression is the numerator and the second is the denominator. When exponentiation is the dominant operation, then the first subexpression is the base and the second is the exponent. We can also define the notion of the possible structural templates for an algebraic expression. For our purposes, a structural template for an expression is another expression containing only variables and operations (i.e. no numbers) that possesses a very particular sort of top-down identicalness to the original expression. More specifically, some of the two expressions least precedent operations and their indicated order of precedence must be exactly the same. For example, returning to the expression 2 x + 3 y 2 , we see that all of the following are possible structural
29 templates for 2 x + 3 y 2 : a + b , a + bc , a + bc d , ab + c , ab + cd , ab + cd e , and even simply a . Again, tree diagrams provide a very useful medium for illustrating the possible structural templates of an expression. Consider again the tree diagram of 2 x + 3 y 2 , shown several times below side-by-side with the tree diagram of one of its structures. In each 2 x + 3 y 2 diagram, the subexpressions playing the roles of the templates variables are circled. This circling makes evident the sense in which the original expression is identical to the structural template: Tree diagram for 2 x + 3 y 2 : + * 2 * Tree diagram for a + bc : +
a
^
*
x
3
b
c
y2
Tree diagram for 2 x + 3 y 2 : + * 2 *
Tree diagram for ab + cd : + * ^ *
x
3
a
b
c
d
y2
30 Tree diagram for 2 x + 3 y 2 : + * 2 * Tree diagram for a + b : +
a
^
b
x
3
y2
It is important to emphasize that most expressions have several possible structural templates. Depending upon the context, one might choose a structural template including either more or fewer of its operations, embedding fewer or more of the least precedent operations within variables. Again, determining the possible structural templates for an expression depends directly upon ones ability to parse. For this reason, we might instead refer to a structural template of an expression as a parse of that expression, forming a noun out of the verb. Also, henceforth I will use the term structure to refer to the component of algebraic knowledge that concerns understanding of parsing, subexpressions, and structural templates.
Transforming an expression: Using subexpressions
Now that I have precisely defined all of the structural notions described above, I can proceed to describe the two remaining ways, along with straightforward matching of expression to rule, to transform an expression. Both of these ways to transform an expression depend heavily upon the notion of an expressions subexpressions.
31 The second way to transform expressions utilizes the capacity of a variable in a rule to stand for an entire subexpression of a given expression. Consider again the expression 2( x + 3) . Suppose again that a person wishes to transform this expression. This time, suppose the person selects the expression mn from the Commutative Property of Multiplication mn = nm . Although mn is not a one-to-one match for
2( x + 3) in the way that m(n + p ) was, mn is a structural template for 2( x + 3) . In
this case, 2 is playing the role of m and the entire subexpression x + 3 is playing the role of n. Here is the match in tree diagrams, with the subexpression x + 3 circled to emphasize that we are treating it as a single entity matched with n: Tree diagram for 2( x + 3) : * 2 Tree diagram for mn : *
+
m
n
x
3
After making this structural match, the person would consider the rules second expression in this case nm and substitute 2 and x + 3 for m and n, respectively, obtaining ( x + 3)2 . The result, ( x + 3)2 , is equivalent to the given expression
2( x + 3) .
The third way to transform expressions involves finding a subexpression of the given expression for which a rule expression is a structural template. Consider once more the expression 2( x + 3) . Suppose again that a person wishes to transform this expression. This time, suppose the person selects the expression m + n from the
32 Commutative Property of Addition m + n = n + m . Although m + n is not a structural template for the given expression, it is a structural template for the subexpression x + 3 . In this case, x is playing the role of m and 3 is playing the role of n. In tree diagrams, to see the match one must ignore part of one tree and focus on a subexpression only, which is circled here: Tree diagram for 2( x + 3) : * 2 Tree diagram for m + n :
+
+
x
3
m
n
After making this structural match, the person would turn to the rules second expression in this case n + m and substitute x and 3 for m and n, respectively, obtaining 2(3 + x) . The result, 2(3 + x) , is equivalent to the given expression
2( x + 3) . In tree diagrams, it is easy to see that only a subexpression is affected by
the transformation; a portion of the tree is unchanged by the transformation: Before the transformation: After the transformation:
2
+
2
+
x
3
3
x
33 Clearly, expression transformation is extremely powerful. It allows us to transform all of an expression or just a part of an expression. Moreover, the capacity of variables to stand for subexpressions renders unnecessary the memorization of many individual rules like m(n + p ) = (n + p )m , (m + n)( p + q ) = ( p + q )(m + n) , and so on; the single rule mn = nm suffices to generalize these and infinitely other more specific instances. We can therefore transform all sorts of expressions while memorizing only a relatively limited set of rules.
Using reference as support for expression transformation
Before completing our description of expression transformation, we need to consider the role that referential algebra plays in the process. In our idealized description of how people use algebra, we located expression transformation squarely within its structural aspect. Thus far in our description of expression transformation, we have regarded the rules of algebra as givens, and therefore the process of transforming an expression has been a purely formal process of using and following rules the conventions of operation precedence and the rules of algebra precisely. We have been regarding algebraic expressions as mere symbol strings to be interpreted and operated upon while ignoring their possible referential meanings. It is also possible to incorporate referential interpretations into the process of expression transformation. These interpretations provide support for the formal structural decisions that occur during the transformation process described above.
34 Here is an example from the Core-Plus Mathematics Project of how referential interpretations can be used to support the structural transformation process. Coxford et al. (2003, p. 188) describe a situation involving profit and expenses for production of a CD:
Suppose that when a new band recorded its first album with a major label, it had to deal with these business conditions:
Expenses of $365,000 for the recording advance, video production, touring, and promotion (to be repaid out of royalties)
Income of $0.81 per CD from royalties Income of $0.52 per CD for publishing rights
Letting x represent the number of CDs sold, students form a generalization: for any number x of CDs sold, the expression (0.81x + 0.52 x) 365,000 represents the profit made from these sales. Forming this generalization is clearly an act of referential algebra. Now, suppose the students want to transform this expression. Certainly students could transform the expression to the equivalent 1.33 x 365,000 using the rules of algebra as described above. However, students can also reach the same conclusion by thinking about the situation and forming a new generalization. Specifically, it should be clear to students that while the positive component of the profit can be computed by individually multiplying the number of CDs sold by the royalty income and the publishing rights income and then adding, this profit can also
35 be computed by adding the royalty income per CD and the publishing rights income per CD and then multiplying by the number of CDs sold. Depending upon the experience of the student, such referential reasoning could inform the students thinking about the symbol manipulations. The Teachers Guide explains that students can reach the conclusion about the equivalence of (0.81x + 0.52 x) 365,000 and
1.33 x 365,000 by applying contextual knowledge to make sense of rearrangements
of symbols (p. T188). Elsewhere, the Teachers Guide explains how empirical evidence from tables of values or from graphs can be used as referential support for expression transformation decisions. Although referents can, as illustrated by these examples, be used as supports during expression transformation, this paper deals primarily with student learning of expression transformation as a structural task. Henceforth, whenever I refer to expression transformation, unless I specifically indicate otherwise, I will be describing a purely structural approach to expression transformation that utilizes the known rules of algebra without any further referential support.
Diagnosing student difficulties with expression transformation
Our overall goal in this paper is to derive some instructional strategies to help students overcome some common difficulties that they have when transforming algebraic expressions. In the previous section, I provided a careful description of competent performance of expression transformation. In particular, I uncovered the crucial roles for expression transformation of a procedure called parsing and of a set of structural concepts, especially subexpression and structural template.
36 Now we will begin to diagnose what goes wrong when students transform expressions incorrectly. In other words, we will begin to identify how the idealized description of competent expression transformation so often fails to become reality. Since an understanding of structure is so fundamental for competent expression transformation, I give the following initial diagnosis of student difficulties: The
traditional algebra curriculum fails to provide students with the necessary experiences to develop a full understanding of critical structural notions.
To make this diagnosis credible, I will utilize Sfards (1991) framework for the relationship between processes and objects in mathematics. We will see that Sfards framework nicely captures the relationship between the process of parsing and the objects known as subexpressions. Her framework also provides insight as to how students can eventually come to understand mathematical objects. With Sfards insights in mind, we will examine the traditional algebra curriculum and expose its superficial treatment of the very objects upon which expression transformation is performed.
Reification: Mathematical objects as compressed processes
As we have seen, knowing how to parse an expression is a prerequisite skill for simplifying numeric expressions and for evaluating algebraic expressions. Moreover, understanding the notion of a parse (or structural template) of an expression is a prerequisite concept for transforming algebraic expressions. For the simpler skills, parsing is a procedure to be performed. For the more advanced skill of expression transformation, a parse is a thing to be understood conceptually. What is
37 the relationship between knowing how to parse and understanding the notion of a parse? This question really pertains to the relationship between the procedural
knowledge of how to parse and the conceptual knowledge of what is a parse. To
answer it, we will briefly examine a small part of the extensive literature about the relationship between procedural knowledge and conceptual knowledge in mathematics. While current debates about curriculum might lead one to believe that the two are in opposition, scholars tend to agree that procedural knowledge and conceptual knowledge are both vital and necessary components of mathematical proficiency. For instance, Kilpatrick (1988) argues that some balance needs to be found between meaning and skill (p. 274). Similarly, Rittle-Johnson, Siegler, & Alibali (2001) conclude that competence in a domain requires knowledge of both concepts and procedures (p. 359) and that the two are mutually reinforcing: The relations between conceptual and procedural knowledge are bidirectional and improved procedural knowledge can lead to improved conceptual knowledge, as well as the reverse (p. 360). To cite one more example, Star (2005), in discussing procedural and conceptual knowledge, claims that both are critical components of students mathematical proficiency (p. 406). Sfard (1991) goes one step further. Like the researchers cited above, Sfard regards procedures and concepts as necessary and mutually reinforcing. However, she takes the additional step of claiming that the two types of knowledge (she calls
38 them operational and structural) are inseparable, though dramatically different, facets of the same thing (p. 9). Sfards claim that procedures and concepts are two facets of the same thing hinges upon her careful explanation of the notion of a mathematical object. In mathematics, Sfard explains, processes are usually performed upon one or another sort of thing or object. For instance, addition can be performed upon the objects known as natural numbers, composition can be performed upon the objects known as functions, and so on. According to Sfard, many of these fundamental mathematical objects are themselves simply processes which, through repetitive familiarity, become condensed in the mind of their users into static things or objects. For instance, Sfard argues that while rational numbers are mathematical objects, this objectperception of rational numbers as static entities grows out of and complements a process-perception of rational numbers as the comparing or dividing of two natural numbers. She provides many other examples of mathematical things that possess this process-object duality. According to Sfard, process-perception necessarily precedes objectperception, both in the historical development of mathematics and for the student learning mathematics. One must, Sfard argues, learn how to perform a process before one can step back and look at the result of that process as an entity in its own right. The concept of a rational number as a static thing, for instance, emerges out of the process of forming ratios of natural numbers. Sfard uses the term reification to describe the moment in which one shifts from a process-perception to an object-perception and forms a static, conceptual
39 understanding of a mathematical object. For example, the moment when a person shifts from only being able to regard 5 divided by 12 as a process to also being able to regard the ratio 5:12 as an object is a moment of reification. Sfard uses the terms
interiorization and condensation to describe earlier stages in which the process
gradually condenses for the learner. According to Sfard, interiorization and condensation are gradual processes; however, she regards reification not as a process but rather as a sudden change in perspective, in which a process is finally grasped allat-once as an object. While Sfard regards the process-perception as necessarily preceding the moment of reification and its object-perception, she also regards the object-perception as necessary for an understanding of even more advanced processes. From a historical perspective, Sfard explains, the mental demands of performing more advanced processes on the objects of simpler processes is what has necessitated reification and its ensuing object-perception. Let us continue to use rational numbers as an example. As long as simply comparing natural numbers to one another was the goal, there was no need to conceive of these ratios as objects in their own right. However, once people developed needs to compare, add, and multiply ratios, it became expedient to be able to treat individual ratios as objects hence the reification of rational numbers into independent entities. Mathematical objects, therefore, serve as necessary pivot points, or, as Sfard says, way-stations (p. 29) between one procedure and a more advanced procedure. Sfard regards this process-object-process sequence as repeating itself iteratively in humanitys development of more and more advanced mathematics.
40 Over and over again, she explains, various processes had to be converted into compact static wholes to become the basic units of a new, higher level theory (p. 16). The process repeats itself, she explains, as one newly reified object itself becomes the object of more advanced processes: When we broaden our view and look at mathematics (or at least at its big portions) as a whole, we come to realize that it is a kind of hierarchy, in which what is conceived purely operationally at one level should be conceived structurally at a higher level. Such hierarchy emerges in a long sequence of reification, each one of them starting where the former ends, each one of them adding a new layer to the complex system of abstract notions. (p. 16) Sfard provides the following diagram to illustrate the process-object-process-objectprocess structure of mathematical knowledge:
41
Returning again to the example of rational numbers will help illustrate the meaning of this diagram. Here is how the diagram would depict the relationship between the object of rational number to the simpler processes that produce rational numbers and to the more complex processes for which rational numbers are the objects:
42
Rational numbers (new objects)
Multiplication of rational numbers (new process)
Natural numbers (concrete objects)
Forming ratios (process on concrete objects)
Sfard of course knows that she is oversimplifying the complex history of the emergence of mathematical concepts. Nonetheless, she regards this account as a good approximation of not only the historical development of mathematics but also of its development for the individual learner.
Subexpressions and structural templates as mathematical objects
Sfards theory of reification has profound implications for student learning of algebra. In a 1994 sequel entitled The Gains and Pitfalls of Reification: The Case of Algebra, Sfard & Linchevski explore the implications of the theory of reification for algebra in particular. Sfard and Linchevski begin by showing that algebraic expressions possess the sort of process-object duality described by the theory of reification. They use the
43 expression 3( x + 5) + 1 as an example. On the one hand, this expression describes a computational process to be performed upon a variable x: take x, then add 5, then multiply by 3, then add 1. On the other hand, this expression is an object in and of itself. Sfard and Linchevski mention three different ways to view this expression as an object. One involves viewing it structurally, in other words, viewing it as a mere
string of symbols (p. 88). They note the power of this sort of object-perception for
the tasks of structural algebra: Although semantically empty, the expression may still be manipulated and combined with other expressions of the same type, according to certain well-defined rules (p. 88). The other two ways to treat the expression as an object involve viewing it referentially, either as a name for a certain number or else as a function. Interestingly, although Sfard and Linchevski mention the semantically empty string-of-symbols perspective of conceiving of an expression as an object which is the way most suited for expression transformation in the remainder of the article they focus on the function perspective. For example, they consider student understanding of relatively advanced algebraic tasks such as solving the inequality
x 2 + x + 1 > 0 , for which a function object-perception is particularly appropriate.
After reviewing a variety of empirical evidence, the authors conclude that reification of algebraic expressions into function-objects is particularly elusive. They regard the process-concept duality inherent in algebraic expressions to be very counterintuitive for novices: Our intuition rebels against the operational-structural duality of algebraic symbols (p. 199). In fact, they go so far as to suggest that this particular leap of reification is too difficult for some students to ever make:
44 The data we collected up to this point provided sufficient evidence that reification is inherently very difficult. It is so difficult, in fact, that at a certain level and in certain contexts, a structural approach may remain practically out of reach for some students. (p. 220) Thus, upon applying the theory of reification to the case of algebra, Sfard and Linchevski are discouraged by their findings regarding the potential for student understanding of algebra. I maintain, however, that can draw more encouraging conclusions from an application of the theory of reification to the learning of algebra. This more optimistic perspective, however, involves applying the theory to algebra in a different way that Sfard and Linchevski apply it. Sfard and Linchevski have chosen to focus on the most advanced of all possible object-perceptions of an algebraic expression, namely the perception of an expression as a function. As Herscovics (1989) has shown, however, the concept of function brings along its own set of cognitive obstacles for the novice. Moreover, as we have shown above, expression transformation one of the most central and most notoriously difficult algebraic tasks depends solely on an object-perception of expressions as strings-of-symbols parsed according to known conventions. Moreover, there is a second and perhaps equally important application of the theory of reification to algebra that Sfard and Linchevski do not identify. In addition to regarding algebraic expressions themselves as objects, it is important for students to be able to regard structural templates as objects. In fact, Sfards description of process-concept complementarity perfectly captures the relationship between the
45 process of parsing and the concept of structural template. An expressions structural templates are nothing but the results of parsing that expression. When one parses, one determines possible structural templates. Moreover, an expressions possible structural templates are themselves the objects of the more advanced process of comparing structural templates and forming matching with rule expressions. The mathematical objects known as subexpressions and structural templates therefore occupy the pivot point between two procedures in the sense described by Sfard. To illustrate:
Structural templates (new objects)
Comparing structural templates (new process)
Expressions (objects)
Parsing (process on objects)
This application of the theory of reification to algebra does not involve cognitively complex objects like functions. Instead, the objects here are empty structural
46 objects. Insofar as an understanding of an expressions possible structures is essential to the task of expression transformation, it is therefore worth exploring to what extent the traditional algebra curriculum provides the sort of experiences that are likely to foster reification of the concepts of subexpression and structural template.
PEMDAS: Structure (or lack thereof) in the traditional curriculum
Recall Sfards contention that process-perception necessarily precedes objectperception. One consequence, therefore, of viewing the relationship between parsing and structural templates through the lens of Sfards theory of reification is an implication for learning: students need to parse first in order to arrive later at a full understanding of structure. When we examine traditional algebra curricula, however, we find two striking absences. First, structure is considered only superficially; second, the parsing process that ideally reifies into an object-perception of structural template is also treated superficially. Traditional texts do not typically deal with structural notions explicitly. The word subexpression does not typically appear in a traditional algebra textbook. Nor do the words structural template in the sense we have used it here. Many texts do define specific structural notions such as factor, term, and so on. However, rarely do these texts provide exercises that require the student to discriminate among these structural notions. We have seen that structural notions play a critical role in competent expression transformation. The absence of attention to structural notions is all the more striking because expression transformation is perhaps the most central activity in the traditional algebra textbook. Foerster (1994), for instance, includes
47 chapters on Distributing: Axioms and Other Properties, Some Operations with Polynomials and Radicals, Properties of Exponents, More Operations with Polynomials, Rational Algebraic Expressions, and Radical Algebraic Expressions. Each of these six chapters (out of a total of fourteen) is devoted nearly entirely to expression transformation. Thus, despite the extensive focus of traditional textbooks on expression transformation, students learning algebra from those books might never encounter the structural notions the understanding of which is essential to transfer expressions competently. Moreover, traditional algebra textbooks include only superficial exposure to parsing as a process, making it even more unlikely that students will attain the objectperception of structural template. On the one hand, typical algebra texts do very much require students to perform exercises that demand implicit demonstration of parsing abilities. Simplifying numeric expressions and evaluating algebraic expressions are standard fare in beginning algebra. On the other hand, algebra textbooks typically do not require explicit demonstration of parsing ability, as through insertion of parentheses or drawing expression trees. They also typically do not include exercises designed to push students from the process-perception of parsing to the object-perception of structural template. The parsing instruction traditional algebra texts do include typically falls under the rubric of instruction in the order of operations. Foersters 1994 text is typical in this regard. Section 1-4 of the text is entitled Order of Operations. He introduces the topic as follows:
48 You have learned that symbols of inclusion can be used to tell which operation is to be performed first in an expression. If there are more than three operations, there would be so many parentheses and brackets that the expression would look untidy, like this:
(4 + [(9 3) 6]) + [(5 8) 7] .
To avoid all this clutter, users of mathematics have agreed on an order in which operations are to be performed. Parentheses are used only to
change this order. (p. 18)
Foerster then goes on to state the agreed-upon order of operations as follows (p. 19):
ORDER OF OPERATIONS
If there are no parentheses to tell you otherwise, operations are performed in the following order: 1. Evaluate any powers first. 2. After powers, multiply and divide, in order, from left to right. 3. Last, add and subtract, in order, from left to right. In the exercises following this presentation, Foersters text provides many examples of numerical expressions to be simplified and algebraic expressions to be evaluated. Some teachers (e.g. Schrock & Morrow, 1993), and even some algebra texts, present a mnemonic device for helping students learn the order of operations. In the United States, this mnemonic is usually PEMDAS, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. To remember the
49 acronym, some students learn the phrase Please Excuse My Dear Aunt Sally. In some other English speaking countries, the acronym is either BEMDAS (with B for Brackets), BIMDAS (with I for Indices), or BOMDAS (with O for Of, as in power of or root of). Many teachers, however, have discovered pitfalls of teaching students to rely on PEMDAS and similar mnemonics. Nurnberger-Haag (2003), for instance, notes that parentheses are only one type of grouping symbol that students encounter in algebra, and that the PEMDAS mnemonic overlooks brackets, fraction bars, and other notational parsing indicators: Teaching students about only one special case of grouping symbols is analogous to teaching only a special case of exponents for the second step (such as squaring) (p. 235). Moreover, several teachers (e.g. Rambhia, 2002; Nurnberger-Haag, 2003) point out that the PEMDAS mnemonic seems to imply that addition and subtraction occupy successive levels in the operation hierarchy (and similarly with multiplication and division), when in fact these operation pairs have the same degree of precedence. Rambhia cautions that as a result of PEMDAS-focused instruction, many students come to believe that multiplication is done before division and that addition is more important than subtraction (p. 194). Thus, PEMDAS and similar mnemonic devices hinder as well as assist the learning of order of operations. In critiquing the traditional treatment of parsing, I am critiquing much more than a mnemonic device whose shortcomings are already famous. I have argued that understanding of structural template as a mathematical object is essential to the competent performance of a skill expression transformation that is the primary
50 content of traditional algebra instruction. The use of PEMDAS in classrooms across the United States is only a symptom of a much larger lack of attention to structure. I have argued that the necessary object-perception of structural template can only arise out of extensive experience parsing algebraic expressions, and I have argued that the traditional curriculum does not require students to parse other than implicitly. The net result is the following partial diagnosis for student difficulty transforming expressions: The traditional algebra curriculum fails to provide students with the
necessary experiences to develop a full understanding of algebraic structure.
Diagnoses that support or overlap with mine
Some teachers and researchers who have written about student difficulty transforming expressions have arrived at diagnoses similar to the one I have argued for here. Several teachers have blamed transformation errors on student confusion about the meaning of specific structural notions, especially the notions of term and
factor. Martinez, for instance, in an article entitled Helping Students Understand
Factors and Terms, concludes, after examining many instances of mal-rule behavior, that in each instance the error is caused by students misunderstanding of factors and terms (p. 747). Similarly, Laursen (1978) points out that some errors stem from the confusion of factors and terms: Many of the theorems in elementary algebra relate specifically to either terms or factors, but not to both. For example, (9)(16) = 9 16
51 But 9 + 16 9 + 16 . (p. 195) One more example comes from Grossman (1925), who diagnoses fraction cancellation errors as stemming from students lack of understanding of what a factor is: This is the real source of many of the mistakes in cancelling fractions. A pupil may understand all that has gone before and still cancel wrongly through not understanding how far the division effect of the cancellation of a factor extends. The essence of the idea is that it extends as far as the multiplying effect of the factor itself extends. (p. 107) These teachers identify student confusion about factors and terms as responsible for many student transformation errors, consistent with my claim that the traditional curriculum does not attend sufficiently to structure. Others have identified students difficulty with the process-object duality of algebraic expressions as responsible for student transformation errors. Barnard (2002a), for instance, writes: It can be argued that algebra starts when the things one is talking and thinking about have become mentally manipulable objects. At the heart of many errors is the failure to conceive the objects of manipulation (e.g. , 7 , 2 x + 5 , their own right. (p. 10).
r 2 + 1 ) as meaningful things in
52 He goes on to explain how student inability to perceive expressions as objects can lead to mal-rules like
a2 + b2 = a + b : a 2 + b 2 as one complete
If pupils are able to see an expression like
object, not only will they not feel the need to work it out further (perhaps incorrectly replacing it with a + b ), but also they will be able to move it around in an equation just as easily as they could move around a single letter or number. Stumbling blocks are often caused by the appearance of unsimplifiable expressions that have no meaning for pupils. (p. 11) Booth (1984) also identifies student difficulty treating unsimplifiable expressions as objects. While these diagnoses are similar to mine, my diagnosis goes into more detail about the roles of subexpressions and structural templates in competent expression transformation and the failure of the traditional curriculum to help students understand these structural notions. Still others have diagnosed student difficulties with expression transformation as stemming from students feeling that algebra is entirely lacking in objects. Sfard and Linchevski (1994), in their discussion of reification in algebra, make just such a diagnosis. After asserting that many students fail to grasp algebraic expressions as objects, they conclude that students will have difficulty learning to operate on expressions beyond what they can memorize from a purely procedural perspective: The process of learning is doomed to collapse: without the abstract objects, the secondary processes will remain dangling in the air they will have to be executed on nothing. Unable to imagine the
53 intangible entities which he or she is expected to manipulate, the student [uses] pictures and symbols as a substitute In the absence of abstract objects and their unifying effect, the new knowledge remains detached from the previously developed system of concepts. In these circumstances, the secondary processes must seem totally arbitrary. The students may still be able to perform these processes, but their understanding will remain instrumental. (p. 221) In other words, Sfard and Linchevski attribute student difficulty operating upon algebraic expressions to student failure to perceive these expressions as objects. Chazan (2000) also points to the lack of explicit objects in the traditional algebra curriculum as partly responsible for student difficulties with the subject. Each of these prior diagnoses overlaps with and to some extent supports my diagnosis. However, my diagnosis specifically designates subexpressions and structural templates as the objects about which students are lacking conceptual understanding.
The role of overgeneralization
Thus we far, have identified expression transformation as an area of student difficulty in algebra. We have seen that understanding structure is critical for competent expression transformation. Using Sfards theory of reification, we have identified parsing as an activity whose performance naturally leads to an understanding of structure in algebra. We have diagnosed student difficulties with expression transformation as at least partly due to the traditional curriculums
54 superficial treatment of structure. Finally, we have collected some related diagnoses that support our diagnosis, although ours goes into more detail in its articulation of what it means to understand structure in algebra. If this were the whole story, then I would now proceed to recommend a less superficial treatment of parsing and of structure. However, two related facts remain mostly unexplained. One fact is the striking uniformity of student errors: we have seen that teachers report predictable wrong responses, and lack of understanding of how to proceed correctly does not explain the uniformity in how many students proceed incorrectly in the same way. A second fact is that while some prior writings support my diagnosis, many other teachers and researchers identify a different culprit as responsible for student difficulties with expression transformation, namely a strong tendency for students to overgeneralize. Schwartzman (1986) provides examples of overgeneralization. In an article entitled The A of a B is the B of an A, he writes about how students have difficulty restraining themselves from overgeneralizing the notion of the distributivity of one operation over another. He provides a list of twenty non-equivalencies that students are prone to assume. These include (a + b) n a n + b n ,
n
1 11 + , and a+b a b
a + b n a + n b . All of these examples are variations on true distributive
statements, such as the power of a product is the product of the powers (p. 181). However, they are overgeneralizations in that the student who accepts them is assuming fewer constraints on which operations distribute over which others than actually exist.
55 In this section, we will examine what Matz (1980) and Kirshner & Awtry (2004) have to say about the role of overgeneralization in producing common transformation errors. In the following sections, I will argue that overgeneralization is not merely a strategy that students use in the absence of structural understanding but is something that algebra curricula need to address even if they give structure its due. In other words, I will argue that our initial diagnosis for why students make common algebra errors is only a partial diagnosis and that a full diagnosis needs to anticipate students strong tendencies to overgeneralize.
Matz: Overgeneralizing rule-revision strategies
In her 1980 paper on common algebra errors, Matz sets out to provide an account of the processes that lead to the many universally predictable errors in algebra. Among the initial observable facts that demand explanation, she includes the striking regularity of the answers produced when students make mistakes while learning symbolic algebra. She strives for a theory of error with broad explanatory power: she laments that previous studies of high school algebra errors have been essentially extensive lists (p. 155), such as the one complied by Marquis, and aims to provide an explanation for algebraic behavior that accounts for these diverse errors in a unified framework. According to Matz, students encountering an algebra task often resort to a strategy that she calls revising a rule. Revising a rule involves taking one of the known rules of algebra and modifying it to fit a given problem to which it does not directly apply. Matz subdivides examples of revising a rule into two categories:
56 revising a rule by generalization and revising a rule by linear application. Matz claims that students are prone to resort to these strategies because these strategies have been successful in the past. In fact, sometimes when students use these strategies, they obtain correct results. However, while sometimes applicable, these inductive strategies are not part of the deductive process of transforming expressions using known rules of algebra. When students incorrectly revise a rule, they are overgeneralizing their applicability. Revising a rule by generalization involves taking a known rule, deciding that one of its components (i.e. a number or an operation) is incidental rather than essential to the rule, and then replacing that component with another. Matz describes this technique as follows: Generalization bridges the gap between known rules and unfamiliar problems by in effect revising a rule to accommodate the particular operators and numbers that appear in a new situation (p. 105). Matz provides examples of rule generalization as a successful strategy, such as the fact that minus can be substituted for the plus operator in the distributive law (p. 105). She also provides examples of incorrect rule generalization. These include many of the incorrect distributing errors discussed by Schwartzman (1986) and referred to above, but they also include instances of incorrectly generalizing specific numbers to arbitrary numbers, such as revising ( x a )( x b) = 0 implies x a = 0 or
x b = 0 to the incorrect ( x a )( x b) = k implies x a = k or x b = k .
Revising a rule by linear application involves assuming that if an expression can be transformed in a particular way, then its parts can be transformed in an identical way, regardless of what the parts are. Linearity, Matz explains,
57 describes a way of working with a decomposable object by treating each of its parts independently. An operator is employed linearly when the final result of applying it to an object is gotten by applying the operator to each subpart and then imply combining the partial results. (p. 111) Again, Matz emphasizes that linearly applying an operator or a procedure is often correct: Most of a students previous experience is compatible with a linearity hypothesis. In arithmetic, the immense number of occasions that
students add and use the distributive law very likely reinforces their acceptance of linearity. problems. (p. 111) However, Matz goes on to show how a linearity strategy can lead to incorrect results in algebra. For instance, all of the distributive errors discussed earlier reflect not only incorrect generalization but also an erroneous linearity assumption. Moreover, Matz regards fraction cancellation errors as also falling under the rubric of errors flowing from false linearity assumptions: Cancellation errors also fit neatly into this theoretical framework. Errors of the form
AX + BY A+ B X +Y
This trend continues with early algebra
can be reproduced using the extrapolation-by-iteration strategy. Here the (iterated) base rule is probably
58
AX A. X
According to this derivation, students notice two instances of the base rule in the new problem. This leads them to linearly decompose the expression, iteratively cancel, and then simply compose the partial results. (p. 118) Thus, according to Matz, students frequently revise rules by assuming that they can be applied linearly in more sorts of situations than those in which they actually can be applied. Over and over again, Matz emphasizes that students are drawn toward these two rule-revision strategies generalization and linear application because they have prior experience of these techniques yielding correct results. In arguing that these two strategies account for much errant algebraic behavior, she points not only to the fact that generalization and linear application describe the errors themselves but also to the fact that student successes with these strategies make their incorrect extension plausible: They are descriptively adequate in that we can use them to reproduce common errors. But in addition to their purely descriptive value these techniques are the obvious ones since they are methods that
worked well for the student in prior mathematical experience. Both
linearity and generalization have this characteristic: they are useful, often encountered techniques that apply correctly in many situations. (p. 129)
59 Thus, Matz diagnoses common transformation errors as resulting from strong student tendencies to overgeneralize the context of application of two strategies generalization and linear application that worked successfully for them in the past.
Kirshner & Awtry: Overgeneralizing memorable visual sequences
Like Matz, Kirshner & Awtry (2004) explain many diverse common transformation errors with a single theory. Also like Matz, they identify overgeneralization as the heart of the problem. However, whereas Matz bases her account on overgeneralization of rule-rewriting strategies, Kirshner and Awtry base theirs on overgeneralization of particularly memorable visual sequences. Kirshner and Awtrys study of transformation errors begins with an observation about the appearances of algebraic rules in standard printed notation. The authors notice that some rules possess a certain visual coherence that makes the left- and right-hand sides of the equations appear naturally related to one another (p. 229). Kirshner and Awtry borrow the term visual salience from the psychology of perception to describe this quality of some algebraic rules. They provide the following examples of rules which they deem visually salient and rules they deem lacking in this quality:
60 Visually salient rules: Non-visually salient rules:
Kirshner and Awtry admit that visual salience cannot be defined rigorously: The quality of visual salience is easy to recognize but difficult to define (p. 229). They liken a visually salient rule to an animation sequence in which distinct visual frames are perceived as ongoing instances of a single scene, allowing us to see the immediate connection between right- and left-hand sides as stemming from a sense that a single entity is being perceived as transformed over time (p. 229). In other words, a visually salient rule is one for which the eye easily perceives a temporal narrative relating the left side of the rule to the right side of the rule: the x was
61 distributed, the fractions were smushed together, and so on. The visually salient rules have a narrative sense to them apart from their truth as generalizations of arithmetic. In contrast, the non-visually salient rules appear to connect two expressions with little obvious visual relationship; their only sense comes from the semantics of arithmetic, which is not visually obvious. After classifying the rules of algebra as either visually salient or non-visually salient, Kirshner and Awtry observe that virtually all of the common transformational mal-rules themselves possess the quality of visual salience. Moreover, they observe that the mal-rules are very similar in appearance to correct visually salient rules. For instance, the mal-rule
a+x a = possesses visual salience (the xs were cancelled), b+ x b ax a = . They bx b
and it is very similar in appearance to the correct visually salient rule
provide the following table to show the visual similarity between visually salient malrules and correct visually salient rules:
62 Thus, unlike Matz, who observes that many transformation errors can be described as overuse of generalization or linearity, Kirshner and Awtry observe that many transformation errors can be described as visual mimicking of correct rules. Kirshner and Awtry conduct an empirical research study to test the hypothesis that students tend to overgeneralize visually salient rules. In the study, students with no previous algebra schooling were taught ten algebra rules, including five visually salient rules and five non-visually salient rules. They were taught the rules purely structurally, without any reference to contextual situations. After instruction, students were given recognition tasks that tested their ability to identify routine applications of the rules and rejection tasks that tested their ability to constrain overgeneralizing the context of application of the given rules (p. 242). (All of these tasks involved only simple one-to-one matches and did not require the use of subexpressions.) The results of the study confirm Kirshner and Awtrys hypothesis that visually salient rules are relatively easy for students to remember but also relatively easy for students to overgeneralize: Percentage correct scores for recognition tasks were significantly higher for visually salient rules than for non-visually-salient rules. Such scores for rejection tasks were significantly lower for the visually salient rules. (p. 242) As a control, other groups of students were taught the same rules using tree diagrams that uniformly lack any of the visual narrative sense that is sometimes present in standard notation. Significantly, the absence of standard algebraic notation affected the results: unlike their peers who learned the rules using standard notation, students
63 who learned the rules using tree notation did not find the visually salient rules easier to recognize or easier to overgeneralize.
The vicious circle of reification
Competent expression transformers certainly know not only the rules of algebra but also know which of their components are essential and which can be generalized away. What educational implications, then, can be derived from Matzs account and from Kirshner & Awtrys account of transformation errors? Should we regard these accounts of students overgeneralizing merely as explanations of how students produce filler in the absence of the necessary structural understanding? If so, then educators could ignore these tendencies to overgeneralizing and assume that they will go away once they have remedied the lack of attention to structure in the traditional algebra curriculum. I now present two reasons why it would be a mistake to dismiss these overgeneralizing behaviors as lacking in educational significance. The first reason stems from a situation that Sfard calls the vicious circle of reification. The second reason flows from Kirshners argument for a connectionist view of mind.
The pragmatic value of reification
Recall, for a moment, Sfards diagram showing the progress of mathematics toward ever-more abstract objects. Recall, too, that mathematical objects, in Sfards scheme, serve as pivot points between a more basic process and a more advanced process. We have already discussed Sfards contention that a student must first
64 understand the simpler process before reification of the process into an object can occur. One might naturally assume, given the layout of Sfards diagram, that learning can and should proceed in a neat, stepwise fashion: process, reified object, new process, new reified object, and so on. If that neat alternation prevailed, then the educational implications for algebra would be that students should first learn how to parse, next attain an object-perception of structural templates, and only then begin to learn to transform expressions by comparing and matching structural templates with rule expressions. However, Sfard posits the existence of something she calls the vicious circle of reification, a situation that makes reification inherently difficult and renders neat sequential learning nearly impossible. According to Sfard, there is an inherent difficulty in advancing up the hierarchy of mathematical understanding. She describes this difficulty as stemming from the circularity that obtains between understanding a mathematical object and understanding the higher processes performed upon that object. On the one hand, reification and its object-perception is a prerequisite for fully understanding the higher process: one cannot truly understand a process if one does not first understand the objects upon which one is performing that process. On the other hand, engagement with a higher process is precisely what motivates reification and its attendant object-perception: the higher process provides the pragmatic value for the object-perception. In other words, Sfard regards understanding a mathematical object and understanding the processes performed
65 upon that object as prerequisites of one another, hence the vicious circularity. Crucially, then, the moment of reification is typically difficult for students to attain: On the one hand, a person must be quite skillful at performing algorithms in order to attain a good idea of the objects involved in these algorithms; on the other hand, to gain full technical mastery, one must already have these objects, since without them the processes would seem meaningless and thus difficult to perform and remember. (p. 32) One implication for student learning is that understanding of mathematical objects must be encouraged simultaneously from two directions: the object-perception can only develop from sufficient experience performing both the more basic process (of which the object is the result) and the more advanced process (which is performed upon the object). For instance, understanding of rational numbers as objects must be encouraged by simultaneously engaging students in the more basic process of dividing two natural numbers (from which rational numbers result) and the more advanced process of comparing ratios (which takes rational numbers as its objects). For convenience, I will based on Sfards diagram speak of the need to induce reification vertically and horizontally.
The vicious circle and algebra: A window for overgeneralization
How does the vicious circle of reification play out in the learning of algebras? Recall that we have been regarding structural templates as mathematical objects. Recall further that we have been regarding these objects as the results of the simpler
66 process of parsing and the objects of the more advanced process of comparing and matching structural templates for expression transformation. The vicious circle of reification implies that conceptual understanding of structural template, on the one hand, and procedural ability to find matches in structure and transform expressions, on the other hand, are prerequisites for another. It is impossible, according to Sfards theory, to proceed sequentially from the process of parsing to the concept of structural template to the process of comparing structure for transforming expressions. As a result, students must necessarily engage not only in the more basic process of parsing but also in the more advanced process of comparing structures before fully attaining an object-perception of structural
template.
This situation creates an opening a window for student tendencies to overgeneralize to interfere with student learning of expression transformation. During the messy time before reification of structural concepts has occurred, students will be engaged in processes like expression transformation requiring a full understanding of those very structural concepts! Lacking this full understanding but still of necessity engaged in these more advanced processes, students will tend to overgeneralize, making common errors. The vicious circle of reification is one factor that limits the possible improvement in student algebra performance: Understanding the objects of algebra is inherently difficult, and strong tendencies to overgeneralize whether in response to past successes, or in response to memorable visual sequences, or both are likely to interfere even given a curriculum that attends adequately to structure.
67
Competing impulses: The connectionist view of mind
There is a second reason that instruction cannot ignore student tendency to overgeneralize. In a series of papers, Kirshner (1989, 1993, 2001, 2004, 2006) builds a case for looking at the mind of the algebra learner from the perspective of a school of thought in cognitive psychology called connectionism. Connectionism regards the mind as inherently ill-suited to formal reasoning tasks, like those involved in expression transformation, and stubbornly inclined to incorporate formally irrelevant information, such as visual patterns, into its decision-making process. In another series of papers, cognitive scientists Landy & Goldstone (2007a, 2007b, 2007c) support Kirshners connectionist perspective on algebra learning. Their research indicates that as novices work toward an understanding of algebraic structure, that understanding will necessarily be in competition with non-rational impulses, such as overgeneralizing tendencies.
Kirshner on the role of spacing in parsing decisions
When Kirshner makes his fullest case for connectionism in a 2006 paper, he cites both his and Awtrys 2004 study discussed above and a 1989 study about parsing. Kirshners 1989 study is motivated by an observation about algebraic notation: in standard printed algebra, there is a correlation between the degree of precedence of an operation and the type of spacing used to indicate that operation. While students are encouraged to memorize PEMDAS as a rule, they likely get a
68 silent assist from the spacing around operations in standard algebra notation. Specifically, Kirshner observes that the least precedent operations (addition and subtraction) are indicated by wide spacing; the next least precedent operations (multiplication and division) are indicated by a closer horizontal or vertical juxtaposition; and the operations with the highest precedence (exponentiation and root-taking) are indicated by diagonal juxtaposition (p. 276). He provides the following table to illustrate the distinctive spacing conventions of each operation level:
Thus, as mentioned parenthetically earlier in this paper, Kirshner challenges the notion that the exponents before multiplication before addition convention is completely non-notational; rather, spacing provides clues about an operations level in the precedence hierarchy. Kirshners observation that operation levels correspond with distinctive visual characteristics (p. 276) causes him to question commonsense assumptions about student parsing abilities. As we have seen, parsing is a prerequisite skill for evaluating algebraic expressions. Common sense would seem to indicate that a student who repeatedly demonstrates success at evaluating algebraic expressions must therefore know the rules of operation precedence. Kirshner, however, hypothesizes that while some students who can evaluate algebraic expressions correctly may
69 actually know the rules of precedence, others may depend upon the visual spacing cues of standard notation to make correct parsing decisions. This latter group may use spacing cues in the same manner that parentheses are meant to be used: as visually present indicators of how to parse. To test this hypothesis, Kirshner conducts an experiment involving a nonstandard nonce notation. In this nonce notation, capital letters indicate operations in place of the usual symbols +, , and so on. For example, 3A5 means
3 + 5 . The experiment involves both a spaced nonce and an unspaced nonce
notation. The spaced nonce notation is designed to mimic standard notation by correlating proximity of symbols with precedence of operation. Kirshners table illustrates these alternative notations:
Kirshner hypothesized that if students spontaneously use spacing cues to make parsing decisions, then many students would correctly parse expressions presented in the spaced nonce notation, which mimics the visual assist of standard algebra notation, yet be unable to parse expressions presented in the unspaced nonce notation, even though both nonce notations technically contain all the information algebraically needed to parse: It was reasoned that the ability to correctly parse algebraic expressions presented in the unspaced nonce notation would indicate the presence
70 of propositionally based syntactic knowledge. Conversely, inability to transfer competent behaviors from standard notation to the nonce setting would indicate a dependence on the surface cues of ordinary notation. (p. 277) Indeed, Kirshners results did show that a significant number of students had more difficulty with the unspaced nonce notation than with the spaced nonce notation: Almost all the subjects participating in the study were able to evaluate expressions such as 1 + 3 x 2 , for x = 2 , when presented in standard notation. It proved, however, to be significantly more difficult to transfer this ability to the unspaced nonce notation than to the spaced nonce notation. These two notations differ only in the spacing of the symbols, the latter notation having been devised, specifically, to mimic spacing features of ordinary notation. Thus it seems necessary to conclude that for some students the surface features of ordinary notation provide a necessary cue to successful syntactic division. (p. 282) Kirshner therefore infers that the way operations are spaced on the printed page in standard algebraic notation functions as a notational parsing cue for some students. Moreover, he infers that for some students, spacing is a necessary cue: their ability to order operations according to convention depends not upon declarative knowledge of the conventional rules but rather upon having this visually present spacing cue, and they are unable to parse correctly without it. Because the visual crutch is embedded in the way students usually encounter algebra problems, there is no way to
71 discriminate between the student who really understands the structure and the stduent who is using this crutch.
Kirshner on connectionism
Although Kirshners two research experiments (1989, 2004) pertain to different skills, he draws similar conclusions from the two studies. In both studies, Kirshner concludes that successful performance of a skill (evaluating expressions in the one, transforming expressions in the other) does not necessarily indicate mastery of the formal rules that constitute true competence. In both cases, novices rely upon visual features of standard written algebra (spacing in the one, memorable animationlike visual sequence in the other) to make successful decisions. In both cases, introducing new notations that lack these helpful visual features (the unspaced nonce in the one, the tree notation in the other) is shown to reduce student ability to perform the skill, even though all of the technical information needed to perform the skill is still present in the alternate notation. In reflecting upon these findings, Kirshner adopts a connectionist view of cognition that rejects the analogy of the mind to computer. Connectionism does not view the mind as a neat and orderly machine with a centralized rule-processing apparatus: Connectionist psychology posits dramatic redundancy and a superabundance of active elements, in contrast to the neat, linear processes of rulebased systems (2001, p. 90). Connectionism, Kirshner explains, considers cognition to be spread out rather than centralized: In analogy to the neurology of the brain, connectionism asserts that cognition is parallel and distributed, rather than serial and
72 digital (p. 90). Kirshner uses an analogy to inputs and outputs to explain the connectionist view of how the mind does work: Typically connectionist systems include input nodes corresponding to features of the domain to be mastered and output nodes related to actions that can be taken or decisions that can be reached, as well as hidden units that intermediate between input and output nodes (2006, p. 7). According to the connectionist view, these different inputs are competing at all times, and the relative weight of their input not a formal rule process determines which input or inputs win out and result in an output: When a certain threshold of activation is reached, the node sends signals to those other nodes to which it is connected.
Connectionism models cognitive skills as weighted correlations among a large number of input, output, and intermediate nodes. centralized rule based program runs the show. (p. 7) Connectionism therefore sees the mind as ill-suited for sequential ruleprocessing tasks and well-suited for tasks involving making judgments based on many related and competing sets of input criteria. Kirshner (2001) explains the connectionist view of what the human mind does best: The primary cognitive functions are pattern matching and associative memory, not logic or rule following. Connectionism notices that the long chains of extended reasoning that serial digital computers do best, are hardest for humans. Things that humans do best, like recognizing faces in different situations and from different angles, are the most No
73 difficult feats to simulate on serial computers, but the easiest to implement in connectionist architectures. (p. 90). A connectionist view does not regard a persons understanding of formal rules as non-factors in the persons cognition, but merely as one of many parallel factors competing to produce action: It is too extreme to argue that rules play no role in competent performance, but it is an ancillary role informing cognition rather than constituting it (p. 95). In particular, Kirshner sees connectionism as dovetailing nicely with his empirical observations about how students learn algebra. Kirshners two research studies both suggest that some student algebra behavior can be explained as responses to visual features of standard printed algebra, despite the fact that the visual appearance of algebraic expressions is a mere accident of our notation and not inherent to the structural content of algebra. For Kirshner (2001), connectionist theories incorporate such observations naturally: The connectionist framework seems, in general terms, to afford the possibility of an alternative account of algebraic symbol skills that is more faithful to our observation as educators that students work in algebra is non-reflective and pattern-based (p. 90). Moreover, connectionism explains the stubbornness with which students cling to visual approaches to algebra skill acquisition, for it asserts that learning always is grounded in perception and pattern matching as embedded in practices, not in abstraction and rule following (p. 95).
74
Landy & Goldstones research on formally irrelevant distractions
Kirshner, as we have seen, posits a connectionist understanding of cognition. Connectionism helps to explain Kirshners findings that students seem to spontaneously utilize visual regularities and memorable visual features of algebraic notation when learning algebra, despite the fact that such visual cues are not part of the formal, rule-based apparatus for making decisions in algebra. In a series of recent papers, Landy & Goldstone (2007a, 2007b, 2007c) describe a set of experiments designed explicitly to test the role of formally irrelevant visual cues in the performance of algebraic tasks. In particular, their paper How Abstract Is Symbolic Thought? (2007a) has substantial implications for our consideration of how students learn to transform expressions. In this paper, Landy and Goldstone describe four research experiments, each designed to measure the effect of a formally irrelevant visual distracter on a persons ability to determine whether an expression has been transformed correctly or incorrectly. The first of these four experiments gives a sense of the gist of their work. In this experiment, spacing was the manipulated visual feature. Subjects were asked to judge the correctness of equivalences like the following:
75 Note that in some of these equations, such as the very first one, spacing has been manipulated so that the very wide spacing is correlated with less precedence (as in standard algebra notation), while in other equations, such as the very last one, spacing has been manipulated so that the very wide spacing is correlated with more precedence. Subjects were timed on their responses, and subjects were informed immediately of any incorrect responses. Landy and Goldstone found, like Kirshner, that subjects tended to use wide spacing as an indication of lower operation precedence, even when the wide spacing was around multiplication. In other words, they found that spacing, while irrelevant from a formal perspective, nonetheless influenced subjects syntactic judgments: The physical spacing of formal equations has a large impact on successful evaluations of validity (p. 724). While the first study involved manipulating spacing a formally irrelevant factor that authentically plays a role in standard algebra notation the remaining three studies involved manipulating more contrived visual factors. While also formally irrelevant, these other visual factors do not typically arise as distracters in actual algebra usage. The manipulated visual feature in the second experiment was an oval-shaped region in the background of the equations:
76
The manipulated visual feature in the third experiment was the internal structure of the rearranged terms:
The manipulated visual feature in the fourth experiment was alphabetical proximity of the variables:
77
Thus, Landy and Goldstone go quite a bit further than Kirshner. They consider the effects of a variety of formally irrelevant factors on peoples parsing decisions. Their overall findings support Kirshners connectionist perspective. Repeatedly, they conclude that formally irrelevant features can distract people who otherwise make correct parsing decisions into making incorrect ones. They conclude that a reasoners syntactic interpretation may be influenced by notational factors that do not appear in formal mathematical treatments (p. 721). Like Kirshner, they deem these findings significant because of how they challenge standard assumptions about how people make decisions in rule-based mathematical environments. It is standard, they explain, to assume that when students operate with good faith in a rule-based environment like structural algebra, they make all decisions based only on their understanding of the rules of the domain: Cognitive conceptions of abstract formal interpretation generally follow formal logics by assuming that reasoners explicitly represent rules of combination, and apply those rules to symbolic expressions (2007c). Those who assume students learn algebra solely by learning and applying rules will also, by implication, regard mistakes as evidence of misunderstandings of the rules. Landy and Goldstone see their results as disproving this assumption that people who perform algebra tasks in good faith make their decisions based only on their understanding of the rules:
78 Fundamentally these results challenge the conception that human reasoning with formal systems uses only the formal properties of symbolic notations, and that errors are driven by misunderstandings of those properties. Instead, people seem to use whatever regularities formal or visual, rule-based or statisticalare available to them, even on an entirely formal task such as arithmetic. The engagement of visual features and processes indicates that formal reasoning shares mechanisms with the diagrammatic and pictorial reasoning processes with which it is normally contrasted. (2007c) Put another way, Landy and Goldstone join Kirshner in concluding that student performance on algebra tasks is best modeled not by computer-like rule-following machines but rather by the sort of associative reasoning captured by connectionist frameworks of mind. More impressively, Landy and Goldstone demonstrate that formally irrelevant features can persist in influencing algebraic decision-making even when the person making the decisions actually knows the correct rules. Their interviews with study participants reveal that some participants realized that they were affected by formally irrelevant features and that participants knew that responding on the basis of space, alphabetic formality, and similarity of notation were incorrect, but they continued to be influenced by these factors (2007a, p. 730). Furthermore, participants persisted in using formally irrelevant features in their decision-making even while receiving feedback during the experiment itself:
79 One might have argued that participants were influenced by grouping only because they believed that they could strategically use superficial grouping features as cues to mathematical parsing. However, constant feedback did not eliminate the influence of these superficial cues. This suggests that sensitivity to grouping is automatic or at least resistant to strategic, feedback-dependent control processes. Grouping continued to exert and influence even when participants realized, after considerable feedback, that it was likely to provide misleading cues to parsing. (p. 730) Landy and Goldstone cite other psychological research on non-mathematical rulebased domains that also shows that people may use perceptual cues instead of rules even when they know that the rules should be applied (p. 731). Ultimately, their findings indicate that a persons knowledge of algebraic structure competes with other inputs during algebraic decision-making, even when those other inputs are irrelevant from a formal perspective. Transformation errors are not necessarily symptoms of lack of structural understanding but rather of the fact that structural understanding competes for attention alongside formally irrelevant visual features. Their research therefore implies that students strong tendency to overgeneralize is not just in play when students do not understand necessary structural concepts.
Other examples of competition in algebra performance
Other instances of competition between formally relevant and formally irrelevant features support these conclusions. We will examine two such instances.
80 Wong (1997) provides one example of structure in competition with other factors. She observes that students who can perform a transformation task involving only variables sometimes have difficulty performing a structurally identical task involving both numbers and variables. For instance, Wong observes that some students who successfully transform (hk) n into h n k n will consistently err when h is replaced with a number, transforming (2a m ) n into 2a mn , (2 x 3 ) 4 into 2x12 , and so on. Her general conclusion is that students who have learned to transform algebraic expressions according to some standard procedures will sometimes fail to do the transformation correctly when the familiar letters are replaced by numbers, despite the fact that the replacement leaves the structure of the expression unchanged (p. 286). Wong explicitly links her findings to a connectionist framework, noting the importance of the degree of strength between the connections of information items in learning situations, and concludes that students sometimes fail to activate the appropriate information items in their mind (p. 289). Linchevski & Livneh (2002) describe situations in which structure competes with specific number combinations for student attention. In a study, they found that certain biasing number combinations can override student structural knowledge and lure students into parsing errors. For instance, they find that students who repeatedly parse expressions of the form m n + p correctly are somewhat more prone to parse this expression incorrectly when presented with 267 30 + 30 . In this example, the repetition of 30 draws student attention to the addition first, despite what students know about how to parse expressions with this structure generally. The authors conclude that the particular number combination in the expression competes with the
81 algebraic structure for the students attention. While from a structural point of view, the particular numbers in an addition expression are irrelevant, in practice the particular numbers involved can lead to a greater or lesser frequency of particular parsing errors. Student knowledge and understanding of structure competes with other stimuli for student attention.
Instructional strategies
Earlier, we attributed student difficulty transforming expressions to their insufficient experience parsing and to insufficient exposure to the structural notions that underlie the expression transformation process. However, we then saw that several teachers and researchers attribute the uniformity and persistence of common transformation errors to student tendencies to overgeneralize. We asked whether overgeneralization is merely a strategy that students adopt when they lack structural understanding or whether it has deeper educational implications. We have now seen two reasons why overgeneralization merits educational consideration: (1) Because of the vicious circle of reification, students should start operating on subexpressions and structural templates before they attain a reified object-perception of those objects, opening a window for overgeneralizing tendencies to influence student decisionmaking; (2) Connectionism suggests that student understanding of structure is not allor-nothing but rather in competition with other impulses, especially with the impulse to incorporate formally irrelevant visual cues into algebraic decision-making. What would instructional strategies for improving student ability to learn expression transformation look like? From the conclusions about the centrality of
82 parsing reached earlier in this paper, we can derive the following instructional principle: Algebra curricula need to give explicit attention to parsing and to structure. However, from the conclusions about overgeneralizing and associative reasoning reached more recently in this paper, we can modify the instructional principle so as to incorporate our findings about student receptivity to the visual: Algebra curricula
need to give explicit attention to parsing and to structural notions in ways that will make structure a strong competitor for perceptual salience among the many impulses competing for student attention.
Instructional strategies for helping students achieve the process-perception of parsing
As we have discussed repeatedly, the process-perception of parsing necessarily precedes the object-perception of parsing: students need to parse before they can attain an object-perception of parsing as creating structural template. Since students ultimately need to attain the object-perception of parsing in order to transform expressions, curricula ought to make certain that students learn how to parse. However, as discussed earlier, traditional algebra texts treat parsing superficially, presenting the order of operations and then testing student ability to parse only implicitly through activities like numeric expression simplification. My first instructional proposals, therefore, are for activities and exercises that ask students to parse expressions explicitly. For instance, students could be required to draw expression trees:
83
Exercise Set A
Draw an expression tree for each expression.
1. 2 x + 5 3. 3 x 2 + 5 x + 2 2. 2( x + 5) 4. 4( x + 2)(2 x + 3) + 3( x + 1) 2
Such exercises reveal explicitly for the teacher the extent to which students understand the parsing conventions. While Exercise Set A requires students to resort to an alternative tree notation to parse an expression, an instructional strategy that I call surgery provides students with the opportunity to parse expressions physically and visually right on their standard worksheets. Here is how the surgery approach to parsing involves students in actively breaking expressions into their component parts. First, I tell students that they have two knives: the Addition Knife and the Multiplication Knife. The Addition Knife is the primary one and is used for underlining, and the Multiplication Knife is secondary and is used for slashing. The rule for using the Addition Knife is as follows: start underlining from the beginning of the expression, and start a new underline for each plus or minus sign that is outside of grouping symbols. For instance, if I told students to underline the expression 3 x 2 + 5 x + 2 using their Addition Knife, I would expect the following result:
3x + 5 x + 2
2
Similarly, if I told students to slash apart the first underline using their Multiplication Knife, I would expect the following result:
84
3x
2
In essence, the underlining provides visual support to the role of addition and subtraction as the least precedent of all operations, and the slashing provides visual support to the role of multiplication as more precedent than exponentiation. Performing surgery on an expression reinforces correct parsing decisions and helps students avoid defaulting to a left-to-right order of operations. For instance, recall Example A, which asked students to simplify 2 + 5(4 6) 2 . Here is how a student might proceed to write the work using underlining and slashing to parse the expression visually:
Example A
Simplify 2 + 5(4 6) 2 .
85 The underlines effectively serve as grouping symbols, making it very unlikely for students accidentally to proceed left to right and obtain the incorrect 3(4 6) 2 .
Instructional strategies for inducing reification of structure vertically
Thus far, I have proposed activities for helping students achieve procedural mastery of parsing. Procedural mastery of parsing necessarily precedes an objectperception of structural notions. In this subsection and in the next one, I propose instructional strategies for helping students achieve this object-perception. Here I propose activities that lead toward object-perception of structural template and subexpression by gradually helping students to compress the results of the parsing processes into objects. Earlier, we have called this inducing reification vertically. In the next subsection I will propose activities that urge students toward reification by illustrating the pragmatic value of an object-perception of structure. We have called this inducing reification horizontally. First, it is worth noting that the surgery approach to parsing discussed above already goes a long way toward helping students transition from parsing as process to parse as object. When students simplify numeric expressions, as in Example A, without underlining and slashing, parsing for them is just a decision-making process used to determine what to do first, second, and so on. When students underline and slash in the process of the same simplification task, they transform parsing into an activity that makes subexpressions visible. Students gradually are able to transition from thinking of underlining as something they do to thinking of the underlines or terms as things they can see. Underlining and slashing, therefore, not only help
86 students make correct parsing decisions; they also moBiosecurity WorkshopDevelopments among international scientific and health bodies on dual use issueMaria Jose Espona National Defense School, Argentina8 10 december 2006Dual use definition:Materials, technology and knowledge that have both
Biosecurity WorkshopDevelopments among international scientific and health bodies on dual use issueMaria Jose Espona National Defense School, Argentina8 10 december 2006Dual use definition:Materials, technology and knowledge that have both
|
Vectors: A New Kind of Animal
Imagine we are walking down the street after a tasty supper. We look to our left, and we sight a minotaur ambling down the street with a cup of joe in hand. Distracted, we trip over an overgrown root breaking through the sidewalk. Clumsy us. We...
Please purchase the full module to see the rest of this course
Purchase the Points, Vectors, and Functions Pass and get full access to this Calculus chapter. No limits found here.
|
Globalshiksha has come up with LearnNext Uttarakhand Board Class 10 CDs for Maths and Science. This CD contains the entire syllabus for Uttarakhand Board Class 10 Mathematics and Science for the current year. Included lessons are in audio and visual format. Solved examples, practice workout, experiments, tests and many more tests related to Maths and Science. It also includes various set of visual tools and activities on each Lesson with Examples, Experiments, Summary and workout, which is benefit for the students. Students can understand the concepts well; clear all doubts with ease through this Educational compact disk and get score in the exams.
This multimedia comes with a useful Exam Preparation like Lesson tests usually 20-30 minutes in duration, which will help you to evaluate the understanding of each lesson and Model tests usually 150-180 minutes in duration, that cover the whole subject on the lines of final exam pattern and package that can help you sharpen your preparation for exams, identify your strengths and weaknesses and know answer to all tests with a thorough explanation, overcome exam fright and get scores in exams.
|
Mathematical Challenges VI Problems and solutions from Years 2003 to 2006 In the years of Mathematical Challenges covered by this book, questions were set at four levels: Primary, Junior, Middle and Senior...
Can You Prove It? Mathematics which is proof-orientated develops pupils' ability to reason logically and deepens their understanding of mathematical concepts...
|
Mathematics Faculty: Curriculum Overview 2012-13
The grid below gives an overview of the curriculum for this academic year.
Year 7
Students cover work from all strands of Mathematics each term. They consolidate and improve both their mental and written methods for addition, subtraction, multiplication and division of whole numbers, decimals, integers and fractions. They begin to develop both algebraic and geometric reasoning skills and statistical inquiry; studying probability, sequences, functions, graphs, transformations, ratio, percentages, measurement and aspects of discrete Mathematics.
Year 8
Students further develop their skills gained in Year 7 with an increased focus on algebraic manipulation and application to real-life contexts.
Year 9
Each half term has a broad theme to enable students to perceive the links between different topics. In order, these themes are: Fractions, percentages and applications; decimals and approximation; formulae manipulation and measurement; sequences and graphs; geometry and equations and finally angles with trigonometry.
Year 10
All students study the Linear GCSE (Edexcel 1MA0) course; a few students are also entered in Entry Level Mathematics (OCR). Students review and develop their non-calculator numerical skills, measurement, algebraic manipulation, graphical representation and angle properties, statistics and probability. Elements of functional skills are developed, together with the quality of students' written communication.
Year 11
Students continue with the course they began in Year 10. There is greater emphasis on the topics at the higher end of the grades applicable for their tier of entry (either foundation or higher). Topics include trigonometry, circle theorems, vectors, congruence proofs and transformations of functions at Higher level and quadratics, Pythagoras' theorem, transformations, construction and loci, inequalities, measurement and scatter graphs at Foundation level.
Statistics (Year 10 and 11)
Students may study GCSE Statistics as an option at Year 10 and 11. Statistics GCSE extends the handling data and probability elements of GCSE Mathematics. It is a project based course. Each project begins with some kind of data collection task, and this data is then used to illustrate and practice the techniques learnt. Students collect data in a variety of ways; experiments, from the internet and by using a questionnaire. This data is then displayed using statistical graphs and summarised by statistical calculations. ICT programs such as Excel and Fathom are used to analyse data, and pencil and paper techniques are also learnt.
|
Description: In this presentation, the author will briefly introduce the subject of Descriptive Set Theory and the motivation for its study. The author will discuss the idea of a projective set and also define the mathematical notion of a "tree" as an example of a projective set. The author will conclude with a brief mention of a significant result that can be proved using the notion of a tree.
Description: This presentation discusses a research study on the effect of technology on achievement in mathematics in middle school. This presentation discusses the research, which analyzes how these concepts, ideas, and problems have been discussed in the past in order to form a solid platform that will support technology in schools in the future.
Description: This presentation discusses research on mathematics anxiety. The author describes the current state of research and understanding of math anxiety, expounds upon this information with independent research conducted at UNT, evaluates this research, and suggests a plan for improved results in mathematics education.
|
This course will make math come alive with its many intriguing examples of algebra in the world around you, from bicycle racing to amusement park rides. You'll develop your problem solving skills as you learn new math concepts. Need a little extra help? Interested in an application? Click on the chapter links below to get lesson help or explore application and career links.
|
This module introduces you to the mathematical notation and techniques relevant to studying engineering at undergraduate level. The emphasis is on developing the skills that will enable you to analyse and solve engineering problems. You cover algebraic manipulation and equations, the solution of triangles, an introduction to vectors and an introduction to probability and descriptive statistics.
|
Product Description
From the Back Cover
Master MATLAB®!
If you want a clear, easy-to-use introduction to MATLAB®, this book is for you! The Third Edition of Amos Gilat's popular MATLAB®, An Introduction with Applications requires no previous knowledge of MATLAB and computer programming as it helps you understand and apply this incredibly useful and powerful mathematical tool.
Thoroughly updated to match MATLAB®'s newest release, MATLAB® 7.3 (R2007b), the text takes you step by step through MATLAB®'s basic features—from simple arithmetic operations with scalars, to creating and using arrays, to three-dimensional plots and solving differential equations. You'll appreciate the many features that make it easy to grasp the material and become proficient in using MATLAB®, including:
Sample and homework problems that help you hands-on practice solving the kinds of problems you'll encounter in future science and engineering courses
New coverage of the Workspace Window and the save and load commands, Anonymous Functions, Function Functions, Function Handles, Subfunctions and Nested Functions
By showing you not just how MATLAB® works but how to use it with real-world applications in mathematics, science, and engineering, MATLAB®, An Introduction with Applications, Third Edition will turn you into a MATLAB® master faster than you imagined.
About the Author
Amos Gilat, Ph.D., is a Mechanical Engineering Professor at the Ohio State University. Dr. Gilat's research has been supported by the National Science Foundation, NASA, FAA, Department of Energy, Department of Defense, and various industries.
If you do not know anything about MATLAB, this is the book you should have at the first step. It teaches you every basic steps and how to apply them to real engineering or mathematical problems in an interactive environment. It has very good screen shots and real world problems to show how to use MATLAB. It reinforces the concepts with quality exercise questions. It is very easy to read and understand. It is absolutely a beginner book not for an advance user.
20 of 21 people found the following review helpful
5.0 out of 5 starsThe perfect introductory text for MATLABDec 7 2005
By shuttledude - Published on Amazon.com
Format:Paperback
If you are completely new to MATLAB then you will find no better book to guide you through the basics. It is perfectly suited for teaching yourself several basic but still very interesting and useful programming techniques. Topics are presented to the reader in an order carefully determined to produce maximum benefit and knowledge. The book is short and very readable, with many example programs.
In short: if you want a FIRST introductory textbook for MATLAB, you can't beat this book. And it covers the latest version (Release 14).
21 of 23 people found the following review helpful
2.0 out of 5 starsNot all bad, but it does fall shortJan 20 2009
By Neurofox - Published on Amazon.com
Format:Paperback
Not a bad book per se - if you are an absolute beginner with Matlab.
However, for the asking price, there are some rather glaring deficiencies.
Here are the major ones:
1. It doesn't go far enough. After working through the examples, it leaves one not entirely self-sufficient and confident.
2. There are quite a few copy-editing mistakes, particularly in the current edition.
3. Feels too generic. In a nutshell, one can't shake the feeling that this represents a somewhat annotated help file. As the in-built Matlab help is getting better and better all the time, the need for such a generic book becomes questionable.
4. Outdated. This book is 3 releases behind the current version of Matlab - and it shows. Some basic features that were introduced in the more recent releases of Matlab are - naturally - not discussed here.
Recommendation: Release a new edition that catches up with the evolution of Matlab, fixes the copyediting mistakes, goes farther and includes more distinctive examples.
|
Mathematics for 3D Game Programming and Computer Graphics
Cover
Details
Author:
Eric Lengyel
Year Published:
2001
ISBN:
1584500379
Description
This invaluable resource teaches the mathematics that a game programmer needs to develop a professional-quality 3D engine. The book starts at a fairly basic level such as vector geometry, modern algebra, and physics, and then progresses to more advanced topics. Particular attention is given to derivations of key results, ensuring that the reader is not forced to endure "gaps" in the theory. The book discusses applications in the context of the OpenGL architecture, and assumes a basic understanding of matrix algebra, trigonometry, and calculus.
|
...Students learn the rules how to work with fractional variables, graphing and solving two or more variable equations,inequalities, and inequalities. Students learn to construct, solve and check real world problems. They also learn how to manipulate an equation in terms of another variable Emphasis on logic proofs
|
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:
A Maths Dictionary for IGCSE
This may prove a useful reference. Note that teachers and students should refer to the syllabus as the definitive source for notation and content.
Author: R E Jason Abdelnoor
ISBN: 9780748781966
Published in 2007.
Edition: 2
Published by Nelson Thornes, UK [New window]
Cambridge IGCSE Mathematics Core and Extended Coursebook (with CD-ROM)
This highly illustrated coursebook has been written by an experienced author and an IGCSE Maths teacher to cover the complete Cambridge IGCSE Mathematics (0580) syllabus. Core and Extended material is combined in one book, offering a one-stop-shop for all students whatever their capabilities.
The extended material is clearly marked, and useful hints are included in the margins for Core students needing more support, leaving the narrative clear and to the point. Students doing the Extended course are given access to the parts of the Core syllabus that they need without having to use an additional book.
Answers to book exercises and exam questions are at the back of the book, with additional exercises and interactive questions on the CD-ROM.
Author: Morrison, K and Hamshaw, N
ISBN: 9781107606272
Published in 2012.
Published by Cambridge University Press, UK More information on Cambridge IGCSE Mathematics Core and Extended Coursebook (with CD-ROM) [New window]
Cambridge IGCSE Mathematics Core Practice Book
A practice tool that will successfully accompany any Cambridge IGCSE Mathematics (0580) textbook. This 'Core Practice Book' offers a wealth of questions with hints and tips to reinforce skills and learning. Comprehensive and targeted exercises ensure plenty of practice for the classroom, independent learning and revision.
Author: Morrison, K and Dunne, L
ISBN: 9781107609884
Published in 2012.
Published by Cambridge University Press, UK More information on Cambridge IGCSE Mathematics Core Practice Book [New window]
Cambridge IGCSE Mathematics Extended Practice Book
An endorsed practice tool that will successfully accompany any Cambridge IGCSE Mathematics (0580) coursebook. Comprehensive and targeted exercises ensure plenty of practice for the classroom, independent learning and revision.
Author: Morrison, K and Dunne, L
ISBN: 9781107672727
Published in 2013.
Published by Cambridge University Press, UK More information on Cambridge IGCSE Mathematics Extended Practice Book [New window]
Cambridge IGCSE Mathematics Second Edition updated with CD
This second edition, written especially to support the University of Cambridge International Examinations IGCSE Mathematics (0580) syllabus, is now in full colour and includes a student's CD-ROM. The text is ldeal for students following the Extended Curriculum, International contexts are used throughout to aid understanding and ensure this text is relevant to students everywhere.
Author: Pimental, R and Wall, T
ISBN: 978-1444123159
Published in 2011.
Edition: 2
Published by Hodder Education, UK [New window] More information on Cambridge IGCSE Mathematics Second Edition updated with CD [New window]
Cambridge IGCSE Maths Student Book
'Collins Cambridge IGCSE Maths' provides all the material you need for both the Core and Extended sections of the syllabus (0580) in the one handy book.
Author: Pearce, C
ISBN: 9780007410187
Published in 2011.
Edition: 1
Published by Collins Educational, UK [New window] More information on Cambridge IGCSE Maths Student Book [New window]
Core Mathematics for Cambridge IGCSE
Core Mathematics for Cambridge IGCSE provides both a two-year course leading to the Cambridge IGCSE Mathematics Core Level examination from University of Cambridge International Examinations and the firt year of a two-year course leading to the Extended Level examination. The book completely covers the syllabus for Cambridge IGCSE Mathematics Core Level. It has been designed to work through sequentially either as a classroom textbook or for self-study.
Author: Simpson, A
ISBN: 9780521727921
Published in 2010.
Edition: 1
Published by Cambridge University Press, India [New window] More information on Core Mathematics for Cambridge IGCSE [New window]
Core Mathematics for Cambridge IGCSE
Starting from basic principles and written with the international student in mind, 'Core Mathematics for Cambridge IGCSE' provides a complete course matching the core level content of the Cambridge IGCSE 0580 Mathematics syllabus.
Author: Haighton, J, Manning, A, McManus, G, Thornton, M and White, K
ISBN: 9781408516508
Published in 2012.
Published by Nelson Thornes, UK More information on Core Mathematics for Cambridge IGCSE [New window]
Core Mathematics for Cambridge IGCSE - Teacher's Resource Kit (with CD)
This new Teacher's Resource Kit offers expert support for your Cambridge IGCSE teaching. The Teacher's Guide includes lesson plans and worksheets, while the Teacher's CD offers a host of customisable worksheets and ready-made editable PowerPoints. Fully endorsed by University of Cambridge International Examinations.
Author: Bettison, I
ISBN: 9780199138739
Published in 2011.
Published by Oxford University Press, UK More information on Core Mathematics for Cambridge IGCSE - Teacher's Resource Kit (with CD) [New window]
|
A must have material for probabilty & statistics If you are studying advanced statistics, you might want to add this as an essential item to your reading list. It's a perfect balance between the mathematical and descriptive approach, but you need to be familiar with a bit of 'A' level maths though. But the book gives good value for money and knowledge.
Terrific Set of Physics Problems I give almost all Schaum's Outlines and Solved Problems Series a perfect 5 stars. They are indispensable in learning how to solve math, science, and engineering problems. This particular Schaum's has 3000 solved elementary physics problems (no subject outlines), with numerous beautiful illustrations. I skimmed the first 32 chapters and then read the remaining 7 chapters carefully; there are ...
very helpful If you know a little calculus, have taken it a while back or need to supplement your text, this book is very helpful. Plenty of problems to go through. Very helpful when trying to review for exams or GRE math subject test. If you have never taken any calculus, the explanations here are in outline form and will not help you understand much. This book covers Calc 1-2-3 in most undergrad ...
Thank you I am a little late in writing this so I'd like to say sorry. The product wasn't what I was looking for but it was exactly what I ordered so that's my fualt. It arrived in great condition, Thank you.
not for kids The thing I like about the Schaum's series is that they don't try to be your friend. If you're going to try to sit down to learn something intricate like geometry, you've got some serious work to do, and the sooner you get to it the better. To this end, there are no pictures in the book (other than geometric diagrams, of course), no blurbs on famous geometers or famous applications of geometry. ...
Worth the cost... This book is a great refresher. It is easy to read and works examples in a logical step-by-step fashion that non-accountants/non-accounting-majors may need. This book makes a great supplement to formal accounting training (introductory accounting coursework). It would also be very helpful for someone who needs an accounting refresher (like business education majors who plan to take the Praxis ...
Great Grade Booster I bought this because I have been struggling in my Physics w/ Calculus course at my university. Quite simply, after purchasing this guide and studying it for a good hour or two before the exam (I had neither gone to class recently nor done the homework for weeks) my grade on the test was the avg grade for the class: 74%. I am fairly convinced that without this guide I would have failed ...
very helpful I found this book to be very helpful.
Pros: Excellent as a supplement or even as a text. Lots of worked out problems, some are even proofs of theorms. The explanations are to the point and easy to understand. Good for math, physics or engineering undergrad majors. There is enough material for 1.5 undergrad semesters of Linear Algebra. Very appropriate for math GRE subject test review.
Cons: ...
Great resource! I used this book to help me learn Spanish on my own and now I use it to assist my students to learn Spanish grammar. It gives clear explanations of verbs and other grammar and it gives good practice. I highly recommend this book.
Confusing Textbooks ...
Comprehensive French Grammar book We purchased this book for our daughter who is attending college. This was on the list of required textbooks. She is thrilled with this book; she says it covers everything in French grammar, even some of the parts she missed along the way. Would recommend this to other college French students.
Just what I was looking for This is a good book, with a fair and well formed assortment of mathematical resources, sepecifically some of the more difficult and common integrals that nobody wants to have to do. At least that's what I use it for. Came in good condition, so that was nice too.
John W. Schaum Piano Course: Pre-A, the Green Book This old favorite course is the best in my opinion. I use it for all my beginning piano students because it is the one I was taught as a child, so I'm most familiar with it. Kids today are different than they were when I was young, so it takes quite a lot of explaining and patience with them, but I think this beginner book does very well in attempting to teach the young student how to begin to ...
One of the most easiest and best Chinese grammar!!! One of the most easiest and best Chinese grammar!!!
Highly recommended by me!
Easy to use, easy to pronounce, easy to handle!
What more can I ask for about it?!
|
From which topic should I start learning mathematics?
From which topic should I start learning mathematics?From which topic should I start learning mathematics?
You can't get past school level without it! Of course it's important! You need to know how to add and multiply before you go into college level!I'll second this.That would be the "learn the grammar and the vocab then learn the language" approach.
Exactly the opposite of what I was advocating: the "immersion" method.Thank you for your time and valuable suggestion.
the blog you linked to has the right idea. but...it's just a guide, everyone's path is different. for example, some people take a shine to analysis, and get all excited discussing measureability and diffeomorphisms, or that weird continuous function that isn't differentiable anywhere. some people like algebra, or number theory, and abhor the sight of an integral sign, or a rational function in z. others like the pure formalism of logic, languages and symbols, and the intricacies of the "subatomic structure of math".
you'll have to discover for yourself, what things you prefer.
calculus is a fairly basic place to start, it's a first-year undergraduate class in most universities. if you find calculus rough going, then back-track and start with "pre-calculus" or even "college algebra" (note: do NOT confuse that with abstract algebra, or what is often just called "algebra", which is a more advanced subject).
if you're already familiar with calculus, you have a lot of freedom in deciding where to go next. you can study...still more calculus (there's about 2-3 years of material, if you really want to know it well). the higher phases of calculus are often called by different names, because of the wide variety of topics available (multivariate calculus, calculus of variations, real analysis, integration and measure theory, just to name a few). related to this is complex analysis, which is similar to multivariate calculus, but has its own unique flavor, due to some singular properties that complex numbers have.
you'll probably want to tackle linear algebra, as well, as (along with calculus) it is one of the "core" subjects, every mathematician knows (at least a little of). again, the well is deep, you can skim it, and just learn the basics of matrices, and the key concepts of vectors and vector spaces, or you can go deeper into dual spaces, tensor analysis, spectral decompositions, all sorts of fun stuff. you might spend as little as a few months, or 2-3 years here (there's lots of interesting things that can be vectors, including some interesting kinds of functions).
another "basic" class is abstract algebra, which covers a lot of ground as well. after doing math for a while, you'll see certain kinds of structures come up again and again. this field (does that count as a pun?) covers structures in their abstract form (a sort of "let's handle all the cases at once" sort of thing). i recommend having some calculus and a little linear algebra before-hand, just so you have some depth of experience, but other people will tell you "dive right in".
and there's more: differential equations, differential geometry, topology, homotopy and homology, category theory, computability theory, statistics, combinatorics, graphs and trees,it's a long list, and it's growing all the time. you won't be able to learn it all, try a little of everything, and gorge yourself on what you enjoy.
while it's good to have a plan, keep in mind, you'll likely change your mind about some things along the way. keep your options open, there's no one "right way" to learn mathI do know my arithmetic and numbers.. :)
I was talking about gaps in algebra...
thanks for your advice..
You can't get past school level without it! Of course it's important! You need to know how to add and multiply before you go into college level
I'm not saying that it isn't important to be able to add and multiply, of course everyone needs to be able to do that. What I'm saying is that being good at it isn't as important (or important at all) at college level- I'm currently (trying) to do mathematical research and my mental arithmetic is really pretty bad.
If the OP was 7 years old, then I'd probably be suggesting that he/she learns to add and multiply, it's an important step, but I was under the impression that this was already a skill that they'd acquired given that they are asking what they should learn to cover college level maths.
And clear guidance as to which subjects should be studied first, and in which order, I think is quite important. You will gain much more studying things in a sensible order (at least, when beginning college level maths). There is a student my friend has supervised who is far ahead of the others, in first year, with a topological and metric spaces book. My friend told him that although it's good to look ahead, there are plenty of subjects that would be better suited for someone just beginning, not only to help them learn the topic faster, but often to put it into context ("enrich, not accelerate" is a nice phrase). Learning real analysis, for example, before metric spaces would seem very sensible.
Telling the OP just to "immerse himself", in my opinion, isn't all that helpful. It's pretty much non-advice- to someone who doesn't know much about mathematics, how are they to pick what they should study first? How are they to know whether or not it will be helpful to them, or is fundamental or specialist?
At a higher level, I'd agree with you. Perhaps a mix of immersion and direct guidance is best. However, when beginning college level maths, I really recommend picking the right subjects to study first. It's not a coincidence that most courses on mathematics all cover the same basic material in the first couple of years.Unfortunately, I just learned all the stuff without any books and just being taught it from lectures and homework assignments.
Hopefully others know some sources. If not, perhaps you could try and look up some lecture notes on first year mathematics modules and homework assignments, and have a go at doing them.
As I said, I have no idea on books, but a quick search gave me this:
I have no idea if it's a good book, but what I'm trying to say is something like this on the foundations of mathematics may be a good start (although definitely not the only way you could start).
And it's not the only thing you'll want to be doing when starting mathematics. As I said, the most important things to focus on are logic/proofs, real analysis and calculus, linear algebra and perhaps some group theory/abstract algebra. How to study these though, and possibly what books might be helpful to you, hopefully others can tell you.
|
To any teacher proficient in algebra 8th grade combining like terms: I seriously need your very noteworthy knowledge. I have several class worksheets for my online Remedial Algebra. I find algebra 8th grade combining like terms could be beyond my potential. I am at a out-and-out loss as far as where I should get started. I have weighed hiring an algebra professor or signing up with a learning center, however, they are unquestionably not low-priced. Any and every alternate suggestion shall be hugely prized!
You are right , there are programs that can help you with study. I think there are a few types that help you solve algebra problems, but I heard that Algebra Buster is the best amongst them. I used the program when I was a student in Algebra 1 for helping me with algebra 8th grade combining like terms, and it never failed me since then. Step by step I understood all the topics, and soon I was able to solve the hardest of the tests on my own. Don't worry; you won't have any problem using it. It was meant for students, so it's very easy to use. Actually you just have to type in the topic that's all .Of course you should use it to learn algebra, not just copy the answers, because you won't improve that way.
Don't worry pal. As what I said, it shows the solution for the problem so you won't really have to copy the answer only but it makes you understand how did the software came up with the answer. Just go to this site and prepare to learn and solve quicker.
I remember having problems with difference of cubes, algebra formulas and conversion of units. Algebra Buster is a truly great piece of algebra software. I have used it through several algebra classes - Intermediate algebra, Algebra 1 and College Algebra. I would simply type in the problem and by clicking on Solve, step by step solution would appear. The program is highly recommended.
|
I find it quite funny that certain models are banned due to the ability for them to carry out certain functions; surely education should be considered training for professionalism - and what engineer (for example) would refuse a calculator on the basis that he might be able to do it in his head?
We've invented the technology to do the sums for us - modern mathematics should teach us how to do it, why it works (including how to do it) and we should be tested on our ability to use the tools available to calculate the answer.
I use an FX-991ES P, and it's not banned. But I can solve algebraic, simultaneous and even logarithmic equations without any rearranging. It goes straight in from the exam question.
But the exam question might be worth say 4 marks. Maybe if I tap it in, get an answer, I'll get 1 mark.
Why does that make sense? In the real world, solutions matter - not each stage of getting to the solution (unless you're running back through because you made a mistake) - so why is education for he real world no based on practices that are practiced (hmm) in the real world?
I could go on and on - and the point of course extends well beyond the use of calculators, in all subjects education should follow the practice of industry: it may be expensive but it's damn worth it if we'd like to continue to be a nation with some of the most outstanding scientists, engineers, and academics.
Calculators is the worst sin though. Why not use what's there?
Perhaps I'll borrow a sine rule from my dad - or my mum's abacus - to use in the exams in January.
I have a TI-83 and i'm pretty sure (but not certain) that it can't perform symbolic integration/equation manipulation, though it does ofcourse integrate between limits, draws graphs and is very handy for FM as it will do sums involving complex numbers i.e. give me the answer to (1+2i)^2 if i type that in.
Is it banned or legal in AQA mathematics? Need a urgent reply please if anyone knows as i plan on using it tomorrow for two mathematics exams!
Today, I was in isolation with a couple of other people because I had S3 and C3 at the same time. The invigilator read a book for the 3 hours of exams. And my graphical calculator's never been checked, even though it has a really obvious USB port on it that can be used for installing programs.
(Original post by Contrad!ction.)Oh okay.
Same deal with my nspire, it has a 'Press-to-Test mode', I've always used it but I've never been questioned about it.
I had spreadsheets stored for Statistics, until I took S1 and read "no retrievable docs" on the front, so I used PTT then and have since.
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.