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Multivariable Calculus Multivariable Calculus is an expansion of Single-Variable Calculus in that it extends single variable calculus to higher dimensions. You may find that these courses share many of the same basic concepts, and that Multivariable Calculus will simply extend your knowledge of functions to functions of several variables. The transition from single variable relationships to many variable relationships is not as simple as it may seem; you will find that multi-variable functions, in some cases, will yield counter-intuitive results. The structure of this course very much resembles the structure of Single-Variable Calculus I and II. We will begin by taking a fresh look at limits and continuity. With functions of many variables, you can approach a limit from many different directions. We will then move on to derivatives and the process by which we generalize them to higher dimensions. Finally, we will look at multiple integrals, or integration over regions of space as opposed to intervals. The goal of Multivariable Calculus is to provide you with the tools you need to handle problems with several parameters and functions of several variables and to apply your knowledge of their behavior. But a more important goal is to gain a geometrical understanding of what the tools and computations mean. Requirements for Completion: In order to complete this course, you will need to work through each unit and all of its assigned materials. You will also need to complete the following: Unit 1 Assignment: Exercise Sets Unit 2 Assignment: Exercise Sets Unit 3 Assignment: Exercise Sets Unit 4 Assignment: Exercise Sets The Final Exam Note that you will only receive an official grade on your Final Exam. However, in order to adequately prepare for this exam, you will need to work through the exercise sets listed above 88.5 hours to complete. This is only an approximation and may take longer to complete about 22 hours to complete. Perhaps you can sit down with your calendar and decide to complete subunit 1.1 (a total of 7 hours) over three days, for example by completing sub-subunit 1.1.1 (a total of 2 hours) on Monday; sub-subunit 1.1.2 (a total of 1 hour) and about half of sub-subunit 1.1.3 (about 2 hours) on Tuesday; the rest of sub-subunit 1.1.3 (about 2 hours) on Wednesday; etc. Tips/Suggestions: As noted in the "Course Requirements," Single-Variable Calculus is a pre-requisite for this course. If you are struggling with the material as you progress through this course, consider taking a break to revisit MA101 Single Variable Calculus I and MA102 Single Variable Calculus II. It will likely be helpful to have a graphing calculator on hand for this course. If you do not own or have access to one, consider using this free graphing calculator. As you read, take careful notes on a separate sheet of paper. Mark down any important equations, formulas, and definitions that stand out to you. It will be useful to use these notes as a review prior to completing the Final Exam. Instructions: Please note that this is an optional resource. Click on "Multivariable Calculus Exploration Applet" link on the left side to upload the applet. This is a dynamic visualization tool for multivariable calculus and can be used throughout the course, especially to visualize three-dimensional objects. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. This unit begins with a discussion of vectors and vector algebra. You will study and then be able to define the geometric properties of the dot product and the cross product. You will learn vector algebra, which will allow you to perform operations on vectors themselves and not just their coordinates. This will help you understand the geometric significance of the operations and computations performed. You will then learn about parameterizations of lines and planes using vector algebra and geometry. This way of thinking, i.e. going back and forth between algebra and geometry, will be very important for the rest of the course. Finally, this unit introduces fundamental concepts of velocity and acceleration using vectors; this will allow us to study these concepts in their natural form. Instructions: Please click on the link above, and read the entire webpage titled "Part 1: Vectors in the Plane." Once you have finished, click on the links to Parts 2-4 at the top of each webpage to move on to the subsequent sections. Read all four parts in their entirety. Note: Microsoft Internet Explorer is recommended when viewing this site so that mathematical symbols are correctly displayed. This reading should take approximately 1 hour to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: On the webpage linked above, click on "Exercises" at the top of the page to link to the problem sets. Please work through exercises 1, 3, 7, 11, 15, 23, and 25. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and then finding section 1.1. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please scroll down the webpage to the "Lecture Notes and Books" heading, and click on the link titled "Math 214 Notes" to open the PDF. Read "Chapter 4: Vectors and Points in Rn" in its entirety (pages 79-87). You may want to save this PDF file to your desktop to easily access again in this course. This reading should take approximately 30 minutes to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click the above link to access Calculus, Applications and Theory. Complete exercises 1, 2, and 5 from "Section 4.4: Exercises with Answers" on pages 88-89. Try to solve each problem independently before checking the answer for each question, which is provided directly below the question. These exercises should take approximately 30 minutes to complete. Note: As printed, Problem 1 has some errors that may affect your answer to Problem 2. An updated version of the problem is provided here (PDF). Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click the link titled "The Inner Product Exercises" above, and then click on the "Exercises" link at the top of the webpage. Complete exercises 1, 9, 17, 23, and 29. Similarly, click on "The Cross Product Exercises" link above, and then select the "Exercises" link at the top of the webpage. Work through exercises 1, 5, 11, 17, 23, and 25. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and scrolling down to find the corresponding sections (1.2 and 1.3). These exercises should take approximately 2 hours to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Read the entire PDF document (3 pages). Please note that this reading is paired with the video lecture posted in the subsection above: please watch the lecture before accessing this reading. Instructions: On the webpage linked above, click on the "Exercises" link at the top of the page to access the problem sets. Complete exercises 1, 7, 11, 19, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and scrolling down to section 1.4. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please read Parts 1-4 in their entirety. Read the first webpage titled "Part 1: Vector-Valued Functions," and then click on the links to Parts 2-4 at the top of the page to access each subsequent section. This reading should take approximately 30 minutes to read and review. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: On the webpage linked above, click on the "Exercises" link at the top of the page to access the question sets. Work on exercises 1, 5, 11, 17, 25, and 29. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.5. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: After opening the webpage "Parametric Curves" linked above, click on links 1-10 under "Examples" at the bottom of the page to access the examples. Each example gives parametric equations and corresponding graphs. Also, try problems 1-6 listed under "Problems" to see if you can draw out the graph from given equations. After working out each problem, click on the graph to see the corresponding curve for the equations. Similarly, click on the above link titled "Parametric Surfaces" and work through examples and problems. Feel free to create and graph your own parametric equations to see what the curves look like. These activities should take approximately 1 hour and 30 minutes to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please access the webpage above and read all four sections in their entirety. Begin by reading the first webpage linked above titled "Part 1: Limits and Derivatives." Then, click on the links to Parts 2-4 at the top of the webpage to access the remaining sections of the reading. This reading should take approximately 30 minutes to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Open the webpage linked above and please select the "Exercises" link at the top of the page to access the questions. Complete exercises 1, 5, 9, 13, 25, and 29. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.6. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please read all four sections in their entirety. Begin by reading the first webpage titled "Part 1: Properties of the Derivative." To access the remaining Parts 2-4, make sure to click on the link for each part at the top of the webpage. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: On the webpage linked above, click on the "Exercises" link at the top of the webpage to access the questions. Work through exercises 3, 5, 9, 13, 21, and 27. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.7. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please open the link posted above and read Parts 1-4 in their entirety. Begin by reading the webpage titled "Part 1: Curvature and the Unit Normal." Then, click on the links for Parts 2-4 at the top of the webpage to continue the reading. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please open the webpage linked above and click on the "Exercises" link at the top of the page to access the questions. Complete exercises 3, 5, 9, 11, and 17. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 1, and then scrolling down to section 1.8. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. For single-variable functions, the derivative of the graph at a specified point is the slope of the tangent line at that point. This is because the dependent variable is only changing with regards to a single independent variable. In this course, we will consider functions of multiple variables. Because these functions have outputs that depend on multiple variables, each variable contributes to the rate of change of the function independently. Thus, we refer to the function's rate of change with respect to a single specified variable as a partial derivative, as it does not describe the entire rate of change of the function. Moreover, for functions of multiple variables, there will no longer be a single tangent line at a specified point. For example, in the case of a function of two variables, we consider a tangent plane as opposed to a tangent line. The concept of a partial derivative is important to study concepts such as tangent spaces, extrema, etc. in the context of functions of 2 or more variables. As in Single-Variable Calculus, Multivariable Calculus studies the maximum and minimum values for given functions due to their numerous applications. Lagrange Multipliers is a method of finding maximum and minimum values for functions subject to constraints that uses partial derivatives; however, the idea of optimization with constraints has no one-variable analogue. Again, you should focus on the geometric meaning of the concepts introduced in this unit. Instructions: Please click on each link above to sections "2.1 Functions of 2 Variables," "2.2 Limits and Continuity," and "2.3 Partial Derivatives." Read all four parts for each individual section. Begin by reading the first webpage for each section linked above, and then select the other Parts at the top of each webpage to continue reading (After finishing Part 4 of "2.3 Partial Derivatives", you will have read 12 Parts total). This reading should take approximately 1 hour and 30 minutes to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the "Functions of 2 Variables Exercises" link above, and at the top of this webpage, click on the "Exercises" link to access the question sets. Work through exercises 7, 9, 15, 19, and 25. Similarly, click on the "Limits and Continuity Exercises" link above, and once on the webpage, select the "Exercises" link at the top of the webpage to access the question sets. Complete exercises 5, 13, 17, and 25. Finally, click on the "Partial Derivatives Exercises" link above, and at the top of this webpage, click on the "Exercises" link to be redirected to the question sets. Try to solve exercises 5, 11, 19, 25, and 29. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 2, and then finding sections 2.1-2.3. These exercises should take approximately 3 hours5 Linearization and the Hessian" and "Section 2.6 The Chain Rule" above. For each section, read Parts 1-4 in their entirety. You may access each part by clicking on the links at the top of each webpage. These exercises should take approximately 3 hours to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click the link titled "2.5 Linearization and the Hessian" posted above, and then select the "Exercises" link at the top of the webpage to access the questions. Try to solve exercises 1, 9, 21, and 27. Then, click on the "2.6 The Chain Rule" link posted above, and select the "Exercises" link at the upper right corner of the webpage to access the questions. Complete exercises 5, 13, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 2, and then scrolling down to find sections 2.5 and 2.6. These exercises should take approximately 2 hours to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please read all four parts of "2.7 Properties of the Gradients." Begin by reading "Part 1: Gradients and Level Curves" linked above, and then select the links for titled "2.7 Properties of the Gradients Exercises." On this webpage, click on the link titled "Exercises" at the upper right corner of the webpage to access the questions. Try to complete exercises 3, 13, and 15. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.78 Optimization" and "Section 2.9 Lagrange Multipliers" found above. Read Parts 1-4 of each section. Make sure to click on the links to each part at the top of each webpage to complete this reading assignment. This reading should take approximately 1 hour to read and review. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the link above titled "2.8 Optimization Exercises", and then select the "Exercises" link at the top right corner of the webpage to access the question sets. Solve exercises 1, 17, and 25. Similarly, click on the link titled "2.9 Lagrange Multipliers Exercises" above, and then click on the link titled "Exercises" at the upper right corner of the webpage to access the questions. Work on exercises 5, 7, and 15. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 2, and scrolling down to sections 2.8 and 2.9. These exercises should take approximately 2 hours to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please read Parts 1-4 of "Section 2.4: Partial Differential Equations" from the link posted above. Begin by reading "Part 1: Partial Differential Equations" linked above, and then click on the links to Parts 2-4 at the top of the webpage. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the "2.4: Partial Differential Equations Exercises" link above, and once on the webpage, click on the "Exercises" link at the top right corner of the webpage to redirect to the exercise sets. Complete exercises 11, 21, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choosing chapter 2, and scrolling down to section 2.4. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Unit 3: Double Integrals and Line Integrals in the Plane In Single-Variable Calculus, you learned that the integral of a function is the area below the graph of the function and over a specified interval. The double integral of a function of two variables is the volume below the graph of the function and over a specified region. In Single-Variable Calculus, you approximated the area under a curve by taking slices of the area. You will now approximate the volume under a function by taking slices of the entire volume. In Single-Variable Calculus, you learned about results such as the area of a region, volume of a solid, and length of a curve using definite integrals. In this unit of Multi-Variable Calculus, we will develop the theory of multiple integrals to determine similar results. You will also learn about Green's Theorem; it defines the relationship between line integrals and double integrals, allowing us to reduce possibly complicated line integrals to a potentially simpler double integral. Please note that Green's Theorem is a two-dimensional case of the more general Stokes' Theorem, which we will discuss in the next unit. Finally, you will learn about flux; it is a scalar quantity important to the fields of mathematics and physics. It is derived from the surface integral over a specified region of a particular vector field. Using what we know about fields and integrals, we can look at the physical interpretations of flux. Instructions: Please open and read Parts 1-4 of both "Section 4.1 Iterated Integrals" and "Section 4.2 The Double Integral." To access each section, click on the links to each part at the top of the webpages. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the "4.1 Iterated Integrals Exercises" link above, and then select the "Exercises' link at the top right hand of the webpage to access the questions. Work through exercises 7, 13, 21, and 23. Once you have completed those exercises, click the link titled "4.2: Double Integrals Exercises" above, and select the "Exercises" link at the top of the webpage to redirect to the question sets. Try to complete exercises 7, 13, 21, and 29. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate sections 4.1 and 4 reading more sections by clicking the relevant links on the bottom of the pages. Instructions: Please read Parts 1-4 of "Applications of the Double Integral" from the link posted above. Begin by reading the webpage titled "Part 1: Mass Density." Then, click on the link to each additional part titled "4.3 Applications of the Double Integral Exercises" above, and once on the webpage, select the "Exercises" link on the upper right corner of the webpage to access the questions. Work through exercises 1, 11, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate section 4.3. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the link above titled "4.5 Integration in Polar Coordinates." Begin by reading the webpage for "Part 1: Change of Variable into Polar Coordinates." Then, click on the link to each additional part at the top of the webpage; read all four parts in their entirety. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click the link titled "Section 4.5 Double Integrals in Polar Coordinates Exercises" found above, and once on the webpage, click on the link titled "Exercises" at the upper right corner of the webpage to access the questions. Solve exercises 5, 11, 21, and 27. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate section 4.54 Change of Variable." Begin by reading "Part 1: Area of the Image of a Region" which the link above will take you to, after you complete Part 1,4.4: Change of Variable Exercises" above, and once on the webpage, select the "Exercises" link to redirect to the question sets. Work on exercises 5, 15, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate section 4 "Section 5.1: Vector Fields." Begin by reading "Part 1: Vector Fields" found above, and then continue on by selecting the links to Parts 2-4 at the top of the webpage. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the "5.1 Vector Fields Exercises" link above, and once on the webpage, click on the "Exercises" link at the top of the webpage to redirect to the exercise sets. Work on questions 1, 5, 17, 21, and 25. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 5, then locate section 5.1. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Read Parts 1-4 of "Section 5.2: Line Integrals." Begin by reading the first webpage which can be found in the link above ("Part 1: Line Integrals over Parameterized Curves"),2: Line Integrals" above, and then select the "Exercises" link at the upper right hand corner of the webpage to access the question sets. Try to solve exercises 5, 9, 21, and 23. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 5, then locate section 5 Section 5.3. Begin by reading "Part 1: Finding Potentials" which can be found through the link above,3: Potentials of Conservative Fields" above, and then, click on the link titled "Exercises" at the upper right corner of the webpage to redirect to the questions. Try to solve exercises 7, 13, 21, and 27. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 5, then locate section 5.34: Green's Theorem." Begin by reading "Part 1: Double Integrals and Boundary Curves" which is linked above, and then select the links to Parts 2-4 at the top of the webpage to continue reading. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click the link titled "Section 5.4: Green's Theorem" above, and then select the link titled "Exercises" at the upper right corner of the webpage to access the questions. Complete exercises 1, 5, 19, 23, and 27. You may check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 5, then locate section 5One common misconception of multiple integrals is that the primary difference is the units of the output of our integration. When we used single integrals, our output was area; when we worked with double integrals, our output was volume. It may be natural to assume that triple integrals give an output of whatever comes next, but this is not the case. The true variation between multiple integrals is not the output, or even the function, but the domain of integration. Single integrals are evaluated over an interval, whereas double integrals are evaluated over a planar region. Triple integrals are similar to other integrals in most regards, but they are evaluated over a 3-dimensional region. After learning about the basics of triple integrals, you will learn about The Divergence Theorem. As with Green's Theorem, the Divergence Theorem is a special case of the more generalized Stokes' Theorem that we will see later in this unit. The Divergence Theorem states that the flux of a vector field is equivalent to the volume integral of the divergence of the specified region. Essentially, this proves that the net flow leaving our region is equal to the total amount of sources (minus sinks) in our region. This brings us to Stokes' Theorem, which generalizes one of the most important theorems in Calculus, the Fundamental Theorem of Calculus. Stokes' Theorem defines a relationship between the integration of various differential forms over manifolds in multiple dimensions. Finally, you will learn a little about Maxwell's Equations. They are a set of four different partial differential equations: Gauss' Law, Gauss' Law for Magnetism, Faraday's Law, and Ampère's Law with Maxwell's Correction. Each equation defines a relationship between electric fields, magnetic fields, density, and various other important mathematical parameters. Instructions: Read Parts 1-4 of "Section 4.6: Triple Integrals." Begin by reading "Part I: Definition of the Triple Integral," and then, and once on the webpage, click the link titled "Exercises" at the upper right corner of the webpage to access the questions. Work through exercises 5, 13, and 21. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate section 4.67: Spherical Coordinates." Begin by reading "Part I: Triple Integrals in Cylindrical Coordinates" linked above. Once you are finished reading Part 1, select the links to each subsequent part at the top of the webpage until you have completed reading all four webpages. Please note that this reading is paired with the video lecture above so please read it after watching the video. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the link above titled "Section 4.7: Spherical Coordinates," and then click on the "Exercises" link at the upper right corner of the webpage to redirect to the question sets. Try to solve the following exercises: 5, 13, 21, and 27. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 4, then locate section 4.75: Surface Integrals." Begin by reading the first webpage linked above, and then select the link to each part "5.5 Surface Integrals" link above, and then select the "Exercises" link at the upper right corner of the webpage to redirect to the questions. Complete exercises 3, 5, 17, and 25. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF); on the main page, choose chapter 5, then locate section 5.5. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Also Available in: HTML and Java (Introduction to a Surface Integral of a Scalar-Valued Function) HTML and Java (Introduction to a Surface Integral of a Vector Field) read more sections by clicking the relevant links on the bottom of the page. Instructions: Open the link posted above and read the entire PDF document (2 pages). Please note that this reading is paired with the video lecture from the subsection above so please read it after watching the video. Instructions: Please read Parts 1-4 of "Section 5.6: The Divergence Theorem." Begin by reading "Part I: The Divergence Theorem" which is linked above, and then continue on by selecting the links to Parts 2-4 found above titled "5.6 The Divergence Theorem Exercises." Once on the webpage, select "Exercises" at the upper right corner of the webpage to access the questions. Try to complete exercises 1, 7, 11, and 19. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF). On the main page, choose chapter 5, then locate section 5.6. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Open and read the entire PDF document (5 pages) posted above. This reading is paired with the video lecture above so please read it after watching the video. Please note that this resource also covers the topic outlined in sub-subunit 4.3.2. Instructions: Please read Parts 1-4 of "Section 5.7: Stokes' Theorem." Begin by reading "Part I: Stokes' Theorem" which is linked above, and then select the links to the remaining parts at the top of the webpage. These exercises should take approximately 1 hour to complete. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: Please click on the link above titled "5.7 Stokes' Theorem Exercises," and then select the "Exercises" link at the upper right corner of the webpage to access the questions. Try to solve exercises 1, 7, 11, 17 and 19. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page; choose chapter 4, then locate section 4.6. This reading should take approximately 30 minutes to study, and work through the notes and the applets. Feel free to work on more examples, or read more sections by clicking the relevant links on the bottom of the page. Instructions: This reading is optional. Please read Parts 1-4 of "Section 5.8: Differential Forms." Begin by reading the first webpage linked above, and then select the links at the top of the webpage to Parts 2-4. This reading should take approximately 30 minutes to study. Terms of Use: Please respect the copyright and terms of use displayed on the webpage above. Instructions: This assessment is optional. Please click on the "5.8 Differential Forms Exercises" link above, and once the webpage has opened, select the "Exercises" link at the top of the webpage to access the questions. Complete exercises 5, 7, 15, 17, and 25. Check the solutions by clicking on the link titled "Answers to Selected Odd Exercises" (PDF) on the main page, choose chapter 5, then locate section 5.8. These exercises should take
The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math 9780691134932 ISBN: 0691134936 Pub Date: 2009 Publisher: Princeton University Press Summary: Steven Strogatz is the Jacob Gould Schurman Professor of Applied Mathematics at Cornell University. His books include the best-selling Sync: The Emerging Science of Spontaneous Order (Hyperion). Strogatz, Steven H. is the author of The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math, published 2009 under ISBN 9780691134932 and 0691134936. Seventeen The C...alculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math textbooks are available for sale on ValoreBooks.com, nine used from the cheapest price of $3.07, or buy new starting at $26.73ISBN-13:9780691134932 ISBN:0691134936 Pub Date:2009 Publisher:Princeton University Press Valore Books is the college student's top choice for cheap The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math rentals, or used and new copies that can get to you quickly.
Essential Matlab for Engineers and Scientists, Fourth Edition Book Description: The essential guide to MATLAB as a problem solving tool This text presents MATLAB both as a mathematical tool and a programming language, giving a concise and easy to master introduction to its potential and power. The fundamentals of MATLAB are illustrated throughout with many examples from a wide range of familiar scientific and engineering areas, as well as from everyday life. The new edition has been updated to include coverage of Symbolic Math and SIMULINK. It also adds new examples and applications, and uses the most recent release of Matlab. · New chapters on Symbolic Math and SIMULINK provide complete coverage of all the functions available in the student edition of Matlab.* New: more exercises and examples, including new examples of beam bending, flow over an airfoil, and other physics-based problems* New: A bibliography provides sources for the engineering problems and examples discussed in the text · A chapter on algorithm development and program design · Common errors and pitfalls highlighted · Extensive teacher support on solutions manual, extra problems, multiple choice questions, PowerPoint slides · Companion website for students providing M-files used within the book
Reviews for Speed and Acceleration Library Media Connection Reviews 2010 November/December This well-written series shows everyday life applications for math and uses colorful illustrations, pictures, and graphs to appeal to students. Each topic in the series is carefully explained for the target audience, with examples to illustrate points. Size of text and font is suitable for this age group with uncluttered pages designed so that text is set off with the illustrations, graphs, or drawings placed vertically. Key words appear in bold. Science specialists and classroom teachers will be glad to have these. The series is a nice replacement for those old, tired, unattractive books that may be gathering dust on library shelves. Bibliography. Glossary. Websites. Recommended. Leslie Greaves Radloff, Teacher/Librarian, Maxfield Technology Magnet, St. Paul, Minnesota ¬ 2010 Linworth Publishing, Inc.
Introduction to the 5th Edition of QU 101 Reader The Individual in. The fifth edition of the QU 101 Reader retains three important features that were first introduced in. landscapes and conceptual locales of our reading, writing, listening and speaking. in the classroom, workplace, and social environment. Wade, An Introduction to Analysis, Fourth Edition Errata and. Wade, An Introduction to Analysis, Fourth Edition. Errata and Addenda real analysis course. (Decimals are treated, although incompletely, in Problem 2.2.9, but the facts needed for the diagonalization argument are not mentioned.) p. Wade, An Introduction to Analysis, Fourth Edition Errata and Wolfweb. Wade, An Introduction to Analysis, Fourth Edition. Errata and Addenda ( Decimals are treated, although incompletely, in Problem 2.2.9, but the facts needed for Introduction to the 5th Edition of QU 101 Reader Quinnipiac. The fifth edition of the QU 101 Reader retains three important features that were first A book with blank pages--open locations--at the end of each reading enables. by many QU 101 teachers and students who will confirm that it is an effective,. An antithesis is… any idea or element of thought excluded from, outside of,
Pre-Statistics Courses A growing number of colleges have begun piloting alternatives to the traditional 3-4 semester developmental Math sequence. The new courses focus not on a review of Algebra concepts and skills, but instead on preparing students for college-level Statistics. Pre-Statistics courses emerged from a widespread recognition that the traditional developmental Algebra sequence is not well-aligned with the study of Statistics. If a student is pursuing a major that includes Statistics rather than Calculus (e.g. fields outside of Science, Technology, Engineering, Math, and Business), the majority of what is covered in elementary and intermediate Algebra courses never comes into play in their college-level Math course. The content is not, in fact, pre-requisite knowledge for the study of Statistics. This misalignment is especially problematic given how many students are lost in long remedial math pipelines. A nationwide study by the Community College Research Center found that among students who begin 3 or more levels below college Math, 90% disappear before ever completing the college-level course. Los Medanos College was the first in the nation to pilot this kind of course with Path2Stats, a one-semester pathway to college Statistics with no minimum placement score. Rather than proceeding step-by-step though all the topics in the traditional math sequence, Path2Stats students engage in statistical analysis from day one. Basic skills remediation occurs in a "just-in-time" fashion, with students reviewing relevant arithmetic and algebraic skills — e.g. calculating percentages, converting ounces to grams — as they are needed for the statistical tasks at hand. Path2Stats students complete college Math at dramatically higher rates that students with comparable starting placements in the traditional sequence. In 2011-12, 7 community colleges are working with the California Acceleration Project to offer redesigned pre-Statistics Courses. They include Diablo Valley College, Cuyamaca College, City College of San Francisco, College of the Canyons, Riverside City College, Moreno Valley College, and Berkeley City College (See video footage of CCSF students talking about their experience.) Some of these courses are open-access; others have arithmetic/pre-Algebra prerequisites. Nationwide, an additional 19 community colleges and 3 state universities are part of the Statway initiative, led by the Carnegie Foundation for the Advancement of Teaching: "a one-year pathway that culminates in college-level statistics…with requisite arithmetic and algebraic concepts taught and applied in the context of statistics." Classroom video: Los Medanos Developmental Math Students Discover an Error in a National Statistics Exam
Maple: What it is for and resources for learning to use it. Posted by: Nicholas O'Brien on November 18, 2010 Maple is a mathematical package that is quite intuitive and easy to use. On the plus side it is very good for calculus, graph theory, and 3D graphic rotations. It also allows easy in-line commenting, and as of Maple 13, an ability to work with and store tabular data. On the down side Maple requires one use strict syntax, and that you load external packages to perform many functions. Additionally, it undergoes big changes version to version so reverse compatibility is a problem. At Bates, Maple is a staple of the math department. Contact Pallavi Jayawant or Eric Towne for more information. You may also seek help with Maple at the Math and Statistics workshop.
Retaining previously successful features, this edition exploits students' access to computers by including many new examples and problems that incorporate computer technology. Historical footnotes trace the development of the discipline.
Problem solving, language and communication, connections within and outside mathematics, and formal and informal reasoning underlie all content areas in mathematics. Students use these processes together with technology and other mathematical tools such as manipulative materials to develop conceptual understanding and solve problems. Components in mathematics are number sense, operations, quantitative reasoning, patterns relationships, algebraic thinking, geometry, spatial reasoning, measurement, probability and statistics along with application of underlying processes and the use of mathematical tools.
Differentiation Teacher Resources Find Differentiation educational ideas and activities Title Resource Type Views Grade RatingTwelfth 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. In this Ares-V cargo rocket learning exercise, students read about this multi-purpose launch vehicle and its rocket boosters. Students are given an equation that relates the acceleration of the rocket to the time and mass of the rocket. They graph the thrust curve and mass curve, they graph the acceleration and they determine the rocket's absolute maximum in a given interval. Twelfth graders investigate logarithm differentiation. In this calculus lesson, 12th graders explore situations in which one would use logarithmic differentiation as an appropriate method of solution. Students should have already studied the chain rule. This video covers the differential notation dy/dx and generalizes the rule for finding the derivative of any polynomial. It also extends the notion of the derivatives covered in the Khan Academy videos, �Calculus Derivatives 2� and �Calculus Derivatives 2.5 (HD).� Note: Additional practice using the power rule for differentiating polynomials (including some with negative exponents) is available to the listener. This video is the 4th in a series of videos that explains derivatives. First, Sal shows another example of differentiating a polynomial and then shows two examples using the chain rule. Sal continues the chain rule in the next video. This video continues with examples of differentiating functions using the chain rule including examples that use negative exponents and more complicated nested parenthesis. Note: Additional practice on using the chain rule is available. In the first example, instead of actually using the quotient rule, Sal rewrites the denominator as a negative exponent and uses the product rule. In subsequent examples, Sal shows, but does not prove, the derivative of several interesting functions including ex, ln x, sin x, cos x, and tan x. Continuing to use the chain rule, Sal shows more examples of finding the derivative, this time, by looking at composite functions. Note: The current set of practice problems titled �Chain rule 1� cannot be solved until one knows how to find the derivative of ex and trigonometric functions 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. Students analyze implicit differentiation using technology. In 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. This lesson plan provides an introduction to integration by parts. It helps learners first recognize derivatives produced by the product rule and then continues with step-by-step instructions on computing these integrals. It also shows integrating special forms with e and trigonometric functions. This resource includes handouts and a practice worksheet. Are your calculus pupils aware that they are standing on the shoulders of giants? This lessonIn this calculus activity, students use integration to solve word problems they differentiate between integration and anti derivatives, and between definite and indefinite integrals. There are 3 questions with an answer key
Wikipedia in English Teaches new Mathematica users some of the important basics of this powerful software tool: defining functions, creating graphs and Notebooks, and applying useful problem-solving techniques. The authors cover 40 functions and use clear language and concise instructions to help readers master the basics.
Guys, I am in need of help on graphing inequalities, distance of points, greatest common factor and triangle similarity. Since I am a beginner to Intermediate algebra, I really want to learn the basics of Pre Algebra fully. Can anyone suggest the best resource with which I can start learning the fundamental principles? I have an exam next week. I remember I faced similar problems with function domain, angle suplements and binomials. This Algebrator is truly a great piece of math software program. This would simply give step by step solution to any math problem that I copied from workbook on clicking on Solve. I have been able to use the program through several Intermediate algebra, College Algebra and Algebra 2. I seriously recommend the program. Thanks you very much for the detailed information. We will surely check this out. Hope we get our assignments completed with the aid of Algebrator. If we have any technical queries with respect to its usage, we would definitely get back to you again. Algebrator is the program that I have used through several algebra classes - Algebra 2, Algebra 2 and Pre Algebra. It is a truly a great piece of math software. I remember of going through problems with lcf, reducing fractions and quadratic formula. I would simply type in a problem homework, click on Solve – and step by step solution to my algebra homework. I highly recommend the program.
аЯрЁБс > ўџ 7 9 ўџџџ 6 Ш, bjbjqPqP 02 : : 0 џџ џџ џџ Є т т т т т т т 6 d ž ž ž 8 ж ъ š О Ж э э э $ t h м Ј ? 9 т э э э э э ? т т л x э э э э ( т т э э э э т т э ЮŸ-ЅЦ ž d э Ž 0 О э " y d " э " т э , э э э э э э э э ? ? н э э э О э э э э š š š ž š š š ž і $ 2 т т т т т т џџџџ Algebra II and ALGEBRA II/TRIG HONORS The standards below outline the content for a one-year course in Algebra II. Students enrolled in Algebra II are assumed to have mastered those concepts outlined in the Algebra I standards. A thorough treatment of advanced algebraic concepts is provided through the study of functions, "families of functions," equations, inequalities, systems of equations and inequalities, polynomials, rational expressions, complex numbers, matrices, and sequences and series. Emphasis will be placed on practical applications and modeling throughout the course of study. Oral and written communication concerning the language of algebra, logic of procedures, and interpretation of results also should permeate the course. These standards include a transformational approach to graphing functions. Transformational graphing uses translation, reflection, dilation, and rotation to generate a "family of graphs" from a given graph and builds a strong connection between algebraic and graphic representations of functions. Students will vary the coefficients and constants of an equation, observe the changes in the graph of the equation, and make generalizations that can be applied to many graphs. Graphing utilities (graphing calculators or computer graphing simulators), computers, spreadsheets, and other appropriate technology tools will be used to assist in teaching and learning. Graphing utilities enhance the understanding of realistic applications through mathematical modeling and aid in the investigation and study of functions. They also provide an effective tool for solving/verifying equations and inequalities. Any other available technology that will enhance student learning should be used. AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers and its subsets, complex numbers, and matrices. AII.2 The student will add, subtract, multiply, divide, and simplify rational expressions, including complex fractions. AII.3 The student will a) add, subtract, multiply, divide, and simplify radical expressions containing positive rational numbers and variables and expressions containing rational exponents; and b) write radical expressions as expressions containing rational exponents and vice versa. AII.4 The student will solve absolute value equations and inequalities graphically and algebraically. Graphing calculators will be used as a primary method of solution and to verify algebraic solutions. AII.5 The student will identify and factor completely polynomials representing the difference of squares, perfect square trinomials, the sum and difference of cubes, and general trinomials. AII.6 The student will select, justify, and apply a technique to solve a quadratic equation over the set of complex numbers. Graphing calculators will be used for solving and for confirming the algebraic solutions. AII.7 The student will solve equations containing rational expressions and equations containing radical expressions algebraically and graphically. Graphing calculators will be used for solving and for confirming the algebraic solutions. AII.8 The student will recognize multiple representations of functions (linear, quadratic, absolute value, step, and exponential functions) and convert between a graph, a table, and symbolic form. A transformational approach to graphing will be employed through the use of graphing calculators. Algebra II and ALGEBRA II/TRIG HONORS (continued) AII.9 The student will find the domain, range, zeros, and inverse of a function; the value of a function for a given element in its domain; and the composition of multiple functions. Functions will include exponential, logarithmic, and those that have domains and ranges that are limited and/or discontinuous. The graphing calculator will be used as a tool to assist in investigation of functions. AII.10 The student will investigate and describe through the use of graphs the relationships between the solution of an equation, zero of a function, x-intercept of a graph, and factors of a polynomial expression. AII.11 The student will use matrix multiplication to solve practical problems. Graphing calculators or computer programs with matrix capabilities will be used to find the product. AII.12 The student will represent problem situations with a system of linear equations and solve the system, using the inverse matrix method. Graphing calculators or computer programs with matrix capability will be used to perform computations. AII.13 The student will solve practical problems, using systems of linear inequalities and linear programming, and describe the results both orally and in writing. A graphing calculator will be used to facilitate solutions to linear programming problems. AII.14 The student will solve nonlinear systems of equations, including linear-quadratic and quadratic-quadratic, algebraically and graphically. The graphing calculator will be used as a tool to visualize graphs and predict the number of solutions. AII.15 The student will recognize the general shape of polynomial, exponential, and logarithmic functions. The graphing calculator will be used as a tool to investigate the shape and behavior of these functions. AII.16 The student will investigate and apply the properties of arithmetic and geometric sequences and series to solve practical problems, including writing the first n terms, finding the nth term, and evaluating summation formulas. Notation will include S№ №a n d a n . A I I . 1 7 T h e s t u d e n t w i l l p e r f o r m o p e r a t i o n s o n c o m p l e x n u m b e r s a n d e x p r e s s t h e r e s u l t s i n s i m p l e s t f o r m . S i m p l i f y i n g r e s u l t s w i l l i n v o l v e u s i n g p a t t e r n s o f t h e p o w e r s o f i . A I I . 1 8 T h e s t u d e n t w i l l i d e n t i f y c o n i c s e c t i o n s ( c i r c l e , e l l i p s e , p a r a b o la, and hyperbola) from his/her equations. Given the equations in (h, k) form, the student will sketch graphs of conic sections, using transformations. AII.19 The student will collect and analyze data to make predictions and solve practical problems. Graphing calculators will be used to investigate scatterplots and to determine the equation for a curve of best fit. Models will include linear, quadratic, exponential, and logarithmic functions. ( Ч Ш Б В Ÿ Ё Ї в г ј љ В Г Z [ o p r h j C I Т! , –, —, ќјюјузуќуЯЫСЫЖуЊујуЊуЊžууЊуvуЊуЊуtуe he7ю he7ю CJ OJ QJ aJ UhЩ#Ю hоg CJ aJ he7ю he7ю 6CJ HaJ he7ю he7ю CJ OJ QJ aJ he7ю he7ю CJ HaJ he7ю he7ю 6CJ aJ he7ю hЩ#Ю CJ aJ he7ю hЩ#Ю ;aJ hЩ#Ю he7ю CJ aJ he7ю he7ю 5CJ aJ he7ю he7ю CJ aJ he7ю he7ю ;aJ he7ю hоg &
I've always wanted to excel in fractions with radicals sum, it seems like there's a lot that can be done with it that I can't do otherwise. I've searched the internet for some good learning resources, and consulted the local library for some books, but all the information seems to be directed to people who already know the subject. Is there any resource that can help new students as well? What is your problem regarding fractions with radicals sum? Can you give me more details on the problems you encountered regarding fractions with radicals sum? I myself had experienced many troubles on my algebra tests. I tried getting a/an math tutor to teach me, but it was too costly. The most convenient way to help you figure out your math problems is by using a decent program. Among all algebra softwares I used, it's the Algebrator that really surpassed my expectations. Aside from giving errors-free answers, it also shows a step-by-step solution that led to the answer. It's really a fine software to learn from but remember to avoid copying answers from it because it would really not help you if you'd just copy the solutions. Use it just to understand how to solve certain algebra problems. Algebrator is the perfect algebra tool to help you with projects. It covers everything you need to be familiar with in trinomials in an easy and comprehensive style. algebra had never been easy for me to grasp but this software made it very easy to understand. The logical and step-by–step approach to problem solving is really a plus and soon you will find that you love solving problems. binomials, radical expressions and multiplying matrices were a nightmare for me until I found Algebrator, which is truly the best algebra program that I have ever come across. I have used it frequently through several math classes – Intermediate algebra, Pre Algebra and Algebra 2. Simply typing in the math problem and clicking on Solve, Algebrator generates step-by-step solution to the problem, and my algebra homework would be ready. I really recommend the program.
More About This Textbook Overview For the modern student like you—Pat McKeague's BEGINNING ALGEBRA, 9E—offers concise writing, continuous review, and contemporary applications to show you how mathematics connects to your modern world. The new edition continues to reflect the author's passion for teaching mathematics by offering guided practice, review, and reinforcement to help you build skills through hundreds of new examples and applications. Use the examples, practice exercises, tutorials, videos, and e-Book sections in Enhanced WebAssign to practice your skills and demonstrate your knowledge. Meet the Author Charles P. "Pat" McKeague earned his B.A. in Mathematics from California State University, Northridge, and his M.S. in Mathematics from Brigham Young University. A well-known author and respected educator, he is a full-time writer and a part-time instructor at Cuesta College. He has published twelve textbooks in mathematics covering a range of topics from basic mathematics to trigonometry. An active member of the mathematics community, Professor McKeague is a popular speaker at regional conferences, including the California Mathematics Council for Community Colleges, the American Mathematical Association of Two-Year Colleges, the National Council of Teachers of Mathematics, the Texas Mathematics Association of Two-Year Colleges, the New Mexico Mathematics Association of Two-Year Colleges, and the National Association for Developmental Education. He is a member of the American Mathematics Association for Two-Year Colleges, the Mathematics Association of America, the National Council of Teachers of Mathematics, and the California Mathematics Council for Community Colleges
reform called for a leaner, meaner calculus curriculum. Instructors needed to spend more time with students, helping them understand what they were doing rather than teaching them all these computations. And the Virginia Tech Math Department jumped right into that stream. Third, Virginia Tech wanted to be known as a leader in using technology. The school instituted a program in which every faculty attended workshops every three or four years and received a high-end computer, configured to their choice, after each workshop. The Math Departmentwas the first guinea pig, and that was when the idea of teaching math in a computer-enhanced environment got its start. The strategy During that first workshop, we were introduced to the computer program Mathematica®. Then, in the spring of 1993, we began using Mathematica® in two of our first-year calculus courses. With only 78 computers and nearly 2,000 students taking the new classes each semester, assessments that included a common final exam showed that students in this new information technology initiative were performing at or above the level of students taking the traditional course. The "IT" students' final grades were half a grade higher that the "traditional" students, and later longitudinal assessments showed that IT students taking other mathematics or engineering courses were doing better than students who had taken traditional math classes. When I became department chair in 1994, the math department was teaching calculus the "Baskin-Robbins" way-if there was any way of teaching calculus, somebody was doing it in the department. I believe in innovations like that. You have to have the big bang, and then after the big bang, and everybody's excited about it, you try to steer, try to figure out what's really working and steer them in that direction. I hired an assessment coordinator in the department, and I issued the edict that every freshman and sophomore course has a common final exam. I was getting all kinds of differing views about what's appropriate for technology-I still get that today-and I decided we need to have some way of judging what students are learning. I also stated that the exams would not be used in faculty evaluations; faculty had always voted down common exams in the past, because they were fearful that they might be evaluated by how students are learning. Now each course has a list of objectives and goals that were set up and improved by the Undergraduate Program Committee, so for every common exam, each question is coded against objectives and goals. Three hundred out of the 1,600 students in the fall took the self-paced math courses, and they did a lot better on the common exam and a lot better in the following course than the traditional students. So that sets the stage for the Math Emporium, because at this time I was teaching 300 students; the math department had control of two labs, one with 40 computers in it and one with 38 computers. Between the self-paced course and engineering calculus, we were teaching classes on those machines from 8 in the morning until 11 at night. There was no room for professors to just go around and do open-ended sessions with them. We also saw that another class, a differential equations class taught to civil engineering students, always outperformed every other differential equations class-we called this class "Green Math," and this class was set in the context of civil engineering and polluted lakes and other kinds of applications. The higher performance was because the math they were learning was set in a relevant context. So the labs were full, and we were teaching calculus to several different audiences: biology and dairy science, business, and architecture, as well as math and engineering students. The business and architecture calculus courses were absolutely terrible, not at all relevant to the students. In all, we calculated that our enrollments were, and still are, 12,000 to 16,000 students per semester. When we told the other deans that we would have to split the business and architecture courses unless we built this Math Emporium idea we were developing, we had all the deans in favor of the project. The core idea was that the best time to teach mathematics is when the student wants to do it. We needed a place-with computers, software, and roving instructors-that is open all day for the students to come in and do their self-paced study. We found an empty department store near campus, and the landlord did the entire infrastructure for us, the T1 and phone lines and such, incorporating the cost into the rent. We signed a 10-year lease with him. The concept was sold to the deans in March, and we moved into the building in August. The University supplied the computers. The learning technology The Math Emporium: The Emporium is open 24 hours a day, seven days a week. The facility holds 500 workstations, with other specialized spaces and equipment included. Instructors are available Monday-Thursday, 9:00 am to 12:00 midnight; Friday, 9:00 am to 4:00 pm; Sunday, 4:00 pm to 11:00 pm. The tutoring lab is staffed by peers and open Sunday-Thursday, 6:00 pm to 9:00 pm. Information Systems takes care of the students, checking them in and out, and also making sure the computers are running. The Math Department takes care of the curriculum, helping the students learn. One faculty member has the teaching assignment of being the Emporium Coordinator, overseeing all the courses and making sure that the trains don't collide into each other. It runs smoothly. A couple of instructors in the Department just teach four courses a semester each for us; they don't have a research assignment. They're out there 40 hours a week, and they are in seventh heaven. It's what they want to do. In total, we have 94 faculty in the department, and during the semester, about 60 faculty have some sort of assignment in the emporium. Students can still move through the course in a traditional way; there's a lecture hall and an area for traditional tutors. Students can go to lecture, do their exercises at the end of the section, and then go to a tutor lab. The tutors look over their work, and the students take their test. But although some students start with the lecture format, because that's the way they were taught, very few students continue that way. They see their peers having successes and they say, "Why am I going to lecture when I can do it this way?" We have a safety net for the traditional students, a focus group that's required, in which their progress is monitored. Each class provides worksheets on the website, with detailed instructions for the students to follow; this includes homework and quiz assignments. Mathematica®: With this well-known math software, students can do numerical computations, extensive simulations and prototyping, and prepare clean reports and presentations. MATLAB®: For our engineering students, the primary software is MATLAB®,, per the engineers' request. This offers a technical computing environment with numeric and symbolic computational abilities; data analysis and visualization; engineering and scientific graphics; modeling, simulation, and prototyping; and programming capabilities. Visit the MATLAB® website. The Geometer's Sketchpad-Geometry Software for Euclidean, Coordinate, Transformational, Analytic, and Fractal Geometry®: We use this software for our architecture students; they wanted a lot more geometry in their sequence. Visit Key Curriculum Press website for more information. Excel: The course for business majors uses Excel®, like crazy-they wanted to learn how to use spreadsheets in a variety of ways. They talk about revenues, incomes, module derivatives; the business people are in love with the course now. We also have an online course, Linear Algebra (Math 1114), which is a course for which we wrote the technology ourselves. The students don't see the technology; they just work through the material online. You can access the worksheets for this course via the Emporium website. The project support We funded the Math Emporium with money from the National Science Foundation and FIPSE grants; but our primary source of money was the University. This was a painful process, because the money for technology had to be carved out of the budget. We also received funds from the University's Center for Excellence in Undergraduate Teaching (CEUT); they give out grants to help faculty renovate courses. There's another body on campus called the Center for Innovative Learning (CIL). Their budget is used to entice faculty to rethink what they're doing with the use of technology, and they have supported us as well. As for the students, there's a lot of leeriness and jeering at first. You have to picture an 18-year-old walking into this building saying: "I'm going to do what?" We spend the first two weeks holding students' hands. We tell them, "First of all, the most important thing you can do at Tech is become an active learner and know how you learn." Then we make them do all kinds of exercises so they see all the resources within the building. At first the students are not real impressed. Some are excited about being on their own, but others feel that we're abandoning them to a sea of computers. By the end of the semester, after they've had successes and such, their attitudes have changed some. To go to the scale we have gone, if the institution is not committed to it, I would tell any department chair to not even think about doing this. Even within the Math Department faculty, I really have critics, a small group that believes this is not the right way and they want to go back to the old way of teaching. You need a group that really believes this is the way for students to learn and it's the faculty's job to protect the students and nourish them and encourage them to do this. You need the group that is focused on student learning and not on "casting pearls among swine," on being the stars of the show. Student learning is a lot different than that. You have to be willing to think on your feet-you walk into this building and you never know what you're going to face. If you have any questions about our project, you can contact me at: olin@vt.edu
Computational Number introduces the vast and fascinating area of computational number theory. It treats algorithms for common number-theoretic problems in an elementary fashion, eliminating the need for an extensive prerequisite of algebra and analysis. The GP/PARI calculator is used throughout to demonstrate the working of arithmetic algorithms. The book contains detailed examples illustrating almost every algorithmic concept discussed. It also includes practical applications of arithmetic algorithms in public-key cryptography. Every chapter ends with ma... MOREny exercises and partial solutions are given in the appendix.
provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives students the information as to how these ...Show synopsisThis manual provides worked-out, step-by-step solutions to the odd-numbered problems in the text. This gives students the information as to how these problems are solved
201795116 / ISBN-13: 9780201795110 Mathematics All Around Pirnot's Mathematics All Around offers the supportive and clear writing style that you need to develop your math skills. By helping to reduce your ...Show synopsisPirnot's Mathematics All Around offers the supportive and clear writing style that you need to develop your math skills. By helping to reduce your math anxiety, Pirnot helps you to understand the use of math in the world around you. You appreciate that the author's approach is like the help you would receive during your own instructors' office hours. The Fifth Edition increases the text's emphasis on developing problem-solving skills with additional support in the text and new problem-solving questions in MyMathLab. Quantitative reasoning is brought to the forefront with new Between the Numbers features and related exercises. Since practice is the key to success in this course, exercise sets are updated and expanded. MyMathLab offers additional exercise coverage plus new question types for problem-solving, vocabulary, reading comprehension, and more. 032192326X / 9780321923264 Mathematics All Around Plus NEW MyMathLab with Pearson eText -- Access Card Package Package consists of 0321431308 / 9780321431301 MyMathLab/MyStatLab -- Glue-in Access Card 0321654064 / 9780321654069 MyMathLab Inside Star Sticker 0321836995 / 9780321836991 Mathematics All Around 5/eHide synopsis ...Show more difficult" for liberal arts students. As a result, students end up with an understanding of and positive attitude toward many different and often challenging mathematical topics." 960 p. Reviews of Mathematics All Around I cannot believe that all of this was crammed into ONE 8 week college course. I'm still getting over the stress of trying to make it through this course. The book is ok if you understand math, but if you don't, you're just going to be more lost than you were before you started. Very confusing stuff
Synopses & Reviews Publisher Comments: This updated book is a self-teaching brush-up course for students who need more math background before taking calculus, or who are preparing for a standardized exam such as the GRE or GMAT. Set up as a workbook, Forgotten Algebra is divided into 31 units, starting with signed numbers, symbols, and first-degree equations, and progressing to include logarithms and right triangles. Each unit provides explanations and includes numerous examples, problems, and exercises with detailed solutions to facilitate self-study. Optional sections introduce the use of graphing calculators. Units conclude with exercises, their answers given at the back of the book. Systematic presentation of subject matter is easy to follow, but contains all the algebraic information learners need for mastery of this subject. Synopsis: back coverSynopsis: This self-teaching brush-up course is designed for students who need more math background before taking calculus, or who are preparing for a standardized exam such as the GRE or GMAT."Synopsis" by Libri, This self-teaching brush-up course is designed for students who need more math background before taking calculus, or who are preparing for a standardized exam such as the GRE or GMAT
Teaching Assistants Recommended Exercises In the list below we only refer to the odd-numbered questions in the textbook. They have solutions at the back of the book. Try to solve as many of them as possible. NOTE: The purpose of the solutions is that you can verify your result after solving the exercise (or looking it up if you are stuck). The learning effect of simply reading a solution without trying to solve the question yourself is close to zero. Section 10.3: 3a+b, 7, 9, 11, 13 Section 12.1: 1-35 Section 12.2: 1-43 Section 12.3: 1-49, 55 Section 12.4: 1-43 Section 12.5: 1-67, 71, 73, 77, 79 Section 12.6: 1-35 Section 13.1: 1, 7-19 Section 13.2: 1-27, 33-41, 47-51 Section 13.3: 1-33, 39 (Ignore the question about N(t) in 17 and 19.) Section 13.4: 1-15, 17a, 19-25, 31 Section 14.1: 1-49 Section 14.2: 3-15 Section 14.3: 1-95 Section 14.4: 1-41 Section 14.5: 1-53 Section 14.6: 1-45, 49-63 Section 14.7: 1-19, 29-35, 39-53 Section 14.8: 1-43 Section 15.9: 1, 5, 7, 9, 11, 13 Multivariable Taylor expansion: 1, 2a, 3a, 4 from the discovery project on page 980, these exercises, and exercise 1 on this page Maple Worksheets Quizzes There will be one quiz every Friday during class time. The lowest quiz score will be dropped when the final grade is calculated. Sep 16: Sections 12.1 and 12.2 up to page 819 including the list "Properties of Vectors" (bottom of page 819), but without the length of a vector (bottom of page 818). For Section 12.2, the relevant exercises are 1-19 (again, without the length of a vector). Sep 23: Rest of Section 12.2, and Section 12.3, plus polar coordinates (beginning of Section 10.3 up to Example 2) and spherical coordinates (beginning of Section 15.9 up to Example 1). (Warning: In class I called "φ" what the book calls "θ" and vice versa.) We are going to review scalar and vector projections at the beginning of the Friday class. Sep 30: Sections 12.4 and 12.5 up to page 843. For Section 12.5, the relevant exercises are 1-21. Oct 7: Rest of 12.5, and Section 12.6. Grading: As explained in class, we will add 2 marks for everyone who wrote this quiz. Nov 18: Section 14.6 Grading: As explained in class, we will add 1 mark for everyone who wrote this quiz. Nov 25: Section 14.7 up to page 975 (the entire page) Dec 2: the rest of Section 14.7, and Section 14.8 WebCT All marks have been uploaded to WebCT. Notes: The "Final" column contains the overall final grade. The result of the final exam is in the column "Final Exam". The quiz marks are out of 6. An "EX" as quiz mark means that you have been excused. If you have not been excused, the mark is 0. The adjustments for the quizzes 4 and 8 mentioned above are included. Also, the lowest quiz mark has been dropped. Both midterm mark and final exam mark are out of 70. The overall final grade has been computed as if there were 63 marks on the final exam. Please contact me if you think that your marks have been entered incorrectly.
For Algebra, Spreadsheets Beat Newer Teaching Tools You already own better algebra-teaching software than any educational software developer is making. I found in teaching elementary algebra to disadvantaged adult learners that the progression from table to graph and then to formula/function/equation might have been the most powerful tool for "selling" algebra I ever encountered. Anyone can see intuitively that for many problems, if you just make a table big enough, trying out all the possible values will lead to a solution. From there, it's a short step to, but what if there are millions of values? Then they're ready to graph it and the answers are right there at the intersections. From there it's just the step to, but what if you need an answer more exact than the line you can draw, or more dimensions than two? Well, by then, they're used to the idea of formula/function/equation as description, and if a point satisfies more than one description, it's a solution. And they've crossed over to doing algebra. The spreadsheet provides a natural bridge from arithmetic procedure to formula/function. At first, students will just input numbers, as if the spreadsheet were a calculator. Then they see that if they input variables, they don't have to type nearly so much, and then that this means having not just this answer this time, but all the answers to all problems of this type, all the time. It's a wide and easy-to-cross bridge from the specific to the general, and from procedure to concept. One of the big changes in mathematics in the last 30 years has been the idea of experimental math, i.e., of exploring how numbers work by setting up numerical processes and looking at the results; it's at the heart of chaos research, for example. Just as the computer has become the equivalent of the telescope or microscope for mathematicians, Excel can be used as an amateur "scope" for exploring numbers, in a way very much analogous to the way countless students have gotten a handle on science by finding a planet in the sky or exploring the ecology of pond water. Among other things, I've used Excel to teach how every fraction is a division, and division is equivalent to multiplying by the inverse. It could easily be used for many other projects beginning even from a very early age in arithmetic. Spreadsheet algebra is such an effective and intuitive idea that it has been re-invented several times in the last 20 years, and some Googling around will turn up immense amounts about it. (Caution: "spreadsheet algebra" is also a term used in advanced mathematics research for a kind of non-linear matrix algebra, so your Googling may very well turn up an article or two that's a bit beyond you. Don't worry, just keep looking!) Although there still needs to be a human being there to guide the student in exploring and using the spreadsheet, as a teaching device for actual algebra (as opposed to a drilling device for standardized tests) the spreadsheet still beats out thousands of purpose-designed products. re: For Algebra, Spreadsheets Beat Newer Teaching Tools They said that about calculators, graph paper, and probably the abacus. The funny thing is that although the spreadsheet does make it possible to avoid learning procedure, you can't use it effectively unless you learn concepts. So it's really a powerful tool for dragging people over the border from the "do your sums this way" to the real kingdom of mathematics
1567506526 9781567506525 Learning and Teaching Number Theory:Number theory has been a perennial topic of inspiration and importance throughout the history of philosophy and mathematics. Despite this fact, surprisingly little attention has been given to research in learning and teaching number theory per se. This volume is an attempt to redress this matter and to serve as a launch point for further research in this area. Drawing on work from an international group of researchers in mathematics education, this volume is a collection of clinical and classroom-based studies in cognition and instruction on learning and teaching number theory. Although there are differences in emphases in theory, method, and focus area, these studies are bound through similar constructivist orientations and qualitative approaches toward research into undergraduate students' and preservice teachers' subject content and pedagogical content knowledge.Collectively, these studies draw on a variety of cognitive, linguistic, and pedagogical frameworks that focus on various approaches to problem solving, communicating, representing, connecting, and reasoning with topics of elementary number theory, and these in turn have practical implications for the classroom. Learning styles and teaching strategies investigated involve number theoretical vocabulary, concepts, procedures, and proof strategies ranging from divisors, multiples, and divisibility rules, to various theorems involving division, factorization, partitions, and mathematical induction. Back to top Rent Learning and Teaching Number Theory 1st edition today, or search our site for Stephen R. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Ablex Publishing Corporation.
Develop Algebraic Thinking 6-8 - MAT ONLINE course will help teachers discover how to support deeper understanding of foundational algebraic concepts in grades 6-8. Teachers will explore growth patterns and functions, variables, linear relationships, and coordinate graphs. Based on the included text and research-based journal articles, teachers will design and implement a unit of grade-level appropriate algebraic thinking activities with their students. All of the readings and activities are built upon the Common Core standards. Teachers may complete this course with or without students. Connect With Testimonial "The information that I have gathered while participating in various courses at Fresno Pacific University has impacted my classroom tremendously. The students are able to learn with a more hands-on approach which has improved their understanding of the material being presented."
1893, Byerly's classic treatise on Fourier's series and spherical, cylindrical, and ellipsoidal harmonics has been used in classrooms for well over a century. This practical exposition acts as a primer for fields such as wave mechanics, advanced engineering, and mathematical physics. Topics covered include: . development in trigonometric series . convergence on Fourier's series . solution of problems in physics by the aid of Fourier's integrals and Fourier's series . zonal harmonics . spherical harmonics . cylindrical harmonics (Bessel's functions) . and more. Containing 190 exercises and a helpful appendix, this reissue of Fourier's Series will be welcomed by students of higher mathematics everywhere. American mathematician WILLIAM ELWOOD BYERLY (1849-1935) also wrote Elements of Differential Calculus (1879) and Elements of Integral Calculus (1881).
Statistics : A First Introductory Statistics-Algebra-based courses. This text offers a straightforward, nuts and bolts, introduction to statistics. The explanations are clear and simple and minimize calculations where possible. A diverse range of applications and examples are presented to make the materials appealing to a wide range of students. The Eighth Edition features new problems, more real data based examples and exercises, and fuller integration of technology. This book offers a straightforward, "nuts and bolts" , introduct... MOREion to statistics. The explanations are clear and simple and minimize calculations where possible. A diverse range of applications and examples are presented to make the materials appealing to a wide range of learners.The Eighth Edition features new problems, more real data based examples and exercises, and fuller integration of technology.For individuals beginning a study of statistics.
Integrated Arithmetic and Basic combination of a basic mathematics or prealgebra text and an introductory algebra text, Integrated Arithmetic and Basic Algebra, Third Edition, provides a uniquely integrated presentation of the material for these courses in a way that is extremely beneficial to students. As opposed to traditional texts that present arithmetic at the beginning and algebra at the end, this text integrates the two whenever possible, so that students see how concepts are interrelated rather than learning them in isolation and missing the "big picture." The ideas and algorithms shared by arithmetic and algebra are introduced in an arithmetic context and then developed through the corresponding algebraic concept. For example, operations with rational numbers and rational expressions are discussed together, whereas most texts discuss operations with rational numbers early on and operations with rational expressions much later. The Jordan/Palow text helps students recognize algebra as a natural extension of arithmetic using variables. This approach aligns directly with NCTM and AMATYC standards, which suggest playing upon students' previous knowledge to teach new concepts.
$ 14.79 14.79 Word problems are the most difficult part of any math course –- and the most important to both the SATs and other standardized tests. This book teaches proven methods for analyzing and solving any type of... $ 12.99 Algebra: A Complete Introduction is the most comprehensive yet easy-to-use introduction to using Algebra. Written by a leading expert, this book will help you if you are studying for an important exam or essay,... $ 14.99 Having trouble understanding algebra? Do algebraic concepts, equations, and logic just make your head spin? We have great news: Head First Algebra is designed for you. Full of engaging stories and practical,... $ 8.79 Preempt your anxiety about PRE-ALGEBRA!Ready to learn math fundamentals but can't seem to get your brain to function? No problem! Add Pre-Algebra Demystified, Second Edition, to the equation and you'll solve... $ 10.29 This book concerns the use of dioid algebra as (max, +) algebra to treat the synchronization of tasks expressed by the maximum of the ends of the tasks conditioning the beginning of another task – a criterion... $ 74.99 Practice Makes Perfect has established itself as a reliable practical workbook series in the language-learning category. Practice Makes Perfect: Algebra, provides students with the same clear, concise approach...
Description of Saxon Advanced Math: Homeschool Kit by Saxon Based on Saxon's proven methods of incremental development and continual review strategies, the Saxon Advanced Math program builds on intermediate algebraic concepts and trigonometry concepts introduced in Algebra 2 and prepares students for future success in calculus, chemistry, physics and social sciences. Kit includes Student Textbook with 125 lessons, proofs and index; Homeschool Packet with test forms; and Answer Key with answers for all tests and student textbook problem sets. (Solutions Manual not included.) Product: Saxon Advanced Math: Homeschool Kit Author: Saxon Edition Number: 2 Edition Description: Teacher Series: Homeschool Advanced Math Binding Type: Other Media Type: Other Weight: 4.96 pounds Length: 11.18 inches Width: 8.64 inches Height: 1.67 inches Publisher: Saxon Publishers Publication Date: January 2005 Subject: Math Curriculum Name: Saxon Learning Style: Auditory, Kinesthetic, Visual Teaching Method: Charlotte Mason, Classical, Traditional, Unit Study, Unschooling There are currently no reviews for Saxon Advanced Math: Homeschool Kit.
Summary:The subject of this class is SUCCESS...what success is and how to achieve it in college. This course is aligned with the textbook: Cornerstones for Community College Success", Sherfield, Robert M., and Patricia G. Moody. 2nd ed. Boston: Pearson Education, 2014. Summary: ... entire course.[Expand Summary] II: Conceptual Explanations" ( and the "Advanced Algebra II: Teacher's Guide" ( collections to make up the entire course.[Collapse Summary] Summary: ... entire course.[Expand Summary].[Collapse Summary] Summary: ... entire course.[Expand Summary] designed for use with the "Advanced Algebra II: Conceptual Explanations" ( and the "Advanced Algebra II: Homework and Activities" ( collections to make up the entire course.[Collapse Summary]
Exploring the relationship between advanced algebra topics and trigonometry, Pre-Calculus is an informative introduction to calculus that challenges students to discover the nature of graphs, nonlinear systems, and polynomial and rational functions. With an emphasis on mathematical reasoning and argument, this advanced course scaffolds rigorous content with clear instruction and an array of scaffolds for learning, providing students with a deep understanding of topics such as matrices, functions, graphing, logarithms, vectors, and conics. The course concludes with a brief introduction to calculus that exposes students to limits, continuity, derivatives, and the Fundamental Theorem of Calculus.
Elementary Algebra-Student's Solutions Manual - 8th edition Summary: When the answer at the back of the book is simply not enough, then you need the Student Solutions Manual. With fully worked-out solutions to all odd-numbered text problems, the Student Solutions Manual lets you "learn by example" and see the mathematical steps required to reach a solution. Worked-out problems included in the Solutions Manual are carefully selected from the textbook as representative of each section's exercise sets so you can follow-along ...show moreand study more effectively. The Student Solutions Manual is simply the fastest way to see your mistakes, improve learning, and get better grades3878.95 +$3.99 s/h Good A Novel Idea Bookstore FL Ocala, FL Paper Back Used-Good $23.87 +$3.99 s/h Good Big Planet Books Burbank, CA 2009-03-09 Paperback Good Expedited shipping is available for this item! $23.87 +$3.99 s/h Good Big Planet Books Burbank, CA 2009-03-09
Algebrainiac: Math Vocabulary Description Hangman and math vocabulary collide in this new app. Choose from three different categories, such as "Algebra", "Geometry" and "Linear Equations". Learn from over 60 different words, complete with it's definition! Fun for students of all ages, and for students who are either just learning, or refreshing their algebraic vocabulary words. RELEASE INFO: The second of our continuing math-based apps...
Analysis of algorithms is really all math. How heavily do you want to get into the topic? The math is usually straight forward, but it uses calculus heavily. If you don't understand calculus, its going to appear opaque. For years, Stanford University used Donald Knuth's Concrete Mathematics as a prerequisite for analysis of algorithms. Any university bookstore will have the textbook that their CS classes use for analysis of algorithms. Its a fairly advanced topic, typically junior or senior year for undergrads, or first year of grad school.
This program is designed to assist Teachers and Home learners to create quick easy to use worksheets. All the worksheets can be printed out, photocopied, and used free of charge. Features include - Added Simplifing/Reducing Ratios. - Added Reducing/Cancelling Down Fractions. - Added Converting Improper Fractiions to Mixed Numbers. - Added Converting Mixed Numbers to Improper Fractions. Please contact Maths Work sheet Generator publisher, geocities.com/win_macro if you have questions or issues regarding this product. Maths Work sheet Generator This program is designed to assist Teachers and Home learners to create quick easy to use worksheets. All the worksheets can be printed out, photocopied, and used free of charge. Graphing Calculator 3D Easy-to-use 3D grapher that plots high quality graphs for 2D and 3D functions and coordinates tables. Graphing equations is as easy as typing them down. Graphs are beautifully rendered with gradual colors and lighting and reflection effects. Das Unit Converter Das Unit Converter is an easy-to-use freeware program that can handle virtually all of your unit conversion needs. It has over 2000 units, over 80 categories, and has the ability to add more (units & categories). Contains no adware, spyware, or junk. DPlot Graphing software for scientists, engineers, and students. It features multiple scaling types, including linear, logarithmic, and probability scales, as well as several special purpose XY graphs and contour plots of 3D data. Equation Wizard The program automatically solves algebraic equations of any order written in any form. Enter your equation and click just one button! Step by step the program will solve the equation, find its roots and describe all its operations. Math Studio Math tool for high school math, middle school math teaching and studying. Function graphing and analyzing, sequence of number, analytic geometry and solid geometry. Selenium Inspector - an open source library for simplifying automated testing for Web applications TeamDev is happy to announce the release of Selenium Inspector, an open source library for simplifying automated testing for Web applications. Selenium Inspector is built on top of the popular Selenium testing framework, and enhances its abilities for component-oriented testing, and simplifies testing of any HTML-based applications as well. Selenium Inspector API allows to create testing... Php Framework based Web Applications Development Strategy by hirephp.com Hirephp.com - an end-to-end e-business solutions and software development company, which combines cutting-edge technology with keen business acumen to deliver high quality, innovative Web solutions and business process automation tools for global enterprises. We provide professional website design services, for companies, businesses and individuals. Hirephp.com is a firm represented... Ruby on Rails based Applications Development Methodology by HiddenBrainscom enbrains.com is continuously receiving admiration of itsclients for its approach of adopting new technologies as company has startedproviding services on Ruby on Rails based Applications Development recently. Companyis having great expertise in the field of web application development for the pastmany years. Our team of expert professionals of the Ruby on Rails is equipped toserve its global... Restore Data From Zip Archives Using Zip Repair Tool You can come across a number of corruption issues while downloading zip files from any website. Download can result in damage or errors in zip archive files. The tendency is always high that you may not be able to extract contents or text of zip files and it becomes too difficult to access. Therefore to resolve and overcome from such unfavorable circumstances, taking help of zip repair tool... Death Clock Launches Advanced Life Expectancy Calculator Wimborne, BH, April 16 2008 -- Leading website publisher MarkMitchell today launched The Death Clock, a new website aimed designed solely forentertainment purposes to attempt to predict the user s date of death by collectingvarious information about the user s life.Users will be asked about certain life prolonging health facts, such as smokinghabits and BMI, an indicator of body fat
Mathematics 123 is a college level algebra course addressing the following topics: Algebraic equations and inequalities, the Cartesian plane, graphs of equations, functions, zeroes of polynomial functions, and systems of equations. A graphing calculator is required. Placement: High School Algebra I, II, and Plane Geometry or equivalent, and satisfactory profile scores III.TEXTBOOK Sullivan, Michael., College Algebra, 2nd Ed., Prentice Hall, 1999 CALCULATOR: Graphing Calculator Required IV.BEHAVIORAL OBJECTIVES The proposed instructional objectives are as follows: The objectives and competencies proposed for this course will be demonstrated as students perform the following activities: Using the properties of real numbers and basic rules of algebra. Solution of linear equations and inequalities and use this knowledge in the solution of verbal problems. Demonstration of a practical knowledge of relations, functions and conic sections. Employment of the concepts of algebra as tools in the solution of applied problems. V.EVALUATION CRITERIA/GRADING SCALE There will be five tests and a comprehensive final exam. The grading scale for determining the course grade and the weights assigned to tests, final examination, and homework are given below. The in class tests and final exam will be graded on a 100 point scale. Make up tests will not be given and the final exam grade will be used as the grade for all tests that are missed. Make up tests will not be given. If all in-class tests have been taken, the lowest score for an in class test will be replaced with the final exam score and the class test average will be computed as in the example below. Homework will be collected randomly at a rate averaging about once a week and given a grade of either pass or fail. The homework score will depend on the percentage of passing grades assigned for collected assignments. Late homeworks will not be accepted. The percentage of passing homework grades will be multiplied by four, rounded, and the result added to the test average. Example: To see how your grade will be calculated, suppose your test scores are 81, 85, 84, 90, and 89, your final exam score is 88 and you received a passing grade on 50% of the homework collected. Since the lowest test grade is dropped (see item 1 under COURSE REQUIREMENTS), your grade would be calculated as follows: 0.16 * [ (81 + 85 + 84 + 90 + 89) ] + 0.16 * 88 = 86.5 86.5 + .504 = 86.5 + 2 = 88.5 = 89 Since 89 is between 80 and 89, inclusive, you would receive a grade of B. Weights Assigned to graded materials: In Class Tests 16.6% Each Comprehensive Final Examination 16.6% Homework 4% Extra Credit Grading Scale: A 90 - 100% Tests 83.3% B 80 - 89% Final Exam 16.6% C 70 - 79% D 60 - 69% F Below 60% VIa.COURSE OUTLINE See attached calendar. * Subject to change by myself for the optimization of instructional assistance. VIb.PROBLEM ASSIGNMENTS: All odd numbered problems are to be completed in the sections assigned as homework unless otherwise stated by the instructor. VIc.READING ASSIGNMENTS: Read each section prior to the presentation of the topic in class. VII.SUPPORT SECTIONS Students enrolled in support sections of Math123, College Algebra ( sections 20 and above ) are required to attend the Math Lab in the Helen Chick Building for an hour per week for further assistance with course materials. The students must contact the director of the Math Lab in room 216B in the Helen Chick building no later than August 28, 1998 to arrange a schedule of attendance. VIII.COURSE REQUIREMENTS Conduct of Course/Classroom Decorum 1. Students are responsible for availing themselves of all class meetings, Tutorial sessions, computer lab sessions, and individual help from the instructor. There are computer software tutorials available for your use in the Helen Chick Building, second floor and SBE 216A. (See the Lab Assistants) Each student should have a textbook and graphing calculator during each class meeting. 2. Students are responsible for maintaining a notebook of problems selected by the instructor. Students are encouraged to include as many additional problems as is possible 3. All tests will be announced prior to their administration. Since the lowest test will be dropped no make-up test will be given. There will be a test given at the end of each chapter, except possibly for chapter 6, and there will be a comprehensive final examination given. 4. Students are expected to enter the classroom on time and remain until the class ends. Late arrivals and early departures will be noted in the record book. The class attendance policy set forth in the 1996-1998 FSU Catalogue will be strictly adhered to. 5. Students must refrain from smoking, eating, and drinking in the classroom. The rights of others must be respected at all times. 6. Students are encouraged to ask questions of the instructor in class and to respond to those posed by the instructor. They should not discourage others from asking or answering questions. Other students often have the same questions on their minds, but are hesitant to ask. 7. Students are expected to complete all class assignments and to spend adequate time on their class work and to read each topic prior to class discussion to insure that the course objectives are met. Two hours of home study is expected for each hour of class. 8. Talking in class between students is strictly unacceptable. Discussions should be directed to the instructor. 9. Extra recitation periods and/or computer lab attendance are mandatory for students whose grades fall below C. They must meet the instructor to arrange for extra activities. 10. Dishonesty on graded assignments will not be tolerated. Students must neither give nor receive help on any work to be graded. The University policy on cheating will be applied to any violations. The minimum penalty will be a grade of zero on the assignment.
Basic Algebra Shape-Up is interactive software that covers creating formulas; using ratios, proportions, and scale; working with integers, simple and multi-step equations. Students are able to track their individual progress. Student scores are kept in a management system that allows teachers and tutors to view and print reports.
Building Mathematical Comprehension Resource Book This book provides a solid foundation for incorporating familiar reading comprehension strategies and relevant research in mathematics instruction to help build students' mathematical comprehension. This stand-alone book demonstrates how to facilitate student learning to build scheme and make connections among concepts. Dig into problem solving and reflect on current teaching practices with this exceptional teacher's guide. The purpose of this book is to help students be successful mathematical solvers. To do this, they should be more comfortable with problem solving themselves. This book presents a strategy-based approach to problem solving and models how to use the strategies effectively with students.
Rates of Change and Limits Calculating Limits Using the Limit Laws Precise Definition of a Limit One-Sided Limits and Limits at Infinity Infinite Limits and Vertical Asymptotes Continuity Tangents and Derivatives 3. Differentiation The Derivative as a Function Differentiation Rules The Derivative as a Rate of Change Derivatives of Trigonometric Functions The Chain Rule and Parametric Equations Implicit Differentiation Related Rates Linearization and Differentials Estimating with Finite Sums Sigma Notation and Limits of Finite Sums The Definite Integral The Fundamental Theorem of Calculus Indefinite Integrals and the Substitution Rule Substitution and Area Between Curves 6. Applications of Definite Integrals Volumes by Slicing and Rotation About an Axis Volumes by Cylindrical Shells Lengths of Plane Curves Moments and Centers of Mass Areas of Surfaces of Revolution and The Theorems of Pappus Work Fluid Pressures and Forces Sequences Infinite Series The Integral Test Comparison Tests The Ratio and Root Tests Alternating Series, Absolute and Conditional Convergence Power Series Taylor and Maclaurin Series Convergence of Taylor Series; Error Estimates Applications of Power Series Fourier Series Double Integrals Areas, Moments and Centers of Mass Double Integrals in Polar Form Triple Integrals in Rectangular Coordinates Masses and Moments in Three Dimensions Triple Integrals in Cylindrical and Spherical Coordinates Substitutions in Multiple Integrals 16. Integration in Vector Fields Line Integrals Vector Fields, Work, Circulation, and Flux Path Independence, Potential Functions, and Conservative Fields Green's Theorem in the Plane Surface Area and Surface Integrals Parametrized Surfaces Stokes' Theorem The Divergence Theorem and a Unified Theory Appendices Mathematical Induction Proofs of Limit Theorems Commonly Occurring Limits Theory of the Real Numbers Complex Numbers The Distributive Law for Vector Cross Products Determinants and Cramer's Rule The Mixed Derivative Theorem and the Increment Theorem The Area of a Parallelogram's Projection on a PlaneGWIWM BOOKS Muskegon, MI 2007 Hardcover Good Cover is slightly worn Cover worn at edges. Spine has lines from use. Thank you for supporting our mission with your purchase.21.48 +$3.99 s/h Good One Stop Text Books Store Sherman Oaks, CA 2007-01-15 Hardcover Good Good. $63.38 +$3.99 s/h VeryGood AlphaBookWorks Alpharetta, GA 032148987
Middle School Math With rigorous support for teachers and real-world contexts that help students understand new ideas, these courses prepare students to be successful in Algebra I and beyond. Middle school mathematics courses for grades 6, 7, and 8 provide powerful foundations in ratios, proportionality, and algebraic and geometric thinking. Students use graphing technology, manipulatives, and other mathematical tools to develop conceptual understanding as they tackle and solve interesting problems. Throughout these courses, students will: Build on their understanding of multiplication and division and equivalent fractions as a basis for understanding ratios and proportional reasoning Begin formal work with expressions and equations as they use variables to represent relationships and solve problems Develop their understanding of variables from two perspectives—as placeholders for specific values and as sets of values represented in algebraic relationships Gain fluency with geometric concepts, such as area, surface area, and volume Talk with us about working together to close the achievement gaps in math and science. For all students. Preparation for high-stakes assessments with both automatically graded and open-response questions Real-time reporting of progress that allows students to take responsibility for their own learning Robust Supports for Teachers In addition to Internet-delivered services, educators and administrators also receive face-to-face seminars, mentoring, and high-quality support materials to manage their demanding workloads, improve their expertise, and dramatically improve outcomes for their students. Online and face-to-face professional development seminars and mentoring directly tied to practice Day-by-day lesson support with advice and classroom strategies, equipping teachers to enact each day of instruction to achieve success for all students Collaboration with Leading Educational Research Center Our mathematics and academic youth programs are developed in collaboration with the Charles A. Dana Center at the University of Texas at Austin. Working with leading educators throughout the country, we have developed mathematics programs for middle school, Algebra I, Geometry, Algebra II, Precalculus, Calculus, and AP Statistics. These programs build on the Dana Center's trusted work with more than 100,000 teachers over the past decade and on the core belief that all students can succeed in mathematics if given the opportunity.
Bonne fête! date de votre anniversaireLes nombres, les jours et les 14 et 15ande des directions'Ardèche 10 et 11 9 8Acknowledgements Tables and charts are a great way to present numerical information in a clear and concise form. This unit explains how to use the Windows calculator to carryNumerical Aerodynamics This course provides a solid theoretical base for various numerical methods used to solve problems in aerodynamics. Results obtained using the methods developed in the course are compared with those from commercially available programs, assessing the advantages and disadvantages of each. Practical classes will be conducted in a fully equipped classroom, so that each student has a computer. During the course, students will actively program in MATLAB. The first few classes of the course are dedi Author(s): Ana Laverón Simavilla,Mª Victoria Lapuerta GonzÃ License information Related content Copyright 2009, by the Contributing AuthorsLicensed under a Creative Commons Attribution - NonCommercial-ShareAlike 2.0 Licence - see - Original copyright The Open University No related items provided in this feed 1.2.3 Working with percentages1 Using tables, percentages and Operating the Windows
You'll begin Stage 1 with the compulsory module Discovering mathematics (MU123)Discovering mathematics::This key introductory Level 1 course provides a gentle start to the study of mathematics. It will help you to integrate mathematical ideas into your everyday thinking and build your confidence in using and learning mathematics. You'll cover statistical, graphical, algebraic, trigonometric and numerical concepts and techniques, and be introduced to mathematical modelling. Formal calculus is not included and you are not expected to have any previous knowledge of algebra. The skills introduced will be ideal if you plan to study more mathematics courses, such as Essential mathematics 1 (MST124). It is also suitable for users of mathematics in other areas, such as computing, science, technology, social science, humanities, business and education.undergraduate.qualification.pathways.Q46-2,module,MU123,,1 (30 credits), which introduces and helps integrate key ideas from statistics, algebra, geometry and trigonometry into your everyday thinking to build your confidence in learning and using mathematics. You'll follow this with three compulsory 30-credit modules: Using mathematics (MST121)Using mathematics::This broad, enjoyable introduction to university-level mathematics assumes some prior knowledge, as described on our MathsChoices website. The course shows how mathematics can be applied to answer some key questions from science, technology, and everyday life. You will study a range of fundamental techniques, including calculus, recurrence relations, matrices and vectors and statistics, and use integrated specialist mathematical software to solve problems. The skills of communicating results and defining problems are also developed. This is not a course for beginners – at the MathsChoices website (mathschoices.open.ac.uk) there are quizzes, sample material and advice to help you determine if this course is right for you.undergraduate.qualification.pathways.Q46-2,module,MST121,,1, Introducing statistics (M140)Introducing statistics::Today, more than ever, statistics is part of our lives. From this key introductory course you will learn how to use basic statistical tools and quantitative methods that are useful in business, government, industry, medicine, the economy, and most academic subjects. Topics covered include: summarising data; examining relationships; randomness and sampling distributions; probability; testing hypotheses; and estimation. Using data from a range of applications, you'll learn practical statistical techniques and fundamental principles, as well as using software and a calculator to analyse data. The skills introduced will be ideal if you plan to study more mathematics courses or if you encounter data in another subject or your daily life.undergraduate.qualification.pathways.Q46-2,module,M140,,1 and Exploring mathematics (MS221)Exploring mathematics::Exploring mathematics builds on the concepts and techniques in Using mathematics (MST121) and uses the same software. It looks at questions underlying some of those techniques, such as why particular patterns occur in mathematical solutions and how you can be confident that a result is true. It introduces the role of reasoning and offers opportunities to investigate mathematical problems. Together with Using mathematics this course will give you a good foundation for higher-level mathematics, science and engineering courses. Even if you don't intend to study further, you will gain a good, university-level understanding of the nature and scope of mathematics.You are advised to be confident with the content of Using mathematics (MST121), or equivalent from elsewhere, before commencing study of this course.undergraduate.qualification.pathways.Q46-2,module,MS221,,1. Together, they provide a broad introduction to various topics in pure and applied mathematics and statistics. You'll develop a variety of skills including problem solving, how to develop and present a mathematical argument and how to represent and interpret statistical data. These are the key skills required for higher level study in mathematics. Stage 2 At Stage 2 you'll develop your skills and knowledge in the following areas: Formal proof, abstract structures, linear algebra, analysis, group theory in Pure mathematics (M208)Pure mathematics::Pure mathematics is one of the oldest creative human activities and this course introduces its main topics. Group Theory explores sets of mathematical objects that can be combined – such as numbers, which can be added or multiplied, or rotations and reflections of a shape, which can be performed in succession. Linear Algebra explores 2- and 3-dimensional space and systems of linear equations, and develops themes arising from the links between these topics. Analysis, the foundation of calculus, covers operations such as differentiation and integration, arising from infinite limiting processes. To study this course you should have a sound knowledge of relevant mathematics as provided by the appropriate Level 1 study.undergraduate.qualification.pathways.Q46-2,module,M208,,1 (60 credits) You'll also continue to broaden your experience in the use of appropriate mathematical software and in explaining and communicating mathematical ideas to others. Stage 3 You'll begin the final stage of this degree with Mathematical thinking in schools (ME620)Mathematical thinking in schools::This course is designed to help you develop your knowledge and understanding of the teaching of mathematics. It is suitable for any Key Stage, and will broaden your ideas about how people learn and use mathematics. There is no formal examination: assessment is based on two tutor-marked assignments and an end-of-module assessment. In order to complete the assessments, you will need access to learners of mathematics. Students on this course have worked with a variety of learners from Key Stage 2 pupils to adults. Places are allocated on a 'first come, first served' basis, so you should register as early as you can.undergraduate.qualification.pathways.Q46-2,module,ME620,,1. This 30-credit module will enhance your skills in communicating mathematical ideas clearly and succinctly and explaining mathematical ideas to others. You'll also focus on how learners' thinking is developed in two of the following 30-credit modules: Developing algebraic thinking (ME625)Developing algebraic thinking::This course is for you if you are interested in developing your knowledge and understanding of the learning of algebra particularly at Key Stages 2–4. It integrates development of the core ideas of algebra with relevant pedagogical constructs and principles, and will extend your awareness of how people learn and use algebra. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. In order to complete the course assessments, you will need access to learners of algebra at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.Q46-2,module,ME625,,1 Developing geometric thinking (ME627)Developing geometric thinking::Develop your knowledge and understanding of the learning of geometry particularly at Key Stages 2–4. This course integrates development of the core ideas of geometry with relevant pedagogical constructs and principles, and will extend your awareness of how people learn and use geometry. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. To complete these assessments, you'll need access to learners of geometry at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.Q46-2,module,ME627,,1 Developing statistical thinking (ME626)Developing statistical thinking::This course will help you develop your knowledge, appreciation and understanding of the learning of statistics particularly at Key Stages 2 to 4. As well as improving your statistical thinking, you'll learn about different teaching approaches, including use of ICT tools such as scientific calculators and computers. There is no formal examination: assessment is based on three tutor-marked assignments and an end-of-module assessment. To complete these assessments, you'll need access to learners of statistics at Key Stages 2–4, which could include adult learners.undergraduate.qualification.pathways.Q46-2,module,ME626,,1 In addition you'll study one more 30-credit module from a range of more advanced mathematical topics currently including: complex analysis, graphs and networks, optimization, groups, metric space theory, number theory, waves, diffusion and variational principles, computer algebra, chaos and simulations, and mathematical methods and fluid mechanics it takes seven years part-time study to complete this qualification, but you can take anything from five (full-time study equivalent) to 16 years
Curriculum & Instruction: Calculus for Teachers EDCI 200 Z02 (CRN: 60947) 3 Credit Hours Jump Navigation About EDCI 200 Z02 This course builds upon prior courses in arithmetic, algebra, and geometry. It is designed to introduce teachers to the branch of mathematics known as calculus in a way that relates calculus to the mathematics taught in the K-8 classroom. Topics include the idea of a limit, the role limits play in K-8 mathematics, and the concept of instantaneous change. Course goals include reinforcing and extending arithmetic, algebra, and geometry knowledge and skills through problem solving involving calculus, and empowering teachers with a deep understanding of how capability in K-8 arithmetic and algebra is foundational for success in higher-level mathematics.
This course is designed to prepare students for advanced study of math at the high school level, including Pre-AP and Pre-IB courses. This course covers all of the concepts of Mathematics 8 with a concentration in Pre-Algebra. The students will use algebra in working with problem-solving, geometry, number theory, linear equations and rational numbers.
Clayton, CA Excel cover the following topics and more in algebra 2: equations and inequalities; linear equations and functions; quadratic equations and factoring; completing the square; quadratic formula; rational functions; exponential and logarithmic functions; sequences and series; trigonometric functio
ClassZone Book Finder. Follow these simple steps to find online resources for your book. the math forum @ drexel university The Math Forum is the comprehensive resource for math education on the Internet. Some features include a K-12 math expert help service, an extensive database of math
Algebra and Trig.: Graphs and Models - Text - 5th edition Summary: The Graphs and Models series by Bittinger, Beecher, Ellenbogen, and Penna is known for helping students ''see the math'' through its focus on visualization and technology. These books continue to maintain the features that have helped students succeed for years: focus on functions, visual emphasis, side-by-side algebraic and graphical solutions, and real-data applications
The Mathematical Funfair 9780521377430 ISBN: 0521377439 Publisher: Cambridge University Press Summary: This book follows the successful Amazing Mathematical Amusement Arcade by the same author. It contains a further 128 puzzles and games designed to challenge people of all ages. The subjects range from matchsticks and coins to chocolate manufacture and expeditions across Dartmoor. The second part of the book contains a commentary giving hints and solutions. Bolt, Brian is the author of The Mathematical Funfai...r, published under ISBN 9780521377430 and 0521377439. Five hundred eight The Mathematical Funfair textbooks are available for sale on ValoreBooks.com, one hundred six used from the cheapest price of $0.01, or buy new starting at $19
Customer Reviews: Very Readable Textbook By Penelope W. Yaqub "Fawzi M. Yaqub" - January 24, 2003 This book is designed to prepare students for upper division math courses-like abstract algebra and advanced calculus-in which mathematical rigor and proofs are emphasized. The authors have made a serious effort to present the material with clarity and sufficient details to make it accessible to students who have completed two courses in calculus. Much of the material covered is fairly standard for such a textbook. Chapters 1-9 are devoted to basic topics from set theory and logic (including four proof techniques: direct proof, proof by contrapositive, proof by contradiction, and mathematical induction), equivalence relations, and functions, as well as a special chapter under the heading, "Prove or Disprove." Chapters 10-13 cover cardinalities of sets and proof techniques applied to results from number theory, calculus, and group theory. In addition, the authors have a web site which includes three additional chapters (Chapters 14-16) dealing with proofs from ring theory, linear... read more A veil has been lifted... By A Student - March 29, 2008 This book pretty much changed my life. My only regret is I didn't find it earlier. What different choices would I have made if I were comfortable with mathematical proofs in high school or early college? I recommend this book to anyone who has an interest in understanding mathematics but for whatever reason never learned or were never taught it by their teachers in school - I'm certain I was never taught these ideas. This book is great for self study and the material should be accessible to most high school students. The book starts off with basic ideas about sets and some logic. The real "Aha!" moment for me was the explanation of implication and biconditional in chapter 2 and how they're used in direct proofs in chapter 3. The examples about even/odd numbers are perfect for someone feeling their way thru new ideas. After reading this book I went back to college as an adult and obtained a BS in mathematics. It's foolish to dwell on what might have been,... read more Book and DVD. A guide to advanced improvisation. This sequel to the best-selling improv book Truth in Comedy is designed to help improv performers move up to the more advanced levels of improvisation ...
Teachers often have too little time to prepare differentiated lessons to meet the needs of all students. Differentiating Instruction in Algebra 1 provides ready-to-use resources for Algebra 1 students. The book is divided into four units: introduction to functions and relationships; systems of linear equations; exponent rules and exponential functions; and quadratic functions. Each unit includes big ideas, essential questions, the Common Core State Standards addressed within that section, pretests, learning targets, varied activities, and answer keys. The activities offer choices to students or three levels of practice based on student skill level. Differentiating Instruction in Algebra 1 is just the resource math teachers need to provide exciting and challenging algebra activities for all students! Showing 2 Reviews: by Andrea Toole on 5/5/2013 A must have resource for Algebra teachers As a junior high math teacher I can say that this is the best resource I have seen for differentiated instruction. The layout of each unit is very thorough. The author outlines each unit with critical vocabulary, unit objectives, launch scenarios, and differentiated lessons and activities. By giving students choices for each lesson, students gain a sense of control over their learning. The activities are engaging and require a lot of mathematical thinking. This book is a must have for every Algebra teacher! by Teresa Boyer on 10/11/2012 Easy to use and Engaging for students As a Special Education Math teacher for the past 10 years, I found Differentiating Instruction in Algebra 1 extremely practical and applicable for students and teachers. The book is laid out in an easy to use manner that any Math educator will find useful. The worksheets, activities and hands on projects are all clearly explained and come with rubrics for easy grading. Additionally, there is a system that allows the teacher to quickly select the appropriate worksheet/activity/activity for the student. Of course the educator view point is important but the most important part of any lesson or educational activity is how students will react. This book has a wonderful variety of activities and lessons that will connect to all learners. The information and activities provide students a connection from math in school to real life. The author provides high interest materials that will engage students. The author has clearly thought about the student engagement aspect as she wrote this book.
MAT 94 Fund of Algebra Course info & reviews This course introduces further applications of algebraic expressions and equations. It includes solutions of linear equations and inequalities, the Cartesian coordinate system, linear equations in two variables and their graphs, systems of linear equations, integer exponents, polynomials, factoring, and an introduction to quadratic equ...
ALEX Lesson Plans Exponential Growth and Decay Description: This lesson on exponential growth and decay involves a variety of ALC (9-12) 3: Use formulas or equations of functions to calculate outcomes of exponential growth or decay. (Alabama) [MA2010] ALT (9-12) 12: Interpret expressions that represent a quantity in terms of its context.* [A-SSE1] [MA2010] ALT (9-12) 21 25: Compare effects of parameter changes on graphs of transcendental functions. (Alabama) Subject: Mathematics (9 - 12) Title: Exponential Growth and Decay Description: This lesson on exponential growth and decay involves a variety ofTitle: Factoring Fanatic Description: ThisStandard(s): [MA2010] AL1 (9-12) 8: Use the structure of an expression to identify ways to rewrite it. [A-SSE2] [MA2010] AL1 (9-12) 8: Use the structure of an expression to identify ways to rewrite it. [A-SSE2) Title: Factoring Fanatic Description: This Thinkfinity Lesson Plans Title: Predicting Your Financial Future Description: In this Illuminations lesson, students use their knowledge of exponents to compute an investment s worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit card payments and debt Subject: Mathematics Title: Predicting Your Financial Future Description: In this Illuminations lesson, students use their knowledge of exponents to compute an investment s worth using a formula and a compound interest simulator. Students also use the simulator to analyze credit card payments and debt. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12Learning ActivitiesThinkfinity Learning Activities Title: Proof Without Words: Completing the Square Description: In this student interactive, from Illuminations, students carry out an interactive, geometric '' proof without words'' for the algebraic technique of completing the square. The page also includes directions and a link to the final solution Subject: Mathematics Title: Proof Without Words: Completing the Square Description: In this student interactive, from Illuminations, students carry out an interactive, geometric '' proof without words'' for the algebraic technique of completing the square. The page also includes directions and a link to the final solution. Thinkfinity Partner: Illuminations Grade Span: 9,10,11,12
Calculus in many ways is the culmination of 17th century European mathematics. Problems in integral calculus (finding complicated areas) and differential calculus (finding instantaneous rates of change and tangents) date back to antiquity, but the genius of Newton and Leibniz was connecting differential and integral calculus with ``The Fundamental Theorem of Calculus''. The calculus is the greatest aid we have to the application of physical truth in the broadest sense of the word. - W. F. Osgood Text This text is written specifically to aid you in understanding the concepts of calculus. Our questions and problems will require you to invoke your understanding rather than to mimic template problems worked in the text, so you should read this text, both before and after each class. (Suggestions and more suggestions about how to read mathematics.) Calculators You should have a graphing calculator available for use in class and on exams. If you are buying a new one, the department recommends the TI-83 or TI-86. You may use other calculators (especially other TIs, Casios, HP or Sharp) as long as you are able to enter a simple program into your calculator and you are comfortable with basic graphing features. Calculators with symbolic algebra capability (e.g. TI-89 or TI-92) will not be allowed during exams. A couple of calculators are on reserve in the library. Exams We will have two in class skills tests, two exams during the semester and a cumulative final exam. The exams during the semester will be given in the evening. The tests and exams are tentatively scheduled for February 20 (in class) March 9 (evening) April 4 (in class) April 25 (evening). The final exam is tentatively scheduled for Saturday, May 20, 10:30 am - 12:30 pm. Please reserve these dates on your calendars. Make-ups will only be given for verifiable emergencies. In particular, make-ups will not be given to accommodate travel plans. Determination of course grade: We will provide you with a number grade on each assignment and on each test, so that you may keep track of your performance. Your instructor will provide you with the details of his/her grading policy. Academic Integrity The academic honesty policy and honor code can be found here. In particular, ``In all academic exercises, examinations, papers, and reports, students shall submit their own work. Footnotes or some other acceptable form of citation must accompany any use of another's words or ideas.'' On each examination students will be required to sign the following pledge: On my honor, I pledge that I have not given, received, or tolerated others' use of unauthorized aid in completing this work. Accessibility Please contact your instructor during the first week of class if you have specific physical, psychiatric, or learning disabilities and require accommodations. All discussions will remain confidential. You can provide documentation of your disability to the Advising Center (204 Johnson Student Union) or call Laurie Bickett (x7027). Class Format We learn by thinking and doing, not by watching and listening. Learning is an active process: it is something we must do, not have done to us. Class time will be a mixture of lectures, discussions, problem solving and presentation of solutions. It is essential that you come to class prepared to do the day's work. In particular, you should read the text and attempt homework before coming to class. Class meetings are not intended to be a complete encapsulation of the course material. You will be responsible for learning some of the material on your own. ``A good lecture is usually systematic, complete, precise -- and dull; it is a bad teaching instrument.'' -- Paul Halmos ``The best way to learn anything is to discover it by yourself... . What you have been obliged to discover by yourself leaves a path in your mind which you can use again when the need arises.'' -- George Polya Prep Problems Preparation problems are meant to help you prepare for classes. Note that preparation problems for a section are assigned at the same time as the reading for that section. This means that you are being asked to read and digest a section and attempt problems before we discuss the material in class. This is intentional. By reading and trying to understand the material before class, you will understand it better after being in class. These problems will often serve as the starting point for class discussions. Prep problems, when assigned, must be done before the beginning of class, by the time deadline given. I hear, and I forget; I see, and I remember; I do, and I understand. -Proverb Homework is assigned for each section. Homework assignments will be collected about twice a week, but you are advised to do the problems from each section right after the class meeting on that section. A selection of the problems turned in will be graded. You are allowed and encouraged to discuss homework and prep problems with others, but (see the College Academic Honesty policy) ultimately you must work the problems and write up the assignment entirely by yourself. As a general rule, you must justify your answers: Explain, or show your work. Let the Golden Rule be your guide in preparing this homework: write it as if you were the person who would have to grade it.
Finite background you need for future courses and discover the usefulness of mathematical concepts in analyzing and solving problems with FINITE MATHEMATICS, 7th Edition. The author clearly explains concepts, and the computations demonstrate enough detail to allow you to follow-and learn-steps in the problem-solving process. Hundreds of examples, many based on real-world data, illustrate the practical applications of mathematics. The textbook also includes technology guidelines to help you successfully use graphing calculators and Microsoft Ex... MOREcel to solve selected exercises. Functions and Lines Functions Graphs and Lines Mathematical Models and Applications of Linear Functions Linear Systems Systems of Two Equations Systems with Three Variables: An Introduction to a Matrix Representation of a Linear System of Equations
Ebook : Exploring Intermediate Algebra: A Graphing Approach Eduspace Student Registration And Enrollment Guide Exploring Intermediate Algebra : A Graphing Approach Intermediate Algebra : A Graphing Approach Summary This free student guide includes the worksheets for the Suggested Activities that are referenced in the Instructor's Annotated Edition. These comprehensive discovery-based exercises, many requiring the graphing calculator, encourage students to interact with the math in a more in-depth way.
Elementary Technical Mathematics: Basic Selectwen/Nelson's ELEMENTARY TECHNICAL MATHEMATICS, Ninth Edition is a well-respected, extremely user-friendly text. It emphasizes essential math skills and consistently relates math to practical applications so students can see how learning math will help them on the job. The applications are drawn from a wide array of technical fields, making the text useful to a broad range of students. Annotated examples and visual images are used to engage students and assist with problem solving. ELEMENTARY TECHNICAL MATHEMATICS ... MOREhelps you develop the math skills so essential to your success on the job! Ewen and Nelson show you how technical mathematics is used in such careers as industrial and construction trades, electronics, agriculture, allied health, CAD/drafting, HVAC, welding, auto diesel mechanic, aviation, and others. The authors include plenty of examples and visuals to assist you with problem solving, as well as an introduction to basic algebra and easy-to-follow instructions for using a scientific calculator. Each chapter opens with useful information about a specific technical career.
Precise Calculator has arbitrary precision and can calculate with complex numbers, fractions, vectors and matrices. Has more than 150 mathematical functions and statistical functions and is programmable (if, goto, print, return, for).
Computational and Applied Mathematics Colleges A program that focuses on the application of a broad range of mathematical and computational methods to modeling, analysis, algorithm development, and simulation for the solution of complex scientific and engineering problems. Includes instruction in numerical analysis, discrete mathematics, operations research, optimization, differential equations, statistics, scientific computation, and applications to specific scientific and industrial topics
This calculator app made its first public demonstration last month at the 12th International Congress on Mathematical Education (ICME). Designed to help dyscalculics, Dyscalculator shows quantities in... More: lessons, discussions, ratings, reviews,... Kanakku is a Web 2.0 application that was designed to be used as a productivity utility. The application is a free online utility that is an advanced calculator with a spreadsheet user interface. Use... More: lessons, discussions, ratings, reviews,... MathPoint is a suite of math tools for students in grades 6 through 12 and college including color graphing, graphing calculator and interactive solving, and an open library for lessons and activit...Tutorial fee-based software for PCs that must be downloaded to the user's computer. It covers topics from pre-algebra through pre-calculus, including trigonometry and some statistics. The software pos... More: lessons, discussions, ratings, reviews,... Lots of real-world data, with descriptions of environmental and mathematical implications; stored in a variety of formats for easy download. Catalogued by mathematical topic and by environmental topi... More: lessons, discussions, ratings, reviews,... Simplesim is suited for modelling of non-analytic relations in systems which are causal in the sense that different courses of events interact in a way that is difficult to see and understand. Exam... More: lessons, discussions, ratings, reviews,... Statcato is a free Java software application developed for elementary statistical computations. Statcato has two main windows: Log, where outputs of computations are displayed; and Datasheet, which coFormulator Tarsia is an editor designed for teachers to create activities in the form of jigsaws, dominos, follow-me cards, etc. for later use in a class. It includes the equation editor for building ... More: lessons, discussions, ratings, reviews,... Geocadabra is dynamic geometry software that supports students learning 2D and 3D geometry, functions and curves (with analysis), and probability. The software was developed in Holland, and is avaiHow do you get started at mathematical modeling? Here is an easy, quick, and engaging activity and a "just add data" interactive Excel spreadsheet to accomplish that first modeling task. The task
Synopses & Reviews Publisher Comments: The book extends the high school curriculum and provides a backdrop for later study in calculus, modern algebra, numerical analysis, and complex variable theory. Exercises introduce many techniques and topics in the theory of equations, such as evolution and factorization of polynomials, solution of equations, interpolation, approximation, and congruences. The theory is not treated formally, but rather illustrated through examples. Over 300 problems drawn from journals, contests, and examinations test understanding, ingenuity, and skill. Each chapter ends with a list of hints; there are answers to many of the exercises and solutions to all of the problems. In addition, 69 explorations invite the reader to investigate research problems and related topics. Book News Annotation: Intended not as a text but to enrich the lives of young mathematicians-to-be who so love the field as to want to read independently in it, and to bring the pleasures of familiarity and fresh discovery into the relaxed hours of more mature mathematicians who can remember being young. Tours both the main elements and many interesting by-ways of an inexhaustible subject. Seven chapters, interrupted at frequent intervals with exercises, problems, invitations to exploration (relevant answers, solutions, hints provided at the end of the book); provides also much of historical and bibliographic interest. Written with sparkling clarity, and from manifestly affectionate seriousness of purpose. (NW)
Essentials of Discrete Mathematics93 FREE About the Book The Second Edition of David Hunter's Essentials of Discrete Mathematics is the ideal text for a one-term discrete mathematics course to serve computer science majors, as well as students from a wide range of other disciplines. The material is organized around five types of mathematical thinking: logical, relational, recursive, quantitative, and analytical. This presentation results in a coherent outline that steadily builds upon mathematical sophistication. Graphs are introduced early and are referred to throughout the text, providing a richer context for examples and applications. Students will encounter algorithms near the end of the text, after they have acquired enough skills and experience to analyze them properly. The final chapter contains in-depth case studies from a variety of fields, including biology, sociology, linquistics, economics, and music.
SJC Student Resources for Guidance on Careers and Graduate Study in Mathematics Career Guidance Mathematics is of central importance to many disciplines, including biological sciences, computer science, economics and finance, statistics, and the social sciences. Hence a wide variety of companies hire mathematicians, including companies in the computer and communication, banks, insurance companies, consulting firms, and all branches of government. Of course, there are career opportunities in teaching mathematics at all levels. On-line Employment Sites The Internet offers one of the best ways to search for employment opportunities for today's graduates. It is also helpful to have your own web page and resume on-line for companies to review. A few of the Internet career sites are linked below: Professional Organizations To find out more information about opportunities, web sites of professional mathematical organizations can be very helpful: The National Council of Teachers of Mathematics The National Council of Teachers of Mathematics is dedicated to improving the teaching and learning of mathematics at all levels. NCTM is the largest nonprofit professional association of mathematics educators in the world. Association for Women in Mathematics Actuarial Careers Actuaries study the mathematics of risk. They employed by insurance companies as well as consulting firms and are often involved in pension planning. The two major professional societies for actuaries: The Society of Actuaries (SOA) and the Casualty Actuarial Society (CAS) have jointly sponsored a web site containing important information about preparing for an actuarial career. Web site BeAnActuary.org Peterson's Guide To Graduate Programs This site will also be of interest to students who are considering graduate school. You can search for information about graduate programs based on a variety of key fields (geographical location, discipline, etc.) About the GRE GRE Website The GRE General Test is primarily a multiple-choice test that most graduate schools use for admission into their graduate programs. You can study for this test and learn some of the "tricks of the trade" that have been developed by educators who have helped thousands of students prepare for this exam, and others similar to it. The Graduate Record Examination Program, also offers 16 Subject Tests (including mathematics), each of which measures achievements in specific fields. Some graduate schools require the subject test.
Math Education for America? analyzes math education policy through the social network of individuals and private and public organizations that influence it in the United States. The effort to standardize a national mathematics curriculum for public schools in the U.S. culminated in 2010 when over 40 states adopted the Common Core State Standards... more... This new, practical book provides an explanation of each of the eight mathematical practices and gives high school educators specific instructional strategies that align with the Common Core State Standards for Mathematics. Math teachers, curriculum coordinators, and district math supervisors get practical ideas on how to engage high school students... more... A Teacher's Guide to Using the Common Core State Standards With Mathematically Gifted and Advanced Learners provides teachers and administrators with practical examples of ways to build a comprehensive, coherent, and continuous set of learning experiences for gifted and advanced students. It describes informal, traditional, off-level, and 21st century... more... This book provides fundamental knowledge in the fields of attosecond science and free electron lasers, based on the insight that the further development of both disciplines can greatly benefit from mutual exposure and interaction between the two communities. With respect to the interaction of high intensity lasers with matter, it covers ultrafast... more... It describes each strategy and clarifies its advantages and drawbacks. Also included is a large sample of classroom-tested examples along with sample student responses. These examples can be used "as is" - or you can customize them for your own class. This book will help prepare your students for standardized tests that include items requiring evidence... more... This practical and easy-to-understand learning tutorial is one big exciting exercise for students and engineers that are always short on their schedules and want to regain some lost time with the help of Simulink.This book is aimed at students and engineers who need a quick start with Simulink. Though it's not required in order to understand how Simulink... more... Topics in Matroid Theory provides a brief introduction to matroid theory with an emphasis on algorithmic consequences. Matroid theory is at the heart of combinatorial optimization and has attracted various pioneers such as Edmonds, Tutte, Cunningham and Lawler among others. Matroid theory encompasses matrices, graphs and other combinatorial entities... more...
Combining theory, methods and instructional activities in one convenient volume, Heddens, Speers and Brahier's Twelfth Edition of "Today's Mathematics" provides a valuable set of ideas and reference materials for actual classroom use. This combined coverage of content and methods creates a long-lasting resource, helping pre-service and in-service teachers see the relationship between what they teach and how they teach. Reflecting recent recommendations from the NCTM Standards, the text emphasizes how to introduce a concept at a given level to expand and reinforce it at successive levels
Homework & Quizzes Homework will not be collected, but the problems on the weekly quizzes will either be chosen from among the homework problems or structured similarly. Thus completing the homework on schedule is in the students' best interests. While working on the homework with classmates is both allowed and encouraged, you need to be able to work out the material yourself, as homework is your best source of preparation for the quizzes and exams. A short quiz will be given in your Tuesday discussion section every week that there is no exam. The two lowest quiz grades will be dropped. Exams There will be three 50 minute midterm exams held during discussion section. Midterms are tentatively scheduled for Tuesday, October 1st, Tuesday, October 29th, and Tuesday, November 26th. If you have a conflict with any of these dates, please contact your instructor immediately. Each midterm will cover material up to the week of the exam. Use of calculators, phones, or any other electronic devices will not be allowed during quizzes or exams. Lecture Notes Slides from lecture will be made available later in the day after each lecture for which they exist. On days when we review the midterms there may be no slides. Important: Slides do not include examples (which are done on the blackboard), nor do they include the detailed explanations of topics which are given during lecture. They are closer to an outline and general guide. They do not replace attending lecture, but merely aid in reviewing the material later. Extra Resources This Trigonometry and Algebra reference card may be useful to students as you work through homework and prepare for exams: Reference Card Students may find the Khan Academy Calculus videos a handy supplement to lecture. Students are especially encouraged to watch the relevant videos if they are unable to attend a lecture: Calculus Videos
0321706064 9780321706065 Using and Understanding Mathematics: Books à la Carte are unbound, three-hole-punch versions of the textbook. This lower cost option is easy to transport and comes with same access code or media that would be packaged with the bound book. Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition increases readers' mathematical literacy so that they better understand the mathematics used in their daily lives, and can use math effectively to make better decisions every day. Contents are organized with that in mind, with engaging coverage in sections like Taking Control of Your Finances, Dividing the Political Pie, and a full chapter about Mathematics and the Arts. This Package Contains: Using and Understanding Mathematics: A Quantitative Reasoning Approach, Fifth Edition, (à la Carte edition) with MyMathLab/MyStatLab Student Access Kit Back to top Rent Using and Understanding Mathematics 5th edition today, or search our site for Jeffrey O. textbooks. Every textbook comes with a 21-day "Any Reason" guarantee. Published by Pearson.
Synopses & Reviews Publisher Comments: Topology continues to be a topic of prime importance in contemporary mathematics, but until the publication of this book there were few if any introductions to topology for undergraduates. This book remedied that need by offering a carefully thought-out, graduated approach to point set topology at the undergraduate level. To make the book as accessible as possible, the author approaches topology from a geometric and axiomatic standpoint; geometric, because most students come to the subject with a good deal of geometry behind them, enabling them to use their geometric intuition; axiomatic, because it parallels the student's experience with modern algebra, and keeps the book in harmony with current trends in mathematics. After a discussion of such preliminary topics as the algebra of sets, Euler-Venn diagrams and infinite sets, the author takes up basic definitions and theorems regarding topological spaces (Chapter 1). The second chapter deals with continuous functions (mappings) and homeomorphisms, followed by two chapters on special types of topological spaces (varieties of compactness and varieties of connectedness). Chapter 5 covers metric spaces. Since basic point set topology serves as a foundation not only for functional analysis but also for more advanced work in point set topology and algebraic topology, the author has included topics aimed at students with interests other than analysis. Moreover, Dr. Baum has supplied quite detailed proofs in the beginning to help students approaching this type of axiomatic mathematics for the first time. Similarly, in the first part of the book problems are elementary, but they become progressively more difficult toward the end of the book. References have been supplied to suggest further reading to the interested student. Synopsis: Synopsis:
Lang's Basic Mathematics is a famous mathematician's look at everything a well-prepared high school student ought to know about math before starting calculus. The exercises are thought-provoking and the solutions are enlightening. There is just enough but not too much drill on each point before moving on, and throughout there is a wonderfully mathematical attitude about the material. Recommended for anyone who has had algebra once and wants to know a lot more about what mathematics is really all about. 4.0 out of 5 starsPreparation for college mathematics from a mathematician's standpoint.Oct. 30 2005 By N. F. Taussig - Published on Amazon.com Format:Paperback Serge Lang's text presents the topics that he feels students should understand before commencing their study of college mathematics. As such, working through this text is a good way for you to supplement what you learned in high school with material that will aid you in studying mathematics in college. Therefore, I particularly recommend it for prospective mathematics majors. The material in the text is well motivated and clearly presented. While Lang explains how to perform routine calculations, he focuses on the underlying structure of the mathematics. The material is developed logically and results are proved throughout the text. However, the presentation of the material is marred by numerous errors, most, but not all, of which are typographical. The problems range from routine calculations to proofs. Many of the problems are challenging and some require considerable ingenuity to solve. Answers to some of the exercises are presented in the back of the text. I should warn you that if you are used to artificial textbook problems in which the correct solution is a "nice" number, you will find that is not the case here. Also, it is useful to read through the problem sets before you begin solving them so that you can do related problems at the same time. The first section of the book covers algebra. Properties of the integers, rational numbers, and real numbers are examined and compared. There is also more routine material on linear equations, systems of linear equations, powers and roots, inequalities, and quadratic equations. A brief discussion of logic precedes a section on geometry. Basic assumptions about distance, angles, and right triangles are used as a starting point rather than Euclid's postulates. This leads to a discussion of isometries, including reflections, translations, and rotations. Area is discussed in terms of dilations. The treatment here is different from that in the high school text Geometry which Lang wrote with Gene Murrow. I found the material on isometries quite interesting. Be aware that the notation and some of the terminology in this section is not standard. The third section of the book covers coordinate geometry. Distance is interpreted in terms of coordinates. This leads to a discussion of circles. Transformations are reinterpreted using coordinates. Segments, rays, and lines are presented using parametric equations. A chapter on trigonometry covers standard topics, but also includes a section on rotations. The section concludes with a chapter on conic sections. Of particular interest is a proof that all Pythagorean triples can be generated from points on the unit circle with rational coordinates. The final section of miscellaneous topics addresses functions, more generalized mappings, complex numbers, proofs by mathematical induction, summations, geometric series, and determinants. The text concludes by demonstrating how determinants can be used to solve systems of linear equations. The eminent mathematicians I. M. Gelfand and Kunihiko Kodaira have also contributed to books intended for high school students. Those of you planning to study mathematics in college would benefit from working through their texts as well. 27 of 28 people found the following review helpful 5.0 out of 5 starsA nice introduction to mathematicsJan. 16 2006 By FePe - Published on Amazon.com Format:Paperback Serge Lang died September 2005, and it was a great loss for many people; he has been a prominent mathematician, who has published many book and articles. He had a very good memory, and it is said that he wrote a book in the course of one weekend on a bet. I don't know if that's true, but you can sense that he feels at home writing about mathematics. Basic Mathematics is suited both for the younger readers who hasn't begun high school yet, and for older readers who needs to refresh their skills. I believe that many people would benefit working through this book before starting in high school, as it will ease and speed up things. The book is structured in a way that it clearly brings the most important of the mathematics which later is to be used. The book has four parts: Algebra, Intuitive Geometry, Coordinate Geometry, and Miscellanous. There are 17 chapter spread over these four parts, which each deals with an important mathematical subject. Of mention are "Functions", "Operations on Points", "Distance and Angles", and "Linear Equations". It's mainly basic mathematical subjects, which are dealt with in an "advanced way", so the author doesn't look down on his reader. Nothing is dwelt upon, but nothing important isn't absent either. Exercises is included in nearly all sections, so that the reader can train himself in both a manipulative and a theoretical level. Som sections has many exercises (which can be tough at times), while some has only three or four. The difficulty is raised, of course, but if you just do the exercises, you'll notice how well the book is structured in that basic techniques are used later in more advanced subjects. Recommended. 19 of 21 people found the following review helpful 4.0 out of 5 starsExcellent overviewMarch 5 2000 By A Customer - Published on Amazon.com Format:Paperback
MATH 2633 This is an archive of the Common Course Outlines prior to fall 2011. The current Common Course Outlines can be found at Credit Hours 4 Course Title Calculus III Prerequisite(s) MATH 2432 with a "C" or better Corequisite(s)None Specified Catalog Description This course includes the study of vectors, solid analytical geometry, partial derivatives, multiple integrals, line integrals, and applications. Expected Educational ResultsGeneral Education OutcomesCourse Content 1. Vectors 2. Partial Derivatives 3. Multiple Integrals 4. Line Integrals ENTRY LEVEL COMPETENCIES Upon entering this course the student should be able to do the following: 1. Investigate limits using algebraic, graphical, and numerical techniques. 2. Investigate derivatives using the definition, differentiation techniques, and graphs. 3. Apply the derivative as a rate of change, optimize functions, use Newton's Method, and sketch curves. 4. Define the definite integral and approximate definite integrals using Riemann sums. 5. State and apply the Fundamental Theorem of Calculus. 6. Graph and use parametric equations. 7. Evaluate integrals using techniques of integration. 8. Use integrals to solve application problems. 9. Solve separable differential equations and apply to elementary applications. 10. Differentiate and integrate algebraic, trigonometric, exponential, logarithmic, and inverse trigonometric functions. Differentiate implicit functions. 11. Investigate the convergence of series and apply series to approximate functions and definite integrals. 12. Apply polar representations including graphs, derivatives, and areas. Assessment of Outcome Objectives I. COURSE GRADE The course grade will be determined by the individual instructor using a variety of evaluation methods such as tests, quizzes, projects, homework, and writing assignments. These methods will include the appropriate use of graphing calculators or PC software as required in the course. A comprehensive final examination is required which must count at least one-fourth and no more than one-third of the course grade. The final examination will include items that require the student to demonstrate ability in problem solving and critical thinking as evidenced by detailed, worked-out solutions. II. COLLEGE WIDE ASSESSMENT This course will be assessed according to the college wide/mathematics department schedule. The assessment instrument will include a set of appropriate questions to be a portion of the final exam for all students taking the course. An out of class project may be an assessment instrument as well. III. USE OF ASSESSMENT FINDINGS The Calculus Committee or a special assessment committee appointed by the Chair of the Math, Computer Science, and Engineering Executive Committee, will accumulate and analyze the results of the assessment and determine implications for curriculum changes. The committee will prepare a report for the Academic Group summarizing its finding.
Extra Examples shows you additional worked-out examples that mimic the ones in your book. These requirements include the benchmarks from the Sunshine State Standards that are most relevant to this course. The benchmarks printed in regular type are required for this course. The portions printed in italic type are not required for this course. understand that numbers can be represented in a variety of equivalent forms, including integers, fractions, decimals, percents, scientific notation, exponents, radicals, absolute value, and logarithms. select and justify alternative strategies, such as using properties of numbers, including inverse, identity, distributive, associative, and transitive, that allow operational shortcuts for computational procedures in real world or mathematical problems. 6. Measure quantities in the real world and use the measures to solve problems. MA.B.1.4.1 use concrete and graphic models to derive formulas for finding perimeter, area, surface area, circumference, and volume of two and three dimensional shapes, including rectangular solids, cylinders, cones, and pyramids. solve real world and mathematical problems involving estimates of measurements, including length, time, weight/mass, temperature, money, perimeter, area, and volume and estimate the effects of measurement errors on calculations. 9. Visualize and illustrate ways in which shapes can be combined, subdivided, and changed. calculate measures of central tendency (mean, median, and mode) and dispersion (range, standard deviation and variance) for complex sets of data and determine the most meaningful measure to describe the data. analyze real world data and make predictions of larger populations by applying formulas to calculate measures of central tendency and dispersion using the sample population data and using appropriate technology, including calculators and computers.
50030224675 May have highlighting or writing and wear to cover. May not include supplemental items like CDs or access codes. We are a tested and proven company with over 900,000 ...satisfied customers since 1997. Choose expedited shipping (if available) for much faster delivery. Delivery confirmation on all US orders.Read moreShow Less More About This Textbook Overview Written in a clear and concise style, this book offers all the review, drill and practice students need to develop proficiency in algebra. In a lecture-format class, each section of the book can be discussed in a forty-five- to fifty-minute class session. In a self-paced situation, the "Practice Problem" in the margins the student to become actively involved with the material before working the problems in the "Problem Set
Faculty & Staff King News While math majors will certainly be captivated by the non-traditional Mathematics course, King's Cryptology class is an increasingly popular option for non-math majors needing to fulfill their math requirements. "Cryptology is the science of secret writing such as devising codes and cracking codes," said Dr. Andrew Simoson, professor of Mathematics for King. "The math used in cryptology is called number theory. Many of the successful codes used in security systems are based upon readily understandable mathematical concepts such as clock arithmetic and prime integers. As such, our King College cryptology course is an ideal core course option for our students. We use real mathematics in a fun way to make and break secret messages, all the while learning about basic security ideas used throughout communication networks - from checking out at the grocery store to protecting our accounts in business transactions." A freshman level course with no course pre-requirements, Cryptology was designed to be a non-traditional mathematics course. Many students choosing to take the course may not have an extensive math background or they might plan to major in mathematics. "All students, whatever their discipline, will learn and have fun in the process," said Dr. Bill Linderman, professor of Mathematics for King. "Cryptology is an accessible course. Many math courses are preparing students for the next level of math. For students who only plan to take one or two math courses, Cryptology is a perfect option. Students see a variety of techniques along with practical applications; it is enjoyable and they are learning something new. "The course initially came by way of a King College alumnus, Tom Barr, who wrote the textbook we use, 'Invitation to Cryptology.'" said Linderman. "About twenty years ago, Tom turned his lecture notes for an undergraduate cryptology course into a popular textbook published by Prentice Hall," said Simoson. "We have proudly used his text for approximately ten years at King College. One semester, Tom took a sabbatical and taught the cryptology course at King. Students loved him and his mathematics." Tom Barr graduated from King College in 1979 with a double major in Mathematics and Physics, and then went on to earn his Ph.D. in Mathematics from Vanderbilt University. For many years, he served as professor and chair of Mathematics and Computer Science Department for Rhodes College in Memphis, Tenn. Currently, he is the managing partner for Entreventures Consulting LLC, which provides consulting services on strategy and operations for entrepreneurial ventures based in bioscience, engineering, information technology, and other scientific areas. "While performing research for the textbook, it was wonderful to immerse myself in mathematical literature and with practitioners of the art of Cryptology," said Dr. Tom Barr. "Since first published, the textbook has been utilized at many colleges and universities not only across the United States but also around the world. "It is my belief that everyone has mathematical abilities," continued Barr. "It is unfortunate, particularly in the United States, that many people are conditioned early in life that math is not for everyone. Cryptology opens up an artful and creative side of mathematics to which many can relate, whether or not they believe they may be a math person." The art and science of encryption has been used throughout history. The course begins with a historical overview of encryption techniques that have been used throughout history. Encryption techniques have also been used during war time throughout the ages. Some of the first mentions of the use of encryption are in the Bible. New techniques for encryption are always being developed to prevent the transfer of classified information from getting into the wrong hands. "Typically, many may not initially think of a math course as being fun, but the course is quite engaging and the students who have taken the course love it," said Linderman. "Math is also not typically associated with creativity. However, Cryptology is basically all pattern recognition where one is trying to disguise something. There are many mathematical techniques that can be utilized during encryption. You do need creativity to do something in a clever way." After students learn techniques of encryption, they are given chance to decode messages and put into practice the techniques they have learned. Students also have opportunities to encode messages themselves. Students from a variety of majors other than mathematics have taken the course; particularly both history and political science majors have shown interest, as well as students minoring in Security and Intelligence Studies. They are able to relate encryption to their own fields of study in areas such as war-time encryption methods. Other fields that easily relate to Cryptology include business and technology and of course more advanced levels of mathematics. "Students, who are studying mathematics and are interested in careers in the federal government such as the National Security Agency, would definitely benefit from taking a course in encryption methods," said Linderman. "There is a high demand for mathematicians within federal agencies. "I have had great success with students who have taken the course, they really seem to enjoy it, and it is a fun course," said Linderman. "If students are only going to take one math course at King, this is definitely one to consider." Tom Barr gives this advice to students who are considering embarking on the mathematical exploration of Cryptology, "Enjoy this experience; enjoy what you will be doing in this course. This is an opportunity to embrace; take the risk. You will be amazed at the rewards."
Discrete mathematics, also called finite mathematics or decision mathematics, is the study of mathematical structures that are fundamentally discrete in the sense of not supporting or requiring the notion of continuity. Objects studied in finite mathematics are largely countable sets such as integers, finite graphs, and formal languages. Concepts and notations from discrete mathematics are useful to study or describe objects or problems in computer algorithms and programming languages.״At this point only one of the planned 15 modules is currently available, that on Groups. 'Taking discrete mathematics? Then you need the Wolfram Discrete Mathematics Course Assistant. This app for discrete... see more 'Taking discrete mathematics? Then you need the Wolfram Discrete Mathematics Course Assistant. This app for discrete math--from the world leader in math software--will help you work through your homework problems, ace your tests, and learn discrete math concepts. The Discrete Mathematics Course Assistant solves your specific discrete math problems on the fly, providing answers to a broad range of subjects. - Do function calculations like domain and range, image and preimage, and inverse and growth - Compute logic problems like minimal forms, implications, propositions, and bitwise operations - Calculate set functions like power set, basic set operations, complement, and Venn diagrams - Use the Number Theory section for division, modular arithmetic, prime numbers, special numbers, and integer functions - Do sequence computations like summation, product, and limit of a sequence - Compute permutation and combinatorics questions, including derangements and permutations of list or finite relations and Pascal's triangle - Use the discrete probability section for Bernoulli trial equations and view statistics on coin and dice probabilities or view various distribution given the probability of success - View information on basic, named, or custom graphs in our Graph Theory section'This app costs $4.99
Sets for Mathematics 9780521010603 ISBN: 0521010608 Publisher: Cambridge University Press Summary: In this textbook, categorical algebra is used to build a foundation for the study of geometry, analysis, and algebra. Lawvere, F. William is the author of Sets for Mathematics, published under ISBN 9780521010603 and 0521010608. Three hundred eighty six Sets for Mathematics textbooks are available for sale on ValoreBooks.com, one hundred two used from the cheapest price of $49.87, or buy new starting at $57.3...3
TI-Navigator Robert Kowalczyk Mathematics Department TI-Navigator Activity • Two-hour IMPULSE Calculus II class • First-time user • Goal — to investigate the potential use of TI-Navigator as a teaching and learning tool Warm-up Activity 1. Plot the function f (x)  x 3  25x 2  x  25 in the best viewing window that you can find?   Student Comments • I liked the program, it was cool to see how the class as a whole found answers, and I'm sure it would help someone who didn't understand the questions. • This was a cool program to use and is a good way to find out how other people do with the problem. • Guaranteed everyone did work. Lesson Plan 1. Revolve the region bounded by y = x, y = 0 and x = 1 about the x-axis. • Poll: What does the solid of revolution look like? • Poll: Find the volume of the solid of revolution. Give answer to 3 decimal places. Lesson Plan (cont.) 2. Let m = your team number (1-12). Find the volume of the solid generated by revolving the region bounded by y = mx, y = 0 and x = 1 about the x-axis. • Enter your team number and volume separated by a comma (e.g., 3, 1.234). • Plot the list of points. • Poll: What type of function do the plotted points represent? Lesson Plan (cont.) 3. Let m = your team number (1-12). Find the volume of the solid generated by revolving the region bounded by y = mx2, y = 0 and x = 1 about the x-axis. 4. Find a formula for the solid generated by revolving the region bounded by y = mxn, y = 0 and x = 1 about the x-axis. Student Comments • The experiment involving TI Navigator was very helpful in studying volumes of revolution. It gave a visual aid that made interpreting the trends in information easy and clear. The interactive element also makes you feel more connected to the class. • I especially liked how it collectively can take the students data and graph it to spot mathematical trends. Lesson Plan (cont.) 5. Let m = your team number (1-12). • Poll: What is the x-value of the point of intersection of the two linear functions y = x/m and y = 5 – x/m. Lesson Plan (cont.) • Find the volume of the solid generated by revolving the region bounded by y = x/m, y = 5 – x/m and x = 0 about the x axis. Student Comments • I thought it was a good program. It made it so that everybody could be involved. By making it so the professor knew which group was giving the answers, it lets the professor know who is actually understanding the material. This program also enables the professor to know what the class needs improvement on and even what they should spend more time focusing on. • It was an interesting diversion from a normal class. And added a break to relax and work with less stress. • Great teamwork project. Student Comments • I enjoyed the program. It allowed everyone to workout the problem and respond anonymously helping shy people to answer questions and learn more about what they are doing. • This was a fun way to work as a team to imply the material learned in class. It was magical. • It wasn't bad. It allowed everyone to participate in class with less pressure than being called on. While people were participating everyone had an easier time paying attention because they had to focus on getting the answer and inputting it. It was also a little more fun. Something different compared to a usual lecture. Student Comments • I think that the TI Navigator Eval went well. I thought that this might become the future of math classes all over the world. Now students can compute answers and that professors can get feedback on whether or not students get the concepts of the lessons. It is interesting to see what your classmates know compared to yourself. • I thought this experience was priceless. I learned a lot about a high tech calculator. It was a fun and invaluable experience. I hope to do this many times over. You are the man, keep on teaching!! Student Comments — Technical • It was a good experience but definitely need one calculator per person. Ends up being one person does it & the other doesn't do anything. • My only negative comment is that once you enter an answer, it cannot be replaced without resetting the question; it would be nice to have on record what the first answer was, but be able to redo it, Other than this the program is great!! • I think that it took too much time to use, especially when we had to enter words into a poll question. Student Comments — Negative • I thought it was a good experience, however I don't see that ever being worked into actual classes. I think it would just cause wasted time. It isn't as practical as just having a teacher go over the material. But other than practicality I think every classroom should have some experience with the TI Navigator. • It seemed a little impractical for day-to-day use in class. There was too much setup time. • It was very easy to get off track like putting in answers that make no sense as are just plain wrong. Student Comments — Negative • Thoroughly enjoyable. Temptation to misbehave is high, but as a teaching and a method for instant reinforcement it was a success. • The experience was fun, and very interactive. I like it because it's a useful tool to measure knowledge. However, it consumes more time from learning a new concept in class. • I thought the experience did a good job of showing the ideas using technology. I don't think it is practical for an everyday classroom experience but it was a good change of pace. Student Comments — Negative • Old fashion (verbal) class participation would have worked just as well. Probably with less confusion and goofing off. • It was interesting, but didn't help with knowledge of the material being covered in class. • Nothing beats teaching on
Mathematics Some people study mathematics for its own sake. They find algebra, calculus, geometry and logic interesting and they love a challenge. Others study it because they will work in a mathematics-related field, such as finance, statistics, physics, engineering, chemistry – the list goes on. Whether you are one of these people or not, any study of mathematics, great or small, will strengthen your reasoning skills. At College of DuPage, we serve all of these different types of people with their different goals. Whether the math courses you take at College of DuPage are transferred to a four-year college or university, are refresher courses that improve your basic skills, or used to prepare you for a new job, you can be confident that you will receive a high-quality education from a strong faculty interested in you successfully learning mathematics. Resources: LearningExpress Library is a comprehensive, interactive online learning platform of practice tests and tutorial course series designed to help patrons—students and adult learners—succeed on the academic or licensing tests they must pass. Spotlight "I was talking with a friend of mine who attends UIC, and he was surprised at the topics that we covered in my Calculus II class at COD that he never learned until later classes. The education provided at COD is first-rate and even better in some areas when compared to big universities. I'm getting a quality yet affordable education at College of DuPage." "I am very grateful for the opportunity to take advanced classes with such great professors as Bob Cappetta and James Africh. COD gave me the opportunity to challenge myself and learn more while still a full-time high school student
Each student should get one evaluation to complete peer editing, keeping in confidence with the teacher and this will enable for class participation Evaluation will be based 4 basic concepts of math skills which entail: 1) Understanding and Comprehending- The skills required to complete assignment 2) Strategy and Process- all work that is shown and provided 3) Mathematical Terms and Notation- recognizing and associating all numerical and variable signs 4)Completion and Punctuality- Assignment finished properly and on time. Evaluation Rubric Poor Developing Exceptional Excellent Score Understanding and comprehending Demonstrates that does not understand what is to be expected when solving/creating equations Demonstrates limited knowledge of what is to be expected when solving/creating equations demonstrates knowledge of what is to be expected when solving/creating equations demostrates continual knowledge of what is to be expected when sovling/creating equations /5 Strategy/Procedures Demonstrates little to no work at all when displaying answers Demonstrates limited work when displaying answers Demonstrates most work when displaying answers Demonstrates all work throughly when displaying answer /5 Mathematical Terms/Notation Demonstrates little to no understanding of mathematical signs and equations
97800307297-Algebra This revision designed for a bridge course between arithmetic and elementary algebra, provides thorough coverage of arithmetic while simultaneously introducing algebra skills. The link between arithmetic and algebra is continually shown. Students are encouraged to develop critical thinking skills, strategies they need to succeed in this course and subsequent mathematical courses, and overall confidence with math. A unique, spiral organization provides coverage that details essential skills for each set of numbers - from whole numbers through rational numbers and integers, Chapter opening applications help students see math in the natural world around them, "Group Activities" promote cooperative learning and "State Your Understanding" exercises sharpen students' writing skills. Each example is worked out with a step-by-step, written explanation provided alongside in a "Strategy" column. "Warm-up" problems, immediately following the examples, actively involve students in the learning process. Scientific calculator exercises have been added, in keeping with NCTM guidelines. Realistic 3-D art helps students visualize application problems and mathematical concepts while "Cumulative Review Tests" after Chapters 3, 6, and 9 reinforce students' grasp of the material and allow self-assessment of
QuickMath is an automated service for answering common math problems over the internet. ... see more QuickMath is an automated service for answering common math problems over the internet. Think of it as an online calculator that solves equations and does all sorts of algebra and calculus problems - instantly and automatically! When you submit a question to QuickMath, it is processed by Mathematica, the largest and most powerful computer algebra package available today. The answer is then sent back to you and displayed right there on your browser, usually within a couple of seconds. Best of all, QuickMath is 100% free! GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On... see more GeoGebra is a dynamic mathematics software for education in secondary schools that joins geometry, algebra and calculus. On the one hand, GeoGebra is a dynamic geometry system. You can do constructions with points, vectors, segments, lines, conic sections as well as functions and change them dynamically afterwards.On the other hand, equations and coordinates can be entered directly. Thus, GeoGebra has the ability to deal with variables for numbers, vectors and points, finds derivatives and integrals of functions and offers commands like Root or Extremum.The GeoGebraWiki is a free pool of educational materials for GeoGebra. Everyone can contribute and upload materials there: International GeoGebraWiki - pool of educational materials for GeoGebra and the German GeoGebraWiki The Dynamic Worksheets GeoGebra can also be used to create dynamic worksheets:Pythagorasvisualisation of Pythagoras' theoremLadder against the Wallapplication of Pythagoras' theorem Circle and its Equationconnection between a circle's center, radius and equation Slope and Derivative of a Function (3 sheets)relation between slope, derivative and local extrema of a functionDerivative of a Polynomial interactive exercise to practice finding the derivative of a cubic polynomialUpper- and Lower Sums of a Functionvisualisation of the backgrounds of Riemann's Integral The Math Solutions Newsletter contains over 20 innovative hands-on activities for teaching a range of mathematics concepts in... see more
Common Errors This is a set of errors that really doesn't fit into any of the other topics so I included all them here. Read the instructions!!!!!! This is probably one of the biggest mistakes that students make. You've got to read the instructions and the problem statement carefully. Make sure you understand what you are being asked to do BEFORE you start working the problem Far too often students run with the assumption : "It's in section X so they must want me to ____________." In many cases you simply can't assume that. Do not just skim the instruction or read the first few words and assume you know the rest. Instructions will often contain information pertaining to the steps that your instructor wants to see and the form the final answer must be in. Also, many math problems can proceed in several ways depending on one or two words in the problem statement. If you miss those one or two words, you may end up going down the wrong path and getting the problem completely wrong. Not reading the instructions is probably the biggest source of point loss for my students. Pay attention to restrictions on formulas This is an error that is often compounded by instructors (me included on occasion, I must admit) that don't give or make a big deal about restrictions on formulas. In some cases the instructors forget the restrictions, in others they seem to have the idea that the restrictions are so obvious that they don't need to give them, and in other cases the instructors just don't want to be bothered with explaining the restrictions so they don't give them. For instance, in an algebra class you should have run across the following formula. The problem is there is a restriction on this formula and many instructors don't bother with it and so students aren't always aware of it. Even if instructors do give the restriction on this formula many students forget it as they are rarely faced with a case where the formula doesn't work. Take a look at the following example to see what happens when the restriction is violated (I'll give the restriction at the end of example.) This is certainly a true statement. Since and . Use the above property on both roots. Since Just a little simplification. Since . So clearly we've got a problem here as we are well aware that ! The problem arose in step 3. The property that I used has the restriction that a and b can't both be negative. It is okay if one or the other is negative, but they can't BOTH be negative! Ignoring this kind of restriction can cause some real problems as the above example shows. There is also an example from calculus of this kind of problem. If you haven't had calculus then you can skip this one. One of the more basic formulas that you'll get is This is where most instructors leave it, despite the fact that there is a fairly important restriction that needs to be given as well. I suspect most instructors are so used to using the formula that they just implicitly feel that everyone knows the restriction and so don't have to give it. I know that I've done this myself here! In order to use this formula n MUST be a fixed constant! In other words you can't use the formula to find the derivative of since the exponent is not a fixed constant. If you tried to use the rule to find the derivative of you would arrive at and the correct derivative is, So, you can see that what we got be incorrectly using the formula is not even close to the correct answer. Changing your answer to match the known answer Since I started writing my own homework problems I don't run into this as often as I used to, but it annoyed me so much that I thought I'd go ahead and include it. In the past, I'd occasionally assign problems from the text with answers given in the back. Early in the semester I would get homework sets that had incorrect work but the correct answer just blindly copied out of the back. Rather than go back and find their mistake the students would just copy the correct answer down in the hope that I'd miss it while grading. While on occasion I'm sure that I did miss it, when I did catch it, it cost the students far more points than the original mistake would have cost them. So, if you do happen to know what the answer is ahead of time and your answer doesn't match it GO BACK AND FIND YOUR MISTAKE!!!!! Do not just write the correct answer down and hope. If you can't find your mistake then write down the answer you get, not the known and (hopefully) correct answer. I can't speak for other instructors, but if I see the correct answer that isn't supported by your work you will lose far more points than the original mistake would have cost you had you just written down the incorrect answer. Don't assume you'll do the work correctly and just write the answer down This error is similar to the previous one in that it assumes that you have the known answer ahead of time. Occasionally there are problems for which you can get the answer to intermediate step by looking at the known answer. In these cases do not just assume that your initial work is correct and write down the intermediate answer from the known answer without actually doing the work to get the answers to those intermediate steps. Do the work and check your answers against the known answer to make sure you didn't make a mistake. If your work doesn't match the known answer then you know you made a mistake. Go back and find it. There are certain problems in a differential equations class in which if you know the answer ahead of time you can get the roots of a quadratic equation that you must solve as well as the solution to a system of equations that you must also solve. I won't bore you with the details of these types of problems, but I once had a student who was notorious for this kind of error. There was one problem in particular in which he had written down the quadratic equation and had made a very simple sign mistake, but he assumed that he would be able to solve the quadratic equation without any problems so just wrote down the roots of the equation that he got by looking at the known answer. He then proceeded with the problem, made a couple more very simple and easy to catch mistakes and arrived at the system of equations that he needed to solve. Again, because of his mistakes it was the incorrect system, but he simply assumed he would solve it correctly if he had done the work and wrote down the answer he got by looking at the solution. This student received almost no points on this problem because he decided that in a differential equations class solving a quadratic equation or a simple system of equations was beneath him and that he would do it correctly every time if he were to do the work. Therefore, he would skip the work and write down what he knew the answers to these intermediate steps to be by looking at the known answer. If he had simply done the work he would have realized he made a mistake and could have found the mistakes as they were typically easy to catch mistakes. So, the moral of the story is DO THE WORK. Don't just assume that if you were to do the work you would get the correct answer. Do the work and if it's the same as the known answer then you did everything correctly, if not you made a mistake so go back and find it. Does your answer make sense? When you're done working problems go back and make sure that your answer makes sense. Often the problems are such that certain answers just won't make sense, so once you've gotten an answer ask yourself if it makes sense. If it doesn't make sense then you've probably made a mistake so go back and try to find it. Here are a couple of examples that I've actually gotten from students over the years. In an algebra class we would occasionally work interest problems where we would invest a certain amount of money in an account that earned interest at a specific rate for a specific number of year/months/days depending on the problem. First, if you are earning interest then the amount of money should grow, so if you end up with less than you started you've made a mistake. Likewise, if you only invest $2000 for a couple of years at a small interest rate you shouldn't have a couple of billion dollars in the account after two years! Back in my graduate student days I was teaching a trig class and we were going to try and determine the height of a very well known building on campus given the length of the shadow and the angle of the sun in the sky. I doubt that anyone in the class knew the actual height of the building, but they had to know that it wasn't over two miles tall! I actually got an answer that was over two miles. It clearly wasn't a correct answer, but instead of going back to find the mistake (a very simple mistake was made) the student circled the obviously incorrect answer and moved on to the next problem. Often the mistake that gives an obviously incorrect answer is an easy one to find. So, check your answer and make sure that they make sense! Check your work I can not stress how important this one is! CHECK YOUR WORK! You will often catch simple mistakes by going back over your work. The best way to do this, although it's time consuming, is to put your work away then come back and rework all the problems and check your new answers to those previously gotten. This is time consuming and so can't always be done, but it is the best way to check your work. If you don't have that kind of time available to you, then at least read through your work. You won't catch all the mistakes this way, but you might catch some of the more glaring mistakes. Depending on your instructors beliefs about working groups you might want to check your answer against other students. Some instructors frown on this and want you to do all your work individually, but if your instructor doesn't mind this, it's a nice way to catch mistakes. Guilt by association The title here doesn't do a good job of describing the kinds of errors here, but once you see the kind of errors that I'm talking about you will understand it. Too often students make the following logic errors. Since the following formula is true there must be a similar formula for . In other words, if the formula works for one algebraic operation (i.e. addition, subtraction, division, and/or multiplication) it must work for all. The problem is that this usually isn't true! In this case Likewise, from calculus students make the mistake that because the same must be true for a product of functions. Again, however, it doesn't work that way! So, don't try to extend formulas that work for certain algebraic operations to all algebraic operations. If you were given a formula for certain algebraic operation, but not others there was a reason for that. In all likelihood it only works for those operations in which you were given the formula! Rounding Errors For some reason students seem to develop the attitude that everything must be rounded as much as possible. This has gone so far that I've actually had students who refused to work with decimals! Every answer was rounded to the nearest integer, regardless of how wrong that made the answer. There are simply some problems were rounding too much can get you in trouble and seriously change the answer. The best example of this is interest problems. Here's a quick example. Recall (provided you've seen this formula) that if you invest P dollars at an interest rate of r that is compounded m times per year, then after t years you will have A dollars where, So, let's assume that we invest $10,000 at an interest rate of 6.5% compounded monthly for 15 years. So, here's what we've got Remember that interest the interest rate is always divided by 100! So, here's what we will have after 15 years. So, after 15 years we will have $26,442.01. You will notice that I didn't round until the very last step and that was only because we were working with money which usually only has two decimal places. That is required in these problems. Here are some examples of rounding to show you how much difference rounding too much can make. At each step I'll round each answer to the give number of decimal places. First, I'll do the extreme case of no decimal places at all, i.e. only integers. This is an extreme case, but I've run across it occasionally. It's extreme but it makes the point. Now, I'll round to three decimal places. Now, round to five decimal places. Finally, round to seven decimal places. I skipped a couple of possibilities in the computations. Here is a table of all possibilities from 0 decimal places to 8. Decimal places of rounding Amount after 15 years Error in Answer 0 $10,000.00 $16,442.01 (Under) 1 $10,000.00 $16,442.01 (Under) 2 $60,000.00 $33,557.99 (Over) 3 $24,540.00 $1,902.01 (Under) 4 $26,363.00 $79.01 (Under) 5 $26,457.80 $15.79 (Over) 6 $26,443.59 $1.58 (Over) 7 $26,442.17 $0.16 (Over) 8 $26,442.02 $0.01 (Over) So, notice that it takes at least 4 digits of rounding to start getting "close" to the actual answer. Note as well that in the world of business the answers we got with 4, 5, 6 and 7 decimal places of rounding would probably also be unacceptable. In a few cases (such as banks) where every penny counts even the last answer would also be unacceptable! So, the point here is that you must be careful with rounding. There are some situations where too much rounding can drastically change the answer! Bad notation These are not really errors, but bad notation that always sets me on edge when I see it. Some instructors, including me after a while, will take off points for these things. This is just notational stuff that you should get out of the habit of writing if you do it. You should reach a certain mathematical "maturity" after awhile and not use this kind of notation. First, I see the following all too often, The just makes no sense! It combines into a negative SO WRITE IT LIKE THAT! Here's the correct way, This is the correct way to do it! I expect my students to do this as well. Next, one (the number) times something is just the something, there is no reason to continue to write the one. For instance, Do not write this as ! The coefficient of one is not needed here since ! Do not write the coefficient of 1! This same thing holds for an exponent of one anything to the first power is the anything so there is usually no reason to write the one down! In my classes, I will attempt to stop this behavior with comments initially, but if that isn't enough to stop it, I will start taking points off.
Websites Students can make use of the extensive video library, interactive challenges, and assessments. Topics include algebra, geometry, trigonometry, calculus, and differential equations. Also has videos of working through problems from the California Standards Math Test (STAR) and SAT Math. Ages 13-18
Buy Used Textbook eTextbook We're Sorry Not Available New Textbook We're Sorry Sold Out More New and Used from Private Sellers Starting at $2ocusing on helping students to develop both the conceptual understanding and the analytical skills necessary to experience success in mathematics, we present each mathematical topic in this text using a carefully developed learning system to actively engage students in the learning process. We have tried to address the diverse needs of today's students through a more open design, updated figures and graphs, helpful features, careful explanations of topics, and a comprehensive package of supplements and study aids. Students will benefit from the text's student-oriented approach. We believe instructors will particularly welcome the new Annotated Instructor's Edition, which provides answers in the margin to almost all exercises, plus helpful Teaching Tips.
Geometry Seeing, Doing, Understanding 9780716743613 ISBN: 0716743612 Edition: 3 Pub Date: 2003 Publisher: W H Freeman & Co Summary: Jacobs innovative discussions, anecdotes, examples, and exercises to capture and hold students' interest. Although predominantly proof-based, more discovery based and informal material has been added to the text to help develop geometric intuition. Jacobs, Harold R. is the author of Geometry Seeing, Doing, Understanding, published 2003 under ISBN 9780716743613 and 0716743612. One hundred nineteen Geometry Se...eing, Doing, Understanding textbooks are available for sale on ValoreBooks.com, nine used from the cheapest price of $60.00, or buy new starting at $408.48
Well, it doesn't say "no math", it says "no math background required." Presumably this means they'll be introducing math concepts in this course as well, starting with 8th grade pre-algebra and ending up at advanced calculus. Seems rather ambitious for a 9-part series of PDFs. I mean there was a whole lot of high school physics that didn't need any math whatsoever to understand, but the math simply helped its application. And as a side note, All they layed out was a puzzle in Linear Algebra. Essentially, linear algebra branches off into some complex systems like encryption and game-theory, but in essence the math behind it is not any more complex than using constants to define variables. A lot of us don't. Even Stephen Hawking has said he's not thrilled with math, and develops most of his ideas visually in his head (source: his book Black Holes and Baby Universes). He only uses the math as the final step, to describe what he sees in his head, not because he enjoys it. He only uses the math as the final step, to describe what he sees in his head, not because he enjoys it. Exactly - in order to describe physics you have to use maths. It is certainly possible to teach the basic concepts but if you think you are learning "graduate level" physics you clearly have no idea what graduate level physics is because that requires maths in order to communicate a full understanding even though the understanding in your head will be in "pictures". For example I can simply tell you that in nature every symmetry produces a conserved quantity. You can think about it for a while and perhap this requires latent 'constants' (I can't remember if the laws of super duper symmetry required some things to be constant absolute and some things to be relatively constant) within the system and means that the system must be in a status of continuous flux. I'd say even more than that. My 1st year of physics I found if painfully difficult to learn what the text was trying to teach... in large part because they special cased everything to keep the required level of math down (otoh at the very same time the Feynman Lectures seemed fairly easy to follow by comparison). Then I took a course in ODE's and PDE's and just about everything I had come across in all of 1st year Physics dropped/popped out as simple examples in this one course. It's the other way around: theoretical physics is even more concerned with math than applied physics. There's a reason why theoretical physics is also called mathematical physics. And I guess you confuse working the numbers with math. Furthermore, you can't separate physics and mathematics because the latter is the formers language. Physics use mathematical tools and most of its notation. However, this serves as a means to an end. That being said, you can also follow Leonard Susskind Stanford lectures on Quantum Physics and learn how Einstein's worked out that E=mc^2 with grade 13 math. Actually, I was working on the ATLAS detector that is in place at the LHC when I started writing for Bureau 42 almost 10 years ago. And I don't know how we profit off of something that's free... My philosophy (which is in lesson nine, and probably should have come sooner; lesson one is more focused on why we need quantum mechanics, and the rest develops over time) is that the concepts and ideas of physics are represented by the math, but not defined by them. Math can certainly point out directions to look at and avenues to explore, and indicate connections between ideas we hadn't previously noticed, but as a student, I always found that the worst possible reason for a physics phenomenon was "because the math says so." This is about getting those ideas across for people who want to learn about the ideas. The ideas covered in the last two lessons are not typically introduced before grad school. (Lesson one starts at the high school level, which is all I wanted to assume from my audience.) Will you be a researcher when you're done? No. Will you have a better understanding of popular science articles relating to quantum physics? I certainly hope so. People profit on free things in the same way that Google profits from giving away free search. I did work with NASA 40 years ago and so I guess that makes me correct also. Wikipedia already has good reference and has some people that maintain the reference well. No point in muddying the waters. [wikipedia.org] It is a difficult science and the relationships are modeled with math. Understanding math is not a suggested dependency it is a prerequisite in any curriculum. As some The Bureau 42 authors don't use the site for profits. Most years, ad banner revenue is about the cost of renewing the domain name, and none of us get paid to post our stuff. We just have fun in our spare time. That's where this came from; when doing my M.Sc., I found I enjoyed teaching in labs far more than I enjoyed doing the actual research. That realization and a case of bilateral elbow tendonitis prompted me to switch to education. Now I teach K-12 (along with other tasks) at the private education My philosophy...is that the concepts and ideas of physics are represented by the math, but not defined by them. Correct - maths is the language of physics and just like any language it is used to express ideas and concepts. As such you can certainly, albeit it crudely, explain the concepts in other languages such as English which lack the precision of maths, in much the same way that you lose a lot of the beauty and depth of Shakespeare if the bard is translated into, say, French. Similarly you are fooling yourself, and more importantly your readers, if you think you have communicated those concepts at the graduate You can get basic points in physics across without using math, but in general if you want to get to the interesting bits you have to be willing to write down some equations. For instance, I can tell you that gravity pulls things together, which is the basic idea, but if you want to know why planetary orbits are elliptical or what the escape velocity from earth is, then you have to do some calculus. In quantum mechanics, the math involved gets deep rather quickly. My personal opinion is that you CAN discuss the principles without going into more details. I think it's pretty easy to explain the concept of a Hilbert space with absolutely no knowledge of calculus, because it's just geometry and common sense. It is problematic to teach physics without math, because you can get it horribly wrong. But you can explain graduate level concepts without math, and you can certainly describe the experiments that prove a formula works, even if you don't go through the complicated math involved in connecting the theory, formula and experiment. It took some time to get from quantum physics to the specific heat of metals in the statistical physics course. But I can tell anyone on the street "look, if we measure the way metals conduct heat, we find that they behave in a certain way. we are only able to explain that if we use quantum physics to describe part of the electrons as a gas moving around inside the metal. classical physics fails.", and that should be enough for a basic idea. Oh certainly. I do agree with you. But discussing the concepts and principles is not graduate level physics, it's conceptual physics, which is what you teach to undergraduate poets and business majors. Nothing wrong with it, and it certainly is very important and worthwhile, but it is not graduate level physics, which is intended to prepare you to do actual novel physics research on your own. Well, by "graduate level physics" Wischon understands "final preparation for research", while I understood "concepts that you don't hear about in school till you get your batchelor's degree". He was saying you can't do research without the math, and I was saying that you can understand a lot of the concepts without the math. Now we agree that we're both right. I think it's pretty easy to explain the concept of a Hilbert space with absolutely no knowledge of calculus, because it's just geometry and common sense. I don't know about you, but lacking a background in Physics, I found it very confusing to jump from integration in 3-D over a Hydrogen probability density wavefunction, to suddenly talking about the infinite-dimensional Hilbert function space. Besides, if the students have a problem visualising that if a < b then a+x < b+x, they may also lack th A Hilbert space is a complete vector space with a scalar (dot) product. The "complete" just means that any infinite sequence of items such that the distance between two successive ones goes to zero has a limit (the set of rational numbers is NOT complete). A trivial example is normal Euclidian 3D space. You don't need to explain anything about functions in order to explain Hilbert space, because any Euclidian space is a Hilbert space. When you do know about functions, you just show that any linear differenti Welcome to Slashdot!:-) It can be somewhat harsh sometimes. I was not the AC who called you asshole, just to clear that up. I don't know why my posting was marked funny, maybe I made a mistake in it (getting rusty). I understand and agree with your explanation that the "where" is 3-D space and the superposition of n different orthogonal wavefunctions Psi(x,y,z) = \sum_i^n C_i psi_i(x,y,z) is a "point coordinate" in n-D (function-)space, all I wanted to whine about in my original posting is, that for a st I think it's pretty easy to explain the concept of a Hilbert space with absolutely no knowledge of calculus, because it's just geometry and common sense. I agree, but to understand why and how a Hilbert space important to QM, you need the features of Hilbert spaces that are unlike Euclidean spaces. To see why this is relevant, take the Uncertainty Principle. It can actually be stated for systems described by finite-dimensional Hilbert spaces (for which one could have a nice geometric intuition), but it's not that interesting. The real understanding (at least for me, and I suspect for most people) only comes when you learn the position and momentum operators, But the fact that we can state the Uncertainty principle in a finite dimensional Hilbert space (as you point out) shows that the Uncertainty principle does rely on properties of infinity. It fact in the finite dimensional case it becomes somewhat easier to understand what is going on. Take the spin-1/2 system which is two dimensional. The eigenvectors any of the operators s_x, s_y or s_z form a basis for the state, however each operator's eigenbasis is not parallel to any other operator's eigenbasis. A vect In infinite dimensional cases things are more complicated because there are various subtitles that can arise. But these subtitles are not at the core of the uncertainty principle, merely a technical distraction that needs to be addressed. I disagree that it's merely a distraction. Yes, when teaching the Uncertainty principle for the first time, it may be a good idea to show it for finite dimensional Hilbert space (in fact, I wish it was done this way, it's so much simpler, like you said!). For your example of a spin-1/2 particle: delta(s_x)delta(s_y) >= abs(<[s_x,s_y]>)/2 It's very nice for an introduction, and it can be derived with very simple math, but you can't honestly say it's graduate-level Physics if you can't even do it for My personal opinion is that you CAN discuss the principles without going into more details And that discussion would be as useful as discussing topics like OO-programming principles with someone who has never written a line of code. Or like discussing the issues with MySQL with someone who has never used a database or written a line of SQL. You can make someone think they "understood" the physics, when, in fact, he haven't understood anything. Much like how you "explain" how you fixed a particular tricky bug to the upper management. A driving instructor can teach someone to drive without knowing all the math behind it. They can also do some amount of research, perhaps learning the math as they go along. given that physics is still a theoretical part of science, by not teaching the current application and instead focusing on the more fundamentals you may well be equipping people far better to then go on to push physics in new directions that 'indoctrinated' individuals wouldn't even think of, because they don't even know that there is a box to think outside of. Perhaps you mean Albert Einstein [wikipedia.org]? He was exceptionally gifted in mathematics and physics, from an early age, and studied both at the Polytechnic in Zurich. If you mean to imply that Einstein was just some schmo with only grade-school level ability in maths then you are barking up the wrong tree. You could also say that he was fairly "indoctrinated", in that he had knowledge of current (har-dy-har;) Physics theories, so your implication that ignorance of prevailing theories freed him to embrace novel ideas so, what your saying is, that he most probably had a good idea of how things worked, far beyond what he was taught and most probably before he was taught it? he may well have taken a lot of stuff on board, thought 'interesting' somewhat useful, but best broken at best, so don't do to much with it. lets go do some patent checking instead. and while I'm at it, seeing as it all looks a bit to crap to really bother with the rest of them, I'll just do some stu I did give a small spin example, but it looks like I may have forgotten to submit after preview. in brief: angular momentum. Can be explained to a good degree of understanding without knowing the math. then but in devisions of 1/4, which relates to spin number (could give formula). Then spin direction as other component. then some experimental examples, and some entanglement, and demonstrations. touch a bit on the standard model I suppose, but it's known to be a bit of a fudge etc... so is that really teaching phy What's with all the negative comments? Anyone look at the lecture 1 PDF? Anyone actually do physics for a living? As I write this, I'm staring at a whiteboard drawing of three equations in my den; E=mc^2, E=hc/lambda, r=2GM/c^2. They are there show my 13 year old niece how much energy a human body is equal to, a question she asked after watching K-PAX two nights ago on Netflix. Then she asked how much energy is in a single photon, then she asked how much energy is in a black hole. All questions a little girl might ask had she been exposed to basic ideas in modern physics, aka television. Does she fully understand quantum mechanics, probably not. Does she she understand the jist with her pre-algebra background, sort of. Did she learn something and does she feel 'smarter' now... you betchya! She annoyed my sister for hours about how a tree could power the whole world, or a tiny little bug could drive her car for years. My explanations, her worlds, and now a scientist in the making. My point, you don't need to be able to derive Maxwell from F=ma, as my advisor's advisor did while backpacking across the Rocky Mts., to understand nature at its most simple, what you see is what you get, level. You also don't need to be some bearded mystic holed up in a university to appreciate, understand, or even contribute to our vastly poor knowledge of nature. I don't disagree with you, and I was not intending to claim that the lecture PDFs are not worthwhile. But I stand by my claim that they do not teach graduate level* physics. They may teach the concepts that are dealt with in graduate level physics courses, but a graduate level physics education prepares one to teach or do research, which this sort of physics-without-much-math most certainly does not do. Physics that uses no more math than this is not graduate-level physics. I call bullshit, politely though. Not only can it be done, you've got to understand what you're doing well enough to step out of the higher level math. One of the most spectacular instances teaching I ever witnessed was at Purdue, where a class on relativity for non-science students was held, using nothing more than F = ma and a^2 + b^2 = c^2. Anyone can become an expert and talk expert to other experts and future experts. The outside. Anyone who can understand a field at the expert level but can explain it in non-specialized language without polysyballic words probably understands it far better than those in the specialists' club. An often misstated (but flexible enough to still work) quote from Ernest Rutherford is "An alleged scientific discovery has no merit unless it can be explained to a barmaid." There's people out there doing this thing which 'can't' be done. Go listen to them. I'm not saying that it isn't physics because you aren't using math. I'm not saying it shouldn't be done. I think that explaining physics on a conceptual level to non-scientists is a really, really good plan. But, a graduate level education (in any field) is intended to prepare you to teach and to do novel research. You cannot teach physics, and you certainly can't do novel physics research, if you don't know any more than grade school math. It is simply impossible. So, the people who are creating what I will add to this one of the greatest physicists around, Albert Einstein, did not know the necessary maths when he wrote his first theory. The maths was done for him, though he did later learn to do mathematics. Science as we know it is not about the maths, but being able to produce a solid theory that stands up under scrutiny. Using scientific process helps add weight and often mathematics can provide a calculable way of showing numerical relationships, but if the reasoning for the theory is sound then the The ou It's hit the concepts dealt with at the graduate level, but I left the math out to make those concepts accessible to people who don't have the heavy mathematical background. I'm half way through writing next year's summer school (linear algebra, full mathematical glory, ending with tensors), and the 2012 curriculum will be Einstein's Relativity and have two parts to each lesson. The first part will be all conceptual, like this, and the second part will have all of the math. 2013 will be real analysis, 2014 assessment theory, and years beyond that haven't been pinned down. The "Bureau 42 teaches" link at the side has everything along these lines listed, with links if they've already been posted. Sorry, I did not mean to degrade your wonderful efforts. I think that a well written accounting of conceptual physics is an excellent thing to have. However, I would caution you to take great care not to overstate what your students are receiving. There are already way too many people out there who think that you don't really need math and rigor, that they can do physics if they just think really hard about weird things, and that "the scientific establishment" only uses math in order to maintain some ima Okay, I can see that point. I admit the language used was imprecise; I was trying to balance between describing what I was doing and keeping it short enough to work as a Slashdot snippet. Perhaps I leaned too far one way. The source article specifies "graduate level physics concepts" instead of just "graduate level physics." This was a submission issue, rather than a source material issue. Grade school level math. The most complicated math in the series is this: "if a times b is less than 6, and we measure a to be 2, then b must be less than 3." If you can follow that, you'll be fine. Physics that uses no more math than this is not graduate-level physics. Physics that uses no more math than this is not college-level physics, unless you want to count the first week or two of the not-for-majors version of the 100-level stuff. Even that requires a fairly decent grasp of algebra and trigonometry. You can talk about quite a few concepts in college-level physics provided that you do so in relatively broad terms. But reaching graduate level physics in any honest sense requires quite a bit of advanced math. Further, it is not something you can learn in any real sense Griffiths' text is commonly used, but I wasn't thrilled with it. I'm of the "do the math right or not at all" mentality, and his use of the probability distribution with operators instead of the psi* operator psi proper methodology in the first few chapters forms bad habits with students. It only works because he carefully chooses examples whose operators do not involve derivatives. His electricity and magnetism textbook is fantastic, and his particle text is great, but I'm not happy with his quantum tex Yeah, introductory quantum mechanics is introduced typically in second year, and then more detailed versions including Dirac notation show up in third and fourth year. The graduate level is where relativistic implications are usually taken into account, unless you take senior undergraduate particle physics. They're not trying to train physicists just help laypeople understand. This is precisely what makes it not graduate level physics, because graduate level physics is trying to train physicists. I'm all for teaching people about physics on a layperson sort of level; I think it is a phenomenally great thing to do. I'm not in favor of lying to them about just what it is that they are learning. Car analogy (possibly bad, as always): I think that making people take a driver's ed program so they can get a license is a really good idea. I think that telling them that their drive Logic at his purest form is not dependent on ANY material. Philosophy involve more than logic is dependent on logic, the same way physic involve more than math but is dependent on math. So philosophy is applied logic to idea and existence etc... Logic is not a subset of philosophy, philosophy is USING logic. Otherwise you could declare math a subset of physic. So at the core of EVERYTHING, logic, is there present and the most purest of all material. What they don't tell you is the course is a superposition of a nine-part series, and that you can't know what course you are going to get until you actually open the pdf file, which is a pretty dicey proposition these days. Mathematics is the primary language by which physicists describe the world around us. Discussing post-16th century physics in any other terms is like discussing poetry purely by means of interpretive dance. It's more like discussing modern dance by performing it as a sequence of ballet moves. Or deconstructing poetry. Or using your words instead of your numbers. In the end, mathematics is a means of manipulating facts to reveal other facts in a deterministic manner (even if they're facts about non-deterministic things). If you can't subsequently describe both sets of facts in terms a non-mathematician can understand, you haven't reached a result that non-mathematicians will know about, much less be able to form How do the classes go? Something like this I imagine: "Revenge is a dish best served cold - and it's very cold in the vacuum of space. Around 2.725 Kelvin; which is -270 deg Celcius. That is minus 27 tens, and that's terrible....ly cold." "KAAAAAAAAAAAAAAAAHN!" I read the first lesson, and while it's interesting, so far I'm not impressed. It presents some of the problems with classical physics, but it seems to focus on the wrong problems. The first problem it mentions is that information can't travel faster than the speed of light-- but to address that problem you need more than just introductory quantum mechanics, you need relativistic quantum mechanics, and I just don't think you can get to Dirac's equation in a nine part series without math. Then they ask a question about nuclear physics ("what holds the nucleus together?"), for which, to even understand the question correctly, you need some information that the reader doesn't have yet (for example, what do they mean when they say that the only macroscopic force is electromagnetic? In fact, all the forces you do experience in everyday life actually are electromagnetic in nature... but you need quantum mechanics to really understand that! It sure isn't obvious that the force that keeps you from falling through the ground to the center of the Earth is electromagnetic). And this really isn't fundamental to quantum mechanics, either. Next, the nucleus mass question is, once again, a question of relativity and not quantum mechanics (although at least one that can be answered without resorting to the Dirac equation!). And the final question seems to require addressing the equation of state in ultradense matter at the beginning of the universe! Good luck with explaining that with grade school math. actually, someone who knows the subject can tell when a particular line of though will lead you where there be dragons. and they're usually right. also: "how can I be impressed if what you're saying has no obvious connection to what I understand as reality?" Love is the behavior which is the result of chemical reaction in the brain and the body (neurotransmitter, hormone, neuron state etc...). Love IS based on chemistry , and therefore fully based on electromagnetic force, QM. All our emotion are based on chemistry. A complicated system, surely, one for which we have only superficial model definitively, but in absence of evidence to the contrary, those are definitively system where only biochemistry is at play. It is a response to the assertion that all ordinary forces are electromagnetic. Without Pauli exclusion, there are no solid bodies, but only points neutralized out of the plasma. Then perhaps you should have said this instead. My view is that Pauli exclusion is not even similar to the claim "where the premise of spacetime, the existence of the metric, goes out of range". I believe modern physics claims that the spacetime metric is good down to the so-called Planck scale, which is considerably smaller than the scale of the electron clouds of atoms. Instead Pauli exclusion is a feature of quantum models which differ from classic models not in the spacetime metric, but in the mathemati The thing that didn't impress me (with all the questions being asked and all) is that they state the mass of the nucleus is not equal to the sum of the components and in fact is usually less. Okay, good observation but my immediate question is how did they measure the mass of the components individually and then the nucleus as a whole? Then once I understood that I'd wonder how could the result of this method be affected in ways not originally intended that would could the mass of the nucleus to appear to b The protons have a mass that's relatively easy to measure. The charge is very well known, as is the interaction of moving charges with magnetic fields. If you fire a proton through a magnetic field, it will be accelerated into a circular motion, and the easily-measured radius of the circle (visible in a bubble chamber) will indicate what the mass is. For neutrons, it's much harder. Early measurements at the time were imprecise compared to today's. Now that we better understand the mechanism of radioactiv I was expecting something like an introduction to really basic quantum stuff, like superposition, entanglement, measurement, etc. This can actually be done the right way with very little math, like this excellent series of lectures from Stanford [youtube.com], where you can learn something that is actually right, not just analogies. Instead, based on what's in the first lesson, it looks like it will try to talk about a lot of things, explaining none of t So far, for an introduction, there's no bad explanation. But it seems they're promising to explain a lot more that is reasonable to expect: are they really planning to go all the way up to relativistic QM without math? If not, why bring up relativity at all? There's a lot of QM to explain before getting into that: superposition, entanglement, Bell states (to see what's really weird with entanglement), measurement, uncertainty principle, etc. And that's just the foundation, then (based on Lesson 1) it seems t By abstracting all the mathmatical conjecture. But then, you're left with "A brief history of the universe", and I suppose, tack an exam (of course, abstracting from the math), and you now have a "graduate-level" course. But then, you're left with "A brief history of the universe", and I suppose, tack an exam (of course, abstracting from the math), and you now have a "graduate-level" course. I humbly submit Feynman 1988 [princeton.edu] as a counterexample. Therein, the author describes the basics of quantum electrodynamics using what appears to be little more than grade school mathematics. I write "appears to be" because his presentation amounts to an extremely casual exposition of elementary ideas from rather more advanced mathematics (comp Yes, we know this (see here [wikipedia.org]). But the whole point of complex algebra is to go the other way, namely from geometry and scaling and little arrows to algebra as a way of simplifying calculations and improving understanding. The status quo before the discovery of analytical geometry was Greek style synthetic calculations, which are much too cumbersome in the presence of viable alternatives. As a CS student I have not studied much physics; but I'm a very curious guy so I could not resist to follow the link. Their requirements are: average level intelligence, basic maths and a PDF reader. Sounds like perfect for me... or too perfect? W. Blaine Dowler took his time to write in LaTeX, which automatically made me think it can be trusted - don't ask me why. But, on second thoughts, this doesn't sound right. At back at school we were taught that physics has laws and mathematical models, which are an The paper is simply packed with logical fallacies. Yes, many of these are commonly accepted in the physics community, and are indeed the cause of the current pithy state of physics research, that continues to leap from one absurd conclusion to the next, discarding logic in the process. But is it really a good idea to pollute the minds of the next generation with them? The paper starts with a misconception right from the start: > Nothing, not even information, can travel faster than the speed of light. I would absolutely recommend David J. Griffiths' "Introduction to Quantum Mechanics". It's blue and has a cat on the cover (and a dead cat on the back), hence it is sometimes known to physicists as "the cat book". Multivariable calculus, linear algebra (with a small emphasis on abstract algebra if possible), and diffeq (partial, not just ordinary), are exactly the math that you need to grok everything in Griffiths. It is one of a few standard undergrad (usually sophomore or junior level) texts, and, in m
Math has practical applications at work By Sue Atkinson, Ph.D. College Professor Created: Thursday, June 27, 2013 11:26 a.m. CST Early man discovered math concepts as part of understanding the processes in life, and experimented with applications, finding new uses over time, developing science and engineering, and increasing the sophistication of the math information processing system as the need arose. The U.S. has lost the ability to continue this historical process because of the removal of concepts from the curriculum and teacher training programs the last 50 years. Not only have science and engineering programs become dependent on international students to keep up standards (and hope they stay in the country to work), but training programs have added applied math courses and continue to struggle with student skill sets in math applications
A Complete Textbook of Mathematics for Term -II absolutely based on new pattern of examination CCE for both Formative and Summative Assessments with following key features and worksheets at the end of each chapter for practising the problems based on : True / False \| Fill in the blanks \| Match the Columns \| MCQ's \| Asserstion Reasoning \| Comprehension \| Riddles \| Special Worksheets \| Activities for Lab Manual \| Important Facts \| Ten CBSE Model Papers We are in a unique position to offer you the most comprehensive platform for your advertising needs with over 500,000 students as our members from India. They are a unique audience who are young, educated, ....
8th Grade Mathematics Curriculum Course Description: The 8th Grade Mathematics course is aligned with the Mathematics Core Curriculum of MST Standard 3 of the New York State Learning Standards. It focuses on algebra with a continuation of students' skills in solving linear equations. Course Essential Questions: TBD 8th Grade State Assessment Information: Approximate Percentage of Questions Assessing Each Strand Strand: Percent: Time allotted: Units: Number Sense and Operations: 11% (2 weeks) (Unit 1) Algebra: 44% (9 weeks) (Unit 2, Unit 2A, Unit 5) Geometry: 35% (7 weeks) (Unit 3, Unit 4) Measurement: 10% (2 weeks) (Unit 1) Probability and Statistics: 0% (0 weeks) Additional Information: 8th Grade Formula Sheet 8th Grade Mathematical Language AECSD 8th Grade Mathematics.doc 1 Post-March 7th Grade Performance Indicators The 7th Grade performance indicators below are denoted by the state as post-test. Therefore, students will be responsible for this knowledge of the 8th Grade assessment. Attention should be given to them during the normal course of instruction or during review. 7.A.2 Add and subtract monomials with exponents of one 7.A.3 Identify a polynomial as an algebraic expression containing one or more terms 7.A.4 Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation 7.A.7 Draw the graphic representation of a pattern from an equation or from a table of data 7.A.8 Create algebraic patterns using charts/tables, graphs, equations, and expressions 7.A.9 Build a pattern to develop a rule for determining the sum of the interior angles of polygons 7.A.10 Write an equation to represent a function from a table of values 7.G.5 Identify the right angle, hypotenuse, and legs of a right triangle 7.G.6 Explore the relationship between the lengths of the three sides of a right triangle to develop the Pythagorean Theorem 7.G.8 Use the Pythagorean Theorem to determine the unknown length of a side of a right triangle 7.G.9 Determine whether a given triangle is a right triangle by applying the Pythagorean Theorem and using a calculator 7.M.1 Calculate distance using a map scale 7.M.5 Calculate unit price using proportions 7.M.6 Compare unit prices 7.M.7 Convert money between different currencies with the use of an exchange rate table and a calculator AECSD 8th Grade Mathematics.doc 2 8th Grade Local Math Standards Numbering Key: Local.Grade level.Mathematics strand.standard # e.g. L.8.N.5 (L = local; 8 = 8th Grade; N = Number Sense and Operations; 5 = 5th standard) Number and Operations: L.8.N.5 Estimation Estimate a percent of a quantity in context; justify the reasonableness of an answer using estimation L.8.N.12 Percent, Ratio, Read, write, and identify percents less than 1% and greater than 100%; apply Proportion percents (including tax, percent increase and decrease, simple interest, sale prices, commission, interest rates, and gratuities); use proportions to convert measurements between equivalent units within a given system (metric or customary). L.8.N.13 Power and Roots Use calculation rules for powers for multiplication and division; evaluate expressions with integral exponents Algebra: L.8.A.1 Patterns and Represent data relationships in multiple ways (algebraically, graphically, Representations numerically (in a table), and in words) and convert between forms (e.g. graph a linear equation using a table of ordered pairs); translate between two-step verbal and algebraic statements (expressions, equations, and inequalities). L.8.A.2 Solving Solve multi-step inequalities that include parentheses (distributive property), Equations and variables on both sides of the inequality, and multiplication or division by a Inequalities negative number and graph the solution on a number line; solve systems of linear equations graphically (use only equations in slope-intercept form with integral solutions). L.8.A.3 Expressions numerator); factor a GCF out of a polynomial; factor trinomials (with a = 1 and c having no more than 3 sets of factors). L.8.A.4 Functions Define a function using correct terminology (domain and range); determine if a relation is a function. L.8.A.5 Quadratics Recognize the characteristics of quadratic equations in tables, graphically, algebraically and in words, and distinguish between linear and quadratic equations. Geometry: L.8.G.1 Shapes and Construct the following figures: congruent segment, congruent angle, Figures perpendicular bisector, angle bisector L.8.G.2 Transformations Describe and identify transformations (rotation, reflection, translation, dilation) and Symmetry using proper notation; perform rotations of 90 and 180 degrees, reflections over a line, translations, and dilations of a given figure; identify properties preserved under each transformation L.8.G.4 Points, Lines, Identify pairs of vertical, supplementary, and complementary angles and use and Angles relationship of pairs to find angle measures (including algebraically); determine the relationship between pairs of angles formed when parallel lines are cut by a transversal and use relationships to find missing angle measures (including algebraically). L.8.G.7 Coordinate Given a line on a graph: determine its slope and explain its meaning as a constant Geometry rate of change, and determine and explain the meaning of the y-intercept; graph a line from a table of values or from an equation in slope-intercept form; determine the equation of a line given its slope and y-intercept Problem Solving: AECSD 8th Grade Mathematics.doc 3 L.8.PS.1 Organization Analyze situations (identify the problem, identify and obtain needed information, and generate possible strategies) and organize work to solve problems (e.g. use Auburn Problem Solving Process). L.8.PS.2 Strategies Solve problems using a variety of strategies and representations (e.g. using proportions, solving a similar or simpler problem, working backwards, and finding a pattern) and recognize that while there may be more than one way to solve a problem, different methods have advantages and disadvantages L.8.PS.3 Reflection Estimate possible solutions; examine solution to ensure it is reasonable in context of problem; compare solution to original estimate. Reasoning and Proof: L.8.RP.1 Observe patterns, make generalizations, and form and evaluate conjectures; support or refute statements with valid arguments including the use of mathematical language and counterexamples (if appropriate). Communication: L.8.CM.1 Decode and comprehend mathematics expressed verbally and in (technical) writing; clearly and coherently communicate mathematical thinking verbally, visually, and in writing using appropriate mathematical vocabulary and symbols; organize and accurately label work. Connections: L.8.CN.1 Recognize and use connections among branches of mathematics and real life (e.g., make and interpret scale drawings of figures or scale models of objects, determine profit from sale of yearbooks, use tables, graphs, and equations to show a pattern underlying a function) Representations: L.8.Rep.1 Represent mathematical ideas in a variety of ways (verbally, in writing, pictorally, numerically, algebraically, or with physical objects); switch among different representations; explain how different representations can express the same relationship but may differ in efficiency. AECSD 8th Grade Mathematics.doc 4 Math 8B Unit Sequence and Timeline: Unit 1 Percents and Proportions (N.12) Length: ~ 3 weeks Timeframe: Early September to end of September Unit 2 Algebra (A.3 + prior knowledge) Length: ~ 3 weeks Timeframe: End of September to mid-October Unit 2A Solving Equations Length: ~ 4 weeks Timeframe: Mid-October to Thanksgiving Unit 3 Special Angle Pairs (G.4) Length: ~ 2 weeks Timeframe: Beginning of December Unit 4 Transformational Geometry (G.2) Length: ~ 3 weeks Timeframe: Middle of December to start of January Unit 5 Polynomials (A.4, N.13) Length: ~ 5 weeks Timeframe: Middle of January to End of February (note that mid-term falls in middle of unit) Mid-term Review and Administration Length: ~ 1 week Timeframe: End of January (24th and 25th ?) State Assessment Review and Administration Length: ~ 2 weeks Timeframe: Early to mid-March (State Assessment: 3/14 and 3/15) Unit 6 Linear Equations Part 2 (A.2, A.1, G.7) Length: ~ 4 weeks Timeframe: Mid-March to mid-April Unit 7 Quadratics (A.2) Length: ~ 1 week Timeframe: End of April Unit 8 Constructions (G.1) Length: ~ 1 week Timeframe: Beginning of May AECSD 8th Grade Mathematics.doc 5 This Page Intentionally Left Blank AECSD 8th Grade Mathematics.doc 6 Unit 1 Percents and Proportions Length: ~ 3 weeks Timeframe: Early September to end of September State Standards (Shaded statements are identified as Post-March Indicators): 8.N.3 Read, write, and identify percents less than 1% and greater than 100% 8.N.4 Apply percents to: Tax, Percent increase/decrease, Simple interest, Sale price, Commission, Interest rates, Gratuities 8.M.1 Solve equations/proportions to convert to equivalent measurements within metric and customary measurement systems Note: Also allow Fahrenheit to Celsius and vice versa Local Standards: L.8.N.12 Read, write, and identify percents less than 1% and greater than 100%; apply percents (including tax, percent increase and decrease, simple interest, sale prices, commission, interest rates, and gratuities); use proportions to convert measurements between equivalent units within a given system (metric or customary). L.8.N.5 Estimate a percent of a quantity in context; justify the reasonableness of an answer using estimation Big Ideas: The fractional or decimal equivalent of a percent may be more efficient for solving a problem. Percents show up every day. Essential Questions: What is the difference between 1/2, .5 and 50%? Why would a store offer a 20% off coupon on top of a 10% off sale rather than just a 30% off sale? Prior Knowledge: to understand concept of whole percents from 0% to 100% to solve a proportion Unit Objectives: to read, write, and identify percents less than 1% and greater than 100% to convert among percents, decimals, and fractions to apply percents to solve a variety of problems to use proportions to convert measurements Resources: SFAW 8th Grade Course 3 – Chapter 6.4 – 6.6 Review Template (No Calculators): Adding and subtracting monomials Simplify: 3x + 4x 7x – 5x AECSD 8th Grade Mathematics.doc 7 Simplify: 5x + 3x 5x – x Simplify: 18x – 7x 4x + 2x Solve proportions Solve: 3/2 = x/4 Solve: 18/27 = x/9 Solve: x/6 = 15/18 Convert among %'s, decimals, and fractions. Convert 2/5 to a decimal and a percentage. Convert .45 to a percentage and a fraction. Convert 6% to a decimal and a fraction. The diagram to the right is an approach to help students recall and organize the relationship between parts, wholes, and percents. It represents that a part can be divided by the whole or the % to get the other, while the whole can be Part multiplied by the percent to get the part. For example, for 3/4, the part is 3, the whole is 4 and the percent is 75%. Three divided by 4 is 75%. Three divided by 75% is 4. Whole % Four times 75% is 3. AECSD 8th Grade Mathematics.doc 8 Unit 2 Algebra Length: ~ 3 weeks Timeframe: End of September to mid-October State Standards (Shaded statements are identified as Post-March Indicators): 8.A.1 Translate verbal sentences into algebraic inequalities 8.A.2 Write verbal expressions that match given mathematical expressions 8.A.3 Describe a situation involving relationships that matches a given graph 8.A.4 Create a graph given a description or an expression for a situation involving a linear or nonlinear relationship Local Standards (Stricken text is covered in a separate unit): L.8.A.1 Represent data relationships in multiple ways (algebraically, graphically, numerically (in a table), and in words) and convert between forms (e.g. graph a linear equation using a table of ordered pairs); translate between two-step verbal and algebraic statements (expressions, equations, and inequalities).L.8.N.5 Estimate a percent of a quantity in context; justify the reasonableness of an answer using estimation Big Ideas: A variable represents an amount that can change. Algebraic statements can have real world meaning. Real world situations can be modeled algebraically and graphically. Essential Questions: Why would you want to model a situation algebraically? Prior Knowledge: (may need to be taught '05-'06) to distinguish between an expression and an equation to know key words and concepts for the four operations to use order of operations using the set of real numbers to add, subtract, multiply and divide integers to plot an ordered pair on the coordinate plane Unit Objectives: to know that a variable represents an amount that can change to use a variable to represent an unknown quantity to translate a verbal statement into an algebraic expression, equation, or inequality to translate an algebraic expression, equation, or inequality into a verbal statement to evaluate an algebraic expression to complete a table of values AECSD 8th Grade Mathematics.doc 9 to write an equation from a table of values to plot a set of ordered pairs from a table and draw a line through them to describe a situation presented in a graph (linear or nonlinear) to graph a relationship described by an equation or verbal context Resources: SFAW 8th Grade Course 3 – Chapter 4.1 – 4.3 Review Template (Calculators): Pythagorean Theorem: Find the hypotenuse given the lengths of the legs. 3 4 Integer operations: a) -3 - -4 -3 – 4 -3 + -4 3–4 b) -7 - -5 -7 – 5 -7 + -5 7–5 Unit price problems: If 4 equally priced CDs cost 54.80, what is the unit price? Which is a better deal, 12 oz. of Coca-Cola for $.75 or 20 oz. for $1.00? If 3 lbs. of hamburger is 9.77, what is price per pound? AECSD 8th Grade Mathematics.doc 10 Unit 2A Solving Equations (necessary for '05-'06) Length: ~ 4 weeks Timeframe: Mid-October to Thanksgiving State Standards (Shaded statements are identified as Post-March Indicators): 7.A.4 Solve multi-step equations by combining like terms, using the distributive property, or moving variables to one side of the equation 7.A.5 Solve one-step inequalities (positive coefficients only) (see 7.G.10) 7.G.10 Graph the solution set of an inequality (positive coefficients only) on a number line (See 7.A.5) Local Standards: L.7.A.2 Solve multi-step equations that include parentheses (distributive property) and variables on both sides of the equation; write an equation that represents the pattern from a table of data; solve one-step equations and inequalities and graph the solution set. Big Ideas: Algebra is a tool to model and interpret real situations. Essential Questions: Why is it important to be able to solve equations? How do algebraic properties aid in solving equations? Prior Knowledge: to use distributive property to use order of operations using the set of real numbers to add, subtract, multiply and divide integers to translate a verbal statement into an equation Unit Objectives: to solve and check multi-step equations with parentheses and variables on both sides to write an equation that models a real-world situation and solve it Resources: Review Template (Calculators): Distribute and simplify: Simplify: 4(7 + 3x) + 8x Simplify: -3(8x – 10) – 2x Pythagorean Theorem: Find the length of a leg (include non-perfect squares) 5 3 3 4 AECSD 8th Grade Mathematics.doc 11 Order of Operations: Calculate showing each step: 3 – 5 + 4^2 * 3 5–8/24 5 + 7 – (3 + 4) (4 + 7 3)/5 1 + 3/3 – 1 + 2^3 AECSD 8th Grade Mathematics.doc 12 Unit 3 Special Angle Pairs Length: ~ 2 weeks Timeframe: Beginning of December State Standards (Shaded statements are identified as Post-March Indicators): 8.A.12 Apply algebra to determine the measure of angles formed by or contained in parallel lines cut by a transversal and by intersecting lines 8.G.1 Identify pairs of vertical angles as congruent 8.G.2 Identify pairs of supplementary and complementary angles 8.G.3 Calculate the missing angle in a supplementary or complementary pair 8.G.4 Determine angle pair relationships when given two parallel lines cut by a transversal 8.G.5 Calculate the missing angle measurements when given two parallel lines cut by a transversal 8.G.6 Calculate the missing angle measurements when given two intersecting lines and an angle Local Standards: L.8.G.4 Identify pairs of vertical, supplementary, and complementary angles and use relationship of pairs to find angle measures (including algebraically); determine the relationship between pairs of angles formed when parallel lines are cut by a transversal and use relationships to find missing angle measures (including algebraically). Big Ideas: Essential Questions: Prior Knowledge: to solve an equation to define and draw an angle Unit Objectives: to define and identify vertical, supplementary, and complementary angles to determine the measure of the other angle (including algebraically) in a vertical, supplementary, or complementary pair; given one angle measure to identify alternate interior, alternate exterior, and corresponding angles on parallel lines cut by a transversal to know and use the fact that alternate interior, alternate exterior, and corresponding angles on parallel lines cut by a transversal are congruent to know and use the fact that interior angles on the same side of the transversal of parallel lines are supplementary to determine the measure of missing angles (including algebraically) formed by parallel lines cut by a transversal AECSD 8th Grade Mathematics.doc 13 Resources: Memory device for students to recall complementary angles. To compliment (complement) someone is the right (angle) thing to do. Review Template (No Calculators): Solve equations: 3(x+2) = 5x Common Percents: 1/5, 1/4, 1/3's Set up proportions If $1.00 is equivalent to 16 euros, how many euros would you get if you exchanged $27? On a map, if 1 inch = 16 miles and two cities are 4.5 inches apart of the map, how far apart are the actual cities? AECSD 8th Grade Mathematics.doc 14 Unit 4 Transformational Geometry (G.2) Length: ~ 3 weeks Timeframe: Middle of December to start of January State Standards (Shaded statements are identified as Post-March Indicators): 8.G.7 Describe and identify transformations in the plane, using proper function notation (rotations, reflections, translations, and dilations) 8.G.8 Draw the image of a figure under rotations of 90 and 180 degrees 8.G.9 Draw the image of a figure under a reflection over a given line 8.G.10 Draw the image of a figure under a translation 8.G.11 Draw the image of a figure under a dilation 8.G.12 Identify the properties preserved and not preserved under a reflection, rotation, translation, and dilation Local Standards: L.8.G.2 Describe and identify transformations (rotation, reflection, translation, dilation) using proper notation; perform rotations of 90 and 180 degrees, reflections over a line, translations, and dilations of a given figure; identify properties preserved under each transformation. Big Ideas: Transformations can be found all around us. Essential Questions: Why are designs more pleasing to the eye when they involve transformations? Prior Knowledge: to graph an ordered pair to measure angles and distances to identify lines of symmetry Unit Objectives: to describe and identify transformations (rotation, reflection, translation, dilation) using proper notation to perform rotations of 90 and 180 degrees of a given figure to perform reflections over a line of a given figure to perform translations of a given figure to perform dilations of a given figure to perform rotations of 90 and 180 degrees of a given figure on a coordinate plane to perform reflections over a line of a given figure on a coordinate plane to perform translations of a given figure on a coordinate plane to perform dilations of a given figure on a coordinate plane to identify properties preserved under each transformation Resources: Connected Mathematics – Kaleidoscopes, Hubcaps, and Mirrors AECSD 8th Grade Mathematics.doc 15 Investigation 2 – transformations Investigation 3 – transformation on the coordinate plane Review Template (Calculators): Translate words to algebra: Mary is 4 more than 3 times Julie's age. If Julie's age is x, represent Mary's age? Translate algebra to words: Write a sentence for the following situation: 2x + 4 = 18 Evaluate: Find the area of a circle whose diameter is 7. AECSD 8th Grade Mathematics.doc 16 Unit 5 Polynomials (A.4, N.13) Length: ~ 5 weeks Timeframe: Middle of January to End of February (note that mid-term falls in middle of unit) State Standards (Shaded statements are identified as Post-March Indicators): 8.N.1 Develop and apply the laws of exponents for multiplication and division 8.N.2 Evaluate expressions with integral exponents 8.A.6 Multiply and divide monomials 8.A.7 Add and subtract polynomials (integer coefficients) 8.A.8 Multiply a binomial by a monomial or a binomial (integer coefficients) 8.A.9 Divide a polynomial by a monomial (integer coefficients) Note: The degree of the denominator is less than or equal to the degree of the numerator for all variables. 8.A.10 Factor algebraic expressions using the GCF 8.A.11 Factor a trinomial in the form ax2 + bx + c; a=1 and c having no more than three sets of factors Local Standards: L.8.N.13 Use calculation rules for powers for multiplication and division; evaluate expressions with integral exponentsBig Ideas: The subtraction of polynomials involves distributing an unwritten negative one. You must have like terms to add or subtract polynomials. When operating on polynomials, deal with the coefficients and then with the variables. The rules for operating with whole numbers hold true for variables. Essential Questions: What do you do when you have a subtraction symbol directly in front of parentheses? What is the difference between 2x and x2? How do operations on polynomials differ from operating on numbers? Prior Knowledge: to define a polynomial (must be taught in '05-'06) to add and subtract monomials (must be taught in '05-'06) to add, subtract, multiply, and divide integers to use the distributive property Unit Objectives: to multiply powers to divide powers AECSD 8th Grade Mathematics.doc 17 to evaluate expressions with integral exponents to multiply monomials to divide monomials to divide a polynomial by a monomial (with degree less than the numerator) to add polynomials with integer coefficients to subtract polynomials with integer coefficients to multiply a binomial by a monomial to multiply a binomial by a binomial to factor a trinomial (with a = 1 and c having no more than 3 sets of factors) to factor a GCF out of a polynomial Resources: Review Template (No Calculators): Find the sum of the interior angles of a: Hexagon octagon Solve equations Pythagorean triples Determine if a triangle with sides of 3, 4, and 5 is a right triangle. AECSD 8th Grade Mathematics.doc 18 Unit 6 Linear Equations Part 2 (A.2, A.1, G.7) Length: ~ 4 weeks Timeframe: Mid-March to mid-April State Standards (Shaded statements are identified as Post-March Indicators): 8.A.13 Solve multi-step inequalities and graph the solution set on a number line 8.A.14 Solve linear inequalities by combining like terms, using the distributive property, or moving variables to one side of the inequality (include multiplication or division of inequalities by a negative number) 8.G.18 Solve systems of equations graphically (only linear, integral solutions, y = mx + b format, no vertical/horizontal lines) 8.G.19 Graph the solution set of an inequality on a number line 8.A.17 Define and use correct terminology when referring to function (domain and range) 8.A.18 Determine if a relation is a function 8.A.19 Interpret multiple representations using equation, table of values, and graph 8.G.13 Determine the slope of a line from a graph and explain the meaning of slope as a constant rate of change 8.G.14 Determine the y-intercept of a line from a graph and be able to explain the y-intercept 8.G.15 Graph a line using a table of values 8.G.16 Determine the equation of a line given the slope and the y-intercept 8.G.17 Graph a line from an equation in slope-intercept form (y = mx + b) Local Standards: L.8.A.2 Solve multi-step inequalities that include parentheses (distributive property), variables on both sides of the inequality, and multiplication or division by a negative number and graph the solution on a number line; solve systems of linear equations graphically (use only equations in slope-intercept form with integral solutions. L.8.A.4 Define a function using correct terminology (domain and range); determine if a relation is a function. L.8.G.7 Given a line on a graph: determine its slope and explain its meaning as a constant rate of change, and determine and explain the meaning of the y-intercept; graph a line from a table of values or from an equation in slope-intercept form; determine the equation of a line given its slope and y-intercept. Big Ideas: The line represents the infinite set of ordered pairs that make the corresponding linear equation true. There are an infinite number of solutions to an inequality. Essential Questions: Why do you flip the inequality when you multiply or divide the inequality by a negative number? How do you represent the infinite number of solutions to an inequality? Prior Knowledge: (may need to be taught '05-'06) to solve multi-step equations including parentheses and variable on both sides AECSD 8th Grade Mathematics.doc 19 to solve and graph one-step inequalities to graph points on the coordinate plane Unit Objectives: to solve and check multi-step inequalities with parentheses and variables on both sides to solve and check multi-step inequalities that include negative coefficients to graph the solution of an inequality on a number line to define a function to determine if a relation is a function to identify the domain and range of a function to determine the slope and y-intercept of a line from a graph to explain the meaning of (a) slope as a constant rate of change to explain the meaning of the y-intercept to graph a line from a table of values to graph a line from an equation in slope-intercept form to write the equation of a line given its slope and y-intercept to solve a system of linear equations graphically (equations in slope-intercept form with integral solutions) Resources: Review Template (No Calculators): Square roots: Sqrt 25 + sqrt 49 = Sqrt of 78 is between which two consecutive whole numbers Scientific Notation: Binomial times binomial: AECSD 8th Grade Mathematics.doc 20 Unit 7 Quadratics (A.2) Length: ~ 1 week Timeframe: End of April State Standards (Shaded statements are identified as Post-March Indicators): 8.G.20 Distinguish between linear and nonlinear equations ax2 + bx + c; a=1 (only graphically) 8.G.21 Recognize the characteristics of quadratics in tables, graphs, equations, and situations Local Standards: L.8.A.5 Recognize the characteristics of quadratic equations in tables, graphically, algebraically and in words, and distinguish between linear and quadratic equations. Big Ideas: The graph of a quadratic equation is symmetric. If x2 is the highest power in the equation, then the equation is quadratic and its graph is a parabola. Essential Questions: How can you tell if a relationship is linear or quadratic? Prior Knowledge: (may need to be taught '05-'06) to graph linear equations to use order of operations with real numbers to factor a quadratic equation (with a = 1) Unit Objectives: to identify a quadratic relationship from a table, a graph, an equation or from context to distinguish between linear and quadratic equations to graph a quadratic equation (with a = 1 and domain given) – (if time allows) to find the roots of a quadratic equation graphically and by factoring – (if time allows) Resources: SFAW 8th Grade Course 3 – Chapter ? Review Template (No Calculators): Factor trinomial: Multiply fractions: Add/subtract fractions: AECSD 8th Grade Mathematics.doc 21 This Page Intentionally Left Blank AECSD 8th Grade Mathematics.doc 22 Unit 8 Constructions Length: ~ 1 week Timeframe: Beginning of May State Standards (Shaded statements are identified as Post-March Indicators): 8.G.0 Construct the following using a straight edge and compass: Segment congruent to a segment, Angle congruent to an angle, Perpendicular bisector, Angle bisector Local Standards: L.8.G.1 Construct the following figures: congruent segment, congruent angle, perpendicular bisector, angle bisector Big Ideas: A compass and straightedge are the only tools allowed for constructions. Essential Questions: Why do we still practice constructions? Prior Knowledge: (may need to be taught '05-'06) to define congruent segments, congruent angles, perpendicular, bisector Unit Objectives: to construct congruent segments to construct congruent angles to construct a perpendicular bisector of a segment to construct an angle bisector of an angle Resources: Discovering Geometry? Review Template (No Calculators): Factor GCF: Division of fractions: Special pairs of angles AECSD 8th Grade Mathematics
Next: Limits (An Intuitive Approach) Previous: The Binomial Theorem Chapter 8: Introduction to Calculus Chapter Outline Loading Contents Chapter Summary Image Attributions Description Students explore and learn about limits from an intuitive approach, computing limits, tangent lines and rates of change, derivatives, techniques of differentiation, conceptual basis for integration and The Fundamental Theorem of Calculus.
How I Teach Calculus: A Comedy (Diff Eq) May 21, 2010Shawn Cornally WhenFirst and foremost, it is important to note that "Diff Eq" sounds like an awesome handle for a rapper. If I were going to launch a career into socially conscious hip hop — which I might — this will be my name. So, there's my official statement of legally binding dibs. Here's a sample of some real hip hop, if you still "hate rap" after listening to this, there is no hope for you: The differential equations covered in my book are simple. Pretty much just separation of variables. This is fine with me, because diff eq's are a vessel for me to teach the finer abstractions of integration. Yes, I find the world of nonlinear dynamics as buzz wordy as the rest of you, I even worked in a plasma lab studying nonlinear behavior in fusion plasmas, but hey, these kids are 17 and more worried about prom than nuclear fusion (today, anyway), and I'm more worried about their ability to become un-addicted to the letter x as a variable. My School is Near a Lake = Sailboats So, the Cornally train gets rolling. How can a I make diff eq's real enough without watering down the math or declaring a state of activity mania? Sailboats, I think. I actually stole this from a question in the back of my book (Larson et al.). A word about book problems: All of us have seen the Dan Meyer TED talk by now (You haven't?!), and the idea here is to create problems that require thinking. The book problem that I stole this idea from gives all of the necessary information for solving the "problem." This is the functional equivalent of cutting your students' Achilles tendons and then asking them to run a 40. To further plagiarize Dan, giving students problems with all the info in them is fake and ridiculous. WhenSo, I have to cut the problem down to its core and do my best to present that in an interesting and engaging way. A story usually does the trick, because oration is one of my stronger suits, the other being of the leisure variety that I inherited from my Dad: Uncle Eddy, anyone? You may want to show a video of a sailboat competition, or do whatever else it is that invites your kids. Many of my students live on/near a lake, so this isn't that big of a stretch. Here's what I want: I want the students to derive the equations of motion [a(t), v(t), and x(t)] from first principles. I want them to think about how the wind pushes a sailboat, and I want them to use their basic understanding of physics (F=ma) to go from there. I can't say that aloud, though, or it has become my investigation they then have to do. I have to let it grow, otherwise all I'll get is a picture of how well they follow my directions, not how well their mathematical intuition is developing.1 So, I need to introduce a good question, one that is clear about what we're doing, but not so clear that it maps the whole process out artificially. This process is the math. By using only the asinine problems from the book you are relegating math to a status of recipe following. Do you really want your kids to be the kind of people that won't attempt to make a pizza from scratch for lack of the 1 tsp of anise seed that has little to do with the overall success of the dish? That's the kind of math that is predominantly taught. Barf. Ferdinand M.F. Magellan: I go all Magellan on them. Let's talk sailing and world history for a minute. Let's talk about using wind to get you where you want to go. Let's talk about the prevailing west winds of the Antarctic circumpolar region that help Magellan get around Africa. This is the perfect time for a dual lesson with my Western Civ teacher. She teaches the social effects of Ferdinand's magical mystery tour, I/we hash out the logistics of such a trip. Collaboration is a beautiful thing. How could Magellan have had the cajones to do what he did? How could he have known what kind of time commitment a trip like his would require? These questions are sufficient enough to start us down the correct mathematical path. I cannot stress enough how hard the next half an hour will be for you as a teacher. They must be assisted but not led. They must be supported but not undercut. We eventually settle on creating our own little sailboat system. They had fun and learned alot from the construction, and I secretly enjoy this part more than the math itself. This took about 20 minutes of solid thinking, a Gatorade bottle, 2 spray paint caps, office supply miscellany, and an air track blower: There are countweights behind the sail, and the bottle has ballast water in it. I love real life. The kids then played for a bit. I. Playing with the sailboat has yielded us a common experience to talk about. How does the force from the air change the speed of the boat? Can the boat go faster than the wind speed? These are all student questions that we endeavor to answer. Eventually we come up with the idea that the slower you are compared to the wind speed, the more force you'll feel. Or: This is acceleration. If we want to know anything else about the boat, its speed or position, we're going to have to integrate. The introduction of k was my idea. Why? Units. The kids bought that pretty well. v dub-ya is the wind speed and vb is the speed of the boat. This is just a hop and a skip (jump and you'll overshoot) from: The lesson on diff eq can now begin. I can teach separation of variables from this point. This tends to make students uncomfortable because of the lack of an x. Hence getting into the deeper abstractions of integration. Here's the work for those of you that are here for content (teachers, skip to the end): Separate the variables from each other. vb and t are the only variables, vw and k are numbers. This is now totally separated, and we can integrate over vb on the left and with respect to t on the right: The integral on the left requires substitution (Yay!): This yields us a function for the speed of the boat [vb(t)], assuming a constant wind speed (vw). Talking about these assumptions is always nice. Working in angles and variable wind speed could definitely be an open investigation, if a kid wanted to do it. I wouldn't necessarily make my kids, because my standard is "Diff Eqs by Separation" not "OMG Sailboats Textbook!" The sailboats are just a tool to create a handle for those kids that are not dominantly abstract thinkers yet. I know some of you scoff at all this extra time not spent on practice problems, homework, sharpening pencils and grind-stoning noses, but I'm telling you, some kids just don't have the necessary life experiences to buy into the abstraction of math. Creating these experiences in a meaningful way will not only help that kid grab a hold of something, it will also help them communicate with their peers better due to commonality. The next step is to check to see who understood what. I say, alright kiddos, find the x(t) function. This is my check for understanding, this is may be the first SBG moment of the lesson. Where are they? What do they know? Can they connect kinematics to differential equations? Can they handle e with tact? This is a certainly a formative moment. I know exactly what kind of board time I need to do after I watch them try this exercise. For those of you here for content, the answer is: Just graphing x(t) and v(t) with your kids when you're learning about exponentials is a rich experience. Let alone deriving them. Is this new? Of course not, I stole it from my book. It's the approach that I want to share with all of you. It's avoiding disabling your kids. It's showing them that math has an actual purpose than just being some class they have to take. It makes me genuinely sad to imagine that most of America's children think math is an affliction with no purpose other than to scare them away from STEM careers. Why? Because they can't give a shit about parabolas until they either buy into the beauty of math (a few kids), or they know what parabolas are good for. You can make a stink about h, k, x, and y all you want. You know what your kids are thinking? "Why h and k? Where'd those come from? I have no idea, crap, what random thing should I try and memorize" This gets old, and they don't want you to point at the board again where you've already circled the same equation twenty times. They want motivation, history, humanization, beauty; let the grind stone come naturally, please. 1. Some of you hate it when I say stuff like that. You place extreme value in following directions, because, as you see it, kids can't. You want to teach them how to be good little students and members of society, which is nice, but in your frustration you've forgotten how effing boring school is when you have no idea what the motivation for a lesson is. Or you just think I'm a twit, and that's, like, your opinion, man. [...] Before the spring term ended, I got a little excited reading about SBG and read through the entire finite mathematics text, taking notes and writing down standards. I then created this spreadsheet. Although it is about 3.5 months from being finalized, I'd love some feedback on it. My next step was to read through the book again with a copy of the final exam from last year (along with the formula sheet the students get). I noticed that the final exam reduces nearly the entire section on financial mathematics to inputting data into the formulas on the formula sheet. Do you really want your kids to be the kind of people that won't attempt to make a pizza from scratch for lack of the 1 tsp of anise seed that has little to do with the overall success of the dish? – Cornally [...] Somewhere I have some research on building flexible thinking in the classroom. Math was by far the hardest b/c kids in math get stuck on "right/wrong." I have a built in advantage in science b/c the kids often expect some sort of frustrated exploration period. In math they're taught to get a problem, watch it worked out, and then do practice problems until they get it. You're fighting a lot of history. @Shawn +1 for Kweli. Another +1 for using the exact same walmart storage container I use for boat races in science. Sam, I certainly can't speak for Mr. Cornally but I can talk on how I handle those situations. I try to operate instructionally much the same way Mr. Cornally does: inquiry driven, find a connection to real life and then talk about what we'll need to find it out, etc. But I teach Algebra I and remedial Algebra, so my content isn't as intense as his. Given that, I think you'd recognize that the students in my class often get frustrated even more than those in his, especially in the remedial Algebra. Just the jump from remedial Algebra to Algebra I produces an evolution from "I don't get it" to "I'm struggling with this". I've found that divide to be enormous. You're right, it comes early. The strategies are to relate to each kid in a unique manner. Know each kid. What embarrasses him, what makes him laugh, what are the signs that she's having a bad day, why is she acting out like that? The sooner I know the answers to these questions, the sooner I can pick my individual response each time a situation like you're asking about arises. Corey in my class just recently, when struggling, quietly sat there with is arms folded. Not defiantly, he just didn't know how to proceed He's a sped kid, I know this already. He's an athlete, so I know he's used to receiving a certain type of criticism from coaches. Him playing sports also tells me that he has a bit of a competitive side. So I play off of that. I tease a bit as I walk by, "Corey, atta kid, sitting there with your arms crossed." I say this loud enough so that others hear it but with enough affection that they know I'm not mad, I just want him to TRY AND FAIL. He's exactly the kind of kid you're talking about, waiting for someone to give him the answer. It's what his resource teacher does. But I won't coddle him like that. So I wish I could say that there was a strategy. But it's about caring about each one, caring enough to know how to press the right buttons. Mayra sits at a different table. She's reserved and shy and given the same exact scenario I'd kneel down next to her desk and ask some probing questions, "Why do you have a $4 next to the b? Oh, it's cuz burritos cost $4 each. Can we spend less than 56 on our combinations?" She'd cower if I called her out like I did with Corey. But I also know that Mayra is sad because her previous teacher first semester NEVER took the time out to talk to her one on one like this. I've taken the time to learn about each of them. Not only do they appreciate it but the entire class recognizes it. Each individual student knows that when I walk over to them I'm going to deal with them in a manner that they can handle. "I." That's powerful. And exactly what I'm curious about. What do you do to the kid that gets frustrated, and refuses to think because they're used to getting the answers handed to them? Clearly you don't just give 'em the answers. And when you're working with this kid, what are the other kids in the class doing? And when things come to a standstill — or you go down an approach that doesn't get you closer to your desired goal — how do you convince your kids that what they're doing is meaningful and interesting? Instead of them getting bored because the curiosity factor has depleted. How do you convince kids to "work though frustration" and stay interested/engaged? It has to be more than just the initial buy in to keep 'em motivated when they get stuck. The initial buy in can only get 'em so far. I know you probably have some natural strategies for these moments… If you think it's interesting enough to write about what they look like to you, how YOU feel when you hit them (it can't always be excitement… or is it?), and what resolving them looks like on the ground, I'd be excited to hear more about those things. I will throw out there that I suspect a lot of the way you deal with this is by building a very specific culture in your classroom, early on, which somehow promotes working through frustration, and being engaged. If that be the case, how did you build that culture? @Samjshah: You're totally right that kids come front loaded with an aversion to frustration and thinking. You're also pretty spot on with your assessment of what I do to rid them of those things. At no point, ever, in my room do kids get rewarded for correct answers. Many times we will work on a problem that intentionally doesn't have an answer, just to create a new piece of mathematics for use somewhere else. I get a lot of help by letting kids approach things how I they want to within my parameters. Kind of like the trick you use on a toddler who's a picky eater: "Do you want peas or carrots?" You don't care because the right answer is "vegetable." A few kids have a hard time with me, and even hate my class. They usually come around when they realize that I'm totally serious about this and that their grade reflects how much they think as well as how much math they can vomit on a page. For instance I put the area of a circle by exhaustion on my final. They don't know how to do this, but I just want to know how they'll attempt it. I doubt any will finish the problem, but because we've done things like that in class, they know that I expect clear statements of assumptions and diagrams. I have nothing of value to add — but I'm commenting anyway — except to say that I probably would have stuck with Math long enough to make it to Calculus if I had some sort of time machine that didn't take me back in time but made me into a younger version of myself during the current time. And if that same time machine housed me in Solon
If you want a thorough discussion of exactly how they are used, then just open up some linear algebra books, or do a Google search. Higher math isn't just solving puzzles (In fact, that's not really what math is). People don't just do mathematics to improve their thinking abilities; it certainly helps, but it has many applications. Keep in mind that mathematics need not be applied to anything. Just because you can't use a result of mathematics (at first) for anything practical does not make it useless. ZealScience May21-11 07:08 AM Re: What is advanced-level mathematics used for? There is great deal of uses in physics (also era involves some Chemistry) and engineering. In general relativity a lot of linear algebra and calculus is required. In engineering, mostly calculating some basic mechanics problem. Other than these, Economics uses great deal of calculus to model the market which is very important. You can search more on financial mathematics (not accounting~~boring). DivisionByZro May21-11 12:12 PM Re: What is advanced-level mathematics used for?Unrest May21-11 01:18 PM Re: What is advanced-level mathematics used for? Quote: Quote by DivisionByZro (Post 3314304) A few uses of Linear Algebra:Samuelb88 May21-11 01:48 PM Re: What is advanced-level mathematics used for? Everything has its applications. Of course, applications of some fields are more obvious than others. FishmanGeertz May21-11 08:07 PM Re: What is advanced-level mathematics used for? What about the use of high-level math in areas other than physics, science, and engineering? paulfr May21-11 11:57 PM Re: What is advanced-level mathematics used for? Mathematics models the natural world. So your question is nearly identical to ...... What use is it to learn English ? Furthermore, Logic, one of the foundations of Mathematics, is the link between The Arts & The Sciences. All artforms (nearly) seek to communicate. How better to make your case than with clear precise easy to understand logic ? Be it painting, screenplay, poem, courtroom summation or a novel. Some of the very the best lawyers were good at Math. That is one of the reasons they excel at the Law. Mathematics underlies nearly everything you see around you. But it will not guarantee a good life. That is the province of religion and moral philosophy. ZealScience May22-11 12:15 AM Re: What is advanced-level mathematics used for? Quote: Quote by DivisionByZro (Post 3314655)I think balancing equations is simply simultaneous equations, I mean for rather complicated equations. Matrices are just simple forms of simultaneous equations, they just save your paper and ink. Of course there are many ways of balancing equations, but many of them might not work for all cases. Also, matrices are very useful in doing statistics, though I haven't learn much of statistics, I heard of something called covariant matrix that is used for complicated systems. And statistics can be applied to many areas. May be you can look more on that.micromass May22-11 08:31 AM Re: What is advanced-level mathematics used for?He never said higher math was useless in the general context, simply useless in daily life. There's nothing wrong with the validity of his statement since we take the meaning of "useless" in every day conversation as "generally useless" rather than "completely useless". However, there are few professions that are useful in daily life, such as cooking, etc. so the statement, although basically true, is misleading. gb7nash May22-11 09:55 PM Re: What is advanced-level mathematics used for? Cryptography is a pretty big one. yenchin May22-11 10:16 PM Re: What is advanced-level mathematics used for? Quote: Quote by micromass (Post 3315838)I prefer this: A math professor, a native Texan, was asked by one of his students: "What is mathematics good for?" He replied: "This question makes me sick! If you show someone the Grand Canyon for the first time, and he asks you `What's it good for?' What would you do? Well, you kick that guy off the cliff!":biggrin: Unrest May22-11 10:25 PM Re: What is advanced-level mathematics used for? Quote: Quote by DivisionByZro (Post 3315796) As said many times in this thread, mathematics is all about logical and abstract thinking; it's basically a form of creativity. Now tell me, how useless are the former? Yes, I was one of those people who said that. But I think the OP was concerned about direct applications. Like when you would want to sit down with a pen and paper to write an integral or perform a matrix operation. For most people the answer would be never in their life. Us mathy types would think about it all the time when we hear news stories or write on internet forums, but that's not normal people. If you aren't inclined to analyze things for fun, then knowing how to integrate won't make you do it.
CAT in Mathematics The CUNY Assessment Test in Mathematics (also known as the CAT in Mathematics, or the COMPASS Math test) is an untimed, multiple-choice, computer-based test designed to measure students' knowledge of a number of topics in mathematics. The test draws questions from four sections: numerical skills / pre-algebra, algebra, college algebra, and trigonometry. Numerical skills / pre-algebra questions range from basic math concepts and skills (integers, fractions, and decimals) to the knowledge and skills that are required in an entry-level algebra course (absolute values, percentages, and exponents). The algebra items are questions from elementary and intermediate algebra (equations, polynomials, formula manipulations, and algebraic expressions). The college algebra section includes questions that measure skills required to perform operations with functions, exponents, matrices, and factorials. The trigonometry section addresses topics such as trigonometric functions and identities, right-triangle trigonometry, and graphs of trigonometric functions. No two tests are the same; questions are assigned randomly from the four sections, adapting to your test-taking experience. Placement into CUNY's required basic math courses is based on results of the numerical skills/pre-algebra and algebra sections. The test covers progressively advanced topics with placement into more advanced mathematics or mathematics-related courses based on results of the last three sections of the test. Students are permitted to use only the Microsoft Windows calculator while taking the test. CAT in Mathematics Practice Materials Below are some sample tests and websites containing more samples and information about the CAT in Mathematics and related materials. Special software may be needed to view some of these files; check under our Software section to get themBrownian motion is the most common model for continuous randomness used in probability.
including algebra 1, algebra 2 and
Math for Elementary Teachers II Welcome to Mth126: "Continued study of the mathematical concepts and techniques that are fundamental to, and form the basis for, elementary school mathematics. Topics include: use of probability and statistics to explore real-world problems; representation and analysis of discrete mathematical problems using counting techniques, sequences, graph theory, arrays and networks; use of functions, algebra and the basic concepts underlying the calculus in real-world-applications." News and Updates See the Homework and Handouts table below to download or print the syllabus, calendar, and all homework sets and class notes. Looking for the Mth126 page from a previous semester? Here's a link to mth126fa08. Here's a link to the interactive Normal Probability on Sketchpad document; you need Geometer's Sketchpad to open it, which is available in all GCA labs on campus. Right click and choose "Save as" to download it to your desktop; then open it from there. Homework Solutions: Here are worked solutions (see right) for some of the problems from Ch. 13. There are solutions to problems I did not assign this semester as well as problems I assigned that do not have a solution listed here. Still, I hope these are helpful. If there are others you have questions on, please stop by my office or ask during class.
STANDARD 4.3 PATTERNS AND ALGEBRA: All students will represent and analyze relationships among variable quantities and solve problems involving patterns, functions, and algebraic concepts and processes. Strand A. Patterns: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will: 3. Use inductive reasoning to form generalizations. Strand B. Functions and Relationships: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will: 1. Understand relations and functions and select, convert flexibly among, and use various representations for them, including equations or inequalities, tables, and graphs. 2. Analyze and explain the general properties and behavior of functions of one variable, using appropriate graphing technologies. Slope of a line or curve Domain and range Intercepts Continuity Maximum/minimum Estimating roots of equations Intersecting points as solutions of systems of equations Rates of change Strand C. Modeling: Building upon knowledge and skills gained in preceding grades, by the end of Grade 12, students will: You are a chemist. You are faced with a problem. You must prepare a solution mixing two given solutions. You need to find how much of each given solution should be used to make your new solution. Once you have completed your problem, you will present your solution and the process used to the class. Student Involvement - The students will complete the task individually and as a cooperative group in a whole class group setting Instruction - Activities will be organized and delivered by differentiating the activities or strategies to offer appropriate ways for students to learn, as a teacher-facilitated set of hands-on activitiesin a student booklet during class time Special Education Accommodations - Students with special needs will require the following electronic devices: Calculators Special Education Accommodations - Students with special needs will team up with a student partner. Special Education Accommodations - Students with special needs will require extra processing and response time, written copies of orally presented instructional or assessment materials, and written or photocopied notes of orally presented instruction or assessment materials Use of Resources - The school will provide classroom materials such as pencil, paper, notebooks. Use of Resources - The students will provide classroom materials such as pencils, paper, notebooks and calculators. Customer for Student Work - The student will present their work as evidence of task completion to peers, teachers, and administrators. Assessment of Student Work - The following people will be involved in assessing student work generated to complete the task: The student's teacher and peers. Reporting Results - The assessment results will be reported as a score point on a rubric and as a letter grade. Timeline - The estimated time needed to plan, teach, and score this task is from 8 to 10 class periods. B) Everyone should know what the problem is about and what you are looking for. C) Review IMAX MOVIE PRICES homework problem. Technology for this activity: Calculators Materials for this activity: Notebooks, pencils, and Practical Applications Problem Packet. Student product or performance for this activity: Students will produce yesterday's HW, provide procedures used, and write new procedures in their notebooks. Scoring tool for this activity: Teacher checklist Activity 2: Mixture of Solution Problem (Est. time 10 min) The teacher directs students to read the problem(#6 in packet) aloud. Ask a student to start the process. DO NOT START IT FOR THEM. WAIT for them; they will get it started. What you know: Label unknowns "x" and "y", a skill learned in past lessons. "Thought Process" - from English sentences (the problem) make Mathematical sentences. The teacher will prompt them by asking and reinforcing "What are Math sentences?"Their response will be Math. sentences are EQUATIONS. So let's make equations. Activity 3: Synthesis and Analysis of Problem ( Est. time 20 min) A) Have them reread the problem aloud and ask for the equations. B) WAIT for them to determine what sentences will be used to determine the equations to be used in solving the problem. C) Write 1st equation. x + y = 200. Ask what that represents? The response you are looking for is "quantity" - amount of stuff. D) Ask "what is the 2nd equation?" and "what does it represent?" E) WAIT! They will have trouble with this part which leads us to Activity 4. Activity 4: Side Bar Example (Est. time 10 min) A) Give this problem: you have 6 quarters and 3 dimes. What do you have? Let them answer. B) WAIT!! Let them answer. A majority will respond $1.80. C) Ask them to now think like a 1st grader and the response will be 9 coins. D) Let them see that BOTH answers are correct. E) Have them write equations: 6 + 3 = 9 quantity of stuff 6(.$25) + 3($.10) = $1.80 quality or value of stuff F) Tell them to use this side bar exercise in figuring out what the 2nd equation in their problem will be G) WAIT!!!!! They will come up with it. Activity 5: Using Applications to Solve (Est. time 10 min) A) Ask for both equations: An equation about "stuff" will be: x + y = 200 An equation about "value" will be: 80%(x) + 30%(y) = 62%(200) B) Once you have both equations, change % values to decimal values .8(x) + .3(y) = .62(200) C) Ask how to remove decimals? Their response will be multiply by 100. A skill previously learned. D) With both equations in "workable" (easier terms) form allow them to complete the "thought process" by solving these two equations. a skill they possess from previous lessons. Activity 6: Recap the Problem (Est. time 10 min) A) Once solutions are found ask various students to present their work and give the "thought process" they used in solving this problem B) This will ensure they know the process needed to solve these problems Activity 7: Discussion (Est. time 10 min) A0 Initiate a group discussion of the process in solving these types of Real Life practical application problems B) These discussions should lead to the presentation of a solution box for mixture problems Pizza and Soda Prices. A campus vendor charges $3.50 for one slice of pizza and one medium soda and $9.15 for three slices of pizza and two medium sodas. Determine the price of one medium soda and the price of one slice of pizza. Film Processing. Cord Camera charges $9.00 for processing a 24-exposure roll of film and $12.60 for processing a 36-exposure roll of film. After Jack's field trip he took 17 rolls of film to be developed at Cord Camera. He paid $171 for processing the film. How many rolls of each type were processed? IMAX Movie Prices. There were 322 people at a recent showing of the IMAX 3D movie "300". Admission was $8.75 each for adults and $6.00 each for children. Receipts totaled $2531.50. How many adults and how many children attended? The Butterfly Exhibit. On one day during a weekend, 1630 people visited The Butterfly Exhibit at White River Gardens in Indianapolis, Indiana. Admission was $7 each for adults and $6 each for children. The receipts totaled $11,080. How many adults and how many children visited that day? Printing. A printer knows that a page of print contains 830 words if large type is used and 1050 words if small type is used. A document containing 11,720 words fills exactly 12 pages. How many pages are in large type? In small type? Mixture of Solutions. A chemist has one solution that is 80% acid (that is 8 parts acid and 2 parts water) and one solution that is 30% acid. What is needed is 200L of a solution that is 62% acid. The chemist will prepare it by mixing the two solutions. How much of each should be used? Paint Mixtures. At a local "paint swap," Kari found large supplies of Skylite Pink (12.5% red pigment) and MacIntosh Red (20% red pigment). How many gallons of each color should Kari pick up in order to mix a gallon of Summer Rose (17% red pigment). Candy Mixtures. A bulk wholesaler wishes to mix some candy worth $.45 per pound and some worth $.80 per pound to make 350 lb. of a mixture worth $.65 per pound. How much of each type of candy should be used? Coffee Blends. Cafebucks coffee shop mixes Brazilian coffee worth $19 per pound with Turkish coffee worth $22 per pound. The mixture is to sell for $20 per pound. How much of each type of coffee should be used in order to make a 300 lb. mixture? Test Scores. You are taking a math test in which items in part A are worth 10 points and items in part B are worth 15 points. It takes 3 min. to complete each item of part A and 6 min. to complete each item in part B. The total time allowed is 60 min. and you do exactly 16 questions. How many questions of each part did you complete? Assuming that all your answers were correct, after all you had a GREAT Math Teacher, what was your score? Octane Ratings. In most areas of the United States, gas stations offer three grades of gasoline, indicated by octane ratings on the pumps, such as 87, 89, 93. When a tanker comes and delivers gas, it brings only two grades of gasoline, the highest and the lowest, filling two large underground tanks. If you purchase the middle grade, the pump's computer mixes the other two grades appropriately. How much of 87-octane gas and 93-octane gas should be blended in order to make 18 gal of 89-octane gas? Suntan Lotion. Lisa has a tube of Kinney's suntan lotion that is rated 15 spf and a second tube of Coppertone that is 30 spf. How many fluid ounces of each type of lotion should be mixed in order to create 50 fluid ounces that is rated 20 spf? Extra Credit Phone Rates. Recently, AT&T offered an unlimited long-distance calling plan to anyone in the United States, 24 hours a day, 7 days a week, for $29.95 a month. Another plan charges $.07 a minute all day every day, but costs an additional $3.95 per month. For what number of minutes will the two plans cost the same? Remember "Thought Process."Level 3: BLevel 2: CLevel 1: DLevel 0: F The response is completely incorrect or irrelevant. There may be no response, or the response may state, "I don't know." Notes: Analyze: I will examine the data in my chart to look for trends, contributing factors, and implications of student performance over a series of assessments of the same learning standard. Trends: I will look for improvement relative to previous lessons which included practical applications. Reflect: I will consider two or more of the following stems to reflect on the results and instructional practices I used and others I might benchmark and apply in the future. Then, I'll write a brief summary about my findings, contributing factors, and implications for improvement. As I relate my students' results with my lesson activities, I noticed that having the students understand the "thought process" and perform solving practical application problems has the most promise for becoming a best practice in my classroom because I find that the students have a better retention of algebraic concepts if they have a chance to see, understand, and perform real life practical applications rather than just reading, hearing, and copying them. This connects to previous and subsequent lessons in the chapter on Systems of Equations. The students are becoming familiar with the techniques of problem solving and are using them correctly. Action Plan: I will complete the following TaskBuilder Figure 8 Strategy Action Plan to prepare for my next standards-based task. 1. Plan - My next standards-based task will focus on: Title: Rational Expressions in Algebra Content Area:Algebra (10th and 11th grade) Learning Standard(s):NJCCS Intent:Define the concept of rational expressions and apply it to the simplification of algebraic expressions and practical real life applications. 5. Team or Grade Level Portfolio and School Web Site - I will insert the standards-based instruction or assessment task, results, samples of student work, and summary into my Team or Grade Level Portfolio and upload them to my School's Instructional Web Site on the following dates: Target date for School Instructional Web Site:May 18, 2007
Math Net 2001 Math Net is a valuable resource for anyone in any level math class. It provides students with detailed lessons in Algebra and Trigonometry, as well as a Glossary in which students can look up definitions. A Calculator page provides useful programs available for download to the TI-83 Calculator. Probably the most useful portion of the site is the Functions section. There are fifteen basic functions will detailed illustrations as to how they react to transformations. This is a valuable tool of reference, especially to students who have been away from math for some time.
Home Study Kit--Calculus, Second Edition Understanding the abstractions of calculus requires far more than limited exposure. Through the 148 sequential lessons in this comprehensive text, future mathematicians, scientists, and engineers will incrementally build and reinforce their knowledge through continual practice and review. Following a condensed summary of key algebra, trigonometry, and analytic geometry topics, students explore limits, functions, and the differentiation and integration of variables. Includes test and answer key booklets. 758 pages, hardcover. Wordly Wise 3000, Grade 12 12, Test Booklet with Answer Key This Wordly Wise 3000 Test Booklet, 2nd Ed. accompanies Wordly Wise 3000,Book 12, 2nd Ed. Tests are multiple choice with varied questions such as finding the antonym/synonym or the best word to complete a sentence. Final test questions are based upon an included passage. Line-listed answers are included. 110 pages, softcover. Grade 12. Exploring Creation with Physics Multimedia CD-ROM is designed to be used alongside the sold-separately Exploring Creation with Physics, 2nd Edition textbook. Filled with helpful, creative features that cannot be included in a physical book, this CD offers videos of experiments that students could not perform in the home, illustrating important concepts that would otherwise be difficult to visualize. Animations of example problems are given with Dr. Wile using an animated whiteboard. Pronunciations of unfamiliar words are also given, allowing you to "speak the language" of physics correctly, and with ease! Short Lessons in Art History: 35 Artists & Their Work Short Lessons in Art History is a fantastic introduction to the artists who have shaped the artistic world from Giotto in 1266 to Alexander Calder in 1976. Each artist's life is explored, along with their major works, public reaction, and impact upon the world and art after their deaths. Approximately 3-4 pages are given to each of the 38 artists featured, with black and white images of one of their most famous works. A center section contains full color plates. Glossary words are included at the end of each chapter. 217 pages, softcover. Grades 6
Elementary Algebra ACCUPLACER What is the Purpose of Elementary Algebra ACCUPLACER? The ACCUPLACER is an assessment test that measures the skills and abilities of the students through the subjects that are tested in this exam. While the results of this test allow the academic advisors and counselors of colleges to choose suitable courses for the test takers, they serve another purpose, as well. The scores of this test are used to find out the areas of strength and weakness of each candidate with respect to the subjects of ACCUPLACER. In fact, those students, who are weak or require additional support or assistance in the ACCUPLACER test subjects, are entitled to obtain the same through interactive online aids. Besides the test subjects, ACCUPLACER also takes into account the consistency of academic performance of each student, their goals and interest in studies for assessing the competence of the students and facilitating their admission in college-level courses. There are ten sections of ACCUPLACER in which there are multiple-choice questions that are to be answered by the students except for the essay writing section, in which there is a writing task for the students. The contents of Elementary Algebra ACCUPLACER are divided into three parts that are included in the following points: Operations related to Rational Numbers and Integers: The topics of this portion comprises of calculation with integers and negative rational, using absolute values and ordering. Finding solutions for Inequalities, Equations and Word Problems: The topics of this portion of Elementary Algebra section in ACCUPLACER are those that require the students to find solutions for linear equation and inequalities, verbal problems in algebraic perspective such as geometric reasoning and graphing, quadratic equations by factoring and translating written phrases into algebraic expressions. Operations related to Algebraic Expressions: In this portion of Elementary Algebra ACCUPLACER, the topics consist of addition and subtraction of polynomials and monomials, simplification of algebraic fractions, evaluations of positive rational roots and exponents, factoring, and evaluation of simple formulae and expressions. As far as Elementary Algebra ACCUPLACER test is concerned, it is one of the sections of this test consisting of 12 questions. What is the purpose of this test in ACCUPLACER? Let us analyze the objective of this exam in the following points: The Elementary Algebra ACCUPLACER test measures the ability of the students for performing basic algebraic operations. The questions in this test include problems that are related to fundamental concepts of algebra. Since algebra is a basic component of mathematics, the scores of Elementary Algebra ACCUPLACER demonstrate whether an applicant is suitable for basic-level or advanced courses of mathematics in college. Those students, who are keen to take up college-level courses based on math, get an opportunity to prove their skills through this section of ACCUPLACER. By taking this test, the candidates are able to sharpen their skills for attempting the other sections of ACCUPLACER such as College-level Math in which the algebraic operations consist of questions on topics such as simplification of rational algebraic operations, expanding polynomials, factoring and manipulating roots and exponents. Having knowledge of Elementary Algebra helps the students in pursuing a rewarding career in Mathematics. If a student is keen to pursue his/her academics in mathematics, taking Elementary Algebra ACCUPLACER provides assistance when the scores are declared. For instance, if a student is unable to fare well in this section of ACCUPLACER, he/she can get further opportunities of training and support for better performance when the test is taken again. When it comes to preparation for Elementary Algebra test in ACCUPLACER, the candidates should focus on practice questions as it allows them to analyze their knowledge in this section through various stages of preparation. In the following points a few online prep resources of Elementary Algebra are stated: The Elementary Algebra ACCUPLACER exam prepares the candidates to choose those subjects in college that are based on math and preferred by them. In short, by answering this section of the test and excelling in it, allows them to feel confident for the fulfillment of their academic goals
Mathematics Mathematics evolved from counting, measurement, and the study of shapes and the motion of physical bodies into a harmonious collaboration of beauty and utility, abstraction and application. The study of mathematics trains the student to search for unity and a desire to treat phenomena comprehensively. Mathematics makes the invisible visible through the use of language and the study of abstractions, patterns and relationships, computation and calculation. Mathematicians make important contributions to society by helping to solve problems in such diverse fields as: Students who prepare as high school mathematics teachers have always found employment in Kansas. The bachelor of science degree prepares a person for employment in math related fields or for graduate study. Mathematics graduates find jobs in fields such as: More Information Mathematics at Newman University involves numerical, algebraic and graphical analysis of patterns and problems. The patterns can involve simple counting numbers or real or complex numbers. Problems solved with mathematics can range from statistical questions to questions about 3-dimensional surfaces. The program has connections to a variety of fields of study such as music, computer graphics, probability and many more. Core Mathematics Students should demonstrate a broad understanding of mathematics, demonstrated by competency in the following areas: statistics, calculus, geometry, and algebra. Critical Thinking Students should develop clear analytical thinking skills, as demonstrated by the following. Proof: Students should demonstrate the ability to employ rigorous reasoning to construct logically correct arguments. Problem Solving: Students should demonstrate the ability to formulate problems quantitatively, and interpret the solutions. Communication Students should demonstrate the ability to effectively communicate mathematics. Technology Students should be able to employ technology as one of a variety of problem solving strategies. Ethics Students can distinguish between ethical and unethical uses of data. High School Preparation It is recommended that students complete four years of English, four years of mathematics and four years of science in high school. Related Programs Our Mission Newman University is a Catholic university named for John Henry Cardinal Newman and founded by the Adorers of the Blood of Christ for the purpose of empowering graduates to transform society. Mission, identity and heritage
This book makes quantitative finance (almost) easy! Its new visual approach makes quantitative finance accessible to a broad audience, including those without strong backgrounds in math or finance. Michael Lovelady...
There's a new article by the author Nicholson Baker that is not raising as much a fuss as "Is Algebra Necessary?" from The New York Times last year, probably because it's at Harper's behind a paywall. Also, as I write this children are fleeing from algebra all over magazine stands: The title strikes me as odd, given that the general argument is aimed at all algebra, and the text constantly references algebra as a whole rather than the quite US-specific version of Algebra II. (As far as I'm aware no other country has such a thing; most of them integrate all forms of mathematics together.) What is Algebra II anyway, and where is the cutoff point where Nicholson Baker considers algebra to be too hard to handle? He hints at what he'd like as a replacement, which strikes me as a math appreciation analogous to music appreciation courses: We should, I think, create a new, one-year teaser course for ninth graders, which would briefly cover a few techniques of algebraic manipulation, some mindstretching geometric proofs, some nifty things about parabolas and conic sections, and even perhaps a soft-core hint of the innitesimal, change-explaining powers of calculus. Throw in some scatter plots and data analysis, a touch of mathematical logic, and several representative topics in math history and math appreciation. This seems like a truly random collection of topics, but it sounds like Mr. Baker is basing his list off the notion of using The Joy of x by Steven Strogatz as a possible text. I have had some personal experience with using popular texts; when I team-taught statistics at the University of Arizona we used The Drunkard's Walk, Fooled by Randomness, and Struck by Lightning: The Curious World of Probabilities. While it led to interesting discussions, there just wasn't enough meat to have students do problem solving based solely on the text. The math from the books was purely a passive experience. (We still did just fine by augmenting with our own material.) I still hold forward the absurd idea that students still solve math problems in a math class. If you're designing a freshman mathematics-teaser course, I might humbly suggest Problem Solving Strategies: Crossing the River with Dogs, which has the virtue of steering away from algebra as the sole touchstone for problem solving. Back to Mr. Baker's attempt to define Algebra I: Six weeks of factoring and solving simple equations is enough to give any student a rough idea of what the algebraic ars magna is really like, and whether he or she has any head for it. Mr. Baker himself seems to have a confused idea of what algebra is like, but he's not alone. (See: my prior post on what is algebra and why you might have the wrong idea.) Also of note: some countries don't bother to factor quadratics. (I haven't made a map comparing countries, but it seems to be continental Europe that ignores it and just says "use the quadratic formula".) I could see solving equations managed in six weeks, but in a turgid, just-the-rules style that would be the opposite of what we'd want in this kind of appreciation class in the first place. Just the concept of a variable can take some students a month to wrap their head around, making me disturbed by the notion that six weeks would be enough for a student to find "whether he or she has any head for it". (This isn't even touching the issue of just how much is internal to the student. I've heard dyscalculia estimates of up to 7% of all students, but excluding that group a great deal of the attitude seems culturally specific. Allegedly in Hungary [I don't have a hard research paper or anything, this is just from personal anecdotes] it is quite common for folks to say mathematics was their favorite subject in school and the level of disdain isn't remotely comparable to the US.) They are forced, repeatedly, to stare at hairy, squarerooted, polynomialed horseradish clumps of mute symbology that irritate them The article's invective along these lines makes me wonder again how much the visual aspect to mathematics is the source of hatred, as opposed to the mathematics itself. This picture is from the Adventure Time episode "Slumber Party Panic" and is supposed to represent the ultimate in math difficulty. However, the math symbolism is strewn truly at random and there is no real problem here. I am guessing a good chunk of the audience couldn't even tell the difference and for them, any difficult math problem looks like random symbolic gibberish. This is related to another issue, that of bad writing. Here's Mr. Baker quoting a textbook: A rational function is a function that you can write in the form f(x) = P(x)/Q(x), where P(x) and Q(x) are polynomial functions. The domain of f(x) is all real numbers except those for which Q(x) = 0. I claim the above definition is simply bad writing, and a cursory check of the Internet reveals several definitions I'd peg as clearer in a students-who-don't-like-math-are-trying-to-read sense: "Rational function" is the name given to a function which can be represented as the quotient of polynomials, just as a rational number is a number which can be expressed as a quotient of whole numbers. Godfrey Harold Hardy admittedly goes on from there, but his text is written for mathematicians. High school texts have no such excuse. In a similar vein, the article later quotes a 7th grade Common Core standard: solve word problems leading to equations of the form px + q = r and p(x+q)=r, where p, q, and r are specific rational numbers which I suppose is meant to be horrifying, but in this case the standards are written for the teachers and aim to remove any ambiguity. Here's problems that matches the standard: 1. You bought 3 sodas for 99 cents each and paid 10 cents in tax. How much money did you pay? 2. You bought 4 candies for 1.50 each and paid $6.20. How much was tax? The standards can't simply produce many examples and gesture vaguely. They must be exact. Just because 4th grade specifies that students "Explain why a fraction a/b is equivalent to a fraction (n×a)/(n×b)" does not mean students are using variables to do so. (In case you're curious what it does look like to the 4th graders, Illustrative Mathematics has tasks here and here matching the standard.) There's lots more to comment on, but let me leave off for the moment on this quote, because I'm curious… Math-intensive education hasn't done much for Russia, as it turns out. Fold from upper right to lower left (colors added to both sides of the paper for clarity): Follow up with one more fold: Voila, a proof that the square root of 2 is irrational. (Mind you, there is some reasoning involved, but what's the fun in giving that away? Start with "if the square root of 2 is rational, then there is some isosceles right triangle where the sides are the smallest possible integers.") Teaching is more than telling and explaining, and learning is more than imitating and memorizing. During the last 60 years teachers of mathematics have gradually sensed that, above all else, their pupils should learn the meaning of mathematical terms, principles, operations, and patterns of thought. So in my last post I opined that the optimal mathematics game in the Tiny Games spirit should "incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom." That led to some confusion. Let me try a do-over: During a game, when the primary action of the players is indistinguishable from doing traditional homework or test problems, it is a gamified drill. There's lots of gamified drills. It's easy to do: just take what you normally would do in a math problem review and tack on a game element somewhere (for me it's usually Math Basketball). To be integrated the primary actions of the players will require using mathematics in a way that is linked with the context of the game. That 1-2 Nim requires understanding multiples of 3 is inextricable from the game itself and not interchangeable the same way Math Basketball can be easily switched to Math Darts. The concept here is to have games suited for different settings that can be described in only a few sentences. Could one make an all-mathematics variant — mathematical scrimmages, so to speak? The only games I could think of offhand in the same spirit as Tiny Games were some Nim variants and Fizz-buzz. 1-2 Nim (for two players): Start with a row of coins. Alternate turns with your opponent. On your turn you can take either 1 or 2 of the coins. The person who takes the last coin wins. Fizz-buzz (for a group): Players pick an order. The first player says the number "1″, and then the players count in turn. Numbers divisible by 3 should be replaced by "fizz". Numbers divisible by 5 should be replaced by "buzz". Numbers divisible by both should be replaced by "fizz-buzz". Players who make a mistake are out. Last one in wins the game. Anyone have some more? EXTRA NOTE: One condition I'd add is the games need to work as games and not as glorified practice. "Challenge a friend to factor a quadratic you made" meets the "Tiny" but not the "Game" requirement. EXTRA EXTRA NOTE: Dan Meyer asks "Aside from the counterexample that follows, what are the qualities that make Fizz-Buzz and Nim gamelike and not, say, exerciselike?" In both cases the games incidentally have some mathematics in them and are the sorts of games one might play even outside of a mathematics classroom. Even though Wikipedia claims Fizz-buzz was invented for children to "teach them about division" (?), my first encounter was from The World's Best Party Games. (This still doesn't totally answer the question, I know. A related question is: what is the difference between a puzzle and a math problem?) Edward L. Thorndike's book The Psychology of Arithmetic (1922) is the earliest I've seen containing criticism of the visuals in textbook design. I wanted to share three of the examples. Fig. 47.—Would a beginner know that after THIRTEEN he was to switch around and begin at the other end? Could you read the SIX of TWENTY-SIX if you did not already know what it ought to be? What meaning would all the brackets have for a little child in grade 2? Does this picture illustrate or obfuscate? Fig. 51.—What are these drawings intended to show? Why do they show the facts only obscurely and dubiously? Fig. 52.—What are these drawings intended to show? What simple change would make them show the facts much more clearly? While it is still common (and frankly, necessary) to rail at the limitations of learning mathematics via watching videos, my personal umbrage has more to do with presentation than with educational philosophy. The mathematical video genre is still in its infancy. I am reminded of early films that were, essentially, canned plays. (From L'Assassinat du Duc de Guise in 1908.) Oftentimes in videos teaching mathematics with notation they simply duplicate what could be done on a blackboard, without fully utilizing the medium. However, there are techniques particular to the video format which can strengthen presentation of even mundane notation. For instance, in my Q*Bert Teaches the Binomial Theorem video I made crude use of a split-screen parallel action to reinforce working an abstract level of mathematics simultaneously with a concrete level. For now, I want to focus on applying animation to the notation itself for clarity. The video is chock-full of interesting animated moments, but I want to take apart a small section at 5:43. In particular the video shows some algebra peformed on . Step 1: Multiply the left side by . The variable "falls from the sky" and is enlonged to convey the gravity of motion. Step 2: Once the variable has fallen, the equation "tilts" to show how it is imbalanced. A second falls onto the right side of the equation. Step 3: The equation comes back into balance, and the two variables on the left side of the equal side divide. Step 4: The variables on the right hand side start to multiply, conveyed by a "merge" effect … Step 5: … forming . Here's a much more recent example from TED-Ed: When adding matrices, the positions are not only emphasized by color but by bouncing balls. When mentioning the term "2×2 matrix" meaning "2 rows by 2 columns" the vocabulary use is emphasized by motion across the rows and columns. The second matrix is "translated up a bit" by doing a full animation of the matrix sliding to the position. When the video discusses "the first row" and the "the first column" not only are the relevant numbers highlighted, but they shrink and enlarge as a strong visual signal. When discussing the problem of why matrix multiplication sometimes doesn't work, the "shrink-and-enlarge" signal moves along the row-matched-with-column progression in such a way it becomes visually clear why the narrator becomes stuck at "3 x …." These are work-heavy to make, yes, but what if there was some application customized to create animation with mathematics notation? At the very least, there's a whole vocabulary of cinematic technique that has gone unexplored in the presentation of mathematics. Kickstarter a letter by a "Donald Ross" that some people think is by Mark Twain: Things a Scotsman Wants to Know. It qualifies for the category of "inverse problems nobody will know the answer to unless someone builds a time machine".)

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