for use with fathom ™ and the geometer’s sketchpad® · or the geometer’s sketchpad before,...

Technology Demonstrations

For use with Fathom ™ and The Geometer’s Sketchpad®

SECOND EDITION

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Editor: Heather Dever

Project Administrator: Tamar Wolins

Writers: Larry Copes, Jennifer North Morris, Cindy Clements, Corey Andreasen

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Printer: Versa Press, Inc.

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Publisher: Steven Rasmussen

©2010 by Key Curriculum Press. All rights reserved.

Cover Photo Credits: Background and center images: NASA; all other images: Ken Karp Photography.

Limited Reproduction PermissionThe publisher grants the teacher whose school has adopted Discovering Advanced Algebra, and who has received Discovering Advanced Algebra: An Investigative Approach, Technology Demonstrations as part of the Teaching Resources package for the book, the right to reproduce material for use in his or her own classroom. Unauthorized copying of Discovering Advanced Algebra: An Investigative Approach, Technology Demonstrations constitutes copyright infringement and is a violation of federal law.

Technology Demonstrations CDKey Curriculum Press guarantees that the CD that accompanies this book is free of defects in materials and workmanship. A defective CD will be replaced free of charge if returned within 90 days of the purchase date.

®The Geometer’s Sketchpad, Dynamic Geometry, The Geometer’s Sketchpad logo, and Key Curriculum Press are registered trademarks of Key Curriculum Press. ™The Discovering Mathematics logo and Sketchpad are trademarks of Key Curriculum Press. ™Fathom Dynamic Data and the Fathom logo are trademarks of KCP Technologies. All registered trademarks and trademarks in this book are the property of their respective holders.

Some of the demonstrations in this book were adapted from the following sources: Sketchpad LessonLink, at http://keyonline.keypress.com (Key Curriculum Press, Emeryville, CA, 2009), and Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger (Key Curriculum Press, Emeryville, CA, 2007).

Key Curriculum Press1150 65th StreetEmeryville, CA 94608(510) [email protected]

Printed in the United States of America

10 9 8 7 6 5 4 3 2 1 13 12 11 10 09 ISBN 978-1-60440-012-0

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iii

Lesson numbers here refer to the corresponding lesson in Discovering Advanced Algebra. Not every lesson in the student book is represented here. The technology needed for each demonstration is noted after the title: Fathom Dynamic Data™ Software (Fathom) or The Geometer’s Sketchpad® (Sketchpad).

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .vii

Chapter 0

Lesson 0.1: Multiplying Binomials (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1

Chapter 1

Lesson 1.2: Looking for the Rebound (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

Exploration: Repeat After Me (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5

Lesson 1.5: Life’s Big Expenditures (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6

Chapter 2

Lesson 2.1: Box Plots (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

Lesson 2.3: MP3 Players (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Lesson 2.3: Eating on the Run (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

Exploration: Different Ways to Analyze Data (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . 14

Chapter 3

Lesson 3.2: Balloon Blastoff (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Lesson 3.3: Finding a Line of Fit (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Exploration: A Good Fit? (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

Chapter 4

Lesson 4.3: Lines (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

Lesson 4.4: Parabolas (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

Lesson 4.5: Square Roots (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

Lesson 4.6: Absolute Value (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

Lesson 4.6: Science Fair (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

Lesson 4.7: Circles and Ellipses (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Chapter 5

Lesson 5.2: Power Functions (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

Lesson 5.3: Rational Exponents (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

Lesson 5.5: The Inverse of a Line (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

Lesson 5.6: Exponential and Logarithmic Functions (Sketchpad). . . . . . . . . . 30

Lesson 5.8: Cooling (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

Lesson 5.8: Curve Straightening (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Chapter 6

Lesson 6.6: Linear Programming (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

Contents

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iv

Chapter 7

Lesson 7.1: Free Fall (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

Lesson 7.2: Rolling Along (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

Lesson 7.4: Quadratic Functions (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Lesson 7.7: Higher Degree (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Chapter 8

Lesson 8.1: Bucket Race (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Lesson 8.2: Definitions of Circle and Ellipse (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . 47

Lesson 8.3: Definition of Parabola (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Lesson 8.4: Definition of Hyperbola (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

Exploration: From Circles to the Ellipse (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

Lesson 8.6: Rational Functions (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Lesson 8.6: Inverse Variation (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

Chapter 9

Lesson 9.1: Arithmetic Series Formula (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

Lesson 9.2: Fractals (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

Exploration: A Geometric Series (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

Chapter 10

Lesson 10.1: Sum of Two Dice (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

Exploration: The Coin Toss Problem (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

Lesson 10.2: Independent Events (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

Exploration: A Repeat Performance (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

Lesson 10.4: Expected Value (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

Lesson 10.5: Ordered Lists (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

Lesson 10.7: Sapsuckers (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Chapter 11

Lesson 11.2: Ages (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

Exploration: Is This Normal? (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

Lesson 11.4: Polling Voters (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Exploration: Quality Control (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

Lesson 11.6: SAT Scores (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

Exploration: How Does It Fit? (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

Chapter 11 Review: Normal Curve and Approximation (Sketchpad) . . . . . 81

Chapter 12

Lesson 12.1: Right Triangle Ratios (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

Lesson 12.2: Law of Sines (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

Lesson 12.5: Vector Sums (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

Lesson 12.6: Parametric Equations (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

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Chapter 13

Exploration: Tracing Parent Graphs (Sketchpad). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

Lesson 13.3: Pendulum (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Lesson 13.5: Day Times (Fathom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

Lesson 13.7: Sound Wave (Fathom). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

Exploration: Rose Curves (Sketchpad) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

Teacher’s Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

v

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Introduction

vii

There’s no replacement for demonstrating a definition with hundreds of examples, all in five minutes or less. With the demonstrations provided in this book you can help students discover the dynamic relationships in important advanced algebra concepts. Previous experience with Fathom Dynamic Data™ software or The Geometer’s Sketchpad® is not necessary; you can show properties of the graphs and data sets simply by dragging different parts with the mouse, or by clicking on buttons in the sketch. If you or your students haven’t used Fathom or The Geometer’s Sketchpad before, these demonstrations are a good place to start. They are quick, and provide a strong introduction to the use of technology in mathematics. To use the Fathom demonstrations in this volume, you need Fathom version 2.1, a free update of Fathom version 2. To check which version you have, choose About Fathom from the Help menu (Win) or Fathom menu (Mac). To download and install the update, choose Check for Updates from the Help menu.

You can use these demonstrations in a variety of ways. You or a student can present them with a projetor or students can work individually in a computer lab. The Fathom and Sketchpad™ documents use a large typeface wherever possible so that students can see them from the back of a classroom. In Fathom you can make all text larger by choosing Preferences from the Edit menu (Win) or Fathom menu (Mac). You can hand out the easy-to-follow demonstration worksheets to students, but you might prefer simply to go over the questions as a class. Each demonstration covers a specific investigation, exercise, or concept in Discovering Advanced Algebra and is referred to in the Teacher’s Edition in the Chapter Overview, as well as in the appropriate lesson. You can use the demonstration in place of, or in addition to, the book’s investigation. The Teacher’s Notes in this book provide suggestions on ways to use the demonstrations, as well as answers to the questions posed on the worksheets.

This book includes several demonstrations (in revised form) that originally appeared in Sketchpad LessonLink, at http://keyonline.keypress.com (Key Curriculum Press, Emeryville, CA, 2009), and Exploring Algebra 1 with Fathom by Eric Kamischke, Larry Copes, and Ross Isenegger (Key Curriculum Press, Emeryville, CA, 2007). This volume adapts those demonstrations for use with Discovering Advanced Algebra lessons.

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Discovering Advanced Algebra Technology Demonstrations CHAPTER 0 1©2010 Key Curriculum Press

(continued)

When you multiply binomials using a rectangle diagram, the goal is to arrange tiles to form a rectangle. This rectangle is the solution.

SketchStep 1 Open the document MultiplyingBinomials.gsp to the page

(x�5)(x�3).

Step 2 Press the Show Solution button.

Investigate 1. Explain how the rectangle diagram relates to the solution.

2. How would you change the initial diagram (without the rectangle) to display (x � 2)(x � 4)? Draw the diagram on your paper.

SketchStep 3 Go to the page (x� 2)(x� 4).

Step 4 To make a rectangle, press and hold the Custom tools icon. Choose the tool for the tile you wish to use (x^2, 1 by x, x by 1, or 1). The tile you selected will now be attached to your pointer. Place the tile by matching the upper-left point to a point in the sketch. If you place something incorrectly, choose Undo from the Edit menu and try again.

Hide Example Tiles

Show Solution

(x � 2)(x � 4)x � 2

x�4

x by 1

x2

1 by x

1

Step 5 Continue using the custom tools until the rectangle diagram is complete.

Investigate 3. Draw the finished rectangle on your paper. What expression does the

rectangle diagram represent?

Lesson 0.1 • Multiplying Binomials Sketchpad

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2 CHAPTER 0 Discovering Advanced Algebra Technology Demonstrations

©2010 Key Curriculum Press

Lesson 0.1 • Multiplying Binomials (continued)

SketchStep 6 Go to the Practice 1 page.

Step 7 Use the custom tools to set up other multiplication problems, and then create diagrams to solve them. If you would like to change the length of x in your tiles, press the Show Length of x button. Drag the endpoint at the top of segment x to change the length. Each segment x in your area diagram will change accordingly. To erase your problem, choose Undo from the Edit menu until you are back where you started.

You can also create additional practice pages. Go to a blank practice page in the document. Choose Document Options from the File menu. Go to the Add Page menu and choose Duplicate � Practice.

Investigate 4. Create a rectangle diagram to simplify (2x � 3)(x � 4). Draw this

diagram on your paper.

5. Explain how you could use rectangle diagrams to simplify binomials that include negative values.



When you drop a ball, the rebound height becomes smaller after each bounce. In this investigation you will write a recursive formula for the height of a real ball as it bounces.

ExperimentStep 1 Plug a motion sensor into a USB port on your computer and

open the Fathom document Rebound.ftm. You will see a blank experiment set up. The current value detected by the motion sensor appears in the Distance meter at the top.

Step 2 Have one person hold the motion sensor above the ball, and another person click the Turn Experiment On button. The first person should drop the ball. If the ball drifts, try to follow it and maintain the same height with the motion sensor. If the bounces appear “upside down,” drag the mouse over the Distance slider. Click the �/� button that appears and repeat the procedure. If you do not capture at least six good consecutive bounces, repeat the procedure.

Clean up the graph by following these steps.

Step 3 There may be some “noise” (points that don’t seem to be part of the bouncing pattern), so highlight those points with the mouse (they’ll turn red) and choose Edit � Delete Cases.

Step 4 Drag the Shift slider so the bottom of the bounces is at 0.

Step 5 Select the graph and choose Graph � Rescale Graph Axes.

(continued)

Lesson 1.2 • Looking for the Rebound Fathom




Investigate 1. Find the maximum height for each bounce by moving the cursor over

the maximum points and looking at the values in the status bar in the lower-left corner of the window. Enter the maximum height for each bounce in the case table labeled Height v. Bounce.

2. There is a third variable, or attribute, called Rebound_Ratio defined with the formula

Rebound_ratio � Height

___________ prev(Height)

This gives the rebound ratio for consecutive bounces. There is no value for the first case because there is no previous bounce. Decide on a single value that best represents the rebound ratio for your ball. Use this ratio to write a recursive formula that models your sequence of Rebound_height data.

ExperimentStep 6 Enter this formula into the fourth column,

Predicted_Height : select the column and choose Edit � Edit Formula. You will need to use an if-statement to define the first term and the recursive formula. Type if (Bounce � 0. On the top line at right enter the initial value, and on the bottom line enter the formula using the prev() function. The predicted heights should appear in the Height vs. Bounce graph.

Investigate 3. Compare your experimental data to the terms generated by your

recursive formula. How close are they? Describe some of the factors that might affect this experiment. For example, how might the formula change if you used a different kind of ball or dropped the ball from a different height?

Lesson 1.2 • Looking for the Rebound (continued)



Exploration • Repeat After Me Sketchpad

Arithmetic sequences have an added term, and geometric sequences have a common ratio. Shifted geometric sequences have a common ratio and an added term. In this demonstration you’ll use two sliders to change the value of the common ratio r and the added term d.

SketchStep 1 Open the document RepeatAfterMe.gsp.

Step 2 Note the differences and

2

�2

4

6

8

10

12

14

16

�5 10 155

u(0)

r � 0.88d � 1.14

Arithmetic

Shifted Geometric

Geometric

r

d

similarities among the graphs of the three types of sequences. Drag the endpoint of each slider up and down. Watch the effect each has on the sequences.

Step 3 Predict what the graphs will look like for each value. Then adjust the sliders to check your prediction.

a. d � 1

b. d � 0

c. r � 1

d. r � 0

Step 4 Find point u (0), the initial term of all the sequences. Drag this point around and watch the effects on the graphs.

Investigate 1. Which sequences are affected by the r slider? By the d slider?

2. What does r represent in the graph of the geometric sequence?

3. How does changing the value of r affect the graph of each sequence?

4. How does changing the value of d affect the graph of each sequence?

5. Find any values of r and d that make the graphs of the two sequences the same. If there are no values, explain why not.

a. Shifted geometric and geometric

b. Shifted geometric and arithmetic

c. Geometric and arithmetic




In this demonstration you will use recursion to explore loan balances and payment options. Fathom will be a helpful tool for exploring loans with different details.

You plan to borrow $22,000 from a bank to purchase a new car. You will make a payment every month to the bank to repay the loan, and the loan must be paid off in 5 years (60 months). The bank charges interest at an annual rate of 7.9%, compounded monthly. Part of each monthly payment is applied to the interest, and the remainder reduces the starting balance, or principal.

Investigate 1. What is the monthly interest rate? What is the first month’s interest

on the $22,000? If you make a payment of $350 at the end of the first month, then what is the remaining balance?

2. Record the balances for the first six months with monthly payments of $350. Explain in words how to calculate the new balance from the previous month’s balance. Rewrite this explanation as a recursive rule.

ExperimentStep 1 Open Expenditures.ftm. You will see sliders that can be used to

control the principal, annual interest rate, and amount of payment for the loan. There is also a slider called Monthly_Interest_Rate. The value of this slider is the annual interest rate divided by 12. You’ll also see two summary tables and a graph.

(continued)

Lesson 1.5 • Life’s Big Expenditures Fathom



Lesson 1.5 • Life’s Big Expenditures (continued)

Step 2 Double-click the collection (the box of gold balls) to show the inspector and look at the formula for the attribute Balance. How is this formula similar to the formula you wrote? How is it different from your formula? Why are those differences there?

Investigate 3. Drag the sliders to represent the values in the problem. With monthly

payments of $350, how many months does it take to pay off the loan?

4. Experiment with other values for the monthly payment by dragging the Monthly_Payment slider. What monthly payment allows you to pay off the loan in exactly 5 years (60 months)? Can you adjust the payment so that you pay off the loan in exactly 60 months and have your last payment the same as the others? You may need to expand the scale on the slider to see this: grab the ends of the axes and drag to change the scale.

5. If you pay off the loan in exactly 60 months, how much do you end up paying for the car all together?

6. Drag the Annual_Interest_Rate slider to explore the impact of changing the interest rate. Does a drop of 1% have much effect on the monthly payments? On the total amount paid? How do the monthly payments or time to pay off the loan change if the interest rate is 18%, which is the approximate rate of most credit cards? What is the total amount paid on a card with an 18% interest rate?

7. Adjust the slider values to find the monthly payment for a 30-year home mortgage of $146,000 with an annual interest rate of 7.25%. You will need to add cases to the collection so you have a total of 360 months (30 years): select the collection and choose Collection � New Cases. (Note : Because this greatly increases the number of calculations, the sliders will not work very smoothly. You will need to type new values into the sliders rather than drag.) What are your monthly payments? How much do you end up paying for the house?



In this demonstration, you’ll play some games to see how much you can push around data points without changing their box plots.

Experiment 1. Draw a box plot on your paper for the data set {0, 1, 2, 3, 4, 5, 6, 7, 8,

9, 10, 11, 12}. How much would the box plot change if the 12 became a 13?

2. Open the document BoxPlots.ftm. You will see four collections with a dot plot and a box plot for each. Right now they are all the same. Is the box plot that Fathom created the same as the one you completed by hand? (It might not be. Some books outline methods of graphing box plots that are different from those used by Fathom.) If your plot is different, explain how Fathom calculates the five numbers determining the box plot.

3. What is the mean of the data?

Here are two games to play with the data. The object of both games is to move dots in the dot plot to get the smallest and largest values of the mean.

Game 1 4. Below are the rules of Game 1. Play Game 1 to find the smallest

mean. What strategy did you use in playing the game?

a. You can drag only one dot at a time in the dot plot.

b. You cannot drag a dot past another one in the dot plot.

c. You cannot change any of the five-number summary values (minimum, first quartile, median, third quartile, or maximum). That is, the box plot should always look exactly like the box plot for the original data. The mean is not part of the five-number summary and is free to change. To see the original data, look at the bottom-right graphs.

5. What is the smallest value of the mean? Check your answer against those of others in the class. What numbers give the smallest mean? As another check, calculate the mean by hand and write it as a fraction in lowest terms.

(continued)

Lesson 2.1 • Box Plots Fathom




6. Use the Game 1: Largest Mean graphs to find the largest possible value of the mean. Follow the Game 1 rules. What values will make the largest possible mean? Write the mean of these values as a fraction in lowest terms.

7. What is the difference between the largest and smallest possible values of the mean for Game 1? Express this as a fraction in lowest terms. Devise a shortcut for finding this difference.

8. Compare the completed dot plots for Smallest Mean and Largest Mean. Which of the two graphs has the bigger spread? Justify your answer.

Game 2To play Game 2, follow all the rules of Game 1, except you may change the value of the median. None of the other four values of the five-number summary can change.

9. Use the Game 2: Largest Mean dot plot to play Game 2, finding the largest possible value for the mean under the relaxed rules. What strategy did you use? What is the largest possible value for the mean now? What is the median value when the mean is the largest possible? Is this graph skewed left or skewed right?

10. Without moving any dots, determine the smallest possible mean and the corresponding value of the median for Game 2.

Lesson 2.1 • Box Plots (continued)



(continued)

Shatevia took a random sample of 50 students who own MP3 players at her high school and asked how many songs they have stored. In this demonstration, you’ll use histograms to study her data.

ExperimentStep 1 Open the document MP3Players.ftm. You’ll see a box of gold balls,

called a collection, that holds data about several individuals, or cases.

Step 2 Select the MP3 Players icon and choose New � Case Table from the Object menu. Scroll down to see more data.

Step 3 Choose New � Graph from the Object menu.

Step 4 In the case table, select the heading “Number_of_Songs.” This heading is the name of what Fathom calls an attribute. Drag this attribute from the case table to the horizontal axis of the graph, releasing the mouse over the message “Drop an attribute here.” In the corner of the graph window, click on Dot Plot and choose Histogram from the pop-up menu that appears.

Step 5 Double-click a blank part of the histogram. An inspector appears with information about the histogram. Change the bin width to 50 and close the inspector. Choose Histogram from the pop-up menu again to see all the data.

Lesson 2.3 • MP3 Players Fathom




Step 6 Select the graph and choose Duplicate Graph from the Object menu. Move this histogram so that it doesn’t overlap the first one by selecting it and dragging the top of the frame. Double-click this histogram and change the bin width to 10. Close the inspector.

Investigate 1. From the first histogram, how many of these 50 MP3 players had

about 925 songs?

2. Estimate the range, median, and mean of the data as well as you can from either histogram.

3. Which graph is better at showing the overall shape of the distribution? What is that shape?

4. Which graph is better at showing the gaps and clusters in the data?

5. Why are the bins shorter in one of the histograms than in the other one?

ExperimentStep 7 Move the cursor over the edge of one of the bins. It should become

a double arrow. Drag the cursor to change the bin width.

Step 8 Change the scale of one of the histograms by moving the cursor over an axis and dragging. If you drag the center of an axis, you’ll move it. If you drag either end, you will stretch or shrink it.

Investigate 6. Experiment with other bin widths, and perhaps with the scale of the

vertical axis, until you find settings that you think give you the most information. Describe them.

7. How can you tell if a histogram is showing all 50 MP3 players?

Lesson 2.3 • MP3 Players (continued)



A teenager requires from 1800 to 3200 calories per day, depending on growth rate and level of activity. The diet should include a high level of protein, moderate levels of carbohydrates and fat, and as little sodium, saturated fat, and cholesterol as possible. In this demonstration you’ll use Fathom to help you analyze the nutritional content of some common fast food items.

ExperimentStep 1 Open the document FastFood.ftm.

Step 2 Select the collection icon. Move the pointer up to the tool shelf. Drag the “Table” icon to the work space and drop it to create a new case table.

Step 3 Drag a new summary table and two new graphs from the tool shelf.

Step 4 Choose one attribute. Drag its title from the case table to the right arrow in the summary table. Choose Add Basic Statistics from the Summary menu and expand the summary table.

Step 5 Drag the same attribute from the case table to the horizontal axis of one of the graphs. Choose Box Plot from the pop-up menu.

Step 6 Drag the same attribute from the case table tothe horizontal axis of the other graph. Choose Histogram from the pop-up menu.

Investigate 1. What is the bin width of the histogram? If you want

to adjust it, drag the edge of a bin or double-click the graph and edit the Properties tab.

2. What is the meaning of each basic statistic in the summary table?

3. Which of the basic statistics could you find from the box plot and which from the histogram?

4. Select one of the graphs and choose Plot Value from the Graph menu. Type mean( and close the window. A blue line representing the mean will appear on the graph. What shape do the data have relative to the mean? Plotting the mean on the other graph may help you decide.

5. Using the box plot and histogram, approximate the five-number summary.

6. The data point at the 90th percentile lies in which bin of the histogram?

7. Drag each attribute in turn to make a histogram. Is there a candidate for an outlier? If so, click on its bin in the histogram, go to the case table, and identify what it is and its value. Then drag the attribute to the summary table and determine whether that data value is an outlier.

Lesson 2.3 • Eating on the Run Fathom




Every ten years, the U.S. Census Bureau collects a variety of data about the population. The bureau analyzes microdata—information about individuals—to produce reports that help governments, organizations, businesses, and citizens make decisions. In this demonstration you’ll make a conjecture about microdata from one small part of the country and then use Fathom to test it.

ExperimentStep 1 Open the document CA_includesBerkeley2000.ftm.

Step 2 Select the collection icon and choose New � Case Table from the Object menu. Scroll to see how many people there are and what specific attributes were collected. Before you make your own conjecture, we’ll test the conjecture “The males are older than the females.”

Step 3 Choose New � Summary Table from the Object menu. Drag the attribute sex from the case table to the right arrow in the summary table. Drag the attribute age to the down arrow in the summary table. Choose Add Basic Statistics from the Summary menu. Now choose Add Five-Number Summary from the Summary menu.

Step 4 Choose New � Graph from the Object menu. Drag the attribute age from the case table to the hori-zontal axis. Drag the attribute sex to the vertical axis. Choose Box Plot from the pop-up menu.

Investigate 1. The names of the statistics are listed below the

summary table in the same order that the statistics themselves appear in the table. Which statistics are most helpful in testing the conjecture? What conclusions can you draw?

2. Do the box plots help you support or refute your conjecture? Explain.

3. Choose Histogram from the pop-up menu on the graph. Do the histograms support your conjecture? Does changing the bin width help? (Note: If you select a bin, the entries in that bin will be highlighted in the case table. You can choose Delete Cases from the Edit menu to study the data without outliers.)

4. Make another conjecture, such as “Most people have incomes below $10,000.” If you deleted cases in Question 3, you can restore them by choosing Undo Delete Cases from the Edit menu. To test this conjecture, drag the attribute income to the right arrow in the summary table. To make a new graph, click on the graph, choose Delete Graph from the Object menu, and make a new histogram with the attribute income on the horizontal axis. What can you conclude about your conjecture? Can you explain any patterns you see?

Exploration • Different Ways to Analyze Data Fathom



(continued)

In this demonstration you will launch a rocket and use your motion sensor’s data to estimate the rocket’s speed. Then you will write an equation for the rocket’s distance as a function of time. Choose one person to be the monitor and one person to be the launch controller.

ExperimentStep 1 Make a rocket of paper and tape. Design your rocket so that it can

hold an inflated balloon and be taped to a drinking straw threaded on a string. Color or decorate your rocket if you like. Tape your rocket to the straw on the string.

Step 2 Start Fathom and open the document BalloonBlastoff.ftm. Plug a motion sensor into the USB port on the computer. You will see a blank experiment setup. The current value detected by the motion sensor appears in the Distance meter at the top right.

Step 3 Inflate a balloon, but do not tie off the end. The launch controller should insert it into your rocket and hold it closed. Tie the string or hold it taut and horizontal.

Step 4 Hold the sensor behind the rocket. At the same time the monitor clicks the Turn Experiment On button in Fathom, the launch controller should release the balloon. Be sure nobody’s hands are between the balloon and the sensor. The experiment is set to collect data for 5 seconds, but you can stop it sooner by clicking Turn

Lesson 3.2 • Balloon Blastoff Fathom




Experiment Off. If you do not get good data, repeat the experiment by simply clicking Turn Experiment On again. This will replace the data in Fathom.

Step 5 Notice that the data have appeared in the graph, with time as the independent variable, x. Clean up the graph by selecting any points that are not part of the actual experiment and choosing Edit � Delete Cases.

Step 6 Select the graph and choose Graph � Add Movable Line. Experiment with moving the line. Shift it by dragging it in the middle and rotate it by dragging either side.

Investigate 1. What are the domain and range of your data? Explain.

2. Use the movable line to select four representative points from the rocket data. Explain why you chose these points. (To see the coordinates in Fathom, click the Cases tab in the inspector to go to that panel. When you click on a point in the graph, its coordinates appear in the inspector.)

3. Record the coordinates of the four points and use the points in pairs to calculate slopes. This should give six estimates of the slope. You can use a summary table as a calculator: drag a new summary table into the document and choose Summary � Add Formula. Double-click a formula to edit it.

4. Are all six slope estimates that you calculated in Question 3 the same? Why or why not?

5. What are the mean and median of the slopes? With your group, decide which value best represents the slope of your data (it does not have to be the mean or median). Explain why you chose this value.

6. How does the value you chose compare to the slope of the movable line?

7. What is the real-world meaning of the slope, and how is this related to the speed of your rocket?

Lesson 3.2 • Balloon Blastoff (continued)



Lesson 3.3 • Finding a Line of Fit Fathom

On a barren lava field on top of the Mauna Loa volcano in Hawaii, scientists have been monitoring the concentration of CO2 (carbon dioxide) in the atmosphere continuously since 1959. In this demonstration you’ll use Fathom to help you find a line of fit for the carbon dioxide concentrations in 13 different years.

ExperimentStep 1 Open the document MaunaLoa.ftm. A case table should be visible.

Step 2 Drag a new graph from the tool shelf. Drag the Year attribute to the horizontal axis and the CO2Level attribute to the vertical axis.

Step 3 From the Graph menu, choose Add Movable Line. A line will be drawn. Its equation will appear at the bottom of the graph.

Investigate 1. Select the line at a point away from the center, and drag it to change

the line’s slope. Drag until the line’s direction is approximately the same as that of the points. According to the equation, what is the slope of the line now?

2. Select the line at a point near the center and drag it to translate it vertically. Adjust the direction and height of the line until the number of data points above the line is the same as the number of points below the line. From the equation, what are the slope and y-intercept of the line now?

3. Adjust the line to pass through the data points for 1982 and 2002. What are the slope and y-intercept now? Is this line better or worse for making predictions than the line you drew in Question 2?

4. Using the line from Question 3, predict the CO2 concentration in the year 2050. To see the value for 2050, move the pointer to the horizontal axis until it turns into a hand with the fingers to the right. Drag to the left until the year 2050 appears. Drag the vertical axis similarly. You can read the value by placing the cursor on the line and looking at the bottom left of the Fathom window.




The populations of U.S. states vary, so the numbers of drivers also vary. But is the ratio of the number of drivers to the population fairly constant? In this demonstration you’ll use Fathom to help you analyze state data on the number of drivers relative to the population size.

ExperimentStep 1 Open the document

States - CarsNDrivers.ftm. A collection icon and case table will be visible. Note that data are not available for some states.

Step 2 Make a new graph. Drag theattribute PopThou to the hori-zontal axis and the attribute DriversThou to the vertical axis. (PopThou is the state’s population in thousands; DriversThou is the number of licensed drivers in thousands.)

Step 3 Choose Add Movable Line from the Graph menu. Then choose Make Residual Plot from the Graph menu.

Investigate 1. Change the slope of the movable line by rotating it until its direction

is about the same as the data points. What is the real-world meaning of the slope?

2. Translate the line until the number of data points above the line is the same as the number of points below the line. (More rotation may be needed.) Does it make sense to have the line go through the origin?

3. As you adjust the movable line, note the outliers. How can you tell the outliers from the residual plot?

4. Double-click the point representing each outlier. Use the inspector to see the corresponding states. Which states are outliers? Why might these states have unusually high or low values?

5. With the graph selected, choose Show Squares from the Graph menu. Experiment with the movable line and explain how the squares are drawn.

6. Choose Median-Median Line from the Graph menu. How do its equation and sum of squares compare with those of your movable line?

7. Choose Least-Squares Line from the Graph menu. How does its sum of squares compare with those of the other lines?

Exploration • A Good Fit? Fathom



Lesson 4.3 • Lines Sketchpad

In this demonstration you’ll explore vertical and horizontal translations of the graph of a linear function.

Sketch 4

2

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–4

–5 5

Dragh

Step 1 Open the document Transformations.gsp to the Lines page.

Step 2 Press the Show Horizontal Translation button.

Step 3 Drag the blue point on the h slider to the left and the right.

Investigate 1. Describe how the transformed function changes as you drag h.

2. What happens to the x-intercept as you drag the point to the right? To the left?

SketchStep 4 Press the Show Vertical Translation button.

Step 5 Drag the blue point on the k slider up and down.

Investigate 3. What happens to the y-intercept as you drag the point up? Down?

4. Adjust the h slider to 2 units right and the k slider to 3 units down. What is the equation of this transformed function?

SketchStep 6 Choose Plot New Function from the Graph menu.

Step 7 Type in the equation you found in Question 4. Be sure you use the “*” key for multiplication.

Step 8 Press the Show Transformed Equation button.

Investigate 5. Was your equation from Question 4 correct? Was it in the same

form as the transformed equation? Describe any differences.

6. Explain why f (x) � (x � h) � k might be a better equation to use than f (x) � a � bx when you’re studying transformations.

7. What is the equation of this transformed graph? Does more than one equation describe this function?

4

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Lesson 4.4 • Parabolas Sketchpad

In this demonstration you’ll explore vertical and horizontal translations of the graph of the parent function f (x) � x 2.

SketchStep 1 Open the document Transformations.gsp to the Parabolas page.

Step 2 Press the Show Vertical Translation button.

Step 3 Drag the blue point below the k slider up and down.

Investigate 4

–2–4 4

1. What happens to the vertex as you drag the point up? Down?

2. Describe how the transformed function changes as you drag k.

3. What is the equation of this translated function?


Step 5 Type in the equation you found in Question 3. Be sure you use the “^” key to get an exponent. For example, you would enter x 2 as x^2.

Step 6 Press the Show Transformed Equation button.

Investigate 4. Was your equation from Question 3 correct? Explain any discrepancies.

SketchStep 7 Press the Hide Transformed Equation button.

Step 8 Press the Show Horizontal Translation button.

Step 9 Drag the blue point on the h slider to the right and the left.

Investigate 5. How does the function change as you move h to the right? To the left?

6. Adjust the h slider to 3 units left of the y-axis. Predict the equation of this translated function. Repeat Steps 4–6 to check your equation.

Sketch

4

6

–2–4 2

Step 10 Press the Show Point (h, k) button.

Investigate 7. What point is (h, k)?

8. Translate the function left 4 units and down 2 units. What is the vertex of your new function?

9. What is the equation of the translated graph at right?



(continued)

Lesson 4.5 • Square Roots Sketchpad

In this demonstration you’ll explore vertical and horizontal translations and reflections of the graph of the square root function.

Sketch 4

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–2–2 42

Step 1 Open the document Transformations.gsp to the Square Roots page.

Step 2 Press the Show Reflection Across x-axis button.

Investigate 1. What is the equation for this new graph?

2. Write this equation in terms of f (x).

3. What are the domain and range of this transformed equation?


Step 4 Type in the equation you found in Question 2. Be sure you use parentheses as appropriate.

Step 5 If your equation is not correct, choose Undo Construct Objects from the Edit menu. Repeat Steps 3 and 4 until you find the correct equation.

Step 6 Press the Show Reflection Across y-axis button.


5. Write the equation in terms of f (x).


SketchStep 7 Repeat Steps 3–5 to check your solutions.

Step 8 Press the Show Reflection Across x- and y-axes button.


8. Write the equation in terms of f (x).


SketchStep 9 Repeat Steps 3–5 to check your solutions.

Step 10 Select parameter h. Use the “�” and “�” keys to change the value of h.




Lesson 4.5 • Square Roots (continued)

Step 11 Select parameter k. Use the “�” and “�” keys to change the value of k.

Investigate 10. How does changing h affect the graphs? Explain.

11. How does changing k affect the graphs? Explain.

SketchStep 12 Select parameter h and then parameter k. Choose

Plot As (x, y) from the Graph menu.

Investigate 12. What point is (h, k)?

13. What are the coordinates of this point on the reflected graphs?



Lesson 4.6 • Absolute Value Sketchpad

In this demonstration you’ll explore horizontal and vertical dilations of the graph of the absolute-value function, as well as horizontal and vertical translations and reflections.

Sketch

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Step 1 Open the sketch Transformations.gsp to the Absolute Value page.

Step 2 Press the Show Vertical/Horizontal Dilations button.

Step 3 Select parameter a. Use the “�” and “�” keys to change the value of a.

Step 4 Select parameter b. Use the “�” and “�” keys to change the value of b.

Investigate 1. How does changing the value of a change the function?

2. How does changing the value of b change the function?

3. If you look only at positive values for x, what is the slope of that side of the function?

4. How does the slope relate to a and b?

5. Rewrite the transformed function using f (x), and a and b. Make sure your equation works for both positive and negative values of x.

6. What happens when a equals 0? Why?

7. What happens when a or b is negative?

SketchStep 5 Press the Show Vertical/Horizontal Translations button.



Investigate 8. How does changing h and k affect the translated function? (Hint: You

may need to set the parameters for a and b to numbers close to zero in order to see the graph.)

9. Write the equation in terms of f (x). Press the Show Transformed Equation button to check your work.

10. What point is (h, k)? Press the Show Point (h, k) button to check your work.




Lesson 4.6 • Science Fair Fathom

A panel of science fair judges rate 20 exhibits. The judges decide to adjust the ratings to be generally higher and closer together.

ExperimentStep 1 Open the document ScienceFair.ftm. You should see a collection

icon and case table of ratings.

Step 2 Make two sliders by choosing New � Slider twice from the Object menu. Double-click slider V1 and name it stretch. Double-click slider V2 and name it translate.

Step 3 Create a new attribute called Adjusted. Double-click the collection box and then on the formula column for Adjusted. Enter the formula stretch * Rating � translate.

Step 4 Make a new graph and drag Rating to the horizontal axis. Then drag Adjusted to the horizontal axis and drop it on the plus sign that appears. The graph will split to show you both plots at once.

Step 5 Choose Plot Value from the Graph menu. Type mean( and click OK. Then choose Plot Value again, enter mean() � stdDev(), and click OK. The values of the mean and mean plus standard deviation that appear are for the plot of Adjusted values.

Investigate 1. Adjust the sliders until the Adjusted scores are as close as you can get

them to the Ratings. What are the mean and standard deviation of the scores?

2. Adjust the sliders to make all Adjusted scores 6 points more than the corresponding Ratings. How do the mean and standard deviation change?

3. Readjust the sliders to the positions you found in Question 1, and then stretch the data to make the top score 100. How have the mean and standard deviations changed?

4. Adjust both sliders until the mean is about 90 and the standard deviation is about 5. Describe how you did this and what the stretch and translation are now.



(continued)

In this demonstration you’ll explore transformations of the circle family.

SketchStep 1 Open the sketch Transformations.gsp to the Circles and

Ellipses page.

Step 2 Press the Show Circle Radius Parameter button.

Step 3 Select parameter r. Use the “�” and “�” keys to change the value of r.

Investigate 1. How does changing the value of r change the function?

2. What is the equation for a half circle with radius 5? With diameter 16?

SketchStep 4 Press the Show Vertical/Horizontal Dilations button.

–5 5

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Investigate 3. How does changing a affect the function? Describe this transformation

in terms of f (x).

4. How does changing b affect the function? Describe this transformation in terms of f (x).

5. Are there any values of a or b for which the function is undefined? Explain.

6. What happens when a or b is negative?

SketchStep 7 Press the Show Vertical/Horizontal Translations button.



Step 10 Press the Show Point (h, k) button.

Lesson 4.7 • Circles and Ellipses Sketchpad




Investigate 7. How does changing h and k affect the function?

8. What point is (h, k)?

9. Using a, b, h, k, and r, write the equation for the transformed function.

10. How would you graph a complete circle or ellipse? Explain why circles and ellipses are not graphs of functions.

Lesson 4.7 • Circles and Ellipses (continued)



In this demonstration you’ll explore power functions with integer exponents.

SketchStep 1 Open the document PowerFunctions.gsp to the page Power

Function Animation.

Step 2 Press the Animate Integer Exponent button to change the variable n in the power function.

Investigate 1. Is there a pattern in the relationship between the exponent and the

shape of the graph? Explain.

2. Predict what the graph of f (x) � x 7 will look like. What about f (x) � x 10? f (x) � x 211?

3. Describe the symmetry of the power functions.

SketchStep 3 Go to the page Power Function Slider.

5

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a

n drag

drag

Step 4 Adjust the n slider to change the value of the integer exponent of the power function. To get positive integer exponents, drag the point to the right. To get negative integer exponents, drag the point to the left.

Investigate 4. Test your predictions for f (x) � x 7 and f (x) � x 10. Were your

predictions correct? Why or why not?

5. What happens when the exponent is 0? Explain algebraically why this happens.

6. Describe the behavior of the power function when the exponent is a negative integer.

Lesson 5.2 • Power Functions Sketchpad




In this demonstration you’ll explore power functions with rational exponents.

SketchStep 1 Open the document RationalExponents.gsp to the page Rational

Exponent Animation.

Step 2 Press the Animate Rational Exponent button to change the variable r in the power function. Note that this variable is the denominator of the exponent rather than the exponent itself.

Investigate 1. Is there a pattern in the relationship between the exponent and the

shape of the graph? Explain.

2. Predict what the graph of f (x) � x 1�7 will look like. What about f (x) � x 1�10?

SketchStep 3 Go to the page Rational Exponent Slider.

5

drag

r

n

drag

4

2Step 4 Adjust the sliders to change the value of the denominator, r, or

the numerator, n, of the exponent of the power function. To get positive values, drag the point to the right. To get negative values, drag the point to the left.

Investigate 3. Test your predictions for f (x) � x 1�7 and f (x) � x 1�10. Were your

predictions correct? Why or why not?

4. Describe the behavior of the power function when the numerator is 1 and the exponent is negative.

5. Describe the similarities and differences between power functions with positive integer exponents, and power functions with positive rational exponents when the numerator is equal to 1.

6. Describe the similarities and differences between functions with positive integer exponents, and functions with positive rational exponents when the numerator, n, is positive and the denominator is constant.

7. What happens to the power function when n � 0? Explain algebraically why this happens.

8. Describe the similarities and differences between functions with integer exponents, and functions with rational exponents when the numerator is negative and the denominator is constant.

Lesson 5.3 • Rational Exponents Sketchpad



In this demonstration you’ll explore the inverse of a linear function.

Sketch

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Drag

Drag

slope

y-intercept

Step 1 Open the document LinearInverse.gsp.

Step 2 Press the Show Inverse button.

Step 3 Adjust the y-intercept slider to change the y-intercept of the original function. Adjust the slope slider to change the slope of the original function.

Investigate 1. When you change the y-intercept, what do you notice about the

intercepts of the function and its inverse?

2. When you change the slope, what do you notice about the intersection of the two functions?

SketchStep 4 Select the graphs of the function and the inverse. Choose

Intersection from the Construct menu.

Step 5 Select the intersection. Choose Trace Intersection from the Display menu.

Step 6 Change the y-intercept and slope of the original function.

Investigate 3. What do you notice about the intersection of the two functions?

4. What is the equation of this line?

SketchStep 7 Choose Erase Traces from the Display menu.

Step 8 Press the Show Point on Function and Show Point on Inverse buttons.

Step 9 Change the y-intercept and slope of the original function by adjusting the sliders.

Investigate 5. How do the coordinates of the two points relate?

6. If (�1, 2) is a point on the original function, name a point on the inverse of the function.

Lesson 5.5 • The Inverse of a Line Sketchpad




In this demonstration you’ll explore the behaviors of the exponential and logarithmic functions.

SketchStep 1 Open the document ExponentialLogarithm.gsp to the page

Exponential.



Investigate

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1. Set parameter a equal to 1. Describe the graph for b � 1, b � 1, and 0 � b � 1.

2. What happens to the graph if b � 0? Explain why this happens.

3. Choose a positive value for b. Describe how the value of a affects the graph.

4. How would you translate the function right 2 units and down 3 units? Write an equation for the translated function.

SketchStep 4 Go to the page Logarithmic.



Investigate 5. Set parameter a equal to 1. Describe the graph for different values

of b.

6. Set parameter b equal to 2. Describe how the value of a affects the graph.

7. How would you translate the function right 4 units and down 3 units? Write an equation for the translated function.

Lesson 5.6 • Exponential and Logarithmic Functions Sketchpad



In this demonstration you’ll find a relationship between the temperature of a cooling object and time.

ExperimentStep 1 Plug a temperature probe into the USB port on your computer and

open a new Fathom document.

Step 2 Drag a new collection down from the tool shelf. Then, click the Meter icon, choose Temperature Sensor, and click somewhere to place the meter. The meter should show the current value for the sensor.

Step 3 Drag the plug in the meter to the collection to start an experiment. Double-click the experiment to show the inspector, and set the experiment to record 6 cases per minute for 10 minutes.

Step 4 Heat the end of the probe by placing it in hot water or holding it tightly in the palm of your hand. When it is hot, set the sensor on a table so that the tip is not touching anything and click Turn Experiment On.

Investigate 1. While the data are being collected, predict what the graph of

Temperature as a function of Time will look like as the temperature sensor cools. Sketch a graph of your prediction. Label the axes and mark the scale on your graph. Compare your prediction with others in your group. Do you all agree? Discuss your predictions and see if you change your mind about your prediction.

Lesson 5.8 • Cooling Fathom

(continued)




2. When the experiment is done, create a scatter plot of Temperature versus Time: click the Cases tab in the inspector and drag the attribute names to the axes of the graph. Does the shape of the graph match your prediction? Does it match the prediction of any of your group members? If not, explain why the shape you see does, indeed, make sense.

3. The shape of the graph should look somewhat like an exponential decay function. Explain why the graph seems to be leveling off at a value greater than 0. What appears to be the limiting value?

ExperimentStep 5 Drag a new slider from the tool shelf. Change

the slider name from V1 to Shift. Set the value of Shift to your estimate of the limiting value from Question 3. (Note : You must add units to your slider. For example, if you think the limit is 30, type 30 Celsius. The slider value will read 30°C.)

Step 6 In the inspector for the collection, define a new attribute called Log _Temp. Double-click the formula cell for the attribute and enter the formula log(Temperature – Shift).

Investigate 4. Create a new scatter plot of (Time, Log _Temp). Does this plot look

linear? Explain.

5. Adjust the slider until the plot looks as linear as possible. Select the graph and choose Graph � Least-Squares Line. Adjust the slider if needed to fit the line better. Stretch the slider scale if you need to make finer adjustments. Write the equation for this line, and use it to write an equation for the (Time, Temperature) data. Explain the real-world meaning of the parameters (numerical values) in the equation.

Lesson 5.8 • Cooling (continued)



In clear weather, the distance you can see from an airplane window depends on your height above the Earth.

ExperimentStep 1 Open the document ViewingDistance.ftm. A collection icon and

case table will be visible. Here height is given in meters and viewing distance in kilometers.

Step 2 Add two attributes, logH and logD, to the table.

Step 3 Double-click the collection icon, and enter formulas for the two new attributes. For logH enter log(Height). For logD enter log(Distance).

Step 4 Drag down four graphs. Put the different pairs of Height or logH on the horizontal axes and Distance or logD on the vertical axes.

Investigate 1. Which graph seems the most linear? Use median-median lines,

residual plots, and sums of squares as needed to help you decide.

2. Write the equation for the median-median line of the graph you chose in Question 1. Write each side as a power of 10, and use the properties of logarithms to rewrite the result in a standard form.

3. Does your answer to Question 2 give an exponential equation or some other kind of equation as a model of the original data?

ExperimentStep 5 Graph your equation with the original data by selecting the

(Height, Distance) plot and choosing Plot Function from the Graph menu. Enter the right side of the equation (in terms of Height). When you click Apply or close the editing window, the graph and its equation will appear.

Investigate 4. Does your function fit the data? Adjust the equation until you are

satisfied with the fit.

Lesson 5.8 • Curve Straightening Fathom



The International Canine Academy raises and trains Siberian sled dogs and dancing French poodles. Breeders can supply the academy with at most 20 poodles and 15 Siberian huskies each year. Each poodle eats 2 lb/d of food, and each sled dog eats 6 lb/d. Food supplies are restricted to at most 100 lb/d. A poodle requires 1,000 h/yr of training, whereas a sled dog requires 250 h/yr. The academy cannot provide more than 15,000 h/yr of training time. If each poodle sells for a profit of $200 and each sled dog sells for a profit of $80, how many of each kind of dog should the academy raise in order to maximize profits?

SketchStep 1 Open the document LinearProgramming.gsp to the page

Exercise 6.

Step 2 Press the Show Equations button.

Investigate 1. What do x and y represent?

2. What is the real-world meaning of each equation?

SketchStep 3 Press the Hide Equations button.

Step 4 Press the Show Profit button.

Step 5 Drag point Z around the shaded solution area.

14

16

2

4

6

8

10

12

–25

O A

CD

Z

B

10

Investigate 3. Approximate the coordinates and the profit at each vertex.

4. Which of these coordinates presents the maximum profit? Is this point realistic? Why or why not?

Lesson 6.6 • Linear Programming Sketchpad

(continued)




SketchStep 6 Choose Snap Points from the Graph menu.

Step 7 Drag point Z around the shaded solution area.

Investigate 5. What is the maximum profit? At what point does it occur?

6. What is the solution to the exercise?

Explore More

SketchStep 8 Go to the page Explore More.

Step 9 Press the Show Edit Food Consumption button.

Step 10 Select one of the values. Use the “�” and “�” keys to change the value.

Step 11 Drag point Z around the shaded solution area to find the maximum profit.

Investigate 7. If you kept the amount of daily food the same for poodles at 2 lb/d

and for sled dogs at 6 lb/d, but had money to increase the total food consumption, how many pounds of food per day would maximize your profit?

8. What is the profit at this point?

9. Repeat Steps 9 and 10 for the training hours. Experiment with different values to find the maximum profit.

Lesson 6.6 • Linear Programming (continued)



What function models the height of an object falling due to the force of gravity? In this demonstration you’ll use a motion sensor to collect data, and use the finite differences method to find a function to model the data.

ExperimentStep 1 Plug the motion sensor into the computer. Open the Fathom

document FreeFall.ftm. You will see an empty experiment set up. The Height meter should be displaying the value of the sensor.

Step 2 Scroll your cursor over the Height meter. You will see a ± button appear. Click that button once to reverse the axis. Double-click the Experiment with Height collection to show the inspector.

Step 3 Have one person hold the motion sensor 2 meters above the ground facing downward. A second person holds a playground ball or basketball under the sensor. A third person clicks Turn Experiment On, then the second person releases the ball. The second person may simply drop the ball or give it a slight upward toss. After the ball hits the ground, the third person clicks Turn Experiment Off.

Step 4 Look at the data in the graph, which shows (Time, Height). If you did not get a section showing the drop, repeat the experiment.

Lesson 7.1 • Free Fall Fathom

(continued)




Step 5 Clean up the data in the (Time, Height) graph: select all the points that were collected while the second person was holding the ball or after the ball hit the ground, and choose Edit | Delete Cases. The cases that remain should form a nice, smooth curve. Select the graph and choose Graph | Rescale Graph Axes.

Step 6 Find the differences for Height: Add a new attribute to the table called Diff1 and double-click the formula cell. Enter the formula Height – prev(Height). Drag down a new graph and make a scatter plot of the (Time, Diff1) data.

Step 7 Repeat Step 6 until you find differences that are nearly constant. To check the mean of any differences, select that graph, choose Graph | Plot Function and enter mean( ).

Investigate 1. Write a description of each graph from Step 7 and explain what these

graphs tell you about the data.

2. Based on your results from using finite differences, what is the degree of the polynomial function that models free fall? Write the general form of this polynomial function.

3. Follow the example on page 379 of your book to write a system of three equations in three variables for your data. Solve your system to find an equation to model the position of the ball as a function of the number of seconds it has been falling.

4. Repeat Question 3 using a different set of three points. Do you get the same equations? Explain.

Lesson 7.1 • Free Fall (continued)



This demonstration will give you practice in using the three forms of a quadratic function—general form, vertex form, and factored form—with real data. You’ll find that the form you use guides which features of the data you focus on. Conversely, if you know only a few features of the data, you may need to focus on a particular form of the function.

ExperimentStep 1 Slightly prop up one end of a long table. Place the motion sensor at

the low end of the table and aim it toward the high end. With tape or chalk, mark a starting line 0.5 m from the sensor on the table.

Step 2 Practice rolling the can up the table directly in front of the motion sensor. Start the can behind the starting line. Give the can a gentle push so that it rolls up the table on its own momentum, stops near the end of the table, and then rolls back. Stop the can after it crosses the line and before it hits the motion sensor.

Step 3 Plug the motion sensor into your computer and open a new Fathom document. Click the Meter icon on the object shelf, choose Motion Detector, and drag it into the workspace. The meter should display the current value of the sensor. Drag down a new collection from the object shelf, and drag the plug from the meter to the collection to start an experiment. (The plug appears when you scroll over the meter.)

Step 4 Set the experiment to collect data for 7 seconds, then click Turn Experiment On. When the sensor begins, roll the can up the table.

Lesson 7.2 • Rolling Along Fathom

(continued)




Step 5 Select the collection and make a new case table. There will be two attributes, Time and Distance. Create a new attribute called AdjDistance. This attribute will adjust the distance by subtracting 0.5 m to account for the position of the starting line. Select the column and choose Edit | Edit Formula. Enter the formula Distance – 0.5 m.

Step 6 Drag down a new graph and make a scatter plot of (Time, AdjDistance). To clean up the data, select any points that do not fit the pattern (points before you started the can and after you stopped it), and choose Edit | Delete Cases.

Investigate 1. What shape is the graph of the data points? What type of function

would model the data? Use finite differences to justify your answer. To find the differences in Fathom, make a new attribute in the case table and give it the formula Distance – prev(Distance).

2. Locate the vertex and another point on your graph. Find the coordinates of these points and use them to write the equation of a quadratic model in vertex form. (You can roll the cursor over a point and see its coordinates in the status bar in the lower left corner of the Fathom window.)

3. From your data, find the distance of the can at 1, 3, and 5 seconds. Use these three data points to find a quadratic model in general form.

4. Locate the x-intercepts on your graph. Approximate the values of these x-intercepts. Use the zeros and the value of a from Question 3 to find a quadratic model in factored form.

5. Verify by graphing that the three equations in Questions 2, 3, and 4 are equivalent, or nearly so. To do this, select the graph and choose Graph | Plot Function. In the formula editor, type your equation. Repeat to add the other two functions.

6. Write a few sentences explaining when you would use each of the three forms to find a quadratic model to fit parabolic data.

Lesson 7.2 • Rolling Along (continued)



Lesson 7.4 • Quadratic Functions Sketchpad

In this demonstration you’ll explore how the coefficients a, b, and c affect the quadratic function in the general form f (x) � ax 2 � bx � c.

SketchStep 1 Open the document QuadraticFunction.gsp to the page Animate

Coefficients.

Step 2 Press the Animate a button or adjust the a slider to change the value of a.

Investigate 1. Describe what happens to the graph as the value of a changes.

2. What do you notice when a is a negative value?

3. What point remains constant? Explain why this happens.

SketchStep 3 Press the Animate b button or adjust the b slider to change the

value of b.

Step 4 Press the Animate c button or adjust the c slider to change the value of c.

Investigate 4. Describe what happens to the graph as the value of b changes.

5. Describe what happens to the graph as the value of c changes.

Explore More

SketchStep 5 Go to the page Trace Vertex.

Step 6 Press the Animate a button to change the value of a and trace the vertex. (Choose Erase Traces from the Display menu to erase the trace.)

Investigate 6. What do you notice about the vertex as a changes?

7. Find the equation of this graph in terms of a, b, and c. (Hint: What are the coordinates of the vertex?)

8. Repeat Step 6 for the value of b. What do you notice about the vertex as b changes? What is the equation of this graph?

9. Repeat Step 6 for the value of c. What do you notice about the vertex as c changes? What is the equation of this graph?

4

–2

–4

5

Animate c

Animate b

Animate a

a

b

c

4

–2

–4

5

Animate c

Animate b

Animate a

a

b

c




In this demonstration you’ll explore the characteristics of higher-degree polynomials.

SketchStep 1 Open the document HigherDegree.gsp to the page 3rd Degree.


Step 3 Repeat Step 2 for parameters b and c.

Investigate 1. What values of a, b, and c give the parent function f (x) � x 3?

2. Describe the graph when a � 1, b � 2, and c � 3.

3. Describe the graph when a � 2, b � 2, and c � 3.

4. Are there any values of a, b, and c that reflect the original function across the y-axis? Describe them if so.

SketchStep 4 Press the Show Point B button.

Step 5 Drag point B along function g (x).

Investigate 5. Estimate the local maximum and minimum values when a � 1, b � 2,

and c � 3.

6. What is the local maximum value when a � 2, b � 2, and c � 3?

SketchStep 6 Go to the page 4th Degree and repeat Steps 2–5 for this function.

Investigate 7. What values of a, b, c, and d give the parent function f (x) � x 4?

8. Describe the graph when a � 1, b � �2, c � 3, and d � 0.

9. Find two sets of values of a, b, c, and d that give the function only two roots, at 3 and �2. Describe the differences in the graphs.

10. Approximate the local extremes when a � �2, b � �2, c � 3, and d � 3.

4

2

a � 1.00b � 1.00c � 1.00

f(x) � x3

5–5

4

2

a � 1.00b � 1.00c � 1.00

f(x) � x3

5–5

Lesson 7.7 • Higher Degree Sketchpad

(continued)



SketchStep 7 Go to the page 5th Degree and repeat Steps 2–5 for this function.

Investigate 11. What values of a, b, c, d, and e give the parent function f (x) � x 5?

12. Describe the graph when a � 1, b � �2, c � 3, d � 0, and e � 1.

13. Describe the values of a, b, c, d, and e for a 5th-degree function with only one root.

14. Find the maximum and minimum number of local maximums and minimums possible in a 5th-degree polynomial.

Lesson 7.7 • Higher Degree (continued)



Imagine a race in which you carry an empty bucket from the starting line to the edge of a pool, fill the bucket with water, and then carry the bucket to the finish line. The starting line of the bucket race is 5 m from one end of a pool, the pool is 20 m long, and the finish line is 7 m from the opposite end of the pool, as shown below. In this demonstration you will find the shortest path from point A to a point C on the edge of the pool to point B. That is, you will find the value of x , the distance in meters from the end of the pool to point C, such that AC � CB is the shortest path possible.

5 m

7 m

x 20 – x

B

A

C

SketchStep 1 Open the document BucketRace.gsp. The sketch is set up to

represent the situation in the bucket race, with 1 cm in the sketch representing 1 m. You should see the length of segment x below the segment, and the lengths of segments AC and BC, as well as the total distance walked.

Step 2 Drag point C back and forth along the side of the pool.

Investigate 1. As you drag point C , you should notice that the total distance is

greater near the ends of the pool. Where is the distance the greatest?

2. Locate the position of point C that results in the shortest total distance. What is the value of x at this location? Is there another location that results in the same total distance?

SketchStep 3 Click in white space to be sure nothing is selected. Then, in order,

click on the measurement of segment x, and then the measurement of the total distance. Choose Graph | Plot As (x, y). A grid should appear with a new point at the coordinates (Length of x, Total Distance).

Step 4 With the plotted point selected, choose Measure | Coordinates, then select only this point again and choose Display | Trace Plotted Point.

Lesson 8.1 • Bucket Race Sketchpad

(continued)




Investigate 3. Drag point C again and describe the curve that is traced by the plotted

point. (If you do not see the point being traced, drag the center of the grid to find it.)

4. How does this curve verify your answer to Question 2?

Now imagine that the amount of water you empty out at point B is an important factor in winning the race. This means you must move carefully so as not to spill water, and you’ll be able to move faster with the empty bucket than you can with the bucket full of water. Assume that you can carry an empty bucket at a rate of 1.2 m/s and that you can carry a full bucket, without spilling, at a rate of 0.4 m/s.

Step 5 Choose Measure | Calculate and define a calculation to determine the total amount of time to get from point A to point C. (Recall the formula distance � rate � time.) To enter a measurement in the calculation, click on the measurement in the sketch.

Step 6 Label the time calculation as T1: choose the Text tool and double-click the calculation. Enter the new label and click OK.

Step 7 Repeat to calculate T2, then calculate the total time T1 � T2.

Step 8 Plot the point (x, Total Time), trace this new point, and display its coordinates. You may need to move the center of the grid to bring this new point into view and then choose Display | Erase Traces.

Investigate 5. What formula did you use for T1?

6. Drag point C again and describe the curve that is traced by the newest plotted point. Find the position of point C that minimizes the total time to carry the bucket.

Lesson 8.1 • Bucket Race (continued)



In this demonstration you’ll explore the circle and ellipse, each as a locus of points.

Circles

SketchStep 1 Open the document CirclesandEllipses.gsp to the page Circle

Locus.

Step 2 Press the Animate Point P button.

Step 3 Drag point Change Distance to different locations and repeat Step 2.

Investigate 1. What shape is constructed when you animate point P ?

2. How does the shape change when P changes position?

SketchStep 4 Press the Show Distance button and then the Animate Point P

button.

Investigate 3. What is the relationship between points P and C ?

4. Write a definition for a circle.

Ellipses

SketchStep 5 Go to the page Ellipse Locus.

I

N

K

L

JStep 6 Drag point L around the circle.

Investigate 5. What shape is created when L makes a complete

rotation around point I ?

6. What do you notice about the perpendicular segments in relation to the shape formed?

SketchStep 7 Choose Erase Traces from the Display menu and

then press the Locus � � � Point N button.

Step 8 Drag points J and K to different locations.

Lesson 8.2 • Definitions of Circle and Ellipse Sketchpad

(continued)




Investigate 7. How do points J and K affect the ellipse?

8. What happens when the two circles are the same size?

SketchStep 9 Press the Locus �� Point N button again and then the Show

Construction button.

Investigate 9. Explain how to construct an ellipse given two circles and their

perpendicular diameters.

SketchStep 10 Go to the page Ellipse Foci.

Step 11 Press the Show Circle T button.

Investigate 10. How are circles T and I related? Press the Show Radii button to check

your answer.

SketchStep 12 Press the Show Foci button.

Investigate 11. How were the two foci constructed?

12. If the distance between I and F1 is c, explain why b 2 � c 2 � a 2 for a horizontal ellipse.

Lesson 8.2 • Definitions of Circle and Ellipse (continued)



In this demonstration you’ll explore the parabola as an “envelope” of lines or a locus of points. You’ll also study the relationship among the focus, the directrix, and the parabola.

SketchStep 1 Open the document Parabolas.gsp to the

page Folded Parabola.

Step 2 Press the Animate Point G button. Press the button again to stop the animation.

Step 3 Drag point Focus different distances from line Directrix and repeat Step 2.

Investigate 1. What shape is constructed when you animate

point G ?

2. Describe how the shape changes according to the position of the focus.

3. What do you notice about point H ?

4. Explain how the line that forms this envelope is constructed.

SketchStep 4 Go to the page Locus of Points.

Step 5 Press the Show Parabola Locus button.

Step 6 Drag point G along the directrix.

Investigate 5. What do you notice about points J and G ?

SketchStep 7 Press the Show Measurements button.

Step 8 Drag point G along the directrix.

Step 9 Press the Show Construction button.

Investigate 6. What do you notice about the measurements?

7. Write a definition of a parabola in terms of the focus, point J, and the directrix.

8. Explain how to construct this locus of points.

Lesson 8.3 • Definition of Parabola Sketchpad




In this demonstration you’ll explore the hyperbola as an envelope of lines.

SketchStep 1 Open the document Hyperbolas.gsp to the page

Folded Hyperbola.

Step 2 Press the Animate Point C button. Press the button again to stop the animation.

Step 3 Choose Erase Traces from the Display menu.

Step 4 Drag point C around the circle.

Investigate 1. What shape is constructed when you animate point C ?

2. How far do you have to drag point C to complete theentire shape?

3. Explain how this envelope is constructed.

SketchStep 5 Go to the page Hyperbola Envelope.

Step 6 Press the Show Hyperbola Envelope button.

Step 7 Drag point B closer to circle A and then farther away.

Step 8 Drag point Circle Radius to make a smaller circle and then a larger circle.

Investigate 4. How does the hyperbola change as B gets closer to the circle? If B is

on the circle? If B is inside the circle?

5. How does the radius of the circle affect the hyperbola?

6. How would you change this sketch so that the hyperbola opens vertically?

Lesson 8.4 • Definition of Hyperbola Sketchpad



In this demonstration you’ll see one way to geometrically construct an ellipse.

Constructing the Ellipse

SketchStep 1 Open the document ConicSections.gsp to the page Ellipse.

Step 2 Drag point C along segment AB.

Investigate 1. How does point C affect the circles? A

F1 F2

C B

2. How are the circles and segments AC and CB related?

SketchStep 3 Press the Show Intersections button.

Step 4 With both intersections selected, choose Trace Intersections from the Display menu.

Step 5 Drag point C along segment AB.

Investigate 3. How is the ellipse created?

SketchStep 6 Choose Erase Traces from the Display menu. Then select the two

intersections and deselect Trace Intersections from the Display menu.

Step 7 Press the Show Locus button.

Step 8 Drag point A to change the length of segment AB.

Investigate 4. How does the length of

___ AB affect the ellipse?

5. How does the location of C affect the ellipse?

SketchStep 9 Drag point F1 to different locations.

Investigate 6. How does the ellipse change as the position of F1 changes?

7. How far apart can the foci be before you no longer have an ellipse?

8. What happens when both foci are at the same point?

Exploration • From Circles to the Ellipse Sketchpad

(continued)




Constructing the Parabola

SketchStep 10 Go to the page Parabola.

Step 11 Press the Show Locus button and the Show Measurements button.

Step 12 Drag point B along the directrix.

Investigate 9. Explain how the parabola is created.

Constructing the Hyperbola

SketchStep 13 Go to the page Hyperbola.

Step 14 Drag point C along line AB.

Investigate 10. How does point C affect the sketch?

11. Which segment equals the radius of F1? Of F2?

SketchStep 15 Drag point C so that the circles intersect, and press the Show

Intersections button.

Step 16 With both intersections selected, choose Trace Intersections from the Display menu.

Step 17 Drag point C along line AB. (Make sure to drag both left and right.)

Step 18 Press the Show Locus button and the Show Measurements button.

Investigate 12. Explain how the hyperbola is created.

13. Describe the similarities and differences between this sketch and the sketch of the ellipse.

Exploration • From Circles to the Ellipse (continued)



An important feature of a rational function is its asymptotes. In this demonstration you will explore how each parameter in a rational function affects the graph of the function and, in particular, the asymptotes.

ExperimentStep 1 Open the Fathom document RationalFunctions.ftm. You will see a

graph of a general rational function with linear expressions in both the numerator and denominator. The parameters are controlled by sliders beside the graph. When you open the file, the sliders are

set so that the function is f (x) � 0x � 1 _____ 1x � 0 , or f (x) � 1 _

x . There are also

two sliders showing the equations of the asymptotes. For example, here the horizontal asymptote has equation y � 0 and the vertical asymptote has equation x � 0.

Step 2 Begin by exploring the effect of adjusting slider a. Be sure to use both positive and negative values of a.

Investigation 1. Describe what happens to the graph of the function as you drag

slider a. What is the effect on the horizontal asymptote? On the vertical asymptote? Be specific. Write observations such as “Every time I increase a by one unit, the vertical asymptote . . .” Use appropriate terminology, such as translate and dilate, in your answer.

2. Does changing a have the same effect on the graph if the other parameters are different? Set b � 2 and adjust slider a again. Do the answers to Question 1 change? What if b � �3?

Lesson 8.6 • Rational Functions Fathom

(continued)




3. Set b � 1 again and c � 2. Does changing a have the same effect as before? What if c � �3?

4. Set c � 1 again and d � �3. Does changing a have the same effect as before? What if c � 2 and d � �3?

5. Summarize your conclusions about how parameter a (the coefficient of x in the numerator) affects the graphs and asymptotes of the rational function.

6. In Questions 1–4 you investigated the effect of parameter a on the graph by changing b, c, and d one at a time, then adjusting slider a. Use a similar strategy to investigate the effects of the other three sliders. Be sure to use different combinations of settings for the sliders you are not investigating to see the full story. Answer these questions:

a. Which parameters affect the location of the horizontal asymptote? Why?

b. Which parameters affect the location of the vertical asymptote? Why?

c. Which parameters dilate the graph?

d. What happens when a � c and b � d ? Why?

e. What happens when c � 0? Why?

f. What happens when a � c � 0? Why?

g. What happens when b � d � 0? Why?

7. Write an equation for each asymptote in terms of the function’s parameters.

Lesson 8.6 • Rational Functions (continued)



In this demonstration you’ll explore the hyperbola formed by the parent inverse variation function, y � 1 _

x .

SketchStep 1 Open the document InverseVariation.gsp to

4

2

–4

–4 4

the page Find Foci.

Step 2 Press the Show Hyperbola button and then the Show Vertices button.

Investigate 1. What are the coordinates of the vertices of the

hyperbola?

2. What are the equations of the asymptotes of this hyperbola?

3. What are the equations of the lines of symmetry?

SketchStep 3 Press the Show Box button.

Investigate 4. What are the lengths of the diagonals of this box?

5. If the dimensions of the box are 2a and 2b, find the values of a and b.

6. The foci are c units from the center of the hyperbola, where a 2 � b 2 � c 2. Find the value of c.

7. How would you construct the foci of this hyperbola? Press the Show Foci button to check your answer.

8. What are the coordinates of the foci?

SketchStep 4 Go to the page Locus Definition.

Step 5 Drag point P along the hyperbola.

Step 6 Press the Show Measurements button and then the Show Difference button.

Investigate 9. Explain why the graph of y � 1 _

x satisfies the locus definition of a

hyperbola.

Lesson 8.6 • Inverse Variation Sketchpad



In this demonstration you’ll find a formula for the partial sums of an arithmetic series: u 1 � �u 1 � d � � �u 1 � 2d � � �u 1 � 3d � � · · · .

SketchStep 1 Open the document ArithmeticSeries.gsp. You’ll see a step-shaped

figure that represents terms of an arithmetic sequence.

A

u1

u2

u3

u4

u5

Investigate 1. Write the sequence u 1 , u 2 , u 3 , u 4 , u 5 represented by the figure. What

is the sum of the series?

2. Drag the slider for N until N � 6. What is u6 and the sum of the series now?

SketchStep 2 Press the Show Copy button to see a copy of the step-shaped figure.

Adjust the angle b slider to rotate the figure and then drag point B to slide the two figures together to make a rectangle with no overlaps. (Or, you can press the Make Rectangle button.)

Investigate 3. What are the dimensions of this new rectangle?

4. Use the figure to express the area of the rectangle in terms of the number of rows, n, and the first and last terms of the sequence.

SketchStep 3 Press the Change Sequence button, then drag the sliders for u 1 and d to

change the first term and the common difference. Drag point B to make a new rectangle with no overlaps. Does your area formula still hold?

Investigate 5. Based on what you have discovered, what is a formula for the partial

sum, Sn , of an arithmetic series?

6. Describe the relationship between your formula and the dimensions of your rectangle.

Lesson 9.1 • Arithmetic Series Formula Sketchpad




In this demonstration you will investigate the perimeters and areas of two fractals.

SketchStep 1 Open the document Fractals.gsp to the

E G

F

HD C

BA

page Squares.

Step 2 Choose Select All from the Edit menu. Use the “�” and “�” keys to change the number of iterations of the fractal.

Investigate 1. How is the fractal created?

2. The first entry of the table gives you the side perimeter and area of the original square. What do the subsequent entries give you?

3. How would you find the value of the next perimeter and area? (Hint: What type of triangle is created?)

4. Let s be the initial side length. If the pattern could be repeated forever, what is the sum of the perimeters of the squares?

5. Find the sum of all the areas.

SketchStep 3 Go to the page Sierpinski.

C B

A

Step 4 Choose Select All from the Edit menu. Use the “�” and “�” keys to change the number of iterations of the fractal.

Investigate 6. How is this fractal created?

7. For the perimeters and areas of each iteration, look at the entries for n � 0, n � 1, n � 2, and so on. In the long run, what happens to the perimeter of each of the smaller triangles?

8. In the long run, what happens to the area of each of the smaller triangles?

9. In the long run, what happens to the sum of the perimeters of all the smaller triangles? (Hint: You can’t use the formula for the sum of infinitely many terms.)

10. In the long run, what is the sum of the areas of the smaller triangles?

Lesson 9.2 • Fractals Sketchpad



In this demonstration you will see one way to visualize the sum of an infinite geometric series.

SketchStep 1 Open the document InfiniteSeries.gsp to the page Basic Sequence.

Step 2 Choose Select All from the Edit menu. Use the “�” and “�” keys to change the number of iterations of the model.

Step 3 Drag point A to change the length of ___

AB . Then drag point C to different locations.

C

A B A�

m AB = 4.23 cmCB = 4.20 cm

n0123

m AB4.23 cm2.12 cm1.06 cm0.53 cm

CB4.20 cm6.32 cm7.37 cm7.90 cm

Investigate 1. What do the values of m

___ AB represent?

2. What geometric series do the values of m ___

AB represent? Write your answer in sigma notation.

3. Move point C to coincide with point A. What does column CB represent?

4. According to the table, what is the sum of this series?

5. Use the formula to confirm your sum from Question 4.

SketchStep 4 Go to the page New Ratio.


Exploration • A Geometric Series Sketchpad

(continued)




Investigate 6. How does this model differ from the first sequence?

7. What series does this model represent? Write your answer in sigma notation.

8. What is the sum of this series?

SketchStep 6 Go to the page Large Ratio.


Investigate 9. What series does this model represent? Write your answer in sigma

notation.

10. What is the sum of this series?

11. What conclusions can you make about the sum of a geometric series and the value of the common ratio?

Exploration • A Geometric Series (continued)



What is the probability of rolling a sum of 6 with a pair of dice?

ExperimentStep 1 Open the document TwoDice.ftm. You’ll see a collection with

three attributes and a case table. For this demonstration, you’ll have Fathom generate the data.

Step 2 Double-click the Pair of Dice collection to show the inspector. To simulate random rolls of the die, enter the formula randomPick(1, 2, 3, 4, 5, 6) for Die_1. (Double-click the formula field to enter the formula.) Fathom will generate 300 pieces of data, representing 300 rolls of Die_1.

Step 3 Enter the formula randomPick(1, 2, 3, 4, 5, 6) for Die_2 and then enter the formula Die_1 + Die_2 for their sum.

Step 4 Drag a graph from the shelf and drag Sum to the horizontal axis. Choose Histogram from the pop-up menu to show the number of each of the sums from 2 to 12.

Investigate 1. In the histogram, select the bin for “6” and look at the lower left of

the Fathom window. In your simulation, how many times was the sum of the pair of dice 6?

Lesson 10.1 • Sum of Two Dice Fathom

(continued)




2. The experimental probability for rolling a sum of 6 is the number of times a sum of 6 occurred divided by the total number of rolls of the pair of dice. What is your experimental probability for rolling a sum of 6 with a pair of dice?

3. Choose New Cases from the Collection menu and add 1500 cases. Now what is your experimental probability for rolling a sum of 6 with a pair of dice?

4. Find the theoretical probability for rolling a sum of 6 with a pair of dice. How do your results for 300 rolls compare to the theoretical probability? How do your results for 1800 rolls compare to the theoretical probability?

Lesson 10.1 • Sum of Two Dice (continued)



If a coin is tossed randomly onto a floor of congruent square tiles, what is the probability that it will land entirely within a tile, not touching any edges?

ExperimentStep 1 Open the document CoinToss.ftm. You will see a grid representing

a tiled floor, 100 green or red “coins” on the grid, and sliders that control the sizes (in millimeters) of the tiles, coins, and tile borders.

Step 2 Adjust the different sliders to experiment with changing the grid or the size of the coins. Then adjust the sliders so that the tiles measure 20 mm with no borders and the coin has a 5 mm radius.

Step 3 Drag a new summary table from the tool shelf. Double-click the grid to open the inspector and drag the attribute Coin_In to the summary table. This attribute tells you how many coins landed entirely within a tile. Close the inspector and click the Rerandomize button.

Exploration • The Coin Toss Problem Fathom

(continued)




Investigate 1. Record the number of times the “coin” lands entirely within a tile, not

touching any lines. What is your experimental probability of success?

2. Adjust the various sliders and rerandomize the data to complete the table.

CoinCoin radius

(mm)20 mm

tile30 mm

tile40 mm

tile

40 mm tile with 5 mm

borders

Maxmillian Gold

5 mm

Dime 9 mm

Penny 10 mm

Quarter 12 mm

3. Where must the center of a coin fall in order to have a successful outcome? What is the area of this region for each combination of coin and tiled floor?

4. Use your answers from Question 3 to calculate the theoretical probability of success for each coin and tiled-floor combination. How do these theoretical probabilities compare to your experimental probabilities? If they are significantly different, explain why.

5. Determine a formula that will calculate the theoretical probability of success, given a coin with radius r and a floor of square tiles with side length T and with borders of thickness B.

Exploration • The Coin Toss Problem (continued)



Lesson 10.2 • Independent Events Fathom

How do you decide whether events are dependent or independent? In this demonstration you’ll look at whether being young and being married are independent or not.

ExperimentStep 1 Open the document CA_includesBerkeley2000.ftm. Select the

collection and pull down a case table.

Step 2 Make a new attribute called young. Select the attribute so that the column is highlighted. Choose Edit Formula from the Edit menu. Enter the formula age<30. The attribute will have values true and false.

Step 3 Make a new attribute called married. Enter the formula marital=“Mar”. (Note the quotation marks.) This attribute will also have values true and false.

Step 4 Pull down a summary table. Drag married to the right arrow and young to the down arrow.

Step 5 Choose Add Formula from the Summary menu. Type in rowProportion.

Investigate 1. What do the values of rowProportion represent?

2. Write a probability statement that compares two values of rowProportion: the one in the cell in the true row and column (that is, married and young) with the value of rowProportion for the entire true column (that is, married).

3. Edit the formula for young. Are there any ages for which these two values of rowProportion are the same? If so, the events are independent.

4. Make up other attributes and search for independent events elsewhere among the data.




What are the results if a die is rolled 1000 times?

ExperimentStep 1 Open the document DiceRolls.ftm. You’ll see two collections and

their case tables. For this activity, the computer will create the data.

Step 2 Double-click the Fair Die collection. To have the computer pick random sides of the die to show, enter the formula randomPick(1,2,3,4,5,6) for Roll. The computer will generate 10 pieces of data, representing 10 rolls of the die.

Step 3 Make a histogram of the data.

Investigate 1. Generate different sets of data by choosing Rerandomize from the

Collection menu. Rerandomize several times. In general, how does your histogram change? Describe any common features you notice.

2. Choose New Cases from the Collection menu and add 90 cases. Is the shape of your histogram different than the shape when you had 10 cases?

3. Rerandomize several times. How does your histogram change, in comparison with the changes when you had 10 cases?

4. Add 900 more cases. Rerandomize several times. Compare the shape and changes in your histogram with the shape and changes in histograms with fewer cases.

ExperimentStep 4 Double-click the Loaded Die collection. Enter a formula

for Roll that simulates a loaded (unfair) die. For example,

randomPick(1, 1, 2, 5, 5, 6) will choose 1’s 1 _ 3 of the time, 5’s 1 _ 3 of the time, 3’s and 4’s not at all, and each of the other two sides 1 _ 6 of the time.

Step 5 Make a histogram of the data.

Investigate 5. Experiment with several quantities of data, and then do the same for

other loaded dice. How many cases do you need before you can tell from the histogram whether the die is loaded?

6. The Law of Large Numbers states that if an experiment is repeated many times, the experimental probability will get closer and closer to the theoretical probability. How does the Law of Large Numbers apply to this activity?

Exploration • A Repeat Performance Fathom



How many games does a best-of-seven tournament like the World Series run, on average?

ExperimentStep 1 Open the document WorldSeries.ftm. You’ll see a collection and

a case table. The table shows the seven possible games in a series, the winner of each game, each team’s cumulative wins, and which game is the last game in the series. The computer uses a random-number generator to decide the outcome of each game. The

probability that each team will win each game is 1 _ 2 .

Investigate 1. Rerandomize the data 10 times by choosing Rerandomize from the

Collection menu. (Each time you rerandomize you simulate another series of seven games.) Record the number of games in each series.

ExperimentStep 2 To collect data on various rerandomizations, you collect measures.

Double-click the collection to open the inspector, and click on the Measures tab. Enter the measure name numGames and the formula sum(last).

Step 3 Select the collection and choose Collect Measures from the Collection menu. A new collection called Measures from World Series will appear, with five measures flying from the original collection to it.

Step 4 With the measures collection selected, pull down a case table.

Investigate 2. List the five values of numGames collected.

3. Double-click the measures collection. Check the box beside Replace existing cases. Enter 15 in place of 5 for the number of measures, and click Collect More Measures. List the 15 measures collected this time.

(continued)

Lesson 10.4 • Expected Value Fathom




4. To find the mean of the measures collected, pull down a summary table. Drag the numGames attribute from the measures case table to the down arrow of the summary table. What is the mean of numGames?

5. Change the number of measures to 40 and deselect Animation on. Click Collect More Measures four more times, and record the mean each time. What do you think the theoretical mean of numGames would be? This is the expected number of games in the series.

Explore More

6. Change the probability in the formula for winner to give one team an advantage. From the means that come up in the summary table as you collect more measures, what can you conclude about how this probability affects the expected number of games?

Lesson 10.4 • Expected Value (continued)



Suppose you are giving a small gift to three different friends. If you have three different gifts to pick from, how many ways are there to distribute these three gifts among your three friends?

SketchStep 1 Open the document OrderedLists.gsp. The first page, labeled Make

Lists, has a set of icons grouped in a box. Each icon represents a different gift. The buttons allow you to adjust the number of available gifts, n, from zero to five.

Step 2 Set the number of gifts to three. Select a gift and drag it out of the box. A copy of it will remain. Drag one of each gift and arrange them in a row.

Step 3 Again, drag one of each gift out of the box and arrange them in another row, but this time in a different order. Continue forming rows of the three gifts until you have formed every possible arrangement.

Investigate 1. How many different arrangements were you able to form of three

distinct gifts drawn from a set of three?

2. Without counting, how many different arrangements do you think you could form of four distinct gifts drawn from a set of four?

SketchStep 4 Go to the Check page. Press the Make All Lists button to check

your answer to Question 1 and to display all the possible arrangements. Then press the Reset button, adjust the number of available gifts n and selected gifts r to 4. Press the Make All Lists button to check your answer to Question 2.

Step 5 Go back to the Make Lists page. Press the Reset button. This time, given your set of three gifts, you want to choose only two of them. So, drag two gifts out of the box, and arrange them in a row. Continue forming rows of two gifts until you have formed every possible arrangement.

(continued)

Lesson 10.5 • Ordered Lists Sketchpad




Investigate 3. How many different arrangements were you able to form of two

distinct gifts drawn from a set of three? Check your work on the Check page.

4. Use your sketch to complete this table for different values of n and r.

n � 1 n � 2 n � 3 n � 4 n � 5

r � 1

r � 2

r � 3

r � 3

r � 3

5. Describe any patterns you found in either the rows or columns of the table.

6. Use the patterns you found in the table to write an expression for the number of ways to arrange 3 gifts selected from a group of 10 different gifts.

Lesson 10.5 • Ordered Lists (continued)



Suppose that a hatching yellow-bellied sapsucker has a 0.58 probability of surviving to adulthood. Assume the chance of survival for each egg in a nest is independent of the outcomes for the other eggs. If a nest has 6 eggs, what is the probability that exactly 4 birds will survive to adulthood?

ExperimentStep 1 Open a new document. You’ll have the computer generate the data.

Step 2 Pull down a collection and open the inspector. Enter the attribute survivors with the formula randomBinomial(6, 0.58).

Step 3 Close the inspector, be sure the collection is selected, and pull down a case table. From the Collection menu, choose New Cases and make 100 cases. For each case, the value of survivors tells how many of the 6 birds in the nest survived to adulthood.

Step 4 Pull down a summary table. Drag the survivors attribute to the down arrow and press the Shift key just before you drop. The summary table will report the number of cases with each number of survivors.

Investigate 1. Rerandomize the data 10 times by choosing Rerandomize from the

Collection menu. (Each time you rerandomize you simulate another 100 nests.) How many of the 10 data sets have more nests in which 4 birds survive than any other number of birds?

ExperimentStep 5 To collect measures of 4 survivors for rerandomizations, double-

click the collection to open the inspector, and click on the Measures tab. Enter the measure name count4 and the formula count(survivors = 4).

Step 6 Select the collection and choose Collect Measures from the Collection menu. A collection of measures will appear.

Step 7 Select the measures collection and pull down a case table.

Lesson 10.7 • Sapsuckers Fathom

(continued)




Investigate 2. List the five values of count4 collected.

3. Double-click the measures collection. Check the box beside Replace existing cases. Enter 15 in place of 5 for the number of measures, and click Collect More Measures. List the 15 measures collected this time.

4. To find the mean of the measures collected, drag down a new summary table. Drag the count4 attribute from the measures case table to the down arrow of the summary table. What is the mean of count4 ?

5. Click Collect More Measures four more times, and record the mean each time. What do these numbers signify about sapsuckers?

Explore More

6. Change the formula for count4 to count the number of times there were 2 survivors or 5 survivors, or some other number. From the means that appear in the measures summary table, what can you conclude about probabilities of the survival of various numbers of sapsuckers?

Lesson 10.7 • Sapsuckers (continued)



How do the ages of U.S. presidents and vice presidents compare?

ExperimentStep 1 Open the document Ages.ftm. The collection gives the ages of

U.S. presidents and vice presidents when they first took office. What were the ages of the current vice president and president when they took office? Update the data set to include these values.

Step 2 Pull down a summary table. Drag the PresAge attribute to the down arrow of the summary table. Then drag the VicePresAge attribute to the same down arrow.

Step 3 Choose Add Basic Statistics from the Summary menu, and then choose Add Five-Number Summary from the Summary menu. Choose Add Formula from the Summary menu and enter the formula iqr( ). Drag the bottom right corner of the summary table to see the statistics for both attributes.

Step 4 Make two histograms. Drag the PresAge attribute to the horizontal axis of one and the VicePresAge attribute to the horizontal axis of the other.

Step 5 Create a new attribute called StdP. Enter its formula as

(PresAge – mean(PresAge)) / popStdDev(PresAge)

Do the same with a new attribute called StdVP, replacing PresAge with VicePresAge in its formula.

Investigate 1. Use the summary statistics and the histograms to compare the ages of

presidents and vice presidents. Overall, would you say that presidents or vice presidents are older when first taking office?

2. Drag the StdP attribute to replace the PresAge attribute on its histogram. What does the StdP attribute measure?

3. Drag the StdVP attribute to replace the VicePresAge attribute on its histogram. Why are the values on the graphs centered on 0?

4. Do the StdP and StdVP values compare in the same way as the PresAge and VicePresAge values?

Lesson 11.2 • Ages Fathom




What census data are normally distributed?

ExperimentStep 1 Open the document Populations.ftm. You’ll see collections of

census microdata for western Colorado; Oahu, Hawaii; and Detroit, Michigan.

Step 2 Select one of the collections and pull down a case table and a graph.

Investigate 1. Browse through the case table. What attribute do you think might be

normally distributed?

2. Make a histogram of that attribute. Experiment with various bin widths by double-clicking the horizontal axis or by dragging a bin edge. Then experiment with other attributes or collections until you find some data that are approximately normally distributed. You may also want to separate the histograms into categories, such as male and female. To do this, drag a second attribute to the vertical axis. What normally distributed data do you find?

3. To check the data that you think might be normally distributed, plot the normal curve on them. Drag down a slider and name it Scale. Then select the graph and choose Plot Function from the Graph menu. Enter Scale � normalDensity(x, mean( ), stdDev( )). Then adjust the slider until the curve fits the data as well as possible. Which attributes still appear to be normally distributed?

4. Some census data are skewed because they include many young people. Select one of the collections and choose Add Filter from the Object menu. Type in a condition such as (age > 20) and (age < 65). Because you had the collection selected, younger and older people will be filtered out of all objects describing that collection (such as the case table or graph). Which data are normally distributed if filtered? What filter did you use?

Exploration • Is This Normal? Fathom



How well can a poll predict the outcome of an election?

ExperimentStep 1 Open the document Polling.ftm. You’ll see two collections: poll,

which contains 50 opinions (either “yes” or “no”), and Measures from poll, which contains the results (the percentage who said “yes”) from each of the 100 polls of a random sample of 50 voters. The case table shows the results of one poll, and the histogram shows the distribution of the percentage results from the 100 polls.

Step 2 Select the graph and choose Plot Value from the Graph menu. Enter

mean(result) – 2stdDev(result)

Click OK, then again choose Plot Value and enter

mean(result) + 2stdDev(result)

Investigate 1. What shape is the distribution of results?

2. What do the values you plotted in Step 2 represent?

3. The slider p represents the true percentage of voters in the population who support the proposal. (This value is unknown in real life.) Set p equal to 0.6 and collect more measures. Is the percentage with a “yes” opinion a majority (more than 50) in 95% of these samples? (In other words, does the 95% confidence interval include only values over 50?)

4. Find the range of p for which the 95% confidence interval includes only values over 50%. (Hint: You might use a summary table of the measures collection.)

Lesson 11.4 • Polling Voters Fathom




In this demonstration you’ll simulate biscuit production and experiment with prediction intervals for mean weights.

ExperimentStep 1 Open the document Biscuits.ftm. You’ll see a collection named

Biscuits, a sample of four biscuits and their graph, and a collection of measures from that sample collection. You’ll also see a slider for m which represents the true mean weight of the biscuits.

Step 2 To simulate a production process, you’ll collect the mean weight of four biscuits from 100 production batches. Select the Measures from Sample of Biscuits collection and choose Collect More Measures from the Collection menu.

Step 3 Drag a new graph down from the shelf. Double-click the Measures collection and drag xbar to the vertical axis of the graph. Change the graph to a line plot.

Exploration • Quality Control Fathom

(continued)



Step 4 A 90% confidence interval corresponds to a z-score of 1.645. The same is true for a 90% prediction interval. So you want to consider the interval with endpoints 500 � 9.87, or about 490 g and 510 g. With the graph selected, choose Plot Value from the Graph menu and plot the value 490. Repeat to plot the values 500 and 510.

Investigate 1. Verify that the endpoints of a 90% prediction interval are about 490 g

and 510 g. (Hint: You will need to go back to the original Biscuit collection to discover the standard deviation of the biscuits’ weight.)

2. The graph in Step 4 is called a control chart. A control chart shows data from a process, a center line showing the mean of the data, and upper and lower horizontal lines called control limits. Between those horizontal lines is a 90% prediction interval, and you have 100 values for xbar. How many values of xbar are outside the prediction interval? Is this what you would expect? Explain.

3. Many producer of goods use a z-value of 3. Calculate the new endpoints and reset your endpoints in your plot. Now how many means are outside of the prediction interval?

4. Suppose an adjustment slips and the process starts producing biscuits with a mean weight of 505 g. Change the slider value to 505 g. Select the measures collection and choose Collect More Measures from the Collection menu. How many batches of biscuits went by before you discovered this small problem?

5. Now change the slider value to 510 g. Collect more measures. How many batches of biscuits went by before you discovered this big problem?

Exploration • Quality Control (continued)




In this demonstration you’ll use the least squares line to predict the mean SAT mathematics scores for 2010, using data from the 1990s.

ExperimentStep 1 Open the document SATScores.ftm. These are mean SAT

mathematics scores for various years in the 1990s. (Year 1 is 1991.) Note that year 4 (1994) is missing.

Step 2 Pull down a graph. Drag the year attribute to the horizontal axis and the score attribute to the vertical axis, to create a scatter plot.

Step 3 Choose Least-Squares Line from the Graph menu.

Step 4 Pull down a summary table and drag each attribute to the down arrow to find the mean. Add the values of these means as case 10 in the case table, and select the row to see the corresponding point on the graph.

Investigate 1. How close does the least squares line come to the point (

____ year ,

____ score )?

2. Verify that the slope of the least squares line is r � sy

__ sx � .

3. What mean score does the least squares line predict for the year 2001? How close is that value to the actual mean score of 514?

4. What mean score does the equation predict for the year 2010? How reasonable is that prediction?

Lesson 11.6 • SAT Scores Fathom



In this demonstration you will compare methods for fitting nonlinear models to data for carbon dioxide emissions in eastern Asia.

ExperimentStep 1 Open the document CO2Emissions.ftm. You’ll see a collection, a

case table, and a graph of the data. From the graph, you can see that the data are not linear. To start the modeling process, you have to decide what kind of function might model the data.

Step 2 You will first fit an exponential curve to the data in the form y � a � b x � h . With the graph selected, choose Plot Function from the Graph menu and enter a � b^(year – 1949). Adjust the sliders until the graph seems to fit the data.

Step 3 Select the graph and choose Show Squares from the Graph menu. Below the equation you will see the sum of the squares of the residuals. Continue to make changes with the sliders until you have minimized this sum. This is your least squares exponential model.

Investigate 1. What is the equation for your least squares exponential model? What

is the sum of the squares of the residuals for this model?

ExperimentStep 4 If y � a � b x�1949 is the model, then log(y) � log �a � b x�1949 � �

log(a) � (x � 1949) � log(b). Create a new attribute, YearsSince1949, in the case table. Select the column and choose Edit Formula from the Edit menu. Give the attribute the formula Year – 1949. Then add another attribute, logy, to the case table and give it the formula log(CO2emissions).

Exploration • How Does It Fit? Fathom

(continued)




Step 5 Create a new graph with YearsSince1949 on the horizontal axis and logy on the vertical axis. Choose Least-Squares Line from the Graph menu to find the least squares model for these data.

Investigate 2. What is the equation for your least squares model?

3. Use the least squares model for the linearized data to write an exponential equation to model the original data. Return to the first graph and plot the equation y � a � b x �1949 . How does this model compare with your original exponential model?

4. In this demonstration you used two methods to find a model for nonlinear data. Identify one good feature of each method of regression.

Exploration • How Does It Fit? (continued)



In this demonstration you’ll investigate how the standard deviation affects the equation of the normal curve with mean zero.

SketchStep 1 Open the document NormalCurve.gsp.

Step 2 Select the standard deviation parameter, sigma. Use the “�” and “�” keys to change the value of sigma.

0.8

0.4

0.6

0.2

–0.2–4–6 –2 2

f(x) =

sigma = 1.00

–1

sigma 2 �e

x 2

sigma� � ��

�

Investigate 1. What happens to the normal curve as the standard deviation gets

larger?

2. What happens when the standard deviation equals zero? Explain why this happens.

3. How does a negative standard deviation affect the graph?

SketchStep 3 Press the Show Normal Approximation Curve button.

Step 4 Select parameter sigma. Use the “�” and “�” keys to change the value of sigma.

Investigate 4. This second equation approximates the normal curve without using e.

For which values of the standard deviation is the second equation a good approximation? A poor approximation? Undefined?

Chapter 11 Review • Normal Curve and Approximation Sketchpad



In this demonstration you’ll explore the trigonometric ratios in triangles of different shapes and sizes.

SketchStep 1 Open the document RightTriangleRatios.gsp. You’ll see a right

triangle ABC and the measurement for angle A. Drag point B and observe its effect on the size and shape of the triangle.

Step 2 Experiment with dragging the Length of Hypotenuse slider and observe its effect on the size and shape of the triangle.

Step 3 Press the Move Angle A to 30 Degrees button.

�A = 30ç

CA

Length of Hypotenuse

B

Investigate 1. For ∠A, identify the opposite side, adjacent side, and hypotenuse.

Then press the Show Labels button to check your work.

2. Press the Show Length Measurements button. Use these measurements to calculate the sine, cosine, and tangent ratios for angle A. Then press the Show Ratios button to check your work.

3. If you change the size of the triangle without changing ∠A, do you think sin A will increase, decrease, or stay the same? Give a reason for your prediction.

SketchStep 4 Drag the Length of Hypotenuse slider to change the size of the

triangle and observe its effect on the different ratios.

Step 5 Drag point B and observe its effect on the different ratios.

Investigate 4. What happens to the trigonometric ratios if the triangle is made larger

or smaller without changing the angles?

Lesson 12.1 • Right Triangle Ratios Sketchpad

(continued)




5. As you increase the measurement of ∠A, does the sine ratio increase, decrease, or stay the same? What happens to the cosine and the tangent ratios?

6. Write down the current measurements for ∠A and the three ratios. What is the complement of this angle? What are the sine, cosine, and tangent of this complement? Press the Complement button to check your work.

SketchStep 6 Press the Hide Hypotenuse Measurement button. Drag point B to

change m∠A.

Step 7 Choose Measure | Calculate and click on two appropriate measurements in your sketch to calculate the length of the hypotenuse.

Step 8 Press the Show Hypotenuse Measurement button to check your work.

Investigate 7. What measurements did you choose to calculate the length of the

hypotenuse?

8. What are two other ways you can calculate the length of the hypotenuse using only the measurements in your sketch?

Lesson 12.1 • Right Triangle Ratios (continued)



In this demonstration you’ll explore the relationship between the sines of the angle measures of a triangle and the lengths of the sides.

SketchStep 1 Open the document LawofSines.gsp. You’ll see triangle ABC and

its measurements.

Step 2 Drag the vertices to make m∠A larger than m∠B and observe the measurements.

Investigate 1. Which side is longer, a or b ? Is this always true? Write down the

measurements from three different examples.

2. Based on your discoveries in Question 1, if side c is longer than b, which angle do you think is larger, ∠C or ∠B ? Check your prediction with the sketch.

SketchStep 3 Press the Show Sines and Ratios button. Select the table that

appears. Choose Graph | Add Table Data and choose to add ten entries as the values change.

Step 4 Drag the vertices to change the angles and side lengths. Observe the ratios.

Investigate 3. What do you observe about the ratios? Write your observation as

an equation.

4. Calculate the reciprocal of each ratio. What do you observe? Write an equation.

These equations are both ways of writing the Law of Sines.

Lesson 12.2 • Law of Sines Sketchpad

(continued)




SketchStep 5 Press the Change Triangles button. You’ll see an altitude from ∠A

to segment BC and its label h.

c

a

b

h

C B

A

D

Investigate 5. Use the labels in the blue triangle to write a formula for sin B. Use the

labels in the pink triangle to write a formula for sin C.

6. The length of segment h appears in both formulas. Solve both formulas for h, and set the results equal to each other.

7. What must you do to this equation to complete a proof of the Law of Sines?

8. Is your proof of the Law of Sines still correct, or must you modify it if you drag point A until segment h is outside triangle ABC ? Explain.

Lesson 12.2 • Law of Sines (continued)



A pilot wants to fly directly west from Toledo, Ohio. Her plane can fly at 120 mi/h. The wind is blowing at 25 mi/h from the south. The forces of both the plane and the wind contribute to the actual path of the plane. In this demonstration you’ll explore how to use vectors and their sums to find the actual motion of the plane.

SketchStep 1 Open the document VectorSums.gsp. You’ll see vector p, which

will represent the plane’s contribution to the motion, and vector w, which will represent the wind’s contribution.

80

40

60

20

–20

–40

–100–200 200100

w

Toledo,Ohio

S

N

EW

p

Step 2 Drag the tail points of each of these vectors to the origin.

Investigate 1. If there were no wind, what vector would describe the motion of the

plane after 2 hours? Drag the head of vector p so that it represents this vector.

2. What vector describes the motion of something carried solely by the wind after 2 hours? Drag vector w so that it represents this vector added to vector p.

3. Find the sum of the vectors p and w. Choose Vector from the Custom Tools menu. This tool draws a vector with an arrow at its head. Use it to draw the vector representing the sum. Press the Add p � w button to check your work.

Lesson 12.5 • Vector Sums Sketchpad

(continued)




Investigate 4. Use your answer to Question 3 and the coordinates in the sketch to

find the plane’s position relative to Toledo after 2 hours. State your answer using a distance and an angle.

w

p

p + w

Toledo, Ohio

SketchStep 3 Select the tail point of the resultant vector and then the point at

its head. Choose Measure⎮Coordinate Distance to check your distance measurement.

Step 4 Select three appropriate points in your sketch and choose Measure⎮Angle to check your angle measurement.

Investigate 5. Experiment with your sketch and find another graphical representation

for finding the resultant vector p � w. Explain your representation.

Lesson 12.5 • Vector Sums (continued)



In this demonstration you’ll use parametric equations to simulate two walkers, create their paths, and determine whether their paths intersect.

SketchStep 1 Open the document ParametricEquations.gsp. You’ll see a slider t

for time, coordinates X1 and Y1 for the first walker (in meters), and plotted point P1 showing the walker’s position.

Step 2 Drag the slider for t to trace the path for the first walker.

Show Walker 3

Show Walker 2

Hide Walker 1

Animate t

4

6

2

–2

–4

–6

–8

–5–10 105

t = 2.630.00 5.00

X1 = 3.69Y1 = 3.10

W1

Investigate 1. Write an equation with respect to t for the x-coordinate of the first

walker’s position. Then write an equation with respect to t for the y-coordinate. (Double-click the expressions X1 and Y1 to check your equations).

2. A second walker’s position is given by x (t) � 4.7 and y (t) � 1.2t. Draw a sketch on your paper of the second walker’s motion for 0 � t � 5. (You’ll check your answer in Step 3.)

3. Looking at your sketch in Question 2 and the trace in your sketch, do you think the two paths intersect? Do you think the walkers collide?

SketchStep 3 Press the Show Walker 2 button and then press the Animate t

button to trace out both walkers’ positions with respect to time. Check your sketch from Question 2 against the green trace for the second walker.

Step 4 Press the Animate t button again to stop the animation. This time, slowly drag the slider for t and observe each point’s position with respect to time and to the other walker’s position.

Lesson 12.6 • Parametric Equations Sketchpad

(continued)




Investigate 4. Where is the point of intersection of the two paths?

5. Do the walkers collide? If so, at what time? If not, give the times when each walker reaches the point of intersection.

6. Give real-world meanings for the values of 1.4, 3.1, 4.7, and 1.2 in the equations.

7. A third walker’s position graph is given by x (t) � 5 � 0.8t and y (t) � 1 � 0.6t. Hide Walkers 1 and 2 and show Walker 3. Double-click X3 and Y3 in the sketch and edit them to trace out this walker’s motion for 0 � t � 5. Find an equation for y in terms of x that describes this walker’s motion.

Lesson 12.6 • Parametric Equations (continued)



In this demonstration you’ll explore the graphs of the sine, cosine, and tangent functions.

SketchStep 1 Open the document ParentGraphs.gsp. You’ll see points B and C

on circle A , the line AC , the arc BC , and some measurements.

Step 2 Drag point C around the circle and observe how all four measurements behave.

1.5

1

0.5

–0.5

–1

–1.5

–2 –1

m BC = 0.30� radiansxC = 0.60yC = 0.80246

Slope AC = 1.34

A B

C

�

��

Investigate 1. What is the radius of the circle? What is the measurement of ∠BAC

equivalent to?

2. Where does each measurement reach its maximum and minimum values and what are these maximum and minimum values?

SketchStep 3 Press the Show Plotted y-coordinate button. This point’s

x-coordinate is the measure of arc angle BC and its y-coordinate is the y-coordinate of point C.

Step 4 Drag point C around the circle and examine the trace that appears.

Exploration • Tracing Parent Graphs Sketchpad

(continued)




Investigate 3. What trigonometric function is this? What is this function plot’s

domain?

4. On your paper, draw a sketch of what you think the graph would look like if you plotted the x-coordinate as a function of the arc angle measure. What if you plotted the slope of line AC as a function of the arc angle measure?

SketchStep 5 Press the Show Plotted x-coordinate button. This point’s

x-coordinate is the measure of arc angle BC and its y-coordinate is the x-coordinate of point C.


Step 7 Press the Show Plotted Slope button. This point’s x-coordinate is the measure of arc angle BC and its y-coordinate is the slope of line AC.


Investigate 5. What trigonometric function does the plot traced in Step 6 represent?

In Step 8? What is the function plot’s domain for the plot traced in Step 6? In Step 8?

6. You have seen that the trigonometric functions are periodic. How much of a cycle is shown of each graph by the trace? Explain.

7. For the trace of the slope as a function of the arc angle, which arc measures correspond to a slope of 0? To an undefined slope?

Exploration • Tracing Parent Graphs (continued)



In this demonstration you will transform sinusoidal functions to fit real-world, periodic data.

ExperimentStep 1 Suspend a washer from two strings so that it hangs 10 to 15 cm

from the floor between two tables or desks. Place a motion sensor on the floor about 1 m in front of the washer hanging at rest.

Step 2 Plug the motion sensor into your computer and open a new Fathom document. Click the Meter icon on the object shelf, choose Motion Detector and drag it into the workspace. The meter should display the current value of the sensor. Drag down a new collection from the object shelf, and drag the plug from the meter to the collection to start an experiment. (The plug appears when you scroll over the meter.)

Step 3 Set the experiment to collect data for 2 seconds. Pull the washer back about 20 to 30 cm and let it swing, then click Turn Experiment On.

Step 4 Select the collection and make a new case table. There will be two attributes, Time and Distance. Drag down a new graph and make a scatter plot of (Time, Distance). If you did not get good data, repeat the experiment.

Investigate 1. First you’ll model your data with a sine function: Drag down four

new sliders and name them a, b, h, and k. Select the graph and choose Graph | Plot Function. Enter the equation for the general form of a sine function. Adjust the sliders until the function fits the data as well as possible. What is your sine model? Give real-world meanings for all numerical values in the equation.

2. Change the model to a cosine function and again fit the function to the data. (Double-click the equation to edit it.) What is your cosine model? Which values are different from the sine model? Which are the same? Explain.

Lesson 13.3 • Pendulum Fathom




The number of hours between sunrise and sunset changes cyclically. What sinusoidal function models this data?

ExperimentStep 1 Open the document DayTimes.ftm. The collection gives the

number of daylight hours for various days between December 21, 1995, and July 23, 1996, in New Orleans, Louisiana, which is located at approximately 30° N latitude. December 31 is day zero. The data are plotted, and sliders have been created for the variables a, b, h, and k.

Step 2 Select the graph and choose Plot Function from the Graph menu. Enter either sin(x) + k or cos(x) + k.

Step 3 Adjust the slider for k so that the graph passes through the middle of the data.

Step 4 Edit your function expression to multiply sin(x) or cos(x) by b. Adjust the slider for b so that the graph is the same height as the data.

Step 5 In your function expression, divide x (not sin(x)) by a, and adjust the slider for a to get the same shape as the data. (You can change the horizontal scale by manipulating the hand.)

Step 6 Subtract h from x to get x � h ____

a , and adjust the slider for h.

Step 7 Adjust all sliders to make as good a fit as you can.

Investigate 1. What values of a, b, h, and k give the best fit?

2. How does changing each value alter the graph?

Lesson 13.5 • Day Times Fathom



In this demonstration you will use a microphone to collect sound data, and use these data to explore the frequency of some tones and combinations of tones.

ExperimentStep 1 Plug the microphone into the USB port on the computer and open the

Fathom document SoundWave.ftm. You will see three experiments set up with graphs and sliders. The meter SoundPressureTF1 should display the current value of the microphone. Double-click Experiment with SoundPressureTF1 to open the inspector.

Step 2 Choose a tuning fork. Have one person hold the microphone. Click Turn Experiment On. To ring the tuning fork, rap it sharply on a semisoft surface such as a book or the heel of your shoe. Wait a second or two before placing the tuning fork near the microphone to allow the note to settle.

Step 3 The experiment is set to collect data for only 0.02 second, so it will stop almost immediately. You should see your data appear in the table and as a line scatter plot on the graph. The data should be in the form of a simple sinusoid. If not, repeat the experiment to collect cleaner data.

Lesson 13.7 • Sound Wave Fathom

(continued)




Investigate 1. An equation has been graphed with your data using the sliders A1, B1,

C1, and D1. What do A1, B1, C1, and D1 represent in your graph?

2. Adjust the sliders to find an equation that fits the data as closely as possible. (If you need to adjust the range of a slider, drag its axis.) What equation can be used to represent your data?

3. Explain the methods you used to find the equation.

4. Which of the values in the equation can be changed without affecting the pitch of the note? Explain.

ExperimentStep 4 Delete the meter SoundPressureTF1 and scroll down to see

the setup for Experiment with SoundPressureTF2. The meter SoundPressureTF2 should display the current value of the microphone.

Step 5 Choose a different tuning fork. Repeat Steps 1–3. Use Experiment with SoundPressureTF2 to collect your data.

Investigate 5. Use the sliders A2, B2, C2, and D2 to find an equation to fit the data

as closely as possible. What equation can be used to represent your data?

ExperimentStep 6 Delete the meter SoundPressureTF2 and scroll down to see the

setup for Experiment with SoundPressureTF1_2. The meter SoundPressureTF1_2 should display the current value of the microphone.

Step 7 Repeat Steps 1–3, but this time, strike both tuning forks at the same time and hold them equidistant from the microphone. Click Turn Experiment On. Make sure the tuning forks do not touch. The graph should show a combination of sinusoids, rather than a simple sinusoid.

Step 8 Move the sliders from the previous experiments down to this experiment. Use them to approximate the equation of the new data. Do not change the frequency of either individual equation.

Investigate 6. Why might you need to adjust the values of the individual equations

to get the combined equation?

7. Model the data with an equation that is the sum of two simple sinusoids.

Lesson 13.7 • Sound Wave (continued)



In this demonstration you will investigate the graphs of rose curves.

SketchStep 1 Open the document RoseCurves.gsp to the page Polar Graph.

Step 2 Double-click parameter n and change the value to 1.

Step 3 Select parameter a. Use the “�” and “�” keys on the keyboard to change the value of a.

Investigate 1. How does the value of a affect the graph?

2. How is the diameter of the circle related to the value of a?

3. What value of a would give a circle located to the left of the y-axis with radius 5?

SketchStep 4 Double-click parameter a and change the value to 1.

Step 5 Select parameter n. Use the “�” and “�” keys on the keyboard to change the value of n.

4

–2

2

–4

2–2 4–4

Investigate 4. What values of n create a complete rose curve?

5. What value of n would give a rose curve with 5 petals? 12 petals? 17 petals?

6. In your own words, write a rule for the number of petals. Consider both odd and even values of n.

7. How does the value of a affect the rose curve?

Exploration • Rose Curves Sketchpad

(continued)




SketchStep 6 Press the Show Sine Equation button.

Step 7 Experiment with different values of a and n.

Investigate 8. Explain the similarities and differences between the cosine and

sine graphs.

SketchStep 8 Go to the page Rectangular Graph.

Step 9 Press the Show Graph of y � a cos(nx) button.

Step 10 Experiment with different values of a and n.

Investigate 9. What do the graphs look like when a and n both equal 1?

When a and n both equal 2?

10. Keep the value of n constant. How does the value of a affect the rectangular graph?

11. Keep the value of a constant. How does the value of n affect the rectangular graph?

12. Explain how you can predict the shape and number of petals in the polar graph by looking at the graph of y � a cos nx.

Explore More

SketchStep 11 Go to the page More Polar Curves.

Step 12 Press the Show Cardioids button.

Step 13 Press the Show�Hide buttons and change the value of a to investigate each type of cardioid.

Investigate 13. How do the graphs of r � a(sin � � 1) and r � a(sin � � 1) differ?

14. How do the sine and cosine graphs differ?

15. How does the value of a affect the cardioids?

SketchStep 14 Press the Show Spiral button and change the value of a.

Investigate 16. How does the value of a affect the spiral?

Exploration • Rose Curves (continued)


Discovering Advanced Algebra Technology Demonstrations TEACHER’S NOTES 99©2010 Key Curriculum Press

T E A C H E R ’ S N O T E S

CHAPTER 0

LESSON 0.1 • Multiplying Binomials REQUIRED DOCUMENT

MultiplyingBinomials.gsp

LESSON GUIDE

This demonstration covers multiplying binomials using rectangle diagrams.

INVESTIGATE

1. There is one blue square, which represents 1x 2. There are 8 red rectangles, which represent 8x, and there are 15 yellow squares, which represent 15.

2. x � 2

x�4

3. (x � 2)(x � 4) � x 2 � 6x � 8. See diagram in answer to Question 2.

4. 2x � 3

x�4

5. Possible answer: To include negative values, you can change the color of the negative tiles. In this case, when you’re finding the answer, one negative rectangular tile cancels out one positive rectangular tile. All the unit squares will be either positive or negative, depending on the original expression.

CHAPTER 1

LESSON 1.2 • Looking for the Rebound

REQUIRED DOCUMENT

Rebound.ftm

LESSON GUIDE

This demonstration can replace the investigation in Lesson 1.2. It allows students to gather and explore real data and describe the relationship with an exponential model. Each group will need a motion sensor.

INVESTIGATE

1. Sample data:

2. Discuss with students different ways to select a single value. Students will probably suggest the arithmetic mean. But one concept of “average” is the idea of spreading the total amount evenly among n addends. Here we have factors rather than addends. An average multiplier would be the number that, if we multiplied by the same amount every time, would have the same overall result. Therefore, a better choice might be the geometric mean. Multiply the n ratios and take the n th root. For the sample data both the arithmetic mean and the geometric mean are approximately 0.662, and the Fathom formula is

3. For the sample data, the predicted heights are quite close to the actual heights. Discuss possible reasons for differences between the heights predicted by the formula and the actual heights. Some possibilities are movement of the sensor as the data are being

DAA2TD_012_TN_a.indd 99DAA2TD_012_TN_a.indd 99 1/9/09 4:20:46 PM1/9/09 4:20:46 PM

100 TEACHER’S NOTES Discovering Advanced Algebra Technology Demonstrations


gathered, and uneven conditions of the ball (as it rotates maybe a different side has a different “bounce,” etc.).

EXPLORATION • Repeat After Me REQUIRED DOCUMENT

RepeatAfterMe.gsp

LESSON GUIDE

This demonstration can replace the Exploration Graph of Sequences.

INVESTIGATE

1. r slider: geometric and shifted geometric sequences; d slider: arithmetic and shifted geometric

2. The ratios of successive terms (y-coordinates) in the geometric sequence

3. Geometric: A value of 1 makes the points of the sequences lie on a horizontal line. Smaller and smaller values make the sequence approach the x-axis faster. Values larger than 1 make it increase infinitely. Shifted geometric: A value of 1 makes the points of the sequence coincide with the graph of the arithmetic sequence. Smaller and smaller values make it approach a lower and lower long-run value. Values larger than 1 make it increase infinitely. Arithmetic: r has no effect on the sequence.

4. Arithmetic: A larger value of d makes the graph of the sequence steeper. Shifted geometric: A larger value of d gives a greater long-term value for the sequence. Geometric: d has no effect on the sequence.

5. a. d � 0 b. r � 1

c. No values exist. The arithmetic sequence is always linear; the geometric sequence is always curved.

LESSON 1.5 • Life’s Big Expenditures

REQUIRED DOCUMENT

Expenditures.ftm

LESSON GUIDE

This demonstration can replace the investigation in Lesson 1.5. It allows students to see how recursive formulas can be used in a real-life setting: repayment of loans. They will develop a formula on their own before using Fathom to adjust the formula to explore deeper questions.

INVESTIGATE

1. The monthly interest rate is the annual interest rate divided by 12: 0.079

____ 12 � 0.006583. The first month’s interest is $44.83. Note that the sliders display these values as percentages. After the first month, the balance is $21,794.83.

2. To calculate the balance from the previous month’s balance, take the previous balance times (1 � monthly interest rate), then subtract the payment. The formula is

Balancen � Balanc e n�1 (1 � Monthly Interest Rate) � Monthly Payment

With the numbers for this particular situation, the formula is

Balancen � Balanc e n�1 (1.006583) � 350

The balances for the first six months are $21,794.83; $21,588.30; $21,380.42; $21,171.16; $20,960.53; and $20,748.52.

EXPERIMENT

Step 2 Students should be able to find their formula buried inside two if-statements. The if-statements are there to get the initial and final balances correct. The prev() function specifies using the previous term in the list, or the (n � 1) term. CaseIndex is the equivalent of n.

INVESTIGATE

3. It takes 82 months (6 years and 10 months) to pay off the loan.

4. Any amount from $444.99 to $451.10 will result in exactly 60 payments. $444.99 gets very close to all payments being equal. Students can type values into the Monthly_Payment slider as well as drag it.

5. The total amount paid will be between $26,696.20 (for monthly payments of $444.99) and $26,618.00 (for payments of $451.10). The difference is due to the fact that with higher payments, more of the principal is paid off, resulting in less interest the next month.

6. At 6.9% annual interest, the same monthly payment will pay off the loan in 58 months rather than 60. The total amount will be between $25,957.50 (for monthly payments of $444.99) and $25,891.70 (for payments of $451.10). At 18% annual interest, the



same monthly payments will take approximately 90 months to pay off the loan, and the total will be about $40,000. Alternatively, to pay off the loan in 60 months, you must make monthly payments of about $560. In this case the total amount paid is approximately $33,400.

7. Monthly payments of $996 will get the loan paid off in exactly 360 months. At this payment amount, the total paid for the house is $358,525. This calculation ignores other parts of the monthly payment, such as insurance and taxes. As an extension, you might have students research the current interest rate for a 30-year home mortgage at one of your local banks. How does using this interest rate change their answers?

CHAPTER 2

LESSON 2.1 • Box Plots

REQUIRED DOCUMENT

BoxPlots.ftm

LESSON GUIDE

Using the ability to drag data points in Fathom, students will explore summary data and realize that it does not reflect all the information in a data set. Because Fathom allows students to change data by dragging dots on a dot plot, students can visualize changes that affect the box plot. This demonstration was adapted from an activity in Exploring Algebra 1 with Fathom, by Eric Kamischke, Larry Copes, and Ross Isenegger.

EXPERIMENT

1. Using the convention in the student book, students should draw box plots from the five-number summary 0, 2.5, 6, 9.5, 12. If 12 became 13, the right whisker would become 1 unit longer.

2. Student box plots probably do not look like the one in BoxPlots.ftm (see the worksheet). Fathom calculates the five-number summary as 0, 3, 6, 9, 12; the median is considered part of both the lower half of the data and the upper half of the data. However, the convention used in the student book identifies the first quartile by finding the median of those numbers less than 6.

3. The mean of the data is 6.

GAME 1 4. As soon as 0, 3, 6, 9, or 12 is moved, the box plot

changes shape and the rules are broken. Move the two dots just above 0 down to 0, those just above 3 down to 3, and so on.

5. The smallest mean is about 5.07692. Using the numbers {0, 0, 0, 3, 3, 3, 6, 6, 6, 9, 9, 9, 12}, the smallest value for the mean in lowest terms is

0 � 0 � 0 � 3 � 3 � 3 � 6 � 6 � 6 � 9 � 9 � 9 � 12 __________________________________ 13 � 66

__ 13 .

6. Using the values {0, 3, 3, 3, 6, 6, 6, 9, 9, 9, 12, 12, 12}, the largest possible value for the mean is

0 � 3(3) � 3(6) � 3(9) � 3(12)

____________________ 13 � 90 __ 13 .

7. The difference between the largest and smallest

possible values is 90 __ 13 � 66

__ 13 � 24 __ 13 . Both means have

three each of 3, 6, and 9, so the difference will depend only on the 12; that is, 3(12) � 12, or 24 , will be in the numerator and 13 will be in the denominator.

8. The two dot plots are mirror images of one another with the value 6 as the axis of reflection. Their spreads should be identical.

GAME 2 9. The largest possible value for the mean is 99

__ 13 �

7.61538, and it occurs when the median is 9. Move the data points that are not 0, 3, 9, or 12 as far to the right as possible without moving any dot past its neighbor. Six dots will be at 9, making the median 9. The graph will be skewed left.

10. Again, this situation will be a mirror image of the situation for the largest possible mean. The three dots at 6 would be moved to the left by 3 units. This will make the total go down by 9, from 66 to

57. The smallest mean for Game 2 is 55 __ 13 � 4.38462

and will occur when the median is 3.

LESSON 2.3 • MP3 Players REQUIRED DOCUMENT

MP3Players.ftm

LESSON GUIDE

This demonstration can replace Example A of Lesson 2.3.

INVESTIGATE

1. 18

2. The data lie approximately in the interval 760 to 1020, a range of 260. There are 50 data points; the 25th is in the interval around 930. The mean is the balance point and is more difficult to estimate; it is about 900.

3. The graph with the larger bins is better at showing the overall shape of the distribution, which is sym-metrical.

4. The graph with the smaller bins is better at showing the gaps and clusters in the distribution.




5. The data are put into more bins, so each bin contains fewer data points.

6. Answers will vary. If bins are too small or too large, the shape of the data isn’t visible. A bin width between 20 and 40 is probably best for these data.

7. Add up the heights of the bins.

LESSON 2.3 • Eating on the Run REQUIRED DOCUMENT

FastFood.ftm

LESSON GUIDE

If you focus on Question 7, this demonstration can take the place of the investigation in Lesson 2.3. Questions 1–6 can also serve as a review of Chapter 2.

INVESTIGATE

1. Answers will vary.

2. The function mean() gives the arithmetic mean, the most representative number for many purposes; count() is the number of data points; stdDev() is standard deviation, a measure of spread; stdError() is standard error, an advanced statistic not used until Chapter 11 of Discovering Advanced Algebra; and count(missing()) is a count of the number of missing values.

3. From the box plot you can find none of the basic statistics given in Fathom 2. The value of count() is the sum of the heights of the bins in the histogram. The mean might be approximated from the histogram, as its balance point.

4. All of the graphs except sodium are skewed right to various degrees. The graph for sodium is fairly symmetrical.

5. TotalCalories: 270, 430, 593.5, 747, 1120; TotalFat: 9, 24, 30, 41, 74; SaturatedFat: 2.5, 4.5, 8, 18, 30; Cholesterol: 30, 55, 75, 95, 185; Sodium: 610, 1000, 1280, 1540, 1960; Carbohydrate: 26, 37, 46, 57, 92.

6. Answers will vary. For example, for sodium the point in the 90th percentile has 0.9 � 22 � 19.8 data points below it, so it is the 20th point from the bottom, or the 3rd point from the top. From the default histogram, the 3rd point from the top is in the bin centered on 1850. Students can find how many cases each bin contains by holding the cursor over the bin and reading the information in the lower left of the Fathom window.

7. Candidates for outliers are: TotalCalories: Hardee’s 1/2 LB Six Dollar Burger, Hardee’s 1/2 LB Grilled Sourdough Thickburger; TotalFat: Hardee’s 1/2 LB Six Dollar Burger, Hardee’s 1/2 LB Grilled Sourdough Thickburger, Burger King Double Whopper with Cheese (all satisfy the definition for outliers); SaturatedFat: Hardee’s 1/2 LB Six Dollar Burger, Hardee’s 1/2 LB Grilled Sourdough Thickburger; Cholesterol: Burger King Double Whopper with Cheese (satisfies the definition for outliers); Sodium: no outliers; Carbohydrate: Carl’s Jr. Carl’s Western Crispy Chicken (satisfies the definition for outliers).

EXPLORATION • Different Ways to Analyze Data REQUIRED DOCUMENT

CA_includesBerkeley2000.ftm

LESSON GUIDE

This demonstration can take the place of the Exploration Census Microdata. It uses microdata for Berkeley, California. To use data for another region, choose File � Open Sample Document � Social Science � United States � US Census Files, or use Fathom Help to learn how to Import U.S. Census Data from the Internet. If you’re comfortable using Fathom, you can encourage students to make and test conjectures other than those given here.

INVESTIGATE

1. The mean age of females is greater than the mean age of the males, but the medians are in the opposite order, so the statistics might be said to be inconclusive.

2. The box plots show that there are outliers, which may be affecting the means.

3. The histograms show three females in the highest bin. If you select that bin (turning it red) and scroll through the case table, you’ll see highlighted three females of age 90: numbers 75, 107, and 471. If you choose Delete Cases from the Edit menu, the summary table will change. The means and medians remain in the same order. Deleting the single male outlier does not help settle the conjecture.



4. The summary table for income gives the mean to be about $21,409; but the median is exactly $10,000, so half of the 500 people in this collection have an income below $10,000. A histogram shows so many incomes to be below $10,000 that students may wonder why the mean is so high; point out that the highest salaries are near $200,000, so they bring up the mean dramatically. In fact, a box plot marks many incomes as outliers. Students may also wonder why these incomes are so low. The fact that about 20% of the persons listed reported no income ($0) contributes to the skew.

CHAPTER 3

LESSON 3.2 • Balloon Blastoff

REQUIRED DOCUMENT

BalloonBlastoff.ftm

LESSON GUIDE

This demonstration can replace the investigation in Lesson 3.2. Fathom allows students to easily graph and manipulate multiple lines. Each group will need a motion sensor.

INVESTIGATE

1. For the sample data: domain: 0.0 � x � 0.7; range: 0.13 � y � 2.39. The domain indicates the number of seconds the rocket is in motion. The range indicates the distance from the sensor in meters.

2. Possible answers for sample data: A (0.05, 0.132099); B (0.3, 0.664749); C (0.5, 1.518573); D (0.65, 2.365544)

3. Slope between A and B: 2.1306; slope between A and C: 3.0811; slope between A and D: 3.7224; slope between B and C: 4.2691; slope between B and D: 4.8594; slope between C and D: 5.6465

4. No, the slopes are not the same: the sample data are approximately linear, but not exactly.

5. For the sample data, the mean is 3.95152 and the median is 3.99576. These values are very close. For the sample data, any value around 4 is fine.

6. Students’ values should be fairly close to the slope of the movable line.

7. The slope is the distance traveled over time. The distance of the balloon rocket from the sensor was increasing at approximately 4 m/s.

LESSON 3.3 • Finding a Line of Fit REQUIRED DOCUMENT

MaunaLoa.ftm

LESSON GUIDE

This demonstration can replace the example of Lesson 3.3.

INVESTIGATE

1. Slope: probably between 1.5 and 1.7

2. y-intercept: between �2500 and �3000

3. Approximately CO2Level � �2872 � 1.6207Year. This is close to the equation in the student book.

4. Answers will vary according to the equation found in Question 3. The prediction in the student book is 449.1 ppm.

EXPLORATION • A Good Fit? REQUIRED DOCUMENT

States - CarsNDrivers.ftm

LESSON GUIDE

This demonstration covers the activity of the Exploration Residual Plots and Least Squares.

INVESTIGATE

1. The slope gives the portion of the population that are drivers.

2. Yes; if a state had a population of 0, it would have 0 drivers.

3. Outliers have significantly positive or negative y-values on the residual plot. (California’s large population does not make it an outlier.)

4. Sample answer: Ohio, Florida, and New York are outliers. Much of the population of New York lives in New York City and relies on public transportation.




5. The squares have one side whose length is a residual.

6. Answers will vary, but the lines should be close. The median-median line is DriversThou � 0.678PopThou � 18.

7. The sum of squares of the least squares line (97,050,000) should be less than that of the other two lines. Least squares lines will be studied in Chapter 11.

CHAPTER 4

LESSON 4.3 • Lines REQUIRED DOCUMENT

Transformations.gsp

LESSON GUIDE

This demonstration can be used to explore vertical and horizontal translations of the linear function.

INVESTIGATE

1. The function shifts to the right as you move h to the right, and it shifts to the left as you move h to the left.

2. The x-intercept equals the absolute value of length h.

3. The y-intercept equals the absolute value of length k.

4. f (x) � (x � 2) � 3. Many students may answer f (x) � x � 5.


6. h and k are the number of units the original parent function f (x) � x is translated horizontally and vertically.

7. f (x) � x � 1; f (x) � (x � 2) � 1; f (x) � (x � 2) � 3; f (x) � (x � 3) � 2; and so on. There are infinite equations that describe this function.

LESSON 4.4 • Parabolas

REQUIRED DOCUMENT

Transformations.gsp

LESSON GUIDE

This demonstration can be used to explore vertical and horizontal translations of the parabola function.

INVESTIGATE

1. The vertex shifts up or down by the length of k.

2. The y-intercept equals the value of k.

3. f (x) � x 2 � 2


5. The vertex shifts to the right or left by h units.

6. f (x) � (x � 3)2

7. The vertex

8. (�4, �2)

9. f (x) � (x � 2)2 � 1

LESSON 4.5 • Square Roots REQUIRED DOCUMENT

Transformations.gsp

LESSON GUIDE

This demonstration can be used to explore vertical and horizontal translations and reflections of the parent function f (x) � �

__ x .

INVESTIGATE

1. g (x) � � � __

x

2. g (x) � �f (x)

3. Domain: x � 0; Range: y � 0

4. h (x) � � ___

�x

5. h (x) � f (�x)


7. r (x) � � � ___

�x

8. r (x) � �f (�x)


10. h changes the horizontal orientation of the graphs. Graphs with positive x-values shift horizontally by the value of h. Graphs with negative x-values shift horizontally in the opposite direction. If you look at the original parent function, the equations have been reflected across the y-axis; therefore, the position of h will follow the same reflection.

11. k changes the vertical orientation of the graphs. Graphs with positive y-values shift vertically by the value of k. Graphs with negative y-values shift vertically in the opposite direction. If you look at the original parent function, the equations have been reflected across the x-axis; therefore, the position of k will follow the same reflection.

12. The endpoint of the original parent function and any translated function

13. For the reflection across the x-axis, (h, �k); for the reflection across the y-axis, (�h, k); for the reflection across the x- and y-axes, (�h, �k).



LESSON 4.6 • Absolute Value REQUIRED DOCUMENT

Transformations.gsp

LESSON GUIDE

This demonstration can be used to explore dilations, trans-lations, and reflections of the parent function f (x) � ⏐x⏐.

INVESTIGATE

1. a dilates the function in the horizontal direction.

2. b dilates the function in the vertical direction.


4. slope � b __ a

5. g (x) � bf � x __ a � 6. The transformed equation is undefined because of

division by zero.

7. The graph is reflected across the x-axis.

8. h shifts the function horizontally. k shifts the function vertically.

9. f (x) � bf � x � h _____ a � � k

10. The vertex of the transformed function

LESSON 4.6 • Science Fair REQUIRED DOCUMENT

ScienceFair.ftm

LESSON GUIDE

This demonstration can replace Exercise 13 of Lesson 4.6.

INVESTIGATE

1. The actual mean of Ratings is 83.75, and the standard deviation is about 7.45. Using sliders, students can come close to these numbers.

2. A translation of 6 points changes the mean to about 89.75 and doesn’t change the standard deviation.

3. A stretch of 1.06 changes the mean to about 88.75 and the standard deviation to about 7.88.

4. The stretch is done first, by a factor of 5 ___ 7.45 , or about

0.67. This sets the standard deviation. A translation of about 33.9 points brings the mean to 90.

LESSON 4.7 • Circles and Ellipses REQUIRED DOCUMENT

Transformations.gsp

LESSON GUIDE

This demonstration can be used to explore transforma-tions of the half circle.

INVESTIGATE

1. r changes the radius of the circle.

2. f (x) � � _______

25 � x 2 ; f (x) � � _______

64 � x 2

3. a dilates the function in the horizontal direction; g (x) � f � x __ a � .

4. b dilates the function in the vertical direction; g (x) � bf (x).

5. If a � 0, the function is undefined because of division by zero.

6. If b is negative, the function is reflected. The sign of a has no effect on the function.

7. h shifts the function horizontally. k shifts the function vertically.

8. If the entire ellipse or circle were graphed, (h, k) would be the center.

9. f (x) � b � ____________

r 2 � � x � h _____ a � 2 � k

10. Graph both f (x) and �f (x) on the same axes. For some values of x, there are two values of y.

CHAPTER 5

LESSON 5.2 • Power Functions REQUIRED DOCUMENT

PowerFunctions.gsp

LESSON GUIDE

This demonstration covers the characteristics of the power function, f (x) � ax n.

INVESTIGATE

1. When the exponent is odd, the function increases as x increases and decreases as x decreases. When the exponent is even, the function increases as x either increases or decreases. (This can also be described as concave up.) All the functions have y-intercept 0 and go through the point (1, 1).

2. f (x) � x 7 increases to the right and decreases to the left at a steeper rate than the smaller odd powers. f (x) � x 10 increases to the right and left at a steeper rate than the smaller even powers, and its bottom is flatter. f (x) � x 211 increases to the right and decreases to the left at a steeper rate than the smaller odd powers.

3. The functions with even exponents have reflective symmetry across the y-axis. The functions with odd exponents have rotational symmetry about the origin.





5. The function becomes the line y � 0. This is because f (0) � x 0 � 1.

6. Functions with odd or even exponents approach zero as x gets farther away from zero and never cross the y-axis. Functions with even exponents increase as x approaches zero from the left and right. Functions with odd exponents increase as x approaches zero from the right and decrease as x approaches zero from the left.

LESSON 5.3 • Rational Exponents REQUIRED DOCUMENT

RationalExponents.gsp

LESSON GUIDE

This demonstration explores Exercises 6 and 7 in Lesson 5.3.

INVESTIGATE

1. Both even and odd rational exponents give similar shapes. Functions with even rational exponents are defined only for positive x-values, so the domain of the graph is x � 0. Functions with odd rational exponents are defined for all values of x. The odd functions have rotational symmetry about the origin.

2. f (x) � x 1�7 is similar to the other functions with odd exponents, and f (x) � x 1�10 is similar to the other functions with even exponents. Both are steeper in the center and flatter overall.


4. As x increases, the function approaches zero from the top. As x approaches zero from the right, the function increases. As x approaches zero from the left, the function decreases. As x decreases, the function approaches zero from below. Only functions with odd negative exponents have a left side.

5. Similarities: The y-intercept is at the origin; each graph goes through (1, 1); even- and odd-exponent functions increase as x increases; odd-exponent functions decrease as x decreases. Differences: They are inverses of each other (this will be covered in Lesson 5.5); power functions with integer exponents have a vertical trend; functions with rational exponents have a horizontal trend.

6. Similarities: Each graph shares the points (0, 0) and (1, 1), and the function increases as x increases. Differences: When n is small, the function has a horizontal trend; when n is large, the function has a vertical trend.

7. At n � 0, the function is a horizontal line at y � 1. f (x) � x 0�r � x 0 � 1.

8. Similarities: As x increases, the function approaches zero from the top. As x approaches zero from the right, the function increases. Differences: When the exponent is an integer value, the shape of the function changes. At even integer exponents, as x decreases, the function approaches zero from the top, and as x approaches zero from the left, the function increases. At odd integer exponents, as x decreases, the function approaches zero from the bottom, and as x approaches zero from the left, the function decreases.

LESSON 5.5 • The Inverse of a Line REQUIRED DOCUMENT

LinearInverse.gsp

LESSON GUIDE

This demonstration covers the characteristics of the inverse of a linear function.

INVESTIGATE

1. The y-intercept of the function is equal to the x-intercept of the inverse.

2. The intersection of the two functions seems to follow a linear pattern.

3. It traces a line of symmetry.

4. y � x

5. The x- and y-coordinates are switched.

6. (2, �1)

LESSON 5.6 • Exponential and Logarithmic Functions

REQUIRED DOCUMENT

ExponentialLogarithm.gsp

LESSON GUIDE

This demonstration guides students through a discovery of the effects of a and b on the exponential function, f (x) � ab x, and the logarithmic function, f (x) � a logb x.

INVESTIGATE

1. For b � 1: exponential growth; for b � 1: f (x) � 1, which is a horizontal line; for 0 � b � 1: exponential decay. All the functions have y-intercept 1.

2. If b � 0, the graph disappears because the value of the function is complex.

3. The y-intercept is equal to the value of a. A larger a corresponds to a steeper graph.

4. Replace x with x � 2, and subtract 3 from the function; g (x) � 3 � ab x �2.



5. When the function is defined, it has x-intercept 1. When b � 1, as x increases, the value of the function increases without bound. As x approaches zero from the right, the value of the function decreases without bound. There is an asymptote at x � 0. When b � 1, the function is undefined because when you use the change of base property, the denominator, log 1, is 0. When 0 � b � 1, the function is reflected across the x-axis. (To see this, select the function plot and choose Trace Function Plot from the Display menu.) When b � 0, f (x) � 0 for positive values of x (because when x � 0, log x is undefined). When b � 0, f (x) is undefined because, again using the change of base property, the numerator, log x, is undefined.

6. The x-intercept is always 1. A larger a corresponds to a steeper graph. For increasing a: As x increases, the value of the function increases without bound. As x approaches zero from the right, the value of the function decreases without bound. There is an asymptote at x � 0. For decreasing a: As x increases, the value of the function decreases without bound. As x approaches zero from the right, the value of the function increases without bound. There is an asymptote at x � 0.

7. Replace x with x � 4, and add 3 to the function; g (x) � �3 � a logb (x � 4).

LESSON 5.8 • Cooling

REQUIRED DOCUMENT

None

LESSON GUIDE

This demonstration can replace the investigation in Lesson 5.8. It allows students to gather their own data to see a real-world example of an exponential relationship. Each group will need a temperature sensor.

EXPERIMENT

Step 4 It may be interesting to have some groups warm the temperature sensor in their hand, and other groups in cups of water heated to different temperatures. This would give an opportunity to make conjectures about why some features of the graph are different (different initial temperatures and different rates of cooling) and some features are the same (the same lower limit due to the same room temperature).

INVESTIGATE

1. A whole class discussion of the predictions could help students make mathematical sense of the situation. Most students will probably show a decreasing graph, but some may predict a linear relationship and some may predict curves that

indicate the cooling rate will increase. Some well-placed questions could help students see that, for example, the temperature will not continue to decrease forever. But don’t give too much away!

2, 3. After seeing the true shape of the relationship, students should be able to explain that, as the probe gets closer to room temperature, it will not lose heat as fast. Also, the probe will not drop below room temperature, which is the limiting value of the graph.

4, 5. It might be illuminating to use the probe to check the temperature of the room to see how closely the value of Shift matches. The equation should be of the form:

Temperature � Room temperature � Initial probe temperature � (1 � percent decrease)Time

You may also want to have students create a residual plot for the graph of (Time, Log_Temp): select the graph and choose Graph � Make Residual Plot. A periodic pattern will probably emerge. This pattern is common in time-series data because of the dependence of each temperature reading on the previous temperature reading. If the temperature is above the trend at one moment, it is more likely to be above the trend at the next reading.




LESSON 5.8 • Curve Straightening

REQUIRED DOCUMENT

ViewingDistance.ftm

LESSON GUIDE

This demonstration explores Lesson 5.8, Exercise 9. You might use it as an extension of Example B.

INVESTIGATE

1. The (logH, logD) graph is most linear.

2. logD � 0.499logH � 0.555

Distance � 10 0.555 Height 0.499 � 3.589Height 0.499

3. This is a square root function.

4. The function fits the data well.

CHAPTER 6

LESSON 6.6 • Linear Programming

REQUIRED DOCUMENT

LinearProgramming.gsp

LESSON GUIDE

This demonstration explores Exercise 6 in Lesson 6.6.

INVESTIGATE

1. x represents the number of poodles, and y represents the number of sled dogs.

2. x � 20 At most 20 poodles

y � 15 At most 15 sled dogs

y � 100 � 2p

________ 6 The amount of food consumed

y � 15000 � 1000p

_____________ 250 The number of hours of training

3. The exact coordinates are: O (0, 0), $0 profit; D (0, 15), $1,200.00 profit; C (5, 15), $2,200.00 profit; B (11.82, 12.73), $3,381.82 profit; A(15, 0), $3,000.00 profit.

4. Point B shows the maximum profit. This is not realistic because you can’t have 11.82 dogs or 12.73 dogs. The coordinates must be integers.

5. $3,360.00; (12, 12)

6. 12 poodles and 12 sled dogs

7. 112 lb

8. $3,400


CHAPTER 7

LESSON 7.1 • Free Fall

REQUIRED DOCUMENT

Freefall.ftm

LESSON GUIDE

This demonstration can replace the investigation in Lesson 7.1. It allows students to see how the method of finite differences works with real-world data. Because there is variation in the data, the second difference will not be quite constant. Each group will need a motion sensor.

INVESTIGATE

1. The graph of (Time, Height) shows that the distance from the ball to the sensor is not linearly related to time. The ball speeds up as it falls because vertical difference between the points is increasing. If the ball was initially tossed upward, the graph shows that the ball goes up, decreasing in speed before it starts falling.

The graph of (Time, Diff1) shows that the first differences are not constant: because they decrease in a linear fashion, the second differences are likely to be constant. The graph of (Time, Diff2) shows that the second differences are nearly constant, so the correct model should be a 2nd-degree polynomial function.

2. This may not be immediately obvious to students because the second difference is not exactly a constant. This is the reality when working with real data. But it should be clear that in the first two plots that there is a clear decreasing pattern in the data. However, in the third plot, there will probably be a rather random scatter around a horizontal line. This is as close as real data comes to a constant value. Since the second difference is a constant, the degree of the polynomial will be two. The general form of the polynomial function is h � at 2 � bt � c.

3, 4. Answers will vary. In fact, students will get slightly different equations depending on which three points they select. The theoretical value of a is �4.9 m/s2, which is half the acceleration due to gravity. The value of b is the velocity of the ball at time t � 0. Because students deleted some points, this is probably not the instant that the ball was dropped, so there will most likely be some initial velocity. The value of c is the height of the ball at time t � 0.

DAA2TD_012_TN_b.indd 108DAA2TD_012_TN_b.indd 108 1/9/09 4:22:11 PM1/9/09 4:22:11 PM


LESSON 7.2 • Rolling Along

REQUIRED DOCUMENT

None

LESSON GUIDE

This demonstration can replace the investigation in Lesson 7.2. It allows students to model real data using three forms of a quadratic function. Each group will need a motion sensor.

INVESTIGATE

1. The graph is roughly a parabola, and a quadratic function would model the data.

2. Sample answer: With vertex (3.2, 4.257) and a point (5.2, 0.897), the equation is AdjDistance � �0.84(Time � 3.2)2 � 4.257.

3. Sample answer: Using the points (1, 0.546), (3, 4.256), and (5, 1.493), the equation is AdjDistance � �0.81Time 2 � 5.09Time � 3.74.

4. Sample answer: Using the points (0.9, 0) and (5.5, 0), the equation is AdjDistance � �0.84(Time � 0.9)(Time � 5.5).

5. Students should find that the graphs very nearly coincide.

6. Students should describe that with any three points they can write a general form of the quadratic equation, but with special points, such as the vertex or the x-intercepts, they can use different forms of the equation more efficiently.

LESSON 7.4 • Quadratic Functions

REQUIRED DOCUMENT

QuadraticFunction.gsp

LESSON GUIDE

This demonstration shows how changes in a, b, and c in the general form of the quadratic equation, f (x) � ax 2 � bx � c , affect the graph.

INVESTIGATE

1. The graph is stretched vertically as a gets larger in absolute value and stretched horizontally when a gets closer to zero.

2. The parabola is upside down.

3. The y-intercept is always the constant c because the x 2-term does not affect this point.

4. The graph remains the same shape and points in the same direction, but the vertex moves in a quadratic shape around the y-intercept, which stays the same.

5. The graph translates up and down. The y-intercept is equal to the value of c.

6. The vertex follows a linear path.

7. f (x) � c � 1 _ 2 bx

8. The vertex follows a parabolic path; f (x) � c � ax 2.

9. The vertex follows a constant path; x � � b __ 2a .

LESSON 7.7 • Higher Degree

REQUIRED DOCUMENT

HigherDegree.gsp

LESSON GUIDE

This demonstration explores the characteristics of 3rd-, 4th-, and 5th-degree polynomials.

INVESTIGATE

1. a � b � c � 0

2. There are zeros at x � 1, x � 2, and x � 3. There is one local maximum and one local minimum.

3. There are zeros at x � 2 and x � 3. There is one local maximum and one local minimum.

4. No, you would need a negative scale factor to reflect the graph.

5. Local maximum: about (1.42, 0.38); local minimum: about (2.58, �0.38)

6. (2, 0)

7. a � b � c � d � 0

8. There are zeros at x � �2, x � 0, x � 1, and x � 3. There are two local minimums and one local maximum. The graph is symmetric about x � 1 _ 2 .

9. Possible answer: If a � b � 3 and c � d � �2, there are two double roots and the graph is symmetric about x � 0.5, with two local minimums and one local maximum. If a � b � c � �2 and d � 3, there are still the two x-intercepts, one of which is a triple root, but the graph looks very different, with one local minimum and no local maximum.

10. Local maximum: about (0.5, 39.06); local minimums: (�2, 0) and (3, 0)

11. a � b � c � d � e � 0

12. There are zeros at x � 1, x � �2, x � 3, and x � 0. x � 1 is a double root. There are two local maximums and two local minimums.

13. a � b � c � d � e

14. There are a maximum of two local maximums and two local minimums. When there is only one root, there are no local extremes.




CHAPTER 8

LESSON 8.1 • Bucket Race

REQUIRED DOCUMENT

BucketRace.gsp

LESSON GUIDE

This demonstration can replace the investigation of Lesson 8.1. It allows students to explore an optimization situation inductively, in order to motivate the distance formula calculation.

INVESTIGATE

1. The total distance is the greatest at the right end of the pool.

2. The total distance is shortest at about x � 8.325 m, with a total distance of about 23.324 m. There is only one location that gives this minimum distance.

3. The curve is concave up like the bottom of a bowl, but it is not symmetric.

4. There is one minimum value to the curve, showing that the shortest distance occurs at a unique location. The value of that minimum can be approximated by looking at the coordinates of the point near the minimum.

5. T1 � AC _____ 1.2 cm . Sketchpad does not allow seconds as

units, but leaving the units as centimeters may cause confusion. Dividing by 1.2 cm leaves the quotient unitless.

6. Again the curve is concave up with a single minimum. The minimum time of about 33.75 seconds occurs at x � 17.63 m.

LESSON 8.2 • Definitions of Circle and Ellipse

REQUIRED DOCUMENT

CirclesandEllipses.gsp

LESSON GUIDE

Students look at the locus definitions of circles and ellipses.

INVESTIGATE

1. Circle

2. When P approaches C, the circle becomes smaller. When P moves away from C, the circle becomes larger.

3. CP remains constant.

4. A circle is the locus of points that are a given distance from one point.

5. Ellipse

6. The length of the minor axis is the diameter of the small circle, and the length of the major axis is the diameter of the large circle.

7. J changes the size of the larger circle, which changes the length of the major axis. K changes the size of the smaller circle, which changes the length of the minor axis.

8. The ellipse becomes a circle.

9. Construct a line perpendicular to the major axis through point L. Then construct a line parallel to the major axis through the intersection of segment IL and the small circle. As point L moves around the large circle, point N will construct an ellipse.

N

I

K

L

J

10. The circles have the same radius.

11. They are the intersections of circle T with the major axis.

12. The radius of circle I is a, so the radius of circle T is also a. Therefore TF1 � a. If IF1 � c , then you can construct the right triangle TF1I to show that b 2 � c 2 � a 2.

LESSON 8.3 • Definition of Parabola

REQUIRED DOCUMENT

Parabolas.gsp

LESSON GUIDE

Students explore the parabola as a locus of points or envelope of lines, and the relationship of the focus and directrix to the parabola. The first part of the demon-stration covers the investigation in Lesson 8.3. Students should complete the patty paper construction before they see the sketch.



INVESTIGATE

1. Parabola

2. When the focus is closer to the directrix, the parabola is stretched vertically. When the focus is farther away from the directrix, the parabola is stretched horizontally.

3. Point H is the midpoint of the segment from the focus to the directrix.

4. It is the perpendicular bisector of the segment from the focus to any point on the directrix.

5. They lie on a vertical line that is perpendicular to the directrix.

6. They are always equal.

7. A parabola is the locus of points a given distance from a point (the focus) and a line (the directrix).

8. Start with the construction for the folded parabola. Construct a line perpendicular to the directrix through point G. Construct the intersection of that line and the perpendicular bisector of

___ HG . This

intersection creates the envelope of lines.

LESSON 8.4 • Definition of Hyperbola

REQUIRED DOCUMENT

Hyperbolas.gsp

LESSON GUIDE

Students explore the hyperbola as an envelope of lines. This sketch can first be completed using patty paper.

INVESTIGATE

1. Hyperbola

2. C must travel one full revolution around the circle.

3. Construct a segment from point B to the circle. Construct the perpendicular bisector of the segment. Trace that bisector while animating point C around circle A.

4. As B approaches the circle, the hyperbola is stretched, or narrower. When B is on the circle, the hyperbola disappears. When B is inside the circle, an ellipse is formed.

5. The larger the circle, the more concave the hyper-bola until B is inside the circle, in which case the parabola becomes an ellipse. As the circle gets smaller, the hyperbola shrinks and is very shallow.

6. Move point A or B so that the two points lie on a vertical line. Make sure that point B is outside the circle. (You cannot actually do this with this sketch because of the way it is constructed.)

EXPLORATION • From Circles to the Ellipse

REQUIRED DOCUMENT

ConicSections.gsp

LESSON GUIDE

This demonstration covers the activity and Questions 3 and 4 of the Exploration Constructing the Conic Sections.

INVESTIGATE

1. Point C changes the radii of the circles.

2. AC equals the radius of circle F1 and BC equals the radius of circle F2.

3. It is the locus of points created by the intersections of the two circles.

4. The ellipse becomes more narrow as ___

AB gets shorter. It disappears when the circles no longer intersect.

5. Point C changes the location of the intersections but does not change the shape of the ellipse.

6. As F1 approaches F2, the ellipse becomes more circular. As F1 moves away from F2 , the ellipse becomes narrower.

7. When the distance between the foci is greater than AB, the ellipse disappears.

8. A circle is formed.

9. It is the locus of points created by the intersection of the line perpendicular to the directrix and the perpendicular bisector of segment FB.

10. It changes the radii of the circles.

11. The radius of F1 equals AC and the radius of F2 equals CB.

12. The hyperbola is the locus created from the inter-sections of the circles as point C is dragged along line AB.

13. In the hyperbola construction, the radius of the smaller circle is a part of the radius of the larger circle, whereas in the ellipse construction, the radii combined create the entire segment. Both construc-tions are loci formed by the intersections of circles.

LESSON 8.6 • Rational Functions

REQUIRED DOCUMENT

RationalFunctions.ftm

LESSON GUIDE

This demonstration allows students to dynamically explore the effects of each parameter on the graph of a rational function. The relationships here are rather subtle and you may need to frequently interrupt your students to guide them as needed.




INVESTIGATE

1. When parameter a increases by 1 unit, the graph translates up 1 unit, as does the horizontal asymptote. In fact, the equation of the asymptote is y � a. When parameter a decreases by 1 unit, the graph and horizontal asymptote translate down 1 unit. There is no effect on the vertical asymptote.

2. Changing parameter b to 2 dilates the graph, but the effects of parameter a are the same as in Question 1. If parameter b � �3 (or any negative value), the graph is reflected across the y-axis, and it dilates, but the answers to Question 1 do not change.

3. If c � 2, adjusting slider a still translates the graph and asymptote up and down, but now if a increases by 1 unit, the graph translates up 1 _ 2 unit, and the right-hand side of the equation is now a _ 2 . If c � �3, increasing slider a by 1 unit translates the graph down 1 _ 3 unit. The equation of the horizontal asymptote is � a _ 3 . Again, the vertical asymptote is unaffected by a.

4. If d � �3, the graph and vertical asymptote are translated right 3 units. Adjusting slider a translates the graph as in Question 1, but now it also dilates the graph. If d � �3 and c � 2, the graph is translated left 3 _ 2 , or 1.5 units and dilated. Now increasing a by 1 unit translates the graph up 1 _ 2 unit, as in the first part of Question 3 and still dilates the graph.

5. The main point here is for students to realize that changing parameter a (the linear coefficient of the numerator) translates the graph and horizontal asymptote vertically, and that the magnitude of the translation depends on the value of parameter c. In fact, the magnitude of the translation is a _

c . When

d � 0, adjusting slider a also dilates the graph.

6. a. Parameters a and c affect the location of the horizontal asymptote. As x gets very far from 0, the leading coeffecients dominate the function’s value. For example, consider the function f (x) �

2x � 4 _____ 3x � 5 . If x � 10,000 the value is

20,004 _____ 29,995 , or about

0.666911. This is so close to 20,000

_____ 30,000 , or 2 _ 3 , that we

can almost ignore the 4 and �5. This effect gets more drastic as x gets further from 0. This can be thought of as the “end behavior” of the function.

b. The location of the vertical asymptote is affected by parameters c and d. This is because the vertical asymptote occurs when the function is undefined, which is when the denominator is 0. To find the location of the vertical asymptote, solve the equation c � x � d � 0.

c. All the sliders can dilate the graph. On page 492 in the student textbook is an example of using long division to rewrite the function in a way that makes the transformations more obvious. If we do that with the general function we get:

f (x) � a � x � b ____________________ c � x � d

a _ c

c � x � d ) ___________

a � x � b

___________

a � x � a _ c d

0 � � b � a _ c d �

So f (x) � a _ c �

b � a _ c d _____

c x � d � a _

c �

b � a _ c d ______

c � x � d _ c � . In this

equation, a _ c is the vertical shift,

b � a _ c d _____

c is the scale

factor, and d _ c is the horizontal shift.

Parameters a and c are part of the vertical shift component, so they both translate the graph vertically and affect the location of the horizontal asymptote. All four parameters are involved in the scale factor, though a has an effect only if d � 0. So all four parameters can dilate the graph. Parameters c and d are part of the hori-zontal shift, so they affect the location of the vertical attribute.

d. The graph is a horizontal line at y � 1. It may look like the graph disappears because it is drawn on top of the horizontal asymptote. (If you click on the formula for the function below the graph, the function will be highlighted.) This is because the numerator and denominator of the function are the same, so the value is 1 for any value of x.

e. The graph is a line. The denominator of the function is a constant, so the function is linear, and has equation y � a _

d x + b _

d .

f. The graph becomes a horizontal line: the func-tion is a constant with value b _

d .

g. The graph becomes a horizontal line: the func-tion is a constant with value a _

d .

7. The horizontal asymptote has equation y � a _ c . The

vertical asymptote has equation x � d _ c . Reasons for

both of these equations can be seen by looking at the re-expression in part 6c above.

a _ c is the vertical shift, so the horizontal asymptote is

shifted vertically by this amount.

When x � d _ c , the denominator has value 0, so the

function is undefined. This can also be seen from the answer to Question 6b, because the solution to

the equation c � x � d � 0 is x � d _ c .



LESSON 8.6 • Inverse Variation

REQUIRED DOCUMENT

InverseVariation.gsp

LESSON GUIDE

This demonstration covers Exercise 10 from Lesson 8.6.

INVESTIGATE

1. (1, 1), (�1, �1) 2. x � 0, y � 0

3. y � x and y � �x 4. 4 units

5. a � b � � __

2 6. c � 2

7. Construct a circle centered at the center of the hyperbola, (0, 0), with radius 2. This circle should circumscribe the box.

8. �� __

2 , � � __

2 �, � � __

2 , � __

2 � 9. Wherever point P is, the difference of the distances

between P and the two foci is always constant.

CHAPTER 9

LESSON 9.1 • Arithmetic Series Formula

REQUIRED DOCUMENT

ArithmeticSeries.ftm

LESSON GUIDE

This demonstration can replace the investigation of Lesson 9.1.

INVESTIGATE

1. The terms of the sequence are u1 � 4, u 2 � 7, u 3 � 10, u4 � 13, and u 5 � 16. The sum of the series is 50.

2. The sixth term is 19 and the sum is 69.

3. The dimensions of this rectangle are 6 by 23.

4. Area � n �u 1 � un�

5. Sn � n �u1 � un� __________ 2 or Sn �

n �2u 1 � d(n � 1)� ________________ 2

6. n is the number of rows or terms, u 1 is the length of the first row, and un is the length of the last row. The length of the rectangle is (u1 � u n) and the height or width is n. The formula divides by 2 because the rectangle is twice the partial sum.

LESSON 9.2 • Fractals

REQUIRED DOCUMENT

Fractals.gsp

LESSON GUIDE

This demonstration covers Exercises 11 and 12 in Lesson 9.2.

INVESTIGATE

1. By connecting the midpoints of each successive square

2. The perimeter and area of each created square

3. The side of each created square is the hypotenuse

of a 45°-45°-90° triangle with legs of length 1 _ 2 the

side length of the previous square. So to get the

next perimeter, you multiply the previous perim-

eter by � __

2 ___ 2 . To find the next area, you divide the

previous area by 2.

4. 4s ________

1 � � __

2 ____ 2

5. 2s 2

6. By connecting the midpoints of each shaded triangle from the previous stage and removing the shading from the middle triangle formed

7. It approaches zero. 8. It approaches zero.

9. It approaches infinity. 10. It approaches zero.

EXPLORATION • A Geometric Series

REQUIRED DOCUMENT

InfiniteSeries.gsp

LESSON GUIDE

This demonstration covers the Exploration Seeing the Sum of a Series.

INVESTIGATE

1. The length of ___

AB and its subsequent lengths as constructed.

2. � n�1

�

AB � 1 __ 2 � n�1

3. The sum of the lengths in column m ___

AB

4. Answers will vary according to the length of ___

AB .

5. Answers should be equivalent to the answer for Question 5 and equal to 2AB.

6. The common ratio is larger.

7. � n�1

�

AB � 3 __ 5 � n�1 8. 5 _ 3 AB




9. � n�1

�

AB � 6 __ 5 � n�1 10. Infinity

11. If the ratio is between 0 and 1, the series converges to a specific sum. If the ratio is greater than 1, the series diverges and the sum is infinite.

CHAPTER 10

LESSON 10.1 • Sum of Two Dice

REQUIRED DOCUMENT

TwoDice.ftm

LESSON GUIDE

This demonstration can replace Example A in Lesson 10.1.

INVESTIGATE

1. Sums of 6 should cluster around 41.67. Reasonably likely sums vary between 30 and 54.

2. Experimental probabilities should cluster around 0.1389. Reasonably likely proportions vary between 0.10 and 0.18.

3. Reasonably likely sums of 6 vary between 221 and 279 and should cluster around 250. Reasonably likely proportions vary between 0.12 and 0.15 and should cluster around 0.1389.

4. 5 __ 36 , or 0.13888. . . . The results for the 1800 rolls should be closer to this value, on average, than the results for 300 rolls.

EXPLORATION • The Coin Toss Problem

REQUIRED DOCUMENT

CoinToss.ftm

LESSON GUIDE

This demonstration can replace the Exploration Geometric Probability. For Questions 1 and 2, you might want to have students rerandomize the data several times to see a range of experimental probabilities.

INVESTIGATE

1. Answers will vary. Proportions should cluster around 0.25. Reasonably likely proportions will vary between 0.16 and 0.34.

2. Answers will vary. Reasonably likely values are given in the table. (See table below.)

3. The coin’s center must fall in a concentric square with sides of length T � 2r. The area of these region is (T � 2r)2.

4. Here are the theoretical probabilities. Comparisons to experimental probabilities will vary. (See table below.)

5. The formula is (T � 2r)2

__________ � T � n�1

___ n � B � 2 , where n is the number

of tiles in each row or column. Students might think this deno minator should be (T � 2B)2, and this would be the case if there was only 1 tile. However, because each interior tile shares its borders with its neighboring tiles, using this formula would double-count all of these in terior tile borders and make the denominator too large.

CoinCoin radius

(mm) 20 mm tile 30 mm tile 40 mm tile40 mm with

5 mm borders

Maxmillian 5 0.16 –0.34 0.345 –0.544 0.46 –0.66 0.33 –0.53

Dime 9 0 –0.03 0.08 –0.233 0.21 –0.4 0.148 –0.32

Penny 10 0 0.048 –0.175 0.16 –0.34 0.11 –0.27

Quarter 12 0 0 –0.08 0.09 –0.23 0.057 –0.19

Exploration, Question 2

CoinCoin radius

(mm) 20 mm tile 30 mm tile 40 mm tile40 mm with

5 mm borders

Maxmillian 5 0.25 0.44 0.5625 0.432

Dime 9 0.01 0.16 0.3025 0.233

Penny 10 0 0.11 0.25 0.192

Quarter 12 0 0.04 0.16 0.123

Exploration, Question 4



LESSON 10.2 • Independent Events

REQUIRED DOCUMENT

CA_includesBerkeley2000.ftm

LESSON GUIDE

This demonstration covers Exercise 10 in Lesson 10.2. To use census microdata from another area, choose File⏐Open Sample Document⏐Social Science⏐United States⏐US Census Files, or use Fathom Help to learn how to Import U.S. Census Data from the Internet.

INVESTIGATE

1. The probability of being married given that one is young (or not)

2. The probability of being married if (or given that) one is young is smaller than the probability of being married (in general).

3. The events aren’t really independent until all ages are included (age � 90), but the probabilities get closer as the age increases.

4. Answers will vary. Married and German are pretty close, as are not Spanish (that is, the false value of Spanish) and married.

EXPLORATION • A Repeat Performance

REQUIRED DOCUMENT

DiceRolls.ftm

LESSON GUIDE

This demonstration can replace the Exploration The Law of Large Numbers.

INVESTIGATE

1. In general, histograms change quite a bit with only 10 cases.

2. The histogram’s bins will be closer in height to each other than with 10 cases.

3. The histogram will change less with rerandomizing that it did with fewer cases.

4. The histogram will be flatter (the bins more equal in height) and will change less when you reran-domize with the larger number of cases.

5. Answers will vary. The number of cases needed to see a pattern depends on how much the weights of the different outcomes vary.

6. The more cases we looked at, the closer the histogram was to the theoretical probability, for both fair and loaded dice.

LESSON 10.4 • Expected Value

REQUIRED DOCUMENT

WorldSeries.ftm

LESSON GUIDE

This demonstration can replace Example B of Lesson 10.4.

INVESTIGATE

1. Answers will vary among 4, 5, 6, and 7.

2. Answers will vary among 4, 5, 6, and 7.

3. Answers will vary. They should cluster around 5.8.

4. Answers will vary. The mean might be anywhere between 4 and 7.

5. The theoretical expected value is 5.8125.

6. The probability can be changed by changing the 6 in the formula for winner. The less well matched the teams, the smaller the expected number of games.

LESSON 10.5 • Ordered Lists

REQUIRED DOCUMENT

OrderedLists.gsp

LESSON GUIDE

This demonstration can replace the investigation in Lesson 10.5. It was adapted from a Sketchpad Lesson-Link activity.

INVESTIGATE

1. 6

2. 24. For 4 gifts, there are four different choices for the first gift. For each of these 4 choices there are now 3 gifts left and so 6 different arrangements for these last 3 gifts. Thus, for 4 gifts, there are 4 � 6, or 24 different arrangements.

3. 6

4. Students may notice that if n � 0 or r � 0, Sketchpad gives the number of lists as one, but nothing appears. This is correct: there is only one way to draw zero objects from a set of n objects, which is to draw nothing at all.

n � 1 n � 2 n � 3 n � 4 n � 5

r � 1 1 2 3 4 5

r � 2 2 6 12 20

r � 3 6 24 60

r � 4 24 120

r � 5 120




5. Answers will vary. To find the values for a column, start with n. To get the next numbers in that column, multiply by n � 1, then n � 2, n � 3, . . . , 2, 1. To get a value in the nth column and r th row, start with n. Then multiply by n � 1. Continue multiplying for a total of r terms, where each term is 1 less than the previous term.

6. 10 � 9 � 8 � 720

LESSON 10.7 • Sapsuckers

EXAMPLE DOCUMENT

Sapsuckers.ftm

LESSON GUIDE

This demonstration can replace Example B of Lesson 10.7.

INVESTIGATE

1. Usually 2, 3, or 4

2. Answers will vary. The mean over many cases should be about 30.

3. Answers will vary. The mean should be about 30.

4. The mean should be close to 30.

5. Exactly 4 sapsuckers will survive about 30% of the time.

6. P (0) � 0.005; P (1) � 0.045; P (2) � 0.157; P (3) � 0.289; P (5) � 0.165; and P (6) � 0.038. The most probable number of survivors is 3 or 4.

CHAPTER 11

LESSON 11.2 • Ages

REQUIRED DOCUMENT

Ages.ftm

LESSON GUIDE

This demonstration covers Exercise 12 of Lesson 11.2.

INVESTIGATE

1. The mean and median ages of presidents are slightly larger than those of vice presidents, but the spread of vice presidential ages is greater (as determined by either the standard deviation or the IQR).

2. The number of standard deviations each value is from the mean

3. The values cluster around the mean; that is, they cluster around 0 standard deviations from the mean.

4. Answers may vary. It may be clear to some students that the mean of the vice presidents’ standardized ages is less than that of the presidents’ standardized ages, or that the spread of the vice presidents’ stan-dardized ages is more than that of the presidents’ standardized ages. You might bring out the point that the range of standardized values doesn’t extend far beyond �2 or 2 standard deviations, to lay groundwork for the 68-95-99.7 rule.

EXPLORATION • Is This Normal?

REQUIRED DOCUMENT

Populations.ftm

LESSON GUIDE

This demonstration can replace the Exploration Normally Distributed Data. You might have students look at data from their own area. Choose File⎮Open Sample Document⎮Social Science⎮United States⎮US Census Files, or use Fathom Help to learn how to Import U.S. Census Data from the Internet.

INVESTIGATE

1. Students may conjecture any numerical data.

2. Answers will vary. With an appropriate bin width, eduCode might appear normally distributed, but income is usually not.


4. Answers will vary. If the age filter is put on, the distribution of eduCode may appear even more normal.

LESSON 11.4 • Polling Voters

REQUIRED DOCUMENT

Polling.ftm

LESSON GUIDE

This demonstration can replace Example B in Lesson 11.4.

INVESTIGATE

1. Normal

2. They are the bounds of the interval in which 95% of the percentages lie—that is, the bounds of the 95% confidence interval.

3. For p � 0.6, it would be an extremely rare set of 100 samples in which 95% contained only propor-tions above 50%. Usually the 95% confidence intervals will include proportions down to 44% to 47%.



4. Repeated sampling (collecting measures) with p � 0.65 almost always limits the 95% confidence interval to values above 50%. A summary table shows that the standard deviation of the measures is about 7.3, so their 95% confidence interval is 65 2(7.3), or from 50.4 to 79.6.

EXPLORATION • Quality Control

REQUIRED DOCUMENT

Biscuits.ftm

LESSON GUIDE

This demonstration can replace the Exploration Predic-tion Intervals if time is an issue. The simulation is set up for students to begin collecting measures at the beginning of the demonstration.

INVESTIGATE

1. The standard deviation is 12 g. So the sample standard deviation for samples of size four is 12

___ �

__ 4 ,

or 6 g. The 90% prediction interval is then 500 1.645(6) � 500 9.87.

2. On average, there should be at most 10 values outside the 90% prediction interval.

3. 500 3(6) � 500 18, or from 482 g to 518 g. On average, there should be about 0.27% of the values outside this new prediction interval.

4. If a producer uses a z-value of 3, the probability that the sample means fall outside the control limits is only 1.5% (as opposed to 0.27%). The expected number of batches of four biscuits that would be sampled before getting the first batch that goes

outside of the control limits would then be 1 ____ 0.015 , or

66 batches with a standard deviation of 65.6 batches.

5. If a producer uses a z-value of 3, the probability that the sample means fall outside the control limits is 9.12% (as opposed to 0.27%). The expected number of batches of four biscuits that would be sampled before getting the first batch that goes outside of the control limits would then be 1

____ 0.912 , or 11 batches with a standard deviation of 10.45 batches.

LESSON 11.6 • SAT Scores

REQUIRED DOCUMENT

SATScores.ftm

LESSON GUIDE


INVESTIGATE

1. The line passes through that point (within rounding error). This observation can be checked algebraically.

2. Approaches may vary. The claim can be checked algebraically. Or you can graph the equation of a line with this slope but through a point not on the

least squares line, such as y � 104 � r � sy

__ sx � x , and see

that the two lines are parallel.

3. 514.931. This is within 1 point of the predicted score. To read this value approximately, click on the line and drag the red point. Or, you might create a new data point (case 11) with year equal to 2001, and drag the corresponding point on the graph until it’s on the line. The SAT score can then be read from the case table.

4. 524.211. Students may or may not think the result is reasonable.

EXPLORATION • How Does It Fit?

REQUIRED DOCUMENT

CO2Emissions.ftm

LESSON GUIDE

This demonstration can replace the Exploration Nonlinear Regression. The Fathom document is set up for the students to begin with plotting an exponential model.

INVESTIGATE

1. The values for a could be anywhere between 15,000 and 22,000. The values for b could be between 1 and 1.1. The sum of the squares will be quite large.

2. The equation is log(y) � 0.0247085 � YearsSince1949 � 4.2304.

3. The equation is y � 16,998 � 1.05854 3 x�1949 . Comparisons to the original model will vary.

4. Answers will vary. The first method is quick, easy, and understandable. The second method is more exacting and less error-prone because there is only one least squares line possible for a given set of linearized data.

CHAPTER 11 REVIEW • Normal Curve and Approximation

REQUIRED DOCUMENT

NormalCurve.gsp

LESSON GUIDE

This demonstration covers Take Another Look Activities 2 and 3.




INVESTIGATE

1. The normal curve gets wider and shorter.

2. The normal curve disappears. If the standard devia-tion equals zero, then all the data points must be the same value. Therefore, there would be no normal distribution but rather a collection of the same data values.

3. The normal curve is reflected across the x-axis.

4. Good when the absolute value of the standard deviation is greater than 2; poor when the absolute value is between �

___ 0.5 and 2; undefined when the

absolute value is less than � ___

0.5 .

CHAPTER 12

LESSON 12.1 • Right Triangle Ratios

REQUIRED DOCUMENT

RightTriangleRatios.gsp

LESSON GUIDE

This demonstration can replace Example A of Lesson 12.1. It was adapted from a Sketchpad LessonLink activity.

INVESTIGATE

1. In triangle ABC, segment BC is the side opposite angle A, segment AC is the side adjacent to angle A, and segment AB is the hypotenuse.

2. For m∠A � 30°, sin 30° � 0.5, cos 30° � 0.866, and tan 30° � 0.577.

3. Answers will vary. The important thing is that students make a prediction.

4. The angles for the triangle remain constant as the size of the triangle changes. The resized triangle is similar to the original, so there is a scale factor by which each side has been multiplied to generate the new triangle. Therefore, each ratio must remain constant, because its numerator and denominator have been multiplied by the same factor.

5. As m∠A increases, the sine and tangent both increase and the cosine decreases.

6. The complement of ∠A will be ∠B and the sides will switch roles, or sin(90 � A) � cos A, cos(90 � A) � sin A, and tan(90 � A) � 1

____ tan A .

7. Students should choose either the adjacent measurement and cos A or the opposite measurement and sin A.

8. The three ways are hypotenuse � adjacent

______ cos A ,

hypotenuse � opposite

_____ sin A , and hypotenuse �

� __________________

opposite 2 � adjacent 2 .

LESSON 12.2 • Law of Sines

REQUIRED DOCUMENT

LawofSines.gsp

LESSON GUIDE

This demonstration can replace the investigation in Lesson 12.2. It was adapted from a Sketchpad LessonLink activity.

INVESTIGATE

1. When m∠A is larger than m∠B, side a is always longer than side b. Students should record three different sets of all four values (m∠A, m∠B, a, and b) that they tried.

2. When side c is longer than b, m∠C is larger than m∠B.

3. The ratios remain equal to each other despite changes in the triangle. In equation form, this is the Law of Sines: a

____ sin A � b ____ sin B � c

____ sin C .

4. The reciprocals of equal nonzero values must also be equal. The second form of the Law of Sines is

sin A ____

a � sin B

____ b � sin C

____ c .

5. By construction, the pink and the blue triangles are always right triangles and sin B � h _

c and sin C � h _

b .

6. Solving for h, the two equations are h � c � sin B and h � b � sin C. Setting the right-hand sides equal to each other gives c � sin B � b � sin C.

7. Dividing this result by bc gives sin B ____

b � sin C

____ c . Alterna-

tively, dividing the result by sin B � sin C gives b

____ sin B � c ____ sin C . Both results are correct statements of

the Law of Sines. Once students have written the Law of Sines as the equality of two ratios, ask them to justify adding the third ratio (involving a and sin A) to the equation.

8. Once you drag A so h is outside the triangle, one of the two right triangles involves an exterior angle of the triangle. For instance, if you drag A past B, the right triangle DBA no longer contains ∠CBA but now contains the exterior angle at B, which is supplementary to ∠CBA. Because sin B � sin(180° � B), the Law of Sines is still correct.

LESSON 12.5 • Vector Sums

REQUIRED DOCUMENT

VectorSums.gsp

LESSON GUIDE




INVESTIGATE

1. In rectangular coordinates, p � ��240, 0.

2. In rectangular coordinates, w � �0, 50.

3. In rectangular coordinates, p � w � ��240, 50.

4. The magnitude of p � w is 245.15 and its direction is 168.23°.

5. Instead of starting with the vector p, start with vector w. Then line up the tail of vector p with the tip of vector w. Then the resultant vector w � p will start at the tail of w and end at p’s tip.

LESSON 12.6 • Parametric Equations

REQUIRED DOCUMENT

ParametricEquations.gsp

LESSON GUIDE

This demonstration can replace Exercise 6 in Lesson 12.6.

INVESTIGATE

1. The equations are x (t) � 1.4t and y (t) � 3.1 for 0 � t � 5.

2. The position graph of the second walker is a vertical line segment starting at (4.7, 0) and ending at (4.7, 6).

3. Answers will vary. Students should see that the two paths intersect, but they might not know whether they collide or not.

4. The point of intersection is (4.7, 3.1).

5. No, the first walker gets to the intersection at approximately 3.36 seconds whereas the second walker was at the intersection at approximately 2.58 seconds.

6. The first walker starts 3.1 meters up (north) of the origin. For every second that goes by, this walker goes 1.4 meters farther to the right (east) and stays at the same vertical distance (3.1 m) from the horizontal axis. The second walker starts at 4.7 meters right (east) of the origin. For every second that goes by, this walker goes 1.2 meters higher (north) and stays at the same horizontal distance (4.7 m) from the vertical axis.

7. The path is a line segment from (5, 1) to (1, 4) and the equation can be written y � 1 � 0.75(x � 5), or y � 4.75 � 0.75x.

CHAPTER 13

EXPLORATION • Tracing Parent Graphs

REQUIRED DOCUMENT

ParentGraphs.gsp

LESSON GUIDE

This demonstration can replace the Exploration Circular Functions.

INVESTIGATE

1. The radius of circle A is 1 and the measure of ∠BAC is m�BC .

2. The maximum and minimum values of m�BC are 0 and 2� and occur at 0 and 2�, respectively. The maximum and minimum values of the y-coordinate are 1 and �1 and occur at 0.5� and 1.5�, respectively. The maximum and minimum values of the x-coordinate are 1 and �1 and occur at 0 and �, respectively. The maximum and minimum values of the slope appear in the sketch to be at 0.5� and 1.5�; however, at exactly those values the tangent is undefined.

3. This is the sine function and the domain is [0, 2�].

4. Answers will vary. What is important is that the students make a prediction.

5. The cosine function is traced in Step 6 and the tangent function is traced in Step 8. The domain of the function plot in Step 6 is [0, 2�] and in Step 8 the domain is [0, 2�]\{0.5�, 1.5�} or [0, 0.5�) � (0.5�, 1.5�) � (1.5�, 2�].

6. There is one cycle shown for both the sine and the cosine functions and two cycles shown for the tangent function.

7. The slope measures 0 at 0, �, and 2� and it is undefined at 0.5� and 1.5�.

LESSON 13.3 • Pendulum

REQUIRED DOCUMENT

None

LESSON GUIDE

This demonstration can replace the investigation in Lesson 13.3. Each group will need a motion sensor.




INVESTIGATE

1. Sample answer: Distance � 0.80 � 0.10 �

sin � Time � 1.15 ________ 0.24 � . Students can adjust the scales of

the sliders by dragging the axes or by double-clicking the sliders and entering new ranges. For the sample data, 0.80 is the average distance from the motion sensor; 0.10 is the distance from the average value to the minimum or maximum value of the swing; 1.15 is the number of seconds before the first maximum value occurs; 2�

___ 0.24 , or 1.375, is the number of seconds it takes to complete one cycle; x is measured in radians.

2. Sample answer: Distance � 0.80 � 0.10 �

cos � Time � 1.57 ________ 0.24 � . The only value that changes

from the sine function is h, because the cosine function can be expressed as a sine function with a horizontal translation of � __ 2 .

LESSON 13.5 • Day Times

REQUIRED DOCUMENT

DayTimes.ftm

LESSON GUIDE

This demonstration can motivate the project in Lesson 13.5.

INVESTIGATE

1. There are many possible answers. One is a � 60, b � 2, k � 12, and h � 7 (for cosine) or h � 75 (for sine). Another is a � 60, b � �2, k � 12, and h � 355 (for cosine) or h � 265 (for sine).

2. Changing k shifts the graph vertically, changing b alters the amplitude, changing a alters the period, and changing h affects the phase shift.

LESSON 13.7 • Sound Wave

REQUIRED DOCUMENT

SoundWave.ftm

LESSON GUIDE

This demonstration can replace the investigation in Lesson 13.7. The demonstration is most effective in a quiet environment, as outside noise can distort the data collected. Each group will need a microphone.

As an extension, you might have students select a musical instrument, perhaps a flute, violin, piano, timpani, or their voice, and repeat the experiment. The wave will be too complex to write an equation, but students should be able to identify the fundamental frequency, as well as frequencies of some of the over-tones produced by the instrument.

INVESTIGATE

1. A1 is the amplitude and represents the loudness of the sound; B1 is the period of the function and represents the frequency of the sound; C1 is the phase shift and represents at what point in the sound wave data collection began; and D1 is the vertical translation and represents the sound pressure found in the room before striking the tuning fork.

2. For the sample data, one possible equation is

SoundPressureTF1 � 0.0158 sin [ 2� _____ 0.00229 (Time �

0.343) ] � 0.006. Students should disregard the #Units incompatible# error message in all the graphs.

3. Students will probably use a variety of approaches. Some may simply adjust the sliders randomly until they get a fit, though this can be difficult. Others may apply approaches from the student book. They may average the maximums and average the minimums and use these mean values to find the amplitude and the vertical translation. They can either eyeball these mean values or choose Graph⎮Add Movable Line twice and adjust the lines to pass through the maximums and minimums of the data. They may find the period by averaging the time intervals between minimums, using either the movable lines as reference, or by selecting all the minimum points in the graph, copying them into a new collection, and calculating their differences using the prev( ) function. The periods found this way are quite close to the expected periods, which are the reciprocals of the frequencies. Students may find the phase shift by approximating the first minimums of the function. Again, a movable line is helpful here. This is the most difficult value to get using just the sliders.

4. A1, C1, and D1



5. For the sample data, one possible equation is SoundPressureTF2 �

0.0244 sin [ 2� _____ 0.0305 (Time � 0.0151) ] � 0.007.

6. The graph of the two sounds together will be affected by the loudness (amplitude) and the starting point of the sound wave (phase shift). If each tuning fork varies from the original strike in these ways, an adjustment will need to be made in order to account for these differences in sound. The vertical shift is the sound pressure in the room, and should be the same for all three equations.

7. For the sample data, the equation is approximately SoundPressureTF1_2 �

0.0047 sin [ 2� _____ 0.00229 (Time � 0.0350) ] � 0.003 �

0.00240 sin [ 2� _____ 0.00305 (Time � 0.0148) ] � 0.005.

EXPLORATION • Rose Curves

REQUIRED DOCUMENT

RoseCurves.gsp

LESSON GUIDE

This demonstration covers the activity and Question 1 of the Exploration Polar Coordinates.

INVESTIGATE

1. It changes the diameter of the circle.

2. a is the diameter.

3. �10

4. Integer values

5. n � 5, n � 6, n � 17

6. For an odd number of petals, the number of petals equals n. For an even number of petals, the number of petals equals 2n.

7. a is the length of a petal or the radius of the curve.

8. The graphs are the same, except that the sine curve is rotated by a value depending on n. The values of n and a affect both graphs in the same way. The cosine graph originates from the point (0°, 1) on the horizontal axis. The sine graph originates from the origin.

9. For a � n � 1, the polar graph has diameter, or length, 1 and one petal. The rectangular graph has amplitude 1 and includes one cycle of the cosine curve. For a � n � 2, the polar graph has four petals with length 2. The rectangular graph has amplitude 2 and includes two cycles of the cosine curve.

10. a equals the amplitude of the graph.

11. n equals the number of cycles.

12. The value of a gives you the length of the petals. If n is an integer, it gives you the number of petals (n petals if n is odd and 2n petals if n is even).

13. They have a different starting and ending point.

14. They are rotated 90° from each other.

15. It changes the size of the cardioid.

16. It changes the size of the spiral.


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Please take a moment to provide us with feedback about this book. We are eager to read any comments or suggestions you may have. Once you’ve filled out this form, simply fold it along the dotted lines and drop it in the mail. We’ll pay the postage. Thank you!

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Please list any comments you have about this book.

Do you have any suggestions for improving the student or teacher material?

To request a catalog, or place an order, call us toll free at 800-995-MATH, or send a fax to 800-541-2242. For more information, visit Key’s website at www.keypress.com.

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KEY CURRICULUM PRESS1150 65TH STREETEMERYVILLE CA 94608-9740ATTN: EDITORIAL


for use with fathom ™ and the geometer’s sketchpad® · or the geometer’s sketchpad before,...

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