MATHEMATICS - Kyhn's CAC Pageskyhncac.weebly.com/uploads/1/2/7/4/12743866/module_2_rate_of... · MODULE 2 MATHEMATICS ... Additional Graphs and Materials ... From graphical or tabular

2MODULE

MATHEMATICSRate of Change: Average & Instantaneous

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2014 EDITION

i

Mathematics

MATHEMATICS

Module 2

Rate of Change: Average & Instantaneous

Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.i i

Mathematics

Copyright © 2014 National Math + Science Initiative. All rights reserved.

No part of this publication may be reproduced in any form or by any electronic or mechanical means, including information storage and retrieval systems, without permission in writing from the publisher. Student activity pages may be photocopied for classroom use only.

Printed and bound in the United States of America.

ISBN 978-1-935167-13-6

Grateful acknowledgment is given authors, publishers, and agents for permission to reprint copyrighted material. Every effort has been made to determine copyright owners. In case of any omission, the publisher will be pleased to make suitable acknowledgments in future editions.

Published by:

National Math + Science Initiative 8350 North Central Expressway Suite M-2200 Dallas, TX 75206

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Mathematics


CONTENTS

Belief Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Middle Grades Learner Outcomes . . . . . . . . . . . . . . . . ii

High School Learner Outcomes . . . . . . . . . . . . . . . . . iii

Lessons and Assessments . . . . . . . . . . . . . . . . . . . . . . . 1

Rate of Change Content Progression Chart . . . . . . . 3

Rate of Change Concept Development Chart . . . . . . 4

Finding Pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Road Trip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

Average Rate of Change (a .k .a . Slope) . . . . . . . . . . 27

Calculating Average Rates of Change . . . . . . . . . . . 37

Average Rate of Change vs . Instantaneous Rate of Change . . . . . . . . . . . . . . . . . 47

Walking Piecewise Graphs . . . . . . . . . . . . . . . . . . . 57

Applying Piecewise Functions . . . . . . . . . . . . . . . . 69

Investigating Average Rate of Change . . . . . . . . . . 77

Slopes of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

Sample Quiz Questions . . . . . . . . . . . . . . . . . . . . . 105

Free Response Questions . . . . . . . . . . . . . . . . . . . 117

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A1

Standards for Mathematical Practice . . . . . . . . . . .A3

Additional Graphs and Materials . . . . . . . . . . . . .A11

Graphical Organizer . . . . . . . . . . . . . . . . . . . . . . .A17

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Mathematics

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Mathematics

National Math + ScienceBelief Statements

Accomplished, dynamic teachers are knowledgeable in their content and confident in their abilities to prepare students for higher education. They create classrooms in which students:

• engage intellectually to develop conceptual understanding

• generate their own ideas, questions, and propositions

• interact collegially with one another to solve problems

• employ appropriate resources for inquiry-based learning

Our teacher training program offers meaningful support to teachers as they construct these effective classrooms. Through tested content materials and research-based instructional strategies, our program enables and encourages them to:

• choose significant and worthwhile content and connect it to other knowledge

• use appropriate questioning strategies to develop conceptual understanding

• clarify to students the importance of abstract concepts and “big questions”

• use formative assessments to improve instruction and achieve higher goals

• guarantee equitable access for all students to information and achievement

i i Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics Mathematics

Rate of Change: Average & InstantaneousMiddle Grades

MODULE DESCRIPTION

Middle school teachers examine how concepts involving rate of change progress from sixth grade to calculus. Training begins with a manipulative-rich lesson that leads students to interpret pi as a constant rate of change. As the lessons progress through the vertical strand, participants connect slope and rate of change. They work selected questions from and discuss teaching strategies for model lessons for middle grades in which students interpret speed as a rate of change, calculate average rates of change over specific intervals, and examine graphs to conclude that average rate of change is not the same as instantaneous rate of change. In addition, teachers work through and discuss lessons from Algebra 1 and Geometry or Math 1 and Math 2 in which students apply a difference quotient to calculate average rate of change and are introduced to approximating an instantaneous rate of change from a graph.

LEARNER OUTCOMES

Participants will:• Compare expectations for students from sixth grade math through pre-calculus on the topic of rate of

change to increase vertical alignment.

• Apply deeper content-based knowledge to increase instructional rigor in order to prepare students for high school math courses leading to college-level calculus in an AP* class or university setting.

○ Identify a constant rate of change from a table or graph.

○ Complete a table or sketch a graph using a constant rate of change.

○ Model rate of change using exploratory activities, role play, and motion detectors.

○ Differentiate between average and instantaneous rates of change of a function.

○ Calculate average rate of change using a difference quotient.

• Identify instructional strategies that teachers can use to assist students in developing the habits of mind that are required for college and career readiness.

*Advanced Placement® and AP® are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

Middle Grades Learner Outcomes

i i iCopyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics Mathematics

Rate of Change: Average & InstantaneousHigh School

MODULE DESCRIPTION

High school teachers examine how concepts involving rate of change progress from sixth grade to calculus. Participants begin by reviewing how these concepts are introduced at the middle school level. As the lessons progress through the vertical strand, participants differentiate between average and instantaneous rates of change. They work selected questions from and discuss teaching strategies for high school model lessons in which students apply a difference quotient to calculate average rates of change over specific intervals, approximate an instantaneous rate of change, and calculate slopes of curves that are not functions. In addition, participants model piecewise distance graphs with a motion detector and graph the corresponding speed graphs.

LEARNER OUTCOMES

Participants will:• Compare expectations for students from sixth grade math through pre-calculus on the topic of rate of

change to increase vertical alignment.

• Apply deeper content-based knowledge to increase instructional rigor in order to prepare students for college-level calculus in an AP* class or university setting.

○ Differentiate between average and instantaneous rates of change of a function.

○ Calculate average rate of change using a difference quotient.

○ Model rate of change using exploratory activities, role play, and motion detectors.

○ Estimate instantaneous rate of change using the slopes of secant lines to approach the slope of a tangent line.

○ Apply a formula for instantaneous rate of change at a point on a curve to identify characteristics of the curve.

• Identify instructional strategies that teachers can use to assist students in developing the habits of mind that are required for college and career readiness.

*Advanced Placement® and AP® are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

High School Learner Outcomes

iv Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics


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Mathematics


LESSONS AND ASSESSMENTS

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Mathematics


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Rat

e o

f C

hang

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ont

ent

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n C

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6th

Gra

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7th

Gra

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ebra

1

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met

ry

Alg

ebra

2

Pre-

Cal

culu

s Fr

om g

raph

ical

or t

abul

ar

data

or f

rom

a st

ated

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tuat

ion

pres

ente

d in

pa

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aph

form

, cal

cula

te

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e av

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.

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terv

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ith th

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aver

age

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terv

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ith th

e sa

me

aver

age

rate

of c

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ompa

re a

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tes

of c

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diff

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t in

terv

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n a

tabl

e or

gr

aph.

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on d

iffer

ent

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s in

a ta

ble

or

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h.

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iffer

ent

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a ta

ble

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ate

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s rat

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.

Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.4

Rate o

f Chang

e Co

ncept D

evelop

ment C

hartG

rade 6G

rade 7A

lgebra 1R

aymond is hiking on a trail in R

ocky Mountain N

ational Park in C

olorado. The trail is very dangerous in several places, so he m

ust walk slow

er in those places. The graph indicates his distance, in m

iles, from the starting point

between 0 and 4 hours.

01

23

45

0 1 2 3 4 5 6 7 8 9 10

Time

inhours�t�

Distance in miles �d�

Raym

ond'sHike

What is R

aymond’s average speed betw

een 2 and 4 hours?

Kevin runs cross country races for his high school. C

ross country events are held over open and rough terrain. C

ontestants vary their speed throughout the race, depending on the course. D

uring a recent race, Kevin’s coach recorded

his distance, in meters, and tim

e, in minutes, at three check

points during the race.

24

68

1012

0

1000

2000

3000

Time

inm

inutes�t�

Distancein meters �d�

Kevin'sC

rossCountry

Race

td

00

3.51000

6.12000

11.33000

What is K

evin’s average speed in meters per m

inute between

3.5 and 11.3 minutes? Is his average speed betw

een the three checkpoints constant over the tim

e period from 0 to 11.3

minutes?

Lynn is out for a bike ride around her neighborhood. Her

speed varies throughout the ride. The graph indicates her speed for the first 7 m

inutes of the ride. The vertical scale is intentionally om

itted.

01

23

45

67

Time

inm

inutes�t�

Speed in miles per hour �s�

BikeRide

Arrange the tim

e intervals ,

and so that her respective average rates of change in

speed with respect to tim

e are in decreasing order.

Geom

etryA

lgebra 2Pre-C

alculusThe slope of a function at a particular point is defined as the slope of the line tangent to the function at that point. Lines tangent to the function have been draw

n at the points F, G,

H, and J.

F

H J

G

�3�2�11

23

45

67

89

10x

�3 �2 �1 1 2 3 4 5 6 7 8 9 10 y

Order the values of the slopes of the tangent lines for F, G

, H,

and J from sm

allest to largest

Given the function

what is the

average rate of change over each the following intervals?

a)

b)

c)

By reducing the interval size around the point (2, 0), estim

ate the slope of the tangent line at that point.

For the function f(x)=2x

3−7x

2+7x+

1, the slope function for any point on the curve is given by

Using a graphing calculator, determ

ine the points w

here the slope is zero. Using the x-values w

here the slope is zero, determ

ine the corresponding values for f (x) . W

hat do these y-values represent on the function, f (x) ? Sketch a graph show

ing f (x), m, and your results.

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

LEVELGrade 6 or Grade 7 in a unit involving measurement and circumference

MODULE/CONNECTION TO AP*Rate of Change: Average and Instantaneous

*Advanced Placement and AP are registered trademarks of the College Entrance Examination Board. The College Board was not involved in the production of this product.

MODALITYNMSI emphasizes using multiple representations to connect various approaches to a situation in order to increase student understanding. The lesson provides multiple strategies and models for using those representations indicated by the darkened points of the star to introduce, explore, and reinforce mathematical concepts and to enhance conceptual understanding.

P

G

N A

V

P – Physical V – VerbalA – AnalyticalN – NumericalG – Graphical

Finding PiABOUT THIS LESSON

In this lesson, students engage in a hands-on activity which leads them to discover that, regardless of the size of a circle, the

circumference of the circle divided by its diameter is equal to the value of pi . Students use ribbon to measure the circumference and diameter of circular objects. Precise measurement and reporting is required during this activity. After measuring diameters and circumferences of several circular objects, students record the diameter, circumference,

and Cd

in a table and then construct a graph using

their measurements. Through analyzing the data in both the graphical and numerical forms, students observe the rate of change relationship when circumference is divided by diameter.

OBJECTIVESStudents will

● precisely measure circular objects.● create scatterplots.● determine the relationship of pi to the

circumference and diameter of a circle.

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Mathematics—Finding Pi

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introductiontothespecificskill.

Targeted Standards7.RP.2b: Recognize and represent proportional

relationships between quantities. (b) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. See questions 11, 19, 21

Reinforced/Applied Standards7.RP.2a: Recognize and represent proportional

relationships between quantities. (a) Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. See questions 19-21

7.RP.2d: Recognize and represent proportional relationships between quantities. (d) Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. See questions 10, 20

7.G.4: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. See questions 16-19

6.SP.3: Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. See questions 11-12

COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practicesbasedonprocessesandproficienciesthat are critical for mathematics instruction. NMSI incorporates these important processes andproficienciestohelpstudentsdevelopknowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.5: Use appropriate tools strategically. Students use ribbon to measure the circumferences of various circular objects and to create a graph of circumference versus diameter.

MP.6: Attend to precision. Students determine the value of and consider as a precise unit.

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FOUNDATIONAL SKILLSThe following skills lay the foundation for concepts included in this lesson:

● Measure a diameter● Plotpointsinthefirstquadrant● Write and simplify ratios

ASSESSMENTSThe following types of formative assessments are embedded in this lesson:

● Students summarize a process or procedure.

The following additional assessments are located on our website:

● Rate of Change: Average and Instantaneous – 6th Grade Free Response Questions

● Rate of Change: Average and Instantaneous – 6th Grade Multiple Choice Questions



MATERIALS AND RESOURCES● Student Activity pages● Circular objects such as cans and lids● Four different colors of ribbon for each group● Rulers● Scissors● Butcher paper● Two copies of the tape measure page for each

group to be used prior to the day of the lesson (Warning: Since copiers may change the dimensions of the tape measures on the page, measure a test copy. If the size is incorrect, adjust the scaling on the copier to correct the size.)

● Cellophane tape● Glue● Scientificorgraphingcalculators

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TEACHING SUGGESTIONS

Several days before teaching this lesson, ask each student to bring one circular object from home to be used in the lesson. On the day

before the lesson, give each group two copies of the tape measure page from the end of the lesson. Have the students cut the page along the dotted lines to create 5 strips and then tape the strips together so that the repeated numbers line up with one copy on top of the other copy. These tapes will be used as the vertical and horizontal scales for the graph that the groups construct on butcher paper.

The length of butcher paper needs to be at least 15 centimeters more than the circumference of the largest circular object, and the width of the paper needs to be at least 15 centimeters more than the largest diameter. On the day of the lesson, students should draw a y-axis about 10 centimeters from the left side of the paper and an x-axis about 10 centimeters above the bottom of the paper. The paper tape measures are to be taped onto the graph as the vertical and the horizontal scales for the axes.

On the day the lesson is to be completed, provide each group with a pair of scissors and four pieces of ribbon that are long enough to wrap around the circumference of each object. Provide a ruler or a tape measure marked in centimeters for measuring the diameters and the length of the ribbons that have been wrapped around the circumference of the circular objects.

Demonstrate the procedures for measuring the diameter, for wrapping the ribbon around the object to measure the circumference, for cutting and measuring the ribbon, and for placing the ribbon on the butcher paper graph. Once the students understand the directions, have them work in groups to complete the activity.

A discussion to summarize the learning that takes place in the activity is necessary prior to having students write individual summaries. Students should understandthatthefirstcoordinateofeachpoint

represents the diameter of a circular object and that the second coordinate represents the circumference of the same object. Discuss that the ratio of the change in the circumference to the change in the diameter of their circular objects should be the value of pi. Because of the limitations of measurements, students’ ratios may not always be exactly the same, so calculating the exact decimal value of pi using this method is not possible. Students will enjoy looking at the value of pi as given on a calculator, so be prepared to explain that even a calculator does not show all of the digits of pi. Discuss the fact that pi is a non-terminating, non-repeating decimal – an irrational number. The purpose of this lesson, and the most important concept that students need to understand from this activity, is that the slope of the line is the change in the circumference divided by the change in the diameter. Making the connection between slope and rate of change will benefitstudentsastheyprogressfrommiddleschoolmathematics to algebra.


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9


NMSI CONTENT PROGRESSION CHARTIn the spirit of NMSI’s goal to connect mathematics across grade levels, a Content Progression Chart for eachmoduledemonstrateshowspecificskillsbuildanddevelopfromsixthgradethroughpre-calculusinanaccelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.

The complete Content Progression Chart for this module is provided on our website and at the beginning of the training manual. This portion of the chart illustrates how the skills included in this particular lesson develop as students advance through this accelerated course sequence.

6th Grade Skills/Objectives


Algebra 1 Skills/Objectives

Geometry Skills/Objectives


Pre-Calculus Skills/Objectives

From graphical or tabular data or from a stated situation presented in paragraph form, calculate or compare the average rates of change and interpret the meaning.






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Mathematics NATIONALMATH + SCIENCEINITIATIVE

Finding Pi

Directions for creating the graph for this activity are provided in questions 1 – 9.

1. Using two copies of the centimeter ruler pages provided by your teacher, create two tape measures. Cut each page along the dotted lines and tape the strips together so that the 20 cm mark of the first strip lies exactly under the 20 cm mark of the second strip. Repeat the process of lining up the repeated units until a tape measure 100 cm long is created. The two strips will be used as the vertical and horizontal scales on your butcher paper graph.

2. On the butcher paper construct an x-axis and y-axis. Draw a vertical line for the y-axis 10 centimeters from the left side of the paper and a horizontal line for the x-axis 10 centimeters above the bottom of the paper. Line up one vertical tape just to the left of the vertical axis making sure that 0 on the tape is level with the bottom of the y-axis and then tape it in place. Put the second tape just below the horizontal axis. Line its zero up with the beginning of the x-axis and tape it in place. Ask your teacher to check your graph.

3. Compare the diameter of your circular objects by holding them against one another. Place your circular objects in order from the smallest diameter to the largest diameter. List the names of the objects in this order in the table above question 11.

4. Measure the diameter of the circular objects to the nearest millimeter and record the length of the diameter of each object in the table.

5. Wrap the ribbon around one of the circular objects and then cut the ribbon to the length of the circumference (the distance around the object).

6. Measure the length of the ribbon to the nearest millimeter. Now record this value in the circumference column of the table. Be sure to record it next to the diameter for the same object.

7. For the first object in the chart, locate the x-value on the graph that corresponds to its diameter. Glue the ribbon strip that you used to measure the circumference of that object vertically above the x-value of the diameter. Be sure that it is perpendicular to the x-axis and parallel to the y-axis.

8. Using the scale glued to the graph for the y-axis, check the length of each ribbon to see if it is close to the measurement recorded in the table for the circumference.

9. Repeat the process in steps 7 and 8 with the next object until you have graphed the diameter and circumference of all 4 of your circular objects.

10. Choose a point on the graph. Explain what this specific point means in terms of the circular object it represents. Write this answer on your poster.



Object Diameter Circumference Cd

11. Complete the table by calculating the ratio of the circumference to the diameter, Cd

, of all four of your

objects. Round your answers to the nearest hundredth. Average the four values in the Cd

column and record this average in the space below.

12. What is the name for the measure of central tendency that you just calculated in question 11?

13. The value of is a constant because its value is always the same. Locate the button on your calculator and press enter. Record all the numbers that are displayed on your calculator.

14. What is the number, rounded to two decimal places, that is often used as the value of ?

15. Is there a difference in the value you determined in question 11 and the rounded value we often use for ? If there is, why do you think this difference exists?

16. Write an equation that can be used to determine the value of the constant, , if you know the circumference and the diameter of a circular object.

17. Write an equation that can be used to determine the circumference of a circle if you know the value of the constant, , and the diameter of a circular object.

18. Solve the equation in question 17 for d. In other words, write an equation that can be used to determine the diameter of a circular object if you know the circumference of the object and the value of .



19. Complete the chart with exact answers (answers in terms of ) for the circumference. Plot the coordinates of your five points. Make sure that the diameter is the first coordinate (x-axis) and the circumference is the second coordinate (y-axis). The coordinates of each point will be (diameter, circumference).

20. Connect the points on the graph. Do your points connect in a straight line? If you were to extend your line, would the line pass through the origin (0,0) on your graph? What would the values at the origin represent?

21. For each increase of one unit in your diameter, what happens to the value of the circumference?

22. Write a paragraph describing the activity and your group’s conclusions.

Diameter Circumference12345






Mathematics—

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LEVELGrade 6 in a unit on ratio and rate




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Road TripABOUT THIS LESSON

This lesson presents students with verbal information outlining specific rules for a driving trip. Students use the information

to complete a data table and a graph of distance traveled as a function of time. The lesson reinforces the idea that average rate of change is not necessarily the same as instantaneous rate of change. In addition, students have the opportunity to use dimensional analysis to calculate fuel usage in speed/time scenarios.


● complete a table based on a given scenario.● sketch a graph to represent a situation

involving distance versus time.● determine total distance traveled.● understand that the average rate of change

for an interval does not necessarily indicate constant speed.

● apply dimensional analysis using miles per hour and miles per gallon.

● interpret distance versus time graphs as they relate to the speed of a car.

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Mathematics—Road Trip

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introductiontothespecificskill.

Targeted Standards6.RP.3a: Use ratio and rate reasoning to solve real-

world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (a) Make tables of equivalent ratios relating quantities with whole-number measurements,findmissingvaluesinthetables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios. See questions 2-5, 9-11

Reinforced/Applied Standards6.RP.3d: Use ratio and rate reasoning to solve real-

world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (d) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. See questions 1, 8-9

5.G.2: Represent real world and mathematical problemsbygraphingpointsinthefirstquadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. See question 5


MP.1: Make sense of problems and persevere in solving them. Students must apply all given constraints outlined in the situation and must understand the difference between clock time, coded time, and travel time, in order to complete the table.

MP.6: Attend to precision. Students use dimensional analysis accurately and appropriately.

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● Plot ordered pairs on the coordinate plane● Read and interpret graphs

ASSESSMENTSThe following assessments are located on our website:



MATERIALS AND RESOURCES● Student Activity pages

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To introduce the idea that average rate of change and constant rate of change are not necessarily the same, remind students that a

person can drive a car at an average of 70 miles per hour, but that they will not necessarily maintain a constant speed of 70. Note also that, in this activity, the distance from home is the distance measured along the road rather than the direct line distance from home.

Acting out the scenario before beginning the actual lesson will help the students understand the situation. Ask one student to be the mother who watches the mileage and the time. Ask another student to be the father and ask a third student to be Paul. Instruct the “actors” to walk across the room as the mother tells the driver when to drive and when to take a break. As the scenario takes place, ask the class questions that will lead them to complete the chart and the graph. Students can complete parts of the table as “mother, father, and Paul” model the trip.

In question 6, note that none of the graphs are realistic representations of the position of the car at a given time. Even though the average speed over the two hour time period is 70 mph for each graph, the average speed when calculated over shorter periods of time can exceed 140 mph.

Question 7 provides an opportunity for students to create a more realistic graph of the trip by drawing curved segments on the graph. Asking students to share their sketch provides an opportunity to reinforce the concepts presented in question 6 and to discuss “incorrect” segments where the family travels “back in time.”

Suggestedmodificationsforadditionalscaffoldinginclude the following:2 Discuss the titles for each of the columns in

the table before the student begins to enter the information. Use a number line to represent clock time to help the student with the coded time and to understand the difference between the second and last column. Show the “process” for completing the third column based on the number of hours spent “driving.”

5 Add “clock time” to the table to help the student connect the amount of time after the start the trip with the table in question 2.

10 Provide a table similar to question 2.11 Provide a table similar to question 2.


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Road Trip

Paul Frazier is taking a holiday trip to his grandmother’s house. According to the route that is planned, Paul’s grandmother’s house is 1050 miles from his home.

Paul’s mother has set the following rules for the trip.● She refuses to sit in the car for more than a total of 10 hours a day.● She insists that they take at least a 30 minute break after riding for 2 hours.● She requires that they stop for 1 hour for lunch and for dinner.

1. Paul’s dad drives on the highway at an average rate of 70 miles per hour. If the entire trip is driven on a highway, how many driving hours will be required for the trip?

2. Onthefirstday:● Thetripbeginsat7:00a.m.● Theytakea30minutebreakat9:00a.m.● Theystopat11:30a.m.foraonehourlunch.● After driving for 2 hours, they take a 30 minute afternoon break.● Theystopfrom5:00p.m.to6:00p.m.fordinner.

Complete the table to show when the Frazier family is driving in the car and when they are stopped. The firstthreeentriesarecompleted.

3. At what time do the Fraziers stop for the evening? What is the number of total hours between the time the Fraziers leave home and the time they stop for the evening?

Time Interval(clock time)

Time Interval(Coded time where 0 represents 7:00 am)

Total Miles Driven from the Start of the

Trip to the End

Driving or Stopped

Total Hours in the Car from the Start of the Trip

7:00–9:00 0 hours to 2 hours 140 miles Driving 2 hours9:00–9:30 2 hours to 2.5 hours 140 miles Stopped 2 hours9:30–11:30 2.5 hours to 4.5 hours 280 miles Driving 4 hours



4. HowmanyhoursdidtheFraziersspenddrivinginthecarthefirstday?Howmanyhourswilltheyspend driving in the car the second day? Explain your reasoning.

5. Paul decides to graph their total distance traveled at a given amount of time after they start the trip. He labels the horizontal axis as time and the vertical axis as distance. To make the graph easier to draw, he startswithacodedtimeof0for7:00a.m.Completethetablewiththepreviousinformationwheret is the coded time and d is total distance traveled, and draw Paul’s graph for day one of the trip.

a. What does the ordered pair (2,140) mean in the context of the situation?

b. Describe what was happening between 7.5 hours and 8 hours.

c. During which time interval did the family stop for dinner? Explain your answer.

Time (Hours)

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t d0 02 140

2.5



6. After Paul draws his graph, he realizes that his father does not always drive at exactly 70 miles per hour. Startingandstoppingaswellashavingtospeedupandslowdownfortrafficcausesthecar’sspeedtovary. Each of the graphs correctly represents driving with an average speed of 70 miles per hour for a two hour period.

a. b. c. d.

Explain why each graph represents an average speed of 70 miles per hour.

7. Draw two curved segments on the graph in question 5, one on the interval from 0 to 2 hours and another on the time interval from 8 to 10, to illustrate that the Fraziers drove for 2 hours with an average speed of 70 miles an hour but that they did not drive at a constant speed.

8. The Fraziers intended to drive the remaining 350 miles on the second day of their trip; however, they discovered a route that will be 50 miles shorter than the one they originally planned. The roads are not as good, so they will not be able to average 70 miles per hour. At what average speed must they drive to arrive at Paul’s grandmother’s house in the same amount of time the original route would have taken?



9. Thecaraverages25milespergallonat70milesperhourand30milespergallonat60milesperhour.HowmanygallonsofgasolinewilltheFrazierfamilysavebydrivingthe300milesat60milesperhourinstead of the 350 miles at 70 miles per hour?

10. Using your answer from question 8 and following Mrs. Frazier’s rules for the trip on the new route, at whattimewilltheyarriveattheirgrandmother’shouseiftheystarttheseconddayat7:00?

11. If you were Paul’s dad, which route would you take? Explain.

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LEVELGrade 7 in a unit on rates




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Average Rate of Change (a.k.a. Slope)ABOUT THIS LESSON

In this lesson, students calculate average rates of change over specified intervals and compare different rates. Students use distance-time graphs

and population tables to analyze characteristics such as when a person is traveling at the fastest speed and when a population is increasing or decreasing. Students explain the sign of the rate of change in the context of the population change.

While the lesson focuses on determining average rate of change, it also provides students an early introduction to the idea of slope. The lesson can be extended by having students write equations for the situations given in the graphs and table.


● calculate average rates of change.● construct justifications to compare rates.

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Mathematics—Average Rate of Change (a.k.a. Slope)

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill.

Targeted Standards7.RP.2b: Recognize and represent proportional

relationships between quantities. (b) Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. See questions 1-3

COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practices based on processes and proficiencies that are critical for mathematics instruction. NMSI incorporates these important processes and proficiencies to help students develop knowledge and understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.6: Attend to precision. Students must clearly identify intervals used to compute the average rate of change, interpret positive and negative rates of change, and must include appropriate units.

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● Read and interpret a graph● Understand that rate is equal to distance

divided by time,


● Students engage in independent practice.● Students apply knowledge to a new situation.

The following assessments are located on our website:




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The average rate of change on an interval can be defined as the change in y (the dependent variable) divided by the change in x (the

independent variable) over that interval. In questions 1 and 2, the average rate of change involves the change in distance divided by the change in time, which is usually defined as the average velocity over the time interval. Notice that the questions in the lesson refer to “speed” rather than “velocity.” Since the velocity is always positive in this lesson, the word “speed” is a correct term. “Speed” is actually defined as the magnitude of the velocity (or

). In question 3, the average rate of change calculates the change in population divided by the change in time – an average growth rate for each city over specific time periods. Throughout this lesson, remember that each answer is an average rate of change over a specific interval. No conclusions can be drawn about how the distances or populations are changing at any particular time.

Use a motion detector to demonstrate the relationship and differences between the motion of the cyclist and the distance-time graph. If motion detectors are not available, have students model Al’s ride. It is important that students first have the opportunity to describe verbally what the graph indicates and then to model the scenario physically. Instruct students to use specific quantities and units for distance, time, and speed in their descriptions of the walks.

You may wish to support this activity with TI-Nspire™ technology. See Working with Fractions and Decimals in the NMSI TI-Nspire Skill Builders.

Suggested modifications for additional scaffolding include the following:1c, 2 Set up solutions with units in both the work

and the answer based on the number of hours spent “driving.”


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NMSI CONTENT PROGRESSION CHARTIn the spirit of NMSI’s goal to connect mathematics across grade levels, a Content Progression Chart for each module demonstrates how specific skills build and develop from sixth grade through pre-calculus in an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.














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Average Rate of Change (a.k.a Slope)

1. Al is an avid cyclist. On a recent ride in the country, he traveled at a constant speed throughout the trip. Use the graph of Al’s distance traveled to answer the questions.

a. How far did Al travel in 3 hours?

b. Does the graph represent the path on which Al is traveling? Explain.

c. What is Al’s average speed in miles per hour for the time interval ?

d. What is Al’s average speed in miles per hour for the time interval ?

e. Is Al’s average speed increasing, decreasing, or remaining constant?

1 2 3

6121824303642485460

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Distance vs. Time for Al’s Trip


Mathematics—Average Rate of Change (a.k.a Slope)

2. On another recent trip, there was some road construction. Due to the construction, his speed varied. Use the graph to answer the following questions.

a. What is Al’s average speed for the time interval 0≤ t≤5?

b. What is Al’s average speed for the time interval 0≤ t≤1?

c. What is Al’s average speed for the time interval 1≤ t≤ 2?

d. What is Al’s average speed for the time interval 2≤ t≤ 4?

e. What is Al’s average speed for the time interval 4≤ t≤5?

f. According to the graph, when was Al cycling the fastest? Explain your answer in terms of the graph.

g. According to the graph, when was Al cycling the slowest? Explain your answer in terms of the graph.

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Distance vs. Time for Al’s TripWith Construction



3. The table gives the populations for two cities for five different years.

Year 1990 1995 1998 2000 2002Population in City A 42,000 52,000 62,000 72,000 82,000Population in City B 75,000 70,000 65,000 60,000 55,000

Determine the average rate of change of the population for each city for the given time intervals. Use units in your work and answers.

City A City B

a. 1990 to 2000

b. 1995 to 1998

c. 1995 to 2002

d. What do you notice about the average rate of change of each population? Explain what the average rate of change tells you about each population.



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LEVELGrade 8, Algebra 1, or Math 1 in a unit on slope and average rate of change




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Calculating Average Rates of ChangeABOUT THIS LESSON

This lesson uses a variety of real-world situations to extend beyond the standard calculation of slope as it applies to average

rate of change. It provides an opportunity for students to deepen their understanding of average rate of change by connecting the concept to real life situations. Students read and analyze each question to determine the rate described and then use the difference quotient to determine the average rate of change. In addition to determining values for average rate of change, students must attend to correct use of units and interpret the meaning of the rate for specific time intervals.


● interpret data from real-world scenarios as coordinate pairs.

● use the difference quotient to determine the average rate of change (slope).

● write a sentence interpreting the average rate of change in the context of the situation.

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Mathematics—Calculating Average Rates of Change

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introductiontothespecificskill.Thestarsymbol(★) attheendofaspecificstandardindicatesthatthehigh school standard is connected to modeling.

Targeted Standards (if used in Grade 8)8.F.4: Construct a function to model a linear

relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.★ See questions 1-15

Targeted Standards (if used in Algebra 1)F-IF.6: Calculate and interpret the average

rate of change of a function (presented symbolicallyorasatable)overaspecifiedinterval. Estimate the rate of change from a graph.★ See questions 1-15

COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICEThese standards describe a variety of instructional practicesbasedonprocessesandproficienciesthat are critical for mathematics instruction. NMSI incorporates these important processes and proficienciestohelpstudentsdevelopknowledgeand understanding and to assist them in making important connections across grade levels. This lesson allows teachers to address the following Common Core State Standards for Mathematical Practice.

MP.2: Reason abstractly and quantitatively. Students must determine the independent and dependent quantity of each scenario, calculate the average rate of change, and then interpret the meaning of the slope within the context of the problem situation.

MP.6: Attend to precision. Students must clearly identify independent and dependent quantities, interpret positive and negative rates of change, and include appropriate units.

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● Identify independent and dependent quantities● Calculate the slope between two points


● Students engage in independent practice. The following assessments are located on our website:

● Rate of Change: Average and Instantaneous – Algebra 1 Free Response Questions

● Rate of Change: Average and Instantaneous – Algebra 1 Multiple Choice Questions


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As the class works through the example, have students identify the independent and dependent quantities and write the ordered

pairs to represent the situation. Emphasize the term “difference quotient” and the importance of using units within the calculation. This will help students understand the slope as a rate of change. Discuss the difference between positive and negative slopes and what those slopes could mean in the context of a particular scenario.

Emphasize to students that they are calculating average rates of change, which means that the rate of change may be different at various instants during the interval, sometimes higher than the average and sometimes lower than the average.

As a teaching strategy provide students with a template for organizing their work. Divide a sheet of paper into 3 columns, with the titles “Coordinates,” “Difference Quotient/Answer,” and “Sentence.” Have the student follow these steps for each question. Step 1: Record the coordinates in column 1 and verify the coordinates with the teacher. Step 2: Show the difference quotient and simplify the answer in column 2. Step 3: Write a sentence interpreting slope as an average rate of change in column 3.

Students are directed to write a sentence interpreting the slope as an average rate of change. Two components are essential in writing an appropriate interpretation of these situations: (1) the sentence mustrefertothespecificintervalfortheindependentvariable and (2) the calculated rate of change must beidentifiedasanaveragerate.

Question 8 provides an opportunity for some interesting class discussion since the given quantities lend themselves to more than one interpretation. (See the answer key for a list of possibilities.) Consider allowing the class to brainstorm some of these approaches, work in groups on their solutions,

and then present their solutions to the class. Extend the discussion of the variety of interpretations of this question by asking students to look for the relationship between the rates of change when the identificationoftheindependentanddependentvariables is reversed

Questions 14 and 15 are based on tabular data provided to AP Calculus students in free response questions 1999 AB 3 and 2001 AB 2. Question 14 may be particularly challenging to interpret since it involves calculating an average rate of change of another given rate.

You may wish to support this activity with TI-Nspire™ technology. See Working with Fractions and Decimals in the NMSI TI-Nspire Skill Builders.


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Calculating Average Rates of Change

Instructions: ● Read each problem and determine the coordinates of the two points mentioned. ● Use the coordinates to calculate the slope of a line passing through those two points. Show the

difference quotient that leads to your answer. ● Write a sentence interpreting the slope as an average rate of change. Be sure to include units in

your answer.

Example: While typing her English essay, Tammy noticed that her clock read 12:32 and the word count for herpaperwas568.Whenshefinishedthepaper,herclockread12:48andthefinalwordcountwas1128.

(32,568)and(48,1128)

Duringthetimeintervalfrom12:32to12:48,Tammywastypingatanaveragerateof35wordsperminute.

1. ThepopulationofAustin,Texasin1990was472,000people.Thepopulationin1980was 346,000people.

2. At3o’clock,Sharonpassesmilemarker295onHighway35.At6o’clockshepassesmilemarker475.

3. The value of my new car after 2 years was $11,200. When the car is 6 years old, the value has dropped to $6100.

4. A lab technician is growing a bacteria sample. After one hour, she notes that there are 250 bacteria in the sample. After 3 hours, she notes that there are 1000 bacteria in the sample.

5. Mr. Suarez joined a gym to lose weight. After three weeks of membership, he weighed 189 pounds. When he had been a member for twelve weeks, he weighed 162 pounds.



6. Onhisfifthbirthday,Paulwas42inchestall.Onhisseventhbirthday,hewas48inchestall.

7. In1984,thepriceofaVCRwas$375.In1996,thepricewas$125.

8. DixieleftAustinwithanodometerreadingof12,584milesandafulltankofgasoline.WhenshestoppedtobuygasolineinHouston,herodometerreadingwas12,792miles.Shefilledthetankcompletely with 8 gallons of gasoline and paid $31.12.

9. Daraworksintheclothingdepartmentofalargestore.Whenshebeganhershiftat4p.m.,theregistershowedsalesof$10,550.Whensheclockedoutat9p.m.,theregistershowedsalesof$40,620.

10. At one o’clock in the afternoon, the temperature outside registered 85 degrees. At seven o’clock that evening, the temperature was 61 degrees.

11. Whenanamusementparkopened,thecounterontheturnstileattheentranceread1278.Sevenhourslater,thecounterread3672.

12. The concession stand at the amusement park begins the day with 500 popcorn containers. When the parkcloses,twelvehourslater,aninventoryshowsthereareonly44containersleft.



13. Scott began printing his history paper at 3:15. At 3:20, he found that it had printed 120 of his 150 pages.

14. Therateatwhichwaterflowsoutofapipe,ingallonsperhour,isgivenbyacontinuousfunctionR of time t.Thetableshowstherateasmeasuredevery3hoursfora24-hourperiod.Between3and12hours,thewaterisflowingoutofthepipeatafasterandfasterrate.Determinetheaverageincreaseofthis rate for the 9 hour period.

t(hours)

R(t)(gallons per

hour)0 9.6

3 10.46 10.89 11.212 11.415 11.318 10.721 10.224 9.6

15. Thetemperature,indegreesCelsius,ofthewaterinapondisacontinuousfunctionW of time t. The tableshowsthewatertemperatureasrecordedevery3daysovera15-dayperiod.Basedonthe3-dayintervalsshowninthetable,overwhat3-dayintervalisthewatertemperatureincreasingmostrapidlyandhowfastisitrising?Overwhat3-dayintervalisthewatertemperaturefallingmostrapidlyandhowfast is it dropping?

t (days) W(t) °C

0 20

3 31

6 28

9 24

12 22

15 21


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LEVELAlgebra 2 or Math 3 as a review of tangent lines or in a unit on key features of function graphs

Geometry or Math 2 in a unit including secant and tangent lines

Algebra 1 or Math 1 as an extension of a unit on average rate of change




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Average Rate of Change vs. Instantaneous Rate of ChangeABOUT THIS LESSON

This lesson introduces students to the concept of the slope of a non-linear function. Students use their prior

knowledge of distance-time graphs and slopes of lines to investigate slopes of secant and tangent lines. Students draw a short line segment tangent to the function to approximate the function’s instantaneous rate of change. The lesson begins with a scenario involving constant rate, and then students consider a case where the rate is not constant but the average rate remains the same. Through a series of definitions and questions, students draw conclusions about the implications of positive and negative rates of change, as well as about the relationship between position and velocity functions.


● create distance-time graphs from given information.

● discover the difference between the average rate of change and the instantaneous rate of change of a function.

● approximate the instantaneous rate of change using short tangent line segments.

● interpret the sign and magnitude of the instantaneous rate of change in the context of the situation.

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Mathematics—Average Rate of Change vs. Instantaneous Rate of Change


Targeted StandardsF-IF.6: Calculate and interpret the average

rateofchangeofafunction(presentedsymbolicallyorasatable)overaspecifiedinterval. Estimate the rate of change from a graph.★ See questions 1-4 , 6-8

Reinforced/Applied StandardsF-IF.4: For a function that models a relationship

between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ See questions 5, 8


MP.2: Reason abstractly and quantitatively. Students connect a verbal description, a graph, and a table of values for an object in motion and use the graph and table to interpret specific aspects of the object’s motion.

MP.3: Construct viable arguments and critique the reasoning of others. In question 5, students defend their conclusions about Susan’s possible speeds.

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● Calculate the slope between two points● Interpret slope from a graph


● Students engage in independent practice.● Students apply knowledge to a new situation.




● Rate of Change: Average and Instantaneous – Geometry Free Response Questions

● Rate of Change: Average and Instantaneous – Geometry Multiple Choice Questions



MATERIALS AND RESOURCES● Student Activity pages ● Straightedges

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Begin this lesson by asking two students to simultaneously walk parallel paths across the room, with one student keeping a constant

speed and the other student speeding up and slowing down,sothatbothstudentsarriveatthe“finishline”at the same time. Note that both students walked at the same average speed over the interval of the walk, but that their speeds within the walk’s interval were not always the same. Students should apply these observations to questions 1 – 4.

Questions 5 – 7 introduce the concept that the slope of a non-linear function at a particular point is equal to the slope of a short line segment that is tangent to the function at that point. Students will probably remember that a line tangent to a circle touches the circle in only one point while a secant line intersects the circle at two points. The physical activity of drawing the secant lines and the tangent lines helps the visual and tactile learner understand the differences in the average rate of change and the instantaneous rate of change of non-linear functions.

Questions 5 and 6 foreshadow the Mean Value Theoremfromcalculuswhichreferstothemean(oraverage) rate of change of a function on an interval. This theorem states that, for a smooth continuous function, there must be a point within the interval where the instantaneous rate of change is equal to the average rate of change over that closed interval. So for Susan’s graph, there must be at least one point during the time period where her instantaneous velocity is equal to her average velocity over the entire time period.

After students experience drawing tangent line segments to approximate the rate of change of a function, they apply this skill to analyze the distance-time graph of an object moving along a horizontal line in question 8. The average rate of change of apositionfunctionoveraspecifiedtimeintervalindicates the average velocity over that interval. When calculating the slope over an interval, use the appropriate units to help students develop an

understanding of this concept. To approximate the instantaneous rate of change at a point, students will draw a short tangent line segment and approximate its slope. By considering the sign and magnitude of the instantaneous rate of change in position with respect to time, students answer questions about the direction of the movement and the speed of the object. A positive velocity indicates that the motion isinapositivedirection(awayfromthewallinthisexample). Negative velocity indicates motion in a negativedirection(towardthewallinthisexample).In discussing speed, clarify for students that speed measures how fast the object is moving without regard to its direction and that speed is the absolute value of velocity.

You may wish to support this activity with TI-Nspire™ technology. See Working with Fractions and Decimals in the NMSI TI-Nspire Skill Builders. Suggestedmodificationsforadditionalscaffoldinginclude the following:1 Insertafill-in-the-blankcalculationwithunits

provided for the dimensional analysis required to calculate the distance from home at the end of 20minutes.(Seetheanswerkeyforanexample.)Edit the graph to indicate the starting point .

2 Supply an example of a possible graph of Susan’s drive before having the student draw a different possible graph.

5 Modify a sample graph from question 2 by drawing a secant line between two points to demonstrate that the slope is greater than 35 on that particular interval.


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Recognize intervals of functions with the same average rate of change.



Compare average rates of change on different intervals in a table or graph.



Estimate and/or compare instantaneous rates of change at a point based on the slopes of the tangent lines.



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Average Rate of Change vs. Instantaneous Rate of Change

1. Beginning 2 1

3 miles from home, Jonathan drove away from home at a constant rate for 20 minutes.

If his constant rate is 35 miles per hour, how far is he from home at the end of the 20 minutes? Draw a graph to model his distance from home during the 20 minute time period.

d �miles�

t �minutes�2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

2. Susan, Jonathan’s sister, also drove away from home beginning 2 1

3 miles from home and following the

same path as Jonathan. Susan kept varying her velocity by frequently speeding up and slowing down. She arrived at the same location as Jonathan at the end of 20 minutes. To model Susan’s distance from home during the 20 minute time period, draw a smooth curve without any sharp corners.

d �miles�

t �minutes�2 4 6 8 10 12 14 16 18 20

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3. Calculate the average velocity for both drivers by calculating the change in position divided by the change in time. These two calculations have the same value; explain why this makes sense. Compare your answers to the rate given in question 1.

4. On the graph showing Susan’s position, draw the line segment connecting the point at t = 0 and the point at t = 20. What is the slope of this line segment and what are the units for the slope? How does this slope compare to the slope of the line that modeled Jonathan’s distance from home?

5. If the speed limit over the entire path is 35 miles per hour, did Susan ever drive over the speed limit? Explain your answer by referring to Susan’s graph.

6. For non-linear position functions, the exact velocity at a particular time, called instantaneous rate of change or instantaneous velocity, cannot be calculated precisely without the tools of calculus. However, the velocity can be estimated by approximating the slope of a short line segment drawn tangent to the curve at the particular time. On the graph showing Susan’s distance, locate at least one time when Susan’s instantaneous velocity has the same value as the average velocity. (Position a straightedge on the graph so that it is parallel to the line segment drawn on the curve in question 4. Move the straightedge around on the graph keeping the slope of the straightedge fixed. When the straightedge appears to be tangent to the curve, mark the point(s) and sketch a short segment tangent to the curve. At these point(s), the instantaneous velocity is the same as the average velocity.)



7. Using the function g(x) shown in the graph, draw a small tangent line segment at each labeled point. A small tangent line is drawn at point P as a sample. There is not enough information to draw the segments perfectly, so sketches may vary slightly. Match the slope at each labeled point on the curve with an approximate rate of change value in the table. The slope at each point is called the “instantaneous rate of change” at a point because it is the rate of change at that one instant in time. Hint: The slope may be the same at different places along the graph.

Rate of Change at Point Letter

–2

0

3 p

6

15



8. The graph represents the position x(t) in inches of an object that is moving along a line extending perpendicularly from a wall at a given time, t, measured in seconds. The distance between the object and the wall is indicated on the vertical axis, while time is measured on the horizontal axis.

a. What do the coordinates of B (0.13, 27.5) and D (2, 19) represent in the context of this situation?

b. Mark small tangent line segments on each of the points that are named. Using these tangent segments, for which point(s) is the instantaneous rate of change negative? What do you know about the motion of the object if the instantaneous rate of change is negative?

c. Observing the tangent segments, over which time intervals is the object moving away from the wall? What do the slopes of these line segments mean in the context of the position function?

d. At which point(s) has the object stopped moving? Describe the slope of the tangent line(s) at the point(s).

e. Speed indicates how fast an object is moving without regard to direction. Order the speeds at the following points from least to greatest: D, G, Z. Explain your reasoning.

Point t x(t)

B 0.13 27.5

G 0.45 30

Z 0.87 32

V 1.2 31

D 2.0 19

K 3.02 0

A 3.25 2

C 4.1 70

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LEVELAlgebra 1 or Math 2 in a unit on piecewise linear functions or a unit on analyzing graphs

Algebra 2 or Math 3 as an introductory lesson in a unit including piecewise-defined functions




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Walking Piecewise GraphsABOUT THIS LESSON

This lesson makes connections between distance-time graphs and speed graphs. Students write, graph, and analyze

piecewise linear functions, with the focus on using the rate of change of the distance function to determine the speed graph. The lesson is most effective when using motion detectors to reinforce the connection between distance and speed.


● write a verbal description of the motion that produces a particular distance-time graph.

● create speed piecewise graphs based on distance graphs.

● write piecewise functions from motion graphs.

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Mathematics—Walking Piecewise Graphs

COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENTThis lesson addresses the following Common Core State Standards for Mathematical Content. The lesson requires that students recall and apply each of these standards rather than providing the initial introduction to the specific skill. The star symbol (★) at the end of a specific standard indicates that the high school standard is connected to modeling.

Targeted StandardsF-BF.1: Write a function that describes a

relationship between two quantities.★ See questions 1c–e, 2c–e, 3c–e, 4c–e

F-IF.7b: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.★ See questions 1d, 2d, 3d, 4d

Reinforced/Applied StandardsF-IF.6: Calculate and interpret the average

rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ See questions 1a, 1c, 1e, 2a, 2c, 2e, 3a, 3c, 3e, 4a, 4c, 4e

F-IF.4: For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ See questions 1a, 2a, 3a, 4a

F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ See questions 1a, 1c, 1e, 2a, 2c, 2e, 3a, 3c, 3e, 4a, 4c, 4e


MP.2: Reason abstractly and quantitatively. Students create verbal descriptions from a position-time graph, duplicate the graph by walking with a motion detector, and then write the functions for position and speed.

MP.5: Use appropriate tools strategically. Students use a motion detector to replicate a given graph.

MP.6: Attend to precision. Students use mathematical language, including speed, direction, and time, when describing the scenario represented by the graph.

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● Write linear functions from graphs● Calculate rate of change from a graph


● Students engage in independent practice.






MATERIALS AND RESOURCES● Student Activity pages ● Masking tape● Tape measures● Graphing calculators● CBRs (optional)

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This lesson contains continuous piecewise functions. Note that in the answers, the written equations include the shared endpoints in

each of the domain intervals. (For example, see the answers for part (c) of questions 1 – 4.) Students may argue that including the shared endpoint in both intervals creates an equation that is not a function; however this is simply not true. Teachers should explain that, since evaluating both function pieces at their common x-value produces the same y-value, writing the equation in this way does not contradict the definition of a function which states for each x-value, there is only one y-value. (The Test Development Committee of the College Board has used this approach on past exams. For an example, see 2006 AB/BC 4 Form B.)

The teacher should direct the beginning of this activity. If the students are unfamiliar with the CBR, they should have an opportunity to explore how the CBR functions and the relationship between motion on a horizontal line and the graph generated on the calculator before students begin working on the questions stated in the lesson. Students should discuss what causes the value of the graph to increase and decrease. Using CBRs or acting out the scenario helps provide a physical representation of the piecewise distance function. Even if motion detectors are not available, lead the students through part of the activity by asking them to model a walk that matches one of the distance-time graphs. It is important that students have an opportunity to describe verbally what the graph describes and then to model the scenario physically. Instruct students to use specific quantities and units for distance, time, and speed in their descriptions of the walks. The remainder of the questions should be answered by students working with a partner.

Students will probably notice that the graphs they create on the CBR do not display the sharp corners that appear in the graphs given in the lesson. The sharp corners indicate an instantaneous change in the

speed of the walker which is not physically possible for an actual walker. The CBR graphs will show a rounded “corner,” indicating a smooth transition from one speed to another, rather than an abrupt change. This explains the need for open circles on the endpoints of each piece of the speed graphs. It is not possible for the walker to walk at two different speeds at the same instant, just as it is not physically possible for the walker to immediately jump from a speed of 1 m/sec to 0 m/sec at 2 seconds, as shown in distance graph 1. In reality, any speed graph should be continuous. These “step function” speed graphs are the result of contriving the situation so that the distance graphs could be piecewise linear.This lesson is similar to the format of a science experiment, and, like a science experiment, the results may not be perfect.

Suggested modifications for additional scaffolding include the following:1 Provide the answers to question 1 as a model.

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About using a CBR2:

A CBR2 works much like a radar detector. It sends out a signal that will strike a target and then bounce back to the CBR2. The CBR2 calculates the distance based on the time required for the signal to return and sends the time and distance information to the calculator’s lists. The lists that are used depend on the calculator program that directs the CBR2.

Factors to consider when using a CBR2 with a TI-84:

1. The signal from the CBR2 actually spreads out in a cone-like shape, so make sure that the path in front of the CBR2 is clear of extra furniture and extra people.

2. The target must be in line with the CBR2 and stay in its field of view.

3. Ensure that the CBR2 stays pointed at the target.

4. The target should walk as evenly as possible in order to create a smooth graph.

5. To improve data accuracy, have the target hold a book or a flat piece of cardboard in line with the CBR2.

6. When connected to the TI-84, the CBR2 collects 1 data point every 0.1 seconds for approximately 11 – 12 seconds, so caution the target to walk slowly. The Ranger program shows the time to be 15 seconds; however, this is the maximum x-value on the graphing window and not the time that data is collected. For the data to be shown in real time, the value for the amount of time cannot be changed.

Setting up the equipment for a TI-84:

1. Press APPS; select CBL/CBR and choose RANGER.

2. Choose RANGER

3. Open the face of the CBR2 and slide the button toward the right (the walker).

4. Select 1: SETUP/SAMPLE

5. Match the settings as shown The realtime, display, begin on, smoothing, and units selections are toggle menus.

6. Arrow up to START NOW and press ENTER to begin collecting data.

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About using a CBR2:

A CBR2 works much like a radar detector. It sends out a signal that will strike a target and then bounce back to the CBR2. The CBR2 calculates the distance based on the time required for the signal to return and sends the time and distance information to the calculator’s lists. The lists that are used depend on the calculator program that directs the CBR2.

Factors to consider when using a CBR2 with a TI-Nspire™:

1. The signal from the CBR2 actually spreads out in a cone-like shape, so make sure that the path in front of the CBR2 is clear of extra furniture and extra people.

2. The target must be in line with the CBR2 and stay in its field of view.

3. Ensure that the CBR2 stays pointed at the target.

4. The target should walk as evenly as possible in order to create a smooth graph.

5. To improve data accuracy, have the target hold a book or a flat piece of cardboard in line with the CBR2.

6. When connected to the TI-Nspire™, the collection time and the sampling rate are controlled by the user. The default setting is 20 data points per second for 5 seconds.

Setting up the equipment for a TI-Nspire:

1. Connect the CBR2 to the calculator with a USB cable. A Vernier DataQuest page will automatically open and the CBR2 will begin measuring the position of the nearest object.

2. Change to the graph view to display a position vs. time graph. If two graphs are displayed, press Menu > Graph > Show Graph > Graph 1.

3. Press Menu > Experiment > Collection Setup to configure this time based data collection.

4. Select the green Start button in the lower left corner of the screen to begin collecting data.


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Mathematics—



Walking Piecewise Graphs

1. Use the graph provided to answer the following.

a. Write a description for a walk that will produce the graph.

b. Walk the graph in front of a motion detector following the description written in part (a).

c. Write a piecewise function for the distance-time graph.

d. Draw a graph of the speed based on the distance-time graph.

e. Write a piecewise function for the graph in part (d).

























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time in hours

Ann

'sdi

stan

ce�m

iles�

Ann

'ssp

eed�m

ilesp

erho

ur�

Ann's Ride

1 2 3 4 5 6 7 8 9 10

123456789

10

LEVELAlgebra 1 or Math 2 in a unit on piecewise linear functions or in a unit on analyzing graphs

Algebra 2 or Math 3 as an introductory lesson in a unit on piecewise-defined functions




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Applying Piecewise FunctionsABOUT THIS LESSON

This lesson makes connections between distance-time graphs and speed graphs. Initially, students analyze a distance-time

graph to write its piecewise function and to write and graph its accompanying speed function. The second question presents verbal information about speed in a new scenario and asks students to create graphs and equations for both speed and distance. This lesson is an excellent extension of the lesson, “Walking Piecewise Graphs.”


● create distance and speed piecewise graphs.● write piecewise linear functions from

motion graphs.● evaluate functions.

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Mathematics—Applying Piecewise Functions


Targeted StandardsF-BF.1: Write a function that describes a

relationship between two quantities.★ See questions 1a, 1c, 1e, 2a-b, 2d-e

F-IF.7b: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (b) Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.★ See questions 1c, 2a, 2e

Reinforced/Applied StandardsF-IF.6: Calculate and interpret the average

rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.★ See questions 1a, 1c, 1e, 2a-b, 2d-e

F-IF.5: Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.★ See questions 1a, 1c, 1e, 2a-b, 2d-e

F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. See questions 1b, 2f-g


MP.2: Reason abstractly and quantitatively. Students move fluently between the position-time relationship and the corresponding speed-time relationship. Students represent Ann’s speed graphically and analytically based on information about her distance traveled presented numerically and graphically. They interpret the meaning of the algebraic representation in the problem situation. Likewise, they represent Luke’s speed graphically based on a verbal description, and they work backwards to graph distance and to write the distance equation.

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7 1Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.



● Write linear functions from graphs● Calculate rate of change from a graph

ASSESSMENTSThe following assessments are located on our website:






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This lesson contains continuous piecewise functions. Note that in the questions and answers, the written equations include the

shared endpoints in each of the domain intervals. (For example, see the answers for questions 1a and 2d.) Students may argue that including the shared endpoint in both intervals creates an equation that is not a function; however this is simply not true. Teachers should explain that, since evaluating both function pieces at their common x-value produces the same y-value, writing the equation in this way does not contradict the definition of a function which states for each x-value, there is only one y-value. (The Test Development Committee of the College Board has used this approach on past exams. For an example, see 2006 AB/BC 4 Form B.)

Teachers may wish to begin this lesson by asking students to act out the scenario for “Ann’s Ride,” question 1, or by incorporating CBRs (motion detectors). Question 1 presents the distance data in graphical and tabular form. Students create piecewise functions for distance and speed. To assist students who have had limited exposure to the concept of speed (absolute value of velocity), have them analyze the units when calculating the rate of change for each interval to help them see the connection between speed and the slope in the distance equation. Question 2 increases the rigor by revealing a new scenario where only information about speed in given. Students create the speed graph and function first, and then apply that information to create a graph and function for distance.

Discuss with students that the sharp corners on the distance graph indicate an instantaneous change in the speed – something not physically possible. This explains the need for open circles on the endpoints of each piece of the speed graphs. It is not possible for Ann to ride at two different speeds at the same instant, just as it is not physically possible for Ann to

immediately jump from a speed of 3 mph to 23

mph

at 1 hour, as shown in her distance graph. In reality, any speed graph should be continuous. These “step function” speed graphs are the result of contriving the situation so that the distance graphs could be piecewise linear.

You may wish to support this activity with TI-Nspire™ technology. See Graphing Piecewise Functions in the NMSI TI-Nspire Skill Builders.

Suggested modifications for additional scaffolding include the following:1a, 2b Provide the structure for writing a piecewise

graph by supplying the first equation and interval of the piecewise defined function.

1c, 2e Draw one portion of the piecewise graph for the student.


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Applying Piecewise Functions

1. Ann went on a 10 hour bicycle trip. The graph models the relationship between the time and the total distance traveled by Ann. The table shows selected points on the graph.

t 0 1 4 7 10

d(t) 0 3 5 5 10

a. Write a function ( )d t for her total distance traveled in miles in terms of the time t in hours.

b. Evaluate (3)d . Write an ordered pair to represent the situation and explain its meaning in context of the problem.

c. Draw a graph of Ann’s speed in miles per hour in terms of the time in hours.

d. What is the relationship between the speed and the equations for ( )d t ?

e. Write a function ( )s t for her speed in terms of the time t.



2. Luke drives to his grandmother’s house on Saturday. For the first three hours, he drives at a constant speed of 50 miles per hour. During the next three hours, Luke stops to eat and to buy his grandmother a birthday present. In the last four hours, Luke drives at a constant speed of 60 miles per hour. a. Draw a graph of Luke’s speed in miles per hour in terms of the time in hours.

b. Write a function ( )s t for Luke’s speed in terms of the time t.

c. What is the relationship between the speed and the equations for ( )d t ?

d. Write a function ( )d t for his distance in miles from home in terms of the time t in hours.

e. Draw a graph of his distance from home versus the time.

f. Determine d (2.3) and explain the meaning of the answer.

g. Determine the value of t when d(t) = 322 and explain the meaning of the answer.

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x

y

�5 �4 �3 �2 �1 1 2 3 4 5

�5

�4

�3

�2

�1

1

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LEVELAlgebra 2 or Math 3 in a unit on quadratic functions




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Investigating Average Rate of ChangeABOUT THIS LESSON

This lesson examines the average and instantaneous rates of change of linear and quadratic functions by calculating

the slopes of secant lines and estimating the slopes of tangent lines. First, students consider linear functions and conclude that the slopes of secant lines for any interval of a linear function are equal and that the average and instantaneous rates of change are the same. The second section of the lesson focuses on quadratic functions so that students can observe how secant line slopes change, depending on the interval selected. Students are led to discover a unique property of quadratic functions: the slope of the secant line for any particular interval is equal to the slope of the tangent line at the midpoint of that interval. Students then apply this property to solve a real-world situation. Throughout the lesson, students have opportunities to reinforce their skills in determining function values and calculating slopes.


● determine the slope of a secant line.● estimate the instantaneous rate of change of

a function.● write the equation for a tangent line to

a function.● discover and apply in a real-world situation

a unique property of quadratic functions: the slope of the secant line for any interval is equal to the slope of the tangent line at the midpoint of that interval.

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Mathematics—Investigating Average Rate of Change


Targeted StandardsF-IF.6: Calculate and interpret the average

rateofchangeofafunction(presentedsymbolically or as a table) over a specifiedinterval.Estimatetherateofchange from a graph.★ See questions 1-3, 4b-i, 4k-m, 5b-f, 5i, 6a-b, 6d-e, 7a-b, 7e-g

Reinforced/Applied StandardsF-IF.7a: Graph functions expressed symbolically

and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a)Graphlinearandquadraticfunctions and show intercepts, maxima, and minima.★ See questions 4a-b, 4e, 5a-b, 5e, 7c

A-CED.2: Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★ See questions 4m, 5j, 7c

F-IF.2: Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. See questions 1a, 2a, 4b, 4e, 4g, 4i, 4m, 5b, 5d, 5f, 5j, 7d

S-ID.6a: Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. (a)Fitafunctiontothedata;usefunctionsfittedtodatatosolveproblemsin the context of the data. Use given functions or choose a function suggested bythecontext.Emphasizelinear,quadratic, and exponential models.★ See questions 7c-g

N-Q.1: Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.★ See questions 7e, 7g

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MP.2: Reason abstractly and quantitatively. Students progress from a computational understanding to a verbal generalization then to a real-world application. In question 7, students convert real-world data into a scatterplot, create a regression equation, and then interpret values in terms of the problem situation.

MP.4: Model with mathematics. Students test the car company’s claim by creating a regression function to fit the data and using the model to refute the claim.

MP.5: Use appropriate tools strategically. Students use a graphing calculator to fit a function to data and use the function to predict additional values.

MP.8: Look for and express regularity in repeated reasoning. Students determine that, for quadratic functions, the average rate of change over an interval equals the instantaneous rate of change at the midpoint of that interval, based on repeated calculations, and then use this rule in an applied situation.


● Calculate the slope of a line● Write a linear equation● Sketch graphs of simple quadratic functions


● Students engage in independent practice.● Students apply knowledge to a new situation.● Students summarize a process or procedure.




MATERIALS AND RESOURCES● Student Activity pages ● Straight edges● Coloredpencils(optional)● Graphing calculators

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This lesson offers the advantage of requiring students to practice and apply a variety of essential skills, such as working with function

notation, calculating function values, interpreting interval notation, and computing slopes, while exploring new situations, recognizing patterns, and drawing generalized conclusions. There are also ample opportunities for students to develop graphing calculator skills and expertise. Students should be

encouraged to use the symbol yx

∆∆

to represent the

average rate of change over an interval. The lesson can be easily divided into three separate activities to be completed on three different occasions: questions 1 – 3 address linear functions, questions 4 – 6 use quadratic functions, question 7 applies the conclusions from questions 4 – 6 to a real-world situation.

To avoid rounding errors and emphasize the use of function notation when evaluating the difference quotient, type the function in . From the home

screen type the following command:

or .

For example to calculate the rate of change of

for the interval , enter

then from the home screen

type .

Question 7 is a calculator-active question that provides a real-world application for the skills students have practiced in the earlier questions. Students may need instruction in using the calculator’s regression feature. Rounded values should not be used in subsequent calculations. On the TI84 calculator, enter the x-values in List 1, the y-values in List 2, then use the command “QuadReg L1, L2, Y1” to calculate the regression equation and

to store the equation in the graphing menu. This will avoidtheissueofroundingthecoefficientsintheequation. After storing the regression equation in Y1, use the home screen program to calculate the slope.

You may wish to support this activity with TI-Nspire™ technology. See Storing Values and Expressions and Finding Regression Equations in the NMSI TI-Nspire Skill Builders.

Suggestedmodificationsforadditionalscaffoldinginclude the following:4 Modify the graph to provide the sketch of the

quadraticin(a)andatleastoneofthesecantlinesin(b).

7 Provide a written summary of the calculator procedures that are needed for this question.


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Estimateand/or compare instantaneous rates of change at a point based on the slopes of the tangent lines.



Use and interpret average rate of change as

( ) ( )y f b f ax b a

∆ −=∆ −

Use and interpret average rate of change as

( ) ( )y f b f ax b a

∆ −=∆ −

Use and interpret slopes of secant and tangent lines.


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Investigating Average Rate of Change

For ( )y f x= on the interval [ , ]a b , the average rate of change is ( ) ( )y f b f ax b a

∆ −=∆ −

. This quotient is the

slope of the secant line. In other words, this is the slope calculated between two points on the function f(x). The instantaneous rate of change, the slope of the tangent line at one point, will be explored in this lesson.

1. ( ) 3 1f x x= +

a. Calculate the average rate of change, yx

∆∆

, of the function over each of the given intervals.

i. [–5, –1]

ii. [2, 8]

iii. Choose any different interval.

iv. [0.9, 1]

v. [0.999, 1]

b. What is the instantaneous rate of change at x = 1?

2. 2( ) 33

f x x= − +

a. Calculate the average rate of change, yx

∆∆

, of the function over each of the given intervals.

i. [–6, 3]

ii. [3, 9]

iii. Choose any different interval.

iv. [0.9, 1]

v. [0.999, 1]

b. What is the instantaneous rate of change at x = 1?

3. a. Explain why the answers in question 1 are the same and why the answers in question 2 are the same.

b. Describe an easy method for determining the average rate of change of a linear function.



4. 2( ) 1f x x= +a. Sketch the function by carefully plotting the points at integer values of x.

b. Draw a secant line for each of the following intervals and graphically determine the average rate of change of the function (slope of the secant line) over each interval.

i. [–2, –1]

ii. [–1, 0]

iii. [0, 2]

c. Are the secant lines in part (b) parallel? Do the secant lines in part (b) have the same slope?

d. Based on the answers for part (b), are the average rates of change for a quadratic function constant?



e. Using a colored pencil, draw a secant line for each interval given. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and determine the x-coordinate of the midpoint of each segment. Record your information in the table provided in part (i).

i. [–1, 2]

ii. [0, 1]

f. Are the secant lines in part (e) parallel? Do the secant lines in part (e) have the same slope?

g. Using a colored pencil, draw a secant line for each of the following intervals. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and determine the x-coordinate of the midpoint of each segment. Record your information in the table in part (i).

i. [0.4, 0.6]

ii. [0.49, 0.51]

iii. [0.499, 0.501]

h. Are the secant lines in part (g) parallel? Do the secant lines in part (g) have the same slope?



i. Complete the table including your information from parts (e) and (g).

First Point Second Point Δy ΔxΔyΔx

x-coordinate of the midpoint of the segment

(–1, ____) (2, _____)

(0, _____) (1, _____)

(0.4, ______) (0.6, _______)

(0.49, ________) (0.51, ________)

(0.499, _________) (0.501, _________)

j. Do the coordinates in the table seem to approach a certain point? What is that point?

k. Estimate the instantaneous rate of change (slope of the tangent line) at x = 0.5.

l. At what specific point of ( )f x on [–1, 2] is the instantaneous rate of change of the function equal to the average rate of change of the function on the interval [–1, 2]? For what other intervals given in this question is this same relationship also true?

m. Using your estimate for the instantaneous rate of change at x = 0.5 found in part (k), write the equation of the tangent line through the point . Using a colored pencil, draw this line on your graph.



5. 21( ) ( 2) 64

f x x= − + +

a. Sketch the function by carefully plotting the points at integer values of x.

b. Using a colored pencil, draw a secant line for each given interval. What are the coordinates of the points on the graph of the function where the secant line intersects the curve? Calculate the average rate of change of the function (slope of the secant line) over each interval and calculate the x-coordinate of the midpoint of each segment. Record your information in the table in part (f).

i. [–3, 5]

ii. [–2, 4]

iii. [0, 2]

c. Are the secant lines in part (b) parallel? Do the secant lines in part (b) have the same slope?



d. Using a colored pencil, draw a secant line for each given interval, calculate the average rate of change of the function (slope of the secant line) over each interval, and record your answers in the table in part (f).

i. [0.9, 1.1]

ii. [0.99, 1.01]

iii. [0.999, 1.001]

e. Are the secant lines in part (d) parallel? Do the secant lines in part (d) have the same slope?

f. Complete the table to include your information from parts (b) and (d).

First Point Second Point Δy Δx ΔyΔx

x-coordinate of the midpoint of the segment

(–3, _______) (5, ________)

(–2, ________) (4, ________)

(0, ________) (2, _________)

(0.9, __________) (1.1, _________)

(0.99, ___________) (1.01, __________)

(0.999, __________) (1.001, _________)

g. Do the coordinates in the table seem to approach a certain point? What is that point?

h. Estimate the instantaneous rate of change (slope of the tangent line) at x = 1.



i. At what specific point on [–3, 5] is the instantaneous rate of change of the function equal to the average rate of change of the function?

j. Using your estimate for the instantaneous rate of change at x = 1 found in part (h), write the equation of the tangent line through the point . Using a colored pencil, draw this line on your graph.

6. Fill in the blanks for each statement using the choices provided. Note: Some choices may be used more than once and some may not be used at all.

constant different endpoint lengthmidpoint slope the same zero

a. The average rate of change between two points of a function is the ____________of the secant line.

b. Since the slope of a linear function is ___________, the average rate of change is _____________.

c. For a constant function, the y-coordinate is ____________ for every pair of points selected, so the average rate of change always has a value equal to ____________.

d. The average rate of change for a quadratic function is not _______________ for every pair of points selected.

e. For a quadratic function, the x-value of the point where the average rate of change over a given interval equals the instantaneous rate of change of that interval is the _____________ of the interval.



7. A car company is testing the speed and acceleration of one of its new sports cars. The table shows the distance the car travels when it accelerates from a standstill. Use a graphing calculator to answer the following questions.

Elapsed time in seconds (t) Distance in meters (d)0 010 22514 37518 55020 650

a. Explain why this data is not linear and justify your answer mathematically using the slopes of a pair of secant lines.

b. In the context of the problem, what does dt

∆∆

represent? What is the average rate of change, dt

∆∆

, on

the interval [0, 20]? Indicate appropriate units of measure.

c. Determine the quadratic regression function, ( )R t , for the data and superimpose its graph on a scatterplot of the data. Copy the graph and the data from your calculator onto the grid provided.

Quadratic Regression Equation _______________________________



d. What is (20)R ? Explain the meaning of this value in terms of the problem situation, and explain why this value is different from the value in the table.

e. According to R(t), what is the average rate of change over the 20-second time interval from 0 seconds to 20 seconds? Convert the answer to the nearest whole number in miles per hour and explain its meaning in terms of the problem situation. (1 km = 0.6214 miles)

f. Since the regression function is quadratic, where should the average rate of change be equal to the instantaneous rate of change for the interval [0, 20]?

g. The car company claims the car can accelerate from 0 to 60 mph in 6 seconds. This means that the instantaneous rate of change at 6 seconds must be 60 mph. Prove or disprove this claim by examining the average rate of change over an interval for which 6 seconds is the midpoint.

Note: miles meters60 26.821hour second

≈ .


Mathematics—

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LEVELPre-Calculus or Math 4 in a unit on rational functions




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Slopes of CurvesABOUT THIS LESSON

In this lesson, students are given an equation for the slope of a curve and then asked to describe characteristics of the graph of the curve.

Students apply algebraic skills to simplify, factor, analyze sign tests, and solve algebraic equations to determine intervals where the function is increasing or decreasing and to identify relative minimums and maximums for a function. Students will write equations of lines, solve equations, and determine values for which a function is undefined. In addition, the lesson introduces students to graphing curves that are not functions.


● determine the zeros of a function.● create equations of horizontal and vertical

tangent lines.● analyze values of functions to determine

intervals where the function is positive or negative.

● identify characteristics of the graph using the value of the slope.

● graph non-function curves using parametric equations.

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Mathematics—Slopes of Curves


Targeted StandardsF-IF.4: For a function that models a relationship

between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.★ See questions 1-2, 3a-e, 3g, 4a-c

Reinforced/Applied StandardsA-CED.2: Create equations in two or more variables

to represent relationships between quantities; graph equations on coordinate axes with labels and scales.★ See questions 1b, 2g, 3e-g, 4b, 4e

A-REI.2: Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. See questions 2a, 2c, 3a, 3c-e, 4c

F-IF.7a: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (a) Graph linear and quadratic functions and show intercepts, maxima, and minima.★ See question 2g

F-IF.7c: Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. (c) Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.★ See questions 3f, 4e

A-SSE.2: Use the structure of an expression to identify ways to rewrite it. For example, see

4 4x y− as , thus recognizing it as a difference of squares that can be factored as . See questions 3f, 4d

A-SSE.3a: Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. (a) Factor a quadratic expression to reveal the zeros of the function it defines.★ See question 1a

A-REI.4b: Solve quadratic equations in one variable. (b) Solve quadratic equations in one variable. Solve quadratic equations by inspection (e.g., for 2 49x = ), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a bi± for real numbers a and b. See question 4d

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MP.1: Make sense of problems and persevere in solving them. Students must decide whether to use the slope equation or the curve equation to answer the questions. Students wrestle with implicitly-defined equations, work with techniques to transform them into more familiar forms, and analyze their characteristics.

MP.5: Use appropriate tools strategically. Students use a graphing calculator with parametric capabilities to graph implicitly-defined curves.

MP.6: Attend to precision. Students must check values to determine if solutions are extraneous.

MP.7: Look for and make use of structure. Students see a quadratic expression in terms of y within a polynomial expression in x and y and use the quadratic formula to solve for y.


● Write linear equations● Factor quadratic equations● Determine values where a function is

undefined

ASSESSMENTSThe following formative assessment is embedded in this lesson:

● Students engage in independent practice.


● Rate of Change – Pre-Calculus Free Response Questions

● Rate of Change – Pre-Calculus Multiple Choice Questions

MATERIALS AND RESOURCES● Student Activity pages ● Graphing calculators

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Introduce the lesson to students by sharing some or all of the following information along with the example provided. Note that a piece of

linguine makes an excellent tool for demonstrating how lines tangent to the curve have different slopes at different locations.

To see how TI-Nspire™ technology enables users to graph x as a function of y, see Graphing f(y) Equations in the NMSI TI-Nspire Skill Builders. Note that Graph Trace will enable students to enter y-values to determine the corresponding values of x.

The slope of a curve at a particular point is defined to be the slope of a line tangent to the curve at that point which means that the slopes along the curve will vary. At the points where the tangent line has a positive slope, the curve is increasing. At the points where the slope of the tangent line is negative, the curve is decreasing. A slope of zero means the tangent line is horizontal. To decide if the curve is increasing or decreasing when the slope of the tangent line is zero requires examining the behavior of the curve on either side of the point. A local maximum value will occur at a point where the slope of the curve either equals zero or does not exist and changes from positive to negative; a local minimum value will occur where the slope of the curve equals zero or does not exist and changes from negative to positive. If the tangent line is vertical, then the slope does not exist. (In calculus, vertical tangent lines are often considered to have an infinitely large slope.)

This graph illustrates a relative maximum value with a horizontal tangent line at the marked point.

This graph illustrates a relative minimum value with a vertical tangent line at the marked point.

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The equation for the slope of a curve that is provided for each question is the first derivative of a function. In calculus, students will learn to determine this equation.

Example:For 23 2 4y x x= − + , the slope function m at any point on the curve is 6 2m x= − .

Draw the graph of the parabola and draw the tangent line to the curve at the point (2, 12) .

Calculate the slope of the tangent line at (2, 12) : 6(2) 2 10m = − = . The positive slope value indicates

the function is increasing at the point (2, 12) as indicated in the graph. a. Use the slope function to determine if the curve is

increasing at 0x = .b. Write the equation of the line tangent to the curve

at 1x = − .c. Determine algebraically where the slope of the

tangent line to the curve would be horizontal. Does a maximum or minimum occur there? Since the curve is a parabola, what is the name given to this maximum or minimum point?

Answers:a. When 0x = , 6(0) 2 2m = − = − indicating the

curve is decreasing.b. 2(1) 3( 1) 2( 1) 4 9y = − − − + = and

6( 1) 2 8m = − − = − giving a tangent line equation of 8( 1) 9y x= − + + .

c. 6 2 013

x

x

− =

=

To the left of 13

x = , the slope is 6(0) 2 2− = − .

To the right of 13

x = , the slope is 6(1) 2 4− = . There is a minimum value where the slope equals zero and changes from negative to positive. The minimum point is the vertex of the parabola.

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Determine if a curve is increasing, decreasing, or constant based on the slope of the tangent line.

Determine if a curve has a maximum or minimum based on the change in the slopes of the tangent lines.



Slopes of Curves

1. For 3 24 2 8 13

y x x x= − − + , the slope function m for any point on the curve is given by 24 4 8m x x= − − .a. Determine the points where the slope is zero. Show the algebraic steps.

b. Write the equation(s) of the horizontal tangent line(s) to the curve at the point(s).

c. Determine if the curve has a maximum or minimum value or neither at the point(s). Explain your answer using the values of m.

2. For 23 2 4 10y x y− + = , the slope function m for any point on the curve is given by 1

3 2m

y=

+.

a. Is this slope ever zero? Show why or why not.

b. What does part (a) indicate is true about the graph?

c. When would the slope be undefined? Show the process used to determine the answer.



d. What is x when 23

y = − ?

e. Describe the tangent line to the curve where the slope is undefined.

f. When would the slope be positive? Describe the curve when the slope is positive.

g. On a TI-84, solve the original equation for x in terms of y and then graph the curve using parametric

equations. Let x(t) = equation for x in terms of t instead of y, and let ( )y t t= . On the TI-Nspire™, solve the original equation for x in terms of y and then use the scratchpad to graph ( )x f y= List the equations used and sketch the graph.

h. Verify your answers to part (f) by examining the graph.

3. Consider the curve 3 2 22 6 12 6 1y x y x y+ − + = . The slope function m of the curve at all points is given by

2 2

4 21

x xymx y

−=+ +

.

a. Determine the slope of the curve at the point 1 1,2 2

.

b. Is the curve increasing or decreasing at the point 1 1,2 2

? Explain.



c. Determine the values of x and y where m = 0.

d. Using your answers to part (c), determine the point(s) on the curve where the slope is zero.

e. Write the equations of the horizontal tangent lines to the curve.

f. On the TI-84, solve the equation of the curve for x in terms of y then use parametric equations to graph the curve. Let ( )x t = equation for x in terms of t instead of y, and let ( )y t t= . On the TI-Nspire™, solve the equation of the curve for x in terms of y and then use the scratchpad to graph ( )x f y= List the equations used and sketch the graph.

g. What is the horizontal asymptote of the curve?



4. Consider the curve given by 2 3 6xy x y− = . The slope function m of the curve at any point

is 2 2

3

32x y ymxy x

−=−

.

a. Determine all points on the curve whose x-coordinate is 1.

b. Write an equation for the tangent line at each of the points where the x-coordinate is 1.

c. Determine the x-coordinate of each point on the curve where the tangent line is vertical.

d. Use the quadratic formula to solve for y in terms of x.

e. On the TI-84, use parametric equations to graph the curve. On the TI-Nspire™, use the scratchpad to graph the curve, using the equations from part (d). List the equations used and sketch the graph.



Introduction to the NMSI Mathematics Multiple Choice Quizzes

The National Math and Science Initiative multiple choice questions are modeled after multiple choice questions on the AP* Calculus and Statistics exams. The questions are assigned a course-level designation based on an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.

The grade-level multiple choice quizzes for sixth grade through pre-calculus assess the skills and concepts introduced in each module. These quizzes reflect the module’s Content Progression Chart, which outlines the mathematics imbedded in the activities for each grade level, and the Concept Development Chart, which provides examples of how those concepts or skills might be assessed. Additionally, the quizzes are directly linked to the NMSI posttests for each grade level. Once students have completed the activities, teachers may use the quiz questions to determine student understanding and to prepare students for the level of rigor on the posttests.

When scoring the multiple choice questions, teachers should remember that the quizzes are intended to model the rigor of questioning on AP exams. A suggested scoring guideline, which is also included with the rationales for each quiz, is:

Percent Correct Grade

0 – 29 5030 – 49 6050 – 59 7060 – 69 8070 – 79 90

80 – 100 100

All of these materials – lessons and activities with answer keys, grade-level quizzes with rationales, and free response questions with scoring rubrics and student samples – are available for each module on the NMSI website.


Mathematics


Mathematics

Sample Quiz Questions

1. 6th Grade Module 2 Question 2 Gillian’s family has a pond on their land. She has been monitoring the pond’s depth over the past year. The table indicates the depth of the water, in inches, on the first day of each month.

From month 9 to month 12, what is the average rate of change of the depth of the pond, in inches per month? Round your answer to the nearest inch per month.

A. 2 inchesmonth

B. 3 inchesmonth

C. 8 inchesmonth

D. 23 inchesmonth

E. 117 inchesmonth

Month Depth in inches Month Depth in

inches1 93 7 123

2 98 8 111

3 112 9 106

4 121 10 112

5 129 11 120

6 135 12 129


Mathematics—Sample Quiz Questions

2. 6th Grade Module 2 Question 5 (Calculator) While studying for their math final, Sunjay and his friends notice that the energy drinks they are drinking are in different sized cans. Even though all the cans are right cylinders, the diameter and circumference differ. The boys decide to measure the diameter and circumference of each can and to record the measurements in the table.

Drink Diameter Circumference

can 1 6.5 cm 20.39 cm

can 2 4.3 cm 13.52 cm

can 3 7.1 cm 22.32 cm

What is the arithmetic mean of the ratios of circumference to diameter for the three cans? Round the answer to the nearest ten thousandth of a centimeter.

A. 3.1369 B. 3.1400 C. 3.1416 D. 3.1420 E. 3.1437

3. 7th Grade Module 2 Question 3 On Trevor’s 17th birthday, he received his grandfather’s old truck. To calculate the average miles per gallon, he must divide the number of miles the truck travels by the number of gallons used. The odometer read 143,986 miles when the gas tank was full and 144,261 miles when all of the gas in the 25-gallon tank had been used. What was Trevor’s truck’s gas mileage for that tank to the nearest mile per gallon?

A. 1 milegallon

B. 11 milesgallon

C. miles16gallon

D. 17 milesgallon

E. 25 milesgallon



4. 7th Grade Module 2 Question 5 Students taking their first keyboarding class were asked to record their speed in words per minute at the end of specified weeks during the course. Sharon recorded her information in the table.

Week Speed(words per minute)

1 20

3 30

6 40

8 50

9 55

Which of the following statements is not correct, based on the information in the table?

A. The units for the average rate of change in her speed are words

minute.

B. The average rate of change of Sharon’s speed with respect to time from week 1 to week 3 was 5.

C. The average rate of change of Sharon’s speed with respect to time from week 3 to week 6

was 133

.

D. The average rate of change of Sharon’s speed with respect to time was the same from week 1 to week 3 as it was from week 8 to week 9.

E. The average rate of change of Sharon’s speed with respect to time was the same from week 1 to week 3 as it was from week 6 to week 9.



5. Algebra 1 Module 2 Question 2 In June of 2006, the pre-enrollment number for the incoming freshman class was 328. By the end of their sophomore year, June 2008, the class size was 340. During their junior year, a local business that employs many of the families in the town closed, and several students left; therefore, the class size was 321 in June 2009. By graduation, in June of 2010, the number of students was 315. What was the average rate of change in class size for this group of students from June 2006 to June 2010?

A. students3.25year

−

B. students2.6year

−

C. students2.6year

D. students3.25year

E. students13year



6. Algebra 1 Module 2 Question 6 Throughout the year, football players are required to work on increasing their strength. To help track his progress during his junior year, Javier graphed the maximum weight, in pounds, that he lifted for squats and bench press at the beginning of each testing week. Use the graph and the accompanying table of values to answer the questions.

Use the graph to determine which time intervals, if any, show an increase in Javier’s strength for both squats and bench presses.

I. Between 6 and 10 weeks

II. Between 26 and 30 weeksIII. Between 30 and 36 weeks

A. I only B. II only C. III only D. I and III only E. I, II, and III

Week of training Squat Bench Press

0 250 pounds 135 pounds









7. Geometry Module 2 Question 1 The graph shows a piecewise function that consists of a parabola and three linear segments.

Which of the following statements is not correct?

A. The function is increasing and has a constant rate of change of 3 on the interval 4 5x≤ ≤ . B. The function has an average rate of change that is negative on the intervals 2 0x− ≤ ≤

and 2 3x≤ ≤ .C. The absolute values of the average rates of change for the intervals 5 2x− ≤ ≤ − and 2 0x− ≤ ≤ are

the same.D. The average rate of change on the interval 0 5x≤ ≤ is greater than the average rate of change on the

interval 3 5x≤ ≤ .E. The function has an average rate of change on the interval 5 3x− ≤ ≤ − that is equal to the average

rate of change on the interval 0 2x≤ ≤ .



8. Geometry Module 2 Question 6 Air was pumped into a spherical ball. As the ball was inflated, the measurements were recorded in the table. Time, t, is measured in seconds. The radius, r, is measured in centimeters. The volume, V, is measured in cubic centimeters.

Time, t, in seconds

Radius, r, in centimeters

Volume, V, in cubic

centimeters0 8.0 2144.7

5 9.1 3204.7

15 10.8 5324.7

25 12.1 7444.7

On which of the following time intervals does the average rate of change of the radius with respect to time have the smallest value?

A. 0 seconds to 5 secondsB. 0 seconds to 15 secondsC. 5 seconds to 15 secondsD. 5 seconds to 25 secondsE. 15 seconds to 25 seconds

9. Algebra 2 Module 2 Question 2 The function g(x) is exponential, , where and . On which of the intervals would the average rate of change of g be the smallest?

A.

B.

C.

D.

E.



10. Algebra 2 Module 2 Question 6 (calculator) An account earns interest compounded monthly at a monthly interest rate of 0.5%. Nicholas deposits $5,000 and does not plan to make any additional deposits or withdrawals from the account. The formula, , calculates A , the future value of the account, where the initial (principal) amount deposited, the monthly interest rate, and the number of months that the principal has been invested. What will be the average rate of change in the amount of money in the account with respect to time for the time period from 1 year (12 months) to 3 years (36 months)?

A. $25.19month

B. $28.13month

C. $235.25month

D. $269.06month

E. $832.49month

11. Pre-Calculus Module 2 Question 3

For , the slope equation m at any point on the curve is given by .

What is the equation of the line tangent to f when ?

A.

B.

C.

D.

E.



12. Pre-Calculus Module 2 Question 5 If , where , , , what is the average rate of change of f (x) on the

interval ?

A.

B.

C.

D.

E.



Selected Rationales

1. 6th Grade Module 2 Question 2A. Student calculates the change in depth and divides by 12 for the last month .

B. Student calculates the change in depth and divides by 9 for the first month and rounds the answer.C. Correct. Student calculates the change in depth and divides by the change

in months .

D. Student reports only the change in depth of the water. E. Student calculates the arithmetic mean of the four depths .

6. Algebra I Module 2 Question 6I. The maximum weight for squats at 6 and 10 weeks remained constant, therefore there was no

increase; however, his bench press increased.II. Javier’s maximum weight for squats increased but his maximum for bench press decreased

between 26 and 36 weeks. III. Javier’s maximum weight for squats and bench press increased between 30 and 36 weeks.

A. Student considers a constant value an increase.B. Student selects the interval where the strength for squats increased; however, fails to note that the

strength for his bench press decreased.C. Correct. Student selects the only interval where strength for both the squats and bench

press increased.D. Student considers a constant value an increase.E. Student thinks all should be included that have an increase in either squats or bench press.



7. Geometry Module 2 Question 1A. Student misreads the question and reports the first statement that is true or miscalculates

the rate.B. Student does not realize that both intervals have a negative rate of change.

C. Student does not determine the average rates of change to be

D. Correct. Student determines that the average rate of change on is less than the average rate of change on .

E. Student calculates the rate incorrectly and determines that the specified rates are not equal or student thinks that a parabolic portion does not have an average rate of change.

10. Algebra 2 Module 2 Question 6A. Student uses n values in years instead of months when calculating the average rate of

change, .

B. Correct. Student determines the correct formula, and determines the average

rate of change in A with respect to time, .

C. Student uses the correct formula for A but adds the values in the numerator and denominator of

determining the difference quotient, .

D. Student converts the interest rate to an incorrect decimal form, and uses

n-values in years instead of months .

E. Student converts the interest rate to an incorrect decimal form, and

calculates .





Introduction to the NMSI Mathematics Free Response Questions

The National Math and Science Initiative free response questions are modeled after free response questions on the AP* Calculus and Statistics exams. The questions are assigned a course-level designation based on an accelerated program that enables students to take college-level courses in high school, using a faster pace to compress content. In this sequence, Grades 6, 7, 8, and Algebra 1 are compacted into three courses. Grade 6 includes all of the Grade 6 content and some of the content from Grade 7, Grade 7 contains the remainder of the Grade 7 content and some of the content from Grade 8, and Algebra 1 includes the remainder of the content from Grade 8 and all of the Algebra 1 content.

The free response rubric is a guide to assist the reader, not a detailed solution to the question. Sometimes a method is outlined in the rubric, but another more efficient method may work as well. A student’s correct solution may earn all of the points to be awarded for a particular part of the question, even though the approach does not match the one shown in the rubric. The rubric shows “a way” to work the problem, not “the way” to work the problem.

When scoring the free response questions, teachers should practice “reading with” a student’s error. This statement means that the student is penalized for the error when it first occurs, but the reader then follows the student’s process for full credit in subsequent parts of the question, even when the student continues to use the results of the earlier error. For the free response, the reader should be in the mindset of awarding points, not taking them away. Students start at 0 and can earn up to 9 points rather than starting at 9 points and losing points.

Student directions for the free response questions include the following:● All work for a given part of a question must be shown in the space provided.● Answers do not need to be simplified completely; however, when calculating approximate answers,

do not round intermediate values. Your final answers should be accurate to three places after the decimal point.

● Questions that contain units require units in the answers.● The setup for all mathematical computations and equations is required using mathematical notation

rather than calculator syntax. Intermediate calculations do not have to be shown when determining:the answer to basic arithmetic computations; the zeroes of a function;the maximum/minimum of a function;the intersection point between two functions;a regression equation.

● Part A and B are given equal weight, but parts of a particular question are not necessarily given equal weight.

● During the timed portion for Part A, you may work only on Part A. A calculator may be used on Part A only.

● During the timed portion for Part B, in addition to working on the question in Part B, you may continue to work on Part A without a calculator.


Mathematics—Free Response Questions

Grade 6 2011 Free Response Question - Calculator Allowed

Daisy is preparing to water the flowers in her garden. She places a 6-gallon watering can under the outside faucet and gradually turns on the water so that the rate at which the water is flowing into the watering can is increasing. At one minute, she adjusts the water flow to a steady rate. When the watering can contains 5 gallons of water, she removes it from under the faucet and carries it to the garden where she pours the water along the row of flowers.The graph and the table show the amount of water in the watering can at any given time.

(a) At 2 minutes, how much water is in the watering can? Include units in your answer.

(b) While Daisy carries the watering can from the faucet to the garden, the watering can contains 5 gallons of water. How long does the watering can contain 5 gallons of water? Include units in your answer.

time in minutes

gallons of water

0 0

12

34

1 3

3 5

4 1

25

5 1

20



(c) How fast is the water flowing into the watering can between 1 minute and 3 minutes? Fill in the blank and simplify the answer. Include units in the answer.

Between

12

minute and 1 minute, what is the average rate of change in the number of gallons of

water in the watering can with respect to time? Fill in the blanks and simplify the answer. Include units in the answer.

What is the numerical difference between the rates during the two time periods? State the answer as a decimal and include units in the answer.

(d) While Daisy is pouring the water on the flowers, at what time does the watering can contain exactly 2 gallons of water? Calculate the answer using ratios and proportions. Show the work that leads to your answer, and state the answer in whole minutes and seconds.



Algebra 1 2013 Free Response Question – Calculator Allowed

Marina and Kyoko are participating in a school exercise program in which they walk along the sidewalks around the rectangular block containing the school. The sidewalk along Park Avenue is 748 feet long, and the sidewalk along Royal Lane is 450 feet long. The students meet at the intersection of Park and Royal to begin their walk. Marina will walk east along Park at a constant speed of 5 feet per second. Kyoko will walk north along Royal at a constant speed of 4.5 feet per second. Since Kyoko is younger than Marina, Kyoko is given a 40-foot head start before the timing of their walks begins.

(a) At 90 seconds, what is Marina’s distance from the corner of Park and Royal? Show the work, including the units, that leads to the answer.

At 90 seconds, what is Kyoko’s distance from the corner of Park and Royal? Show the work. including the units, that leads to the answer.

At 90 seconds, what is the straight line distance between Kyoko and Marina? Show the work that leads to the answer, and state the answer to the nearest foot.



(b) Let t = the number of seconds after the timing begins.

What is the equation that represents Marina’s distance, M, measured along the sidewalk from the corner of Royal and Park at any time, t?

What is the equation that represents Kyoko’s distance, K, measured along the sidewalk from the corner of Royal and Park at any time, t?

(c) At what number of seconds will the combined distances measured along the sidewalks from the corner of Royal and Park be equal to the entire distance around the block? Show the work that leads to the answer.

The girls will meet at the time when their combined distances are equal to the distance around the block. When they meet, what is each girl’s distance measured along the sidewalk from the corner of Royal and Park?



Pre-Calculus 2010 Free Response Question - Calculator Allowed

For the function, f (x) = 4x3 2x2 8x +2 , the slope at any point on f (x) can be calculated using the

equation .

(a) What are the zeroes of f (x) ?

(b) At what point(s) on f(x) is the slope equal to zero?

(c) What is the equation of the line tangent to f (x) at the point where f (x) changes from increasing to decreasing?



(d) What is the equation of the line, g(x) , that is tangent to f (x) at x =1.5? One other line, h(x) , is tangent to f (x) and parallel to g(x) . What is the equation of h(x) ?

(e) The absolute maximum value of a function on a closed interval is the greatest y-value on that interval. If f (x) is defined only on the closed interval , what are the absolute maximum and absolute minimum values of the function? Show the analysis that leads to your conclusion.



A1Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics

Appendix

A2 Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.

Mathematics


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Standards for Mathematical Practice

MP.1 - Make sense of problems and persevere in solving them.Mathematically proficient students:

● start by explaining to themselves the meaning of a problem and looking for entry points to its solution.

● analyze givens, constraints, relationships, and goals. ● make conjectures about the form and meaning of the solution and plan a solution pathway rather than

simply jumping into a solution attempt. ● consider analogous problems, and try special cases and simpler forms of the original problem in

order to gain insight into its solution. ● monitor and evaluate their progress and change course if necessary. ● Older students might, depending on the context of the problem,

○ transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need,

○ explain correspondences between equations, verbal descriptions, tables, and graphs, ○ draw diagrams of important features and relationships, graph data, and search for regularity

or trends. ● Younger students might:

○ rely on using concrete objects or pictures to help conceptualize and solve a problem. ● check their answers to problems using a different method, and they continually ask themselves,

“Does this make sense?” ● understand the approaches of others to solving complex problems and identify correspondences

between different approaches.

In assessments, the question:● is designed to take a typical student a long time to solve.● leads to a more difficult problem.● requires a large number of routine and fairly easy steps.● contains several “givens.”● the statement of the problem itself is designed not to allow for jumping in and working the problem

immediately. ● posed using abstract statements that must be parsed carefully before they make sense.● require students to construct their own solution pathway rather than to follow a provided one.● may be unscaffolded so that a multi-step strategy must be autonomously devised by the student.● involve ideas that are currently at the forefront of the student’s developing mathematical knowledge

in a word problem.

Copyright © 2014 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org.A4

Mathematics—Standards for Mathematical Practice

MP.2 - Reason abstractly and quantitatively.Mathematically proficient students:

● make sense of quantities and their relationships in problem situations. ● bring two complementary abilities to bear on problems involving quantitative relationships:

○ the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and

○ the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved.

● use quantitative reasoning that entails habits of creating a coherent representation of the problem at hand:

○ considering the units involved; ○ attending to the meaning of quantities, not just how to compute them; and ○ knowing and flexibly using different properties of operations and objects.

In assessment, the question is designed to:● be contextual so that the student can gain insight into the problem by relating the algebraic form of an

answer or intermediate step to the given context. ● require the use symbolic calculations to generalize a situation and draw conclusions from those

calculations.



MP.3 - Construct viable arguments and critique the reasoning of others.Mathematically proficient students:

● understand and use stated assumptions, definitions, and previously established results in constructing arguments.

● make conjectures and build a logical progression of statements to explore the truth of their conjectures.

● analyze situations by breaking them into cases, and can recognize and use counterexamples. ● justify their conclusions, communicate them to others, and respond to the arguments of others. ● reason inductively about data, making plausible arguments that take into account the context from

which the data arose. ● compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from

that which is flawed, and – if there is a flaw in an argument – explain what it is. ○ Elementary students can construct arguments using concrete referents such as objects, drawings,

diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades.

○ Later, students learn to determine domains to which an argument applies.● listen to or read the arguments of others, decide whether they make sense, and ask useful questions to

clarify or improve the arguments.

In assessments, require students to:● base explanations/reasoning on concrete referents such as diagrams (whether provided in the prompt

or constructed by the student).● construct, autonomously, chains of reasoning that will justify or refute propositions or conjectures.● determine conditions under which an argument does and does not apply. ● distinguish correct explanations/reasoning from that which is flawed, and – if there is a flaw in the

argument – explain what it is.● provide informal justifications.● use of diagrams, words, and/or equations to solve.● reason about key grade-level mathematics.● apply rigorous deductive proof based on clearly stated axioms.● state logical assumptions being used.● test propositions or conjectures with specific examples.● apply a series of logical and well-motivated steps with precise language and terms.



MP.4 - Model with mathematics.Mathematically proficient students:

● apply the mathematics they know to solve problems arising in everyday life, society, and the workplace.

○ In early grades, this might be as simple as writing an addition equation to describe a situation. ○ In middle grades, a student might apply proportional reasoning to plan a school event or analyze a

problem in the community. ○ By high school, a student might use geometry to solve a design problem or use a function to

describe how one quantity of interest depends on another. ● apply what they know are comfortable making assumptions and approximations to simplify a

complicated situation, realizing that these may need revision later. ● identify important quantities in a practical situation and map their relationships using such tools as

diagrams, two-way tables, graphs, flowcharts and formulas.● analyze those relationships mathematically to draw conclusions. ● routinely interpret their mathematical results in the context of the situation and reflect on whether the

results make sense, possibly improving the model if it has not served its purpose.

In assessments, require students to:● apply a known technique from pure mathematics to a real-world situation in which the technique

yields valuable results even though it is not obviously applicable in a strict mathematical sense. ● execute some or all of the modeling cycle: formulate, compute, interpret, validate, and report.● select from a data source, analyze the data and draw reasonable conclusions from it, often resulting in

an evaluation or recommendation.● use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an

unknown quantity.● make assumptions and simplifications.● select from the data at hand or estimate data that are missing.● use reasonable estimates of known quantities in a chain of reasoning that yields an estimate of an

unknown quantity.



MP.5 - Use appropriate tools strategically.Mathematically proficient students:

● consider the available tools when solving a mathematical problem (these tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software).

● are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations.

● High school students: ○ analyze graphs of functions and solutions generated using a graphing calculator. ○ detect possible errors by strategically using estimation and other mathematical knowledge. ○ when making mathematical models, know that technology can enable them to visualize the

results of varying assumptions, explore consequences, and compare predictions with data. ● identify relevant external mathematical resources, such as digital content located on a website, and

use them to pose or solve problems. ● use technological tools to explore and deepen their understanding of concepts.

In assessments, questions involve● making the coordinate plane essential for solving the problem, yet no direction is given to the student

to use coordinates. ● creating circumstances for poor use or misuse of tools.● posing questions that are fairly easy to solve or to answer correctly if a diagram is drawn first, but

very hard to solve or to answer correctly if a diagram is not drawn, yet no direction is given to draw a diagram.

● using formulas or conversions where there is no prompting to use them.● data sets of 15-30 numbers.● using a calculator to test conjectures with many specific cases.● substituting messy numerical values into a complicated expression and find the numerical result.



MP.6 - Attend to precision.Mathematically proficient students:

● try to communicate precisely to others.● try to use clear definitions in discussion with others and in their own reasoning.● state the meaning of the symbols they choose, including using the equal sign consistently and

appropriately. ● are careful about specifying units of measure and labeling axes to clarify the correspondence with

quantities in a problem. ● calculate accurately and efficiently, express numerical answers with a degree of precision appropriate

for the problem context. ○ In the elementary grades, students give carefully formulated explanations to each other. ○ By the time they reach high school, students have learned to examine claims and make explicit

use of definitions.

In assessments, require students to:● use reasoned solving of equations, such as those in which extraneous solutions are likely to be found

and must be discarded. ● solve algebraic word problems in which success depends on carefully defining variables.● present solutions to multi-step problems in the form of valid chains of reasoning, using symbols such

as equal signs appropriately.



MP.7 - Look for and make use of structure.Mathematically proficient students:

● look closely to discern a pattern or structure. ○ Young students, for example, might notice that three and seven more is the same amount as

seven and three more, or they may sort a collection of shapes according to how many sides the shapes have.

○ Later, students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.

○ In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. ○ They recognize the significance of an existing line in a geometric figure and can use the strategy

of drawing an auxiliary line for solving problems. ● step back for an overview and shift perspective. ● see complicated things, such as some algebraic expressions, as single objects or as being composed

of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

In assessments, questions:● can be solved by analyzing parts of figures in relation to one another.● can be solved by introducing auxiliary lines into a figure.● reward seeing structure in an algebraic expression and using the structure to rewrite it for a purpose.● reward or require deferring calculation steps until one sees the overall structure.● assess how aware students are of how concepts link together and why mathematical procedures work

in the way that they do.



MP.8 - Look for and express regularity in repeated reasoning.Mathematically proficient students:

● notice if calculations are repeated, and look both for general methods and for shortcuts. ○ Upper elementary students might notice when dividing 25 by 11 that they are repeating the same

calculations over and over again, and conclude they have a repeating decimal. ○ By paying attention to the calculation of slope as they repeatedly check whether points are on the

line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3.

○ Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series.

● maintain oversight of the process, while attending to the details. ● continually evaluate the reasonableness of their intermediate results.

In assessments, questions require:● repeating calculations to lead to the articulation of a conjecture.● working repetitively with numerical examples leading without prompting to the writing of equations

or functions that describe modeling situations.● recognizing that tedious and repetitive calculation can be made shorter by observing regularity in the

repeated steps.● answers like “multiplying by any number and then dividing by the same number gets you back to

where you started.”● using recursive definitions of functions.● using patterns to shed light on the addition table, the times table, the properties of operations,

the relationship between addition and subtraction or multiplication and division, and the place value system.



Additional Graphs and Materials

Calculating Average Rate of Change Question # __________________

Step 1 – Record the coordinates.

Step 2 – Show the difference quotient.

Step 3 – Simplify the answer.

Step 4 – Write a sentence.


Mathematics—Additional Graphs and Materials



Average Rate of Change Versus Instantaneous Rate of Change:

Incorrect graphs

Additional Graphs and Materials

Average Rate of Change – Versus Instantaneous Rate of Change: Incorrect graphs

t

d

2 4 6 8 10 12 14 16 18 20

2

4

6

8

10

12

14

16

18

20

0








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Rate of Change:

Average & Instantaneous

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Graphical Organizer


Mathematics


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MATHEMATICSRate of Change: Average & Instantaneous

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MATHEMATICS - Kyhn's CAC Pageskyhncac.weebly.com/uploads/1/2/7/4/12743866/module_2_rate_of... · MODULE 2 MATHEMATICS ... Additional Graphs and Materials ... From graphical or tabular