NATIONAL MATH + SCIENCE Mathematics INITIATIVE … · 2014. 9. 8. · Rate of Change: Average and...

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

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LEVELAlgebra 2 or Math 3 in a unit on quadratic functions

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.

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

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.

OBJECTIVESStudents will

● 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

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 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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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.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.

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

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

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

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

The following assessments are located on our website:

● Rate of Change: Average and Instantaneous – Algebra 2 Free Response Questions

● Rate of Change: Average and Instantaneous – Algebra 2 Multiple Choice Questions

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

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TEACHING SUGGESTIONS

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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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

7th Grade Skills/Objectives

Algebra 1 Skills/Objectives

Geometry Skills/Objectives

Algebra 2 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.






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



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



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.

Use and interpret slopes of secant and tangent lines.

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MathematicsNATIONALMATH + SCIENCEINITIATIVE

Investigating Average Rate of Change

Answers

1. a. i. yx

∆∆

= ( 5) ( 1) 14 2 35 1 4

f f− − − − += =− + −

ii. yx

∆∆

= (2) (8) 7 25 32 8 6

f f− −= =− −

iii. Set up varies; answer is 3

iv. yx

∆∆

= (0.9) (1) 3.7 4 30.9 1 0.1

f f− −= =− −

v. yx

∆∆

= (0.999) (1) 2.997 4 30.999 1 0.001

f f− −= =− −

b. 3

2. a. i. yx

∆∆

= ( 6) (3) 7 1 26 3 9 3

f f− − −= = −− − −

ii. yx

∆∆

= (3) (9) 1 3 23 9 6 3

f f− += = −− −

iii. Set up varies; answer is 23

−

iv. yx

∆∆

= (0.9) (1) 20.9 1 3

f f− = −−

v. yx

∆∆

= (0.999) (1) 20.999 1 3

f f− = −−

b. 23

−

3. a. For a linear function, the average rate of change is the same between any two points on the line.

b. The average rate of change is the value of the slope in the equation of the line.

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4. a. See the drawing at the right

b. i. 3yx

∆ = −∆

ii. 1yx

∆ = −∆

iii. 2yx

∆ =∆

c. no; nod. no

e. i. y = 2; y=5; 1yx

∆ =∆

ii. y = 1; y = 2; 1yx

∆ =∆

f. yes; yes

g. i. (0.4) (0.6) 1.16 1.36 10.4 0.6 0.2

y f fx

∆ − −= = =∆ − −

ii. (0.49) (0.51) 1.2401 1.2601 10.49 0.51 0.02

y f fx

∆ − −= = =∆ − −

iii. (0.499) (0.501)0.499 0.501

y f fx

∆ −= =∆ −

1.249001 1.251001 10.002

− =−

h. yes; yes

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i.

First Point Second Point Δy Δx ΔyΔx

x-coordinate of the midpoint of

the segment(–1,2) (2,5) 3 3 1 0.5(0,1) (1,2) 1 1 1 0.5(0.4,1.16) (0.6,1.36) 0.2 0.2 1 0.5(0.49,1.2401) (0.51,1.2601) 0.02 0.02 1 0.5(0.499,1.249001) (0.501,1.251001) 0.002 0.002 1 0.5

j. Thepointsaregettingclosertooneanother.Theyareapproachingthepoint(0.5,1.25).

k. 1

l. 1 5,2 4

;allintervalslistedinparts(e)and(g)

m. 1 52 4

y x = − +

5. a. See the graph of the parabola

b. i. (−3,5.75)and(5,−6.25)

(5) ( 3) 6.25 5.75 35 ( 3) 8 2

y f fx

∆ − − − −= = = −∆ − −

x-value of the midpoint is 1.ii. (–2,6)and(4,–3)

(4) ( 2) 3 6 34 ( 2) 6 2

y f fx

∆ − − − −= = = −∆ − − x-value of the midpoint is 1.

iii. (0,5)and(2,2) (2) (0) 2 5 3

2 0 2 2y f fx

∆ − −= = = −∆ − x-value of the midpoint is 1.

c. yes; yes

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d. i. (0.9) (1.1) 3.8975 3.5975 30.9 1.1 0.2 2

y f fx

∆ − −= = = −∆ − −

ii. (0.99) (1.01)0.99 1.01

y f fx

∆ −= =∆ −

3.764975 3.734975 30.02 2− = −

−

iii. (0.999) (1.001)0.999 1.001

y f fx

∆ −= =∆ −

3.75149975 3.74849975 30.2 2− = −

−

e. yes; yes

f.


x-coordinate of the midpoint of

the segment

(−3,5.75) (5,−6,25) −12 832

− 1

(−2, 6) (4,−3) −9 632

− 1

(0,5) (2,2) −3 232

− 1

(0.9,3.8975) (1.1,3.5975) −0.3 0.232

− 1

(0.99,3.764975) (1.01,3.734975) −0.03 0.0232

− 1

(0.999,3.7514998) (1.001,3.7484998) −0.003 0.00232

− 1

g. yes; 31, 34

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h. 32

−

i. (1,3.75)

j. 3 3( 1) 32 4

y x= − − + See the graph.

6. a. slope b. constant; constant c. the same; zero d. constant e. midpoint

7. a. The average rates of change are not constant; therefore, the function is not linear. Forexample,on[0,10], 22.5d

t∆ =∆

whileon[10,20], 42.5dt

∆ =∆

b. The average rate of change represents the average speed of the car in meters per second.

For[0,20], m32.5sec

dt

∆ =∆

.

c. 2( ) 0.983 12.883 0.263R t t t= + − (Thisisarounded version of the answer.)

d. (20) 650.562R = meters. According to the regression function, the distance that the car hastraveledin20secondsis0.562meters more than the value given in the table. Regression functions model the data, and the data points are not necessarily points on the function.

e. On[0,20], m32.541sec

Rt

∆ =∆

; 32.541 m 1km 60sec 60min 0.6214mi mi73sec 1000m 1min 1hr 1km hr

≈

Onthis20-secondinterval,thecar’saveragerateofchangewithrespecttotime

(speedorvelocity)is mi73hr

.

f. Atthemidpointoftheinterval[0,20],att=10seconds

g. (12) (0) meters24.67812 second

R R− = Thecarhasnotacceleratedto60mphat6sec.

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

Mathematics NATIONALMATH + SCIENCEINITIATIVE

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. Explainwhytheanswersinquestion1arethesameandwhytheanswersinquestion2arethesame.

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

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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 changeofthefunction(slopeofthesecantline)overeachinterval.

i. [–2,–1]

ii. [–1,0]

iii. [0,2]

c. Arethesecantlinesinpart(b)parallel?Dothesecantlinesinpart(b)havethesameslope?

d. Basedontheanswersforpart(b),aretheaverageratesofchangeforaquadraticfunctionconstant?



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 averagerateofchangeofthefunction(slopeofthesecantline)overeachintervalanddeterminethe 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. Arethesecantlinesinpart(e)parallel?Dothesecantlinesinpart(e)havethesameslope?

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? Calculatetheaveragerateofchangeofthefunction(slopeofthesecantline)overeachintervalanddetermine 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. Arethesecantlinesinpart(g)parallel?Dothesecantlinesinpart(g)havethesameslope?



i. Completethetableincludingyourinformationfromparts(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. Estimatetheinstantaneousrateofchange(slopeofthetangentline)atx=0.5.

l. Atwhatspecificpointof ( )f x on[–1,2]istheinstantaneousrateofchangeofthefunctionequaltotheaveragerateofchangeofthefunctionontheinterval[–1,2]?Forwhatotherintervalsgiveninthis question is this same relationship also true?

m. Using your estimate for the instantaneous rate of change at x=0.5foundinpart(k),writetheequation 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 averagerateofchangeofthefunction(slopeofthesecantline)overeachintervalandcalculatethex-coordinateofthemidpointofeachsegment.Recordyourinformationinthetableinpart(f).

i. [–3,5]

ii. [–2,4]

iii. [0,2]

c. Arethesecantlinesinpart(b)parallel?Dothesecantlinesinpart(b)havethesameslope?



d. Using a colored pencil, draw a secant line for each given interval, calculate the average rate of changeofthefunction(slopeofthesecantline)overeachinterval,andrecordyouranswersinthetableinpart(f).

i. [0.9,1.1]

ii. [0.99,1.01]

iii. [0.999,1.001]

e. Arethesecantlinesinpart(d)parallel?Dothesecantlinesinpart(d)havethesameslope?

f. Completethetabletoincludeyourinformationfromparts(b)and(d).


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. Estimatetheinstantaneousrateofchange(slopeofthetangentline)atx = 1.



i. Atwhatspecificpointon[–3,5]istheinstantaneousrateofchangeofthefunctionequaltotheaverage rate of change of the function?

j. Using your estimate for the instantaneous rate of change at x=1foundinpart(h),writetheequationof 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. Theaveragerateofchangebetweentwopointsofafunctionisthe____________ofthesecantline.

b. Sincetheslopeofalinearfunctionis___________,theaveragerateofchangeis_____________.

c. For a constant function, the y-coordinateis____________foreverypairofpointsselected,sotheaveragerateofchangealwayshasavalueequalto____________.

d. Theaveragerateofchangeforaquadraticfunctionisnot_______________foreverypairofpointsselected.

e. For a quadratic function, the x-value of the point where the average rate of change over a given intervalequalstheinstantaneousrateofchangeofthatintervalisthe_____________oftheinterval.



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. Explainwhythisdataisnotlinearandjustifyyouranswermathematicallyusingtheslopesofapairof 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.

QuadraticRegressionEquation _______________________________



d. What is (20)R ?Explainthemeaningofthisvalueintermsoftheproblemsituation,andexplainwhythis value is different from the value in the table.

e. According to R(t),whatistheaveragerateofchangeoverthe20-secondtimeintervalfrom 0secondsto20seconds?Converttheanswertothenearestwholenumberinmilesperhourandexplainitsmeaningintermsoftheproblemsituation.(1km=0.6214miles)

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

g. Thecarcompanyclaimsthecarcanacceleratefrom0to60mphin6seconds.Thismeansthattheinstantaneousrateofchangeat6secondsmustbe60mph.Proveordisprovethisclaimbyexaminingthe average rate of change over an interval for which 6 seconds is the midpoint.

Note: miles meters60 26.821hour second

≈ .

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