NATIONAL MATH + SCIENCE INITIATIVE Mathematics 5 4 1 5 4 1 1 4 5 1 4 5 LEVEL Algebra or Math 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. MODALITY NMSI 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. G N P A V y P Physical V Verbal A Analytical N Numerical G Graphical x Investigating Average Rate of Change ABOUT 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. OBJECTIVES Students 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. i
COMMON CORE STATE STANDARDS FOR MATHEMATICAL CONTENT This 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 Standards F-IF.6: Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. See questions 1-, 4b-i, 4k-m, 5b-f, 5i, 6a-b, 6d-e, 7a-b, 7e-g Reinforced/Applied Standards 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 questions 4a-b, 4e, 5a-b, 5e, 7c A-CED.: 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.: 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, a, 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) Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, 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 ii
COMMON CORE STATE STANDARDS FOR MATHEMATICAL PRACTICE These 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.: 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. FOUNDATIONAL SKILLS The 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 ASSESSMENTS The 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 Free Response Questions Rate of Change: Average and Instantaneous Algebra Multiple Choice Questions MATERIALS AND RESOURCES Student Activity pages Straight edges Colored pencils (optional) Graphing calculators 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. iii
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 y encouraged to use the symbol 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 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 to store the equation in the graphing menu. This will avoid the issue of rounding the coefficients in the equation. 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. Suggested modifications for additional scaffolding include the following: 4 Modify the graph to provide the sketch of the quadratic in (a) and at least one of the secant lines in (b). 7 Provide a written summary of the calculator procedures that are needed for this question. for the interval, enter type. then from the home screen 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, then use the command QuadReg L1, L, Y1 to calculate the regression equation and iv
NMSI CONTENT PROGRESSION CHART In 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. 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 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. 7th Grade 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. Algebra 1 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. Geometry 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. Algebra 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. 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. Estimate and/ or compare instantaneous rates of change at a point based on the slopes of the tangent lines. Estimate and/ or compare instantaneous rates of change at a point based on the slopes of the tangent lines. Estimate and/ 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( a) = b a Use and interpret average rate of change as y f( b) f( a) = b a Use and interpret slopes of secant and tangent lines. Use and interpret slopes of secant and tangent lines. v
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NATIONAL MATH + SCIENCE INITIATIVE Mathematics Investigating Average Rate of Change Answers 1. a. i. ii. y x = f ( 5) f ( 1) 14 + = = 5+ 1 4 y x = f () f (8) 7 = 5 = 8 6 iii. Set up varies; answer is iv. v. y x = f(0.9) f(1).7 4 = = 0.9 1 0.1 y x = f(0.999) f(1).997 4 = = 0.999 1 0.001 b.. a. y f( i. = 6) f() 7 1 = = x 6 9 y f() ii. = f(9) 1 = + = x 9 6 iii. Set up varies; answer is y f(0.9) f(1) iv. = = x 0.9 1 y f(0.999) f(1) v. = = x 0.999 1 b.. 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. vii
4. a. See the drawing at the right y b. i. = x y ii. = 1 y iii. = x c. no; no d. no y e. i. y = ; y = 5; = 1 y ii. y = 1; y = ; = 1 f. yes; yes g. i. ii. iii. y f(0.4) f(0.6) 1.16 1.6 = = = 1 0.4 0.6 0. y f(0.49) f(0.51) 1.401 1.601 = = = 1 0.49 0.51 0.0 y f(0.499) f(0.501) = = 0.499 0.501 1.49001 1.51001 = 1 0.00 h. yes; yes viii
i. First Point Second Point Δy Δx Δy Δx x-coordinate of the midpoint of the segment ( 1, ) (, 5) 1 0.5 (0,1) (1, ) 1 1 1 0.5 (0.4, 1.16) (0.6, 1.6) 0. 0. 1 0.5 (0.49, 1.401) (0.51, 1.601) 0.0 0.0 1 0.5 (0.499, 1.49001) (0.501,1.51001) 0.00 0.00 1 0.5 j. The points are getting closer to one another. They are approaching the point (0.5, 1.5). k. 1 l. m. 1 5, 4 ; all intervals listed in parts (e) and (g) 1 5 y = x + 4 5. a. See the graph of the parabola b. i. (, 5.75) and (5, 6.5) y f(5) f( ) 6.5 5.75 = = = 5 ( ) 8 x-value of the midpoint is 1. ii. (, 6) and (4, ) y f(4) f( ) 6 = = = 4 ( ) 6 x-value of the midpoint is 1. iii. (0, 5) and (, ) y f() f(0) 5 = = = 0 x-value of the midpoint is 1. c. yes; yes ix
d. i. y f(0.9) f(1.1).8975.5975 = = = 0.9 1.1 0. ii. y f(0.99) f(1.01) = = 0.99 1.01.764975.74975 = 0.0 iii. y f(0.999) f(1.001) = = 0.999 1.001.75149975.74849975 = 0. e. yes; yes f. First Point Second Point Δy Δx Δy Δx x-coordinate of the midpoint of the segment (, 5.75) (5, 6,5) 1 8 (, 6) (4, ) 9 6 (0, 5) (, ) (0.9,.8975) (1.1,.5975) 0. 0. 1 1 1 1 (0.99,.764975) (1.01,.74975) 0.0 0.0 1 (0.999,.7514998) (1.001,.7484998) 0.00 0.00 1 g. yes; 1, 4 x
h. i. (1,.75) j. y = ( x 1) + 4 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. d d For example, on [0, 10], =.5 while on [10, 0], = 4.5 t t b. The average rate of change represents the average speed of the car in meters per second. d m For [0, 0], =.5. t sec c. Rt ( ) = 0.98t + 1.88t 0.6 (This is a rounded version of the answer.) d. R (0) = 650.56 meters. According to the regression function, the distance that the car has traveled in 0 seconds is 0.56 meters 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, 0], R = t m.541 sec ;.541 m 1km 60sec 60min 0.614 mi mi 7 sec 1000 m 1min 1hr 1km hr On this 0-second interval, the car s average rate of change with respect to time mi (speed or velocity) is 7. hr f. At the midpoint of the interval [0, 0], at t = 10 seconds g. R(1) R(0) meters = 4.678 The car has not accelerated to 60 mph at 6 sec. 1 second xi
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NATIONAL MATH + SCIENCE INITIATIVE Mathematics Investigating Average Rate of Change y f( b) f( a) For y = f( x) on the interval [ a, b ], the average rate of change is =. This quotient is the b a 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. f( x) = x+ 1 y a. Calculate the average rate of change,, of the function over each of the given intervals. x i. [ 5, 1] ii. [, 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?. f( x) = x+ y a. Calculate the average rate of change,, of the function over each of the given intervals. x i. [ 6, ] ii. [, 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?. a. Explain why the answers in question 1 are the same and why the answers in question are the same. b. Describe an easy method for determining the average rate of change of a linear function. 1
4. f x ( ) = x + 1 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. [, 1] ii. [ 1, 0] iii. [0, ] 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, ] 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 Δx ( 1, ) (, ) (0, ) (1, ) (0.4, ) (0.6, ) (0.49, ) (0.51, ) (0.499, ) (0.501, ) Δy Δx x-coordinate of the midpoint of the segment 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, ] is the instantaneous rate of change of the function equal to the average rate of change of the function on the interval [ 1, ]? 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. 4
5. f x 1 4 ( ) = ( x+ ) + 6 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. [, 5] ii. [, 4] iii. [0, ] c. Are the secant lines in part (b) parallel? Do the secant lines in part (b) have the same slope? 5
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 (, ) (5, ) (, ) (4, ) (0, ) (, ) (0.9, ) (1.1, ) (0.99, ) (1.01, ) (0.999, ) (1.001, ) Δy Δx x-coordinate of the midpoint of the segment 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. 6
i. At what specific point on [, 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 length midpoint 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
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 0 10 5 14 75 18 550 0 650 a. Explain why this data is not linear and justify your answer mathematically using the slopes of a pair of secant lines. d b. In the context of the problem, what does t represent? What is the average rate of change, d t, on the interval [0, 0]? Indicate appropriate units of measure. c. Determine the quadratic regression function, Rt (), 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 8
d. What is R (0)? 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 0-second time interval from 0 seconds to 0 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 = 0.614 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, 0]? 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. miles meters Note: 60 6.81. hour second 9
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