T538 Mathematics Success Grade 8 [OBJECTIVE] The student will compare functions represented algebraically, graphically, with verbal descriptions or in tables and identify functions as linear or non-linear. [PREREQUISITE SKILLS] functions, input, output, rate of change, KEMS Grade 8 Lesson 20 Functions [MATERIALS] Student pages S265 S278 Colored pencils (red, blue and green for each student pair) [ESSENTIAL QUESTIONS] 1. How can you identify the rate of change of a function given a table of values? 2. How can you compare the rate of change with an equation and a graph of a line? 3. How can you determine if a function is linear or non-linear? [WORDS FOR WORD WALL] function, function table, rate of change, linear, non-linear, ratio, coefficient, slope, constant rate of change [GROUPING] Cooperative Pairs (CP), Whole Group (WG), Individual (I) *For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson. [LEVELS OF TEACHER SUPPORT] Modeling (M), Guided Practice (GP), Independent Practice (IP) [MULTIPLE REPRESENTATIONS] SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer, Table, Graph, Algebraic Formula [WARM-UP](IP, I, WG) S265 (Answers on T547.) Have students turn to S265 in their books to begin the Warm-Up. Students will be working with function tables. Give students a few minutes to complete the problems and then review the answers as a whole group. {Algebraic Formula, Verbal Description, Table} [HOMEWORK] Take time to go over the homework from the previous night. [LESSON] [1 2 DAYS (1 day = 80 minutes) M, GP, WG, CP, IP]
Mathematics Success Grade 8 T539 SOLVE Problem (WG, CP, IP) S266, S267 (Answers on T548, T549.) Have students turn to S266 in their books. The students will be completing two SOLVE problems in student pairs. Both of these SOLVE problems are a review of writing functions. The first problem asks students to write the function based on a verbal expression. The second problem will require students to use information from a table to write the function. Students will complete both SOLVE problems with a partner and then review their answers as a whole group. The SOLVE problems will then be extended to introduce the concept of comparing functions. {SOLVE, Graphic Organizer, Verbal Description, Algebraic Formula} Extend the SOLVE Problems Comparing Functions in Table and Equation Form (M, GP, CP, WG) S266, S267, S268 (Answers on T548, T549, T550.) M, GP, CP, WG: Have students turn to S266 in their books. Assign the roles of Partner A and Partner B. Students will identify the components of each function from S266 and S267 in the context of the problem and identify the rate of change for each function. Then they will compare the rates of change to determine which one is greater.{verbal Description, Graphic Organizer, Table, Algebraic Formula} MODELING Extend the SOLVE Problems Comparing Functions in Table and Equation Form Step 1: Have students turn to S268. They will need to refer back to the SOLVE problems on S266 and S267 to complete the table. Partner A, describe the function that represents Nick s Weekly Earnings. (y = 7x + 50) Record. Partner B, describe the function that represents Larissa s Weekly Earnings. (y = 10x + 40.) Record. Partner A, explain what the number in front of the x represents for the function that represents Nick s Weekly Earnings. (The amount earned for every hour worked is $7.00.) Record. Partner B, explain what the number in front of x represents for the function that represents Larissa s Weekly Earnings. (The amount earned for every hour worked is $10.00.) Record. Partner A, explain how we identified the hourly earnings for Nick. (We are given the information of the hourly wage of $7.00. The variable, x represents the number of hours he works each week.) Record. Partner B, explain how we identified the hourly earnings for Larissa. (The hourly wage was found by subtracting the weekly bonus from the pay and then dividing by the number of hours worked using the information in the function table.) Record.
T540 Mathematics Success Grade 8 Step 2: Direct students attention to Question 4. Partner A, what is another word for the amount of money that each person makes per hour? (pay rate) Record. The pay rate per hour is constant in the context of these problems, and when multiplied by the hours worked, will determine the weekly base salary. What term can we use to identify the pay rate? (rate of change). Record. Partner B, which person has an hourly rate that represents a greater rate of change? (Larissa s $10.00 per hour is greater than Nick s $7.00 per hour.) Record. Step 3: Partner A, look at the function that represents Nick s earnings and explain the meaning of the number that is added to the x. (Each week an amount of $50 is added to Nick s earnings for travel expenses.) Record. Partner B, look at the function that represents Larissa s earnings. What does the number added to the x represent? (Each week an amount of $40 is added to Larissa s earnings as a bonus.) Record. Step 4: Let s look at another way to determine the rate of change for Larissa using the table that was given in the SOLVE problem on S267. Partner A, for Larissa, what is the difference in earnings between 40 and 35 hours? (440 390 = $50) Record. Partner B, for Larissa, what is the difference in number of hours between 40 and 35? (40 35 = 5) Record. We can write the relationship between the difference in earnings and the difference in time worked as a ratio. Partner A, what is the difference in earnings over the difference in hours? ( 50 = $10.00 per 1 hour) Record. 5 Partner B, what does this ratio represent? (rate of change) Record. Compare Functions with Verbal Descriptions and Graphs (M, GP, CP, WG) S269 (Answers on T551.) M, WG, CP, GP: Have students turn to S269 in their books. Students will be analyzing and comparing functions represented as verbal descriptions and graphs. Be sure students know their designation as Partner A or Partner B. {Verbal Description, Graph, Graphic Organizer, Algebraic Formula}
Mathematics Success Grade 8 T541 MODELING Compare Functions with Verbal Descriptions and Graphs Step 1: Have student pairs discuss the representations of the functions in the graphic organizer. Partner A, describe the format of Function 1. (It is a verbal description.) Partner B, describe the format of Function 2. (It is represented as a line on a graph.) Partner A, explain how we can use the verbal description to write the function for Function 1. [We know that the input is the x-value and the output is the y-value. The verbal description tells us that we find the output (y) by multiplying 1 times the input and (x) adding 4.] 2 Partner B, explain how we can write Function 1 in the form of an equation. (y = 1 2 x + 4 ) Record. Step 2: Partner A, explain how we can write Function 2, which is represented by the line. (The rate of change of the graph of a line can be found by identifying any two points on the line and finding the rise, or change in y, over the run, or change in x.) Partner B, what is the rate of change of the line? (The line has a rise of 2 and a run of 1. So, the rate of change is 2.) Partner A, which variable has the coefficient of 2 in this function? (the input or the x-value.) Partner B, what information do we still need to write the function? (where it crosses the y-axis) Partner A, where does the line cross the y-axis? (at the point of - 1) Partner B, what is the function that represents the line? (y = 2x 1) Record. Step 3: Partner A, what is the rate of change for Function 1? ( 1 2) Record. Partner B, explain how you determine the rate of change. (It is the coefficient in front of the input variable, which is the x.) Record. Partner A, what is the rate of change for Function 2. (2) Record. Partner B, explain how you determine the rate of change. (Write the ratio as the change in y over the change in x, or rise over run.) Record. Partner A, what is another word for rate of change? (slope) Record. Partner B, which function has the greater rate of change, or slope? (Function 2, because 2 is greater than 1.) Record. 2
T542 Mathematics Success Grade 8 Comparing Rate of Change (M, GP, IP, CP, WG) S270, S271 (Answers on T552, T553.) M, WG, CP, GP: Have students turn to S270 in their books. Students will be analyzing and comparing functions and using that information to identify and determine which function has the greatest rate of change. Be sure students know their designation as Partner A or Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer, Graph, Algebraic Formula, Table} MODELING Comparing Rate of Change Step 1: Have students turn to page S270 and discuss what they see in Column 1. (Possible answers: functions, different representations of functions, graphs, equations, tables) Partner A, describe Function 1. (The output of a function is determined by multiplying the input by five and subtracting three.) Partner B, what is the rate of change of the function? (5) Record. Partner A, explain how you determined the rate of change? (The rate of change is the coefficient of the input value, or the value that is multiplied by the input value.) Record. Step 2: Direct students attention to Function 2. Partner B, identify Function 2. (y = 3x + 7) Partner A, what is the rate of change of the function? (3) Record. Partner B, explain how you determined the rate of change? (The rate of change is the coefficient of the x-value, or the number multiplied by x, which is 3.) Record. Step 3: Direct students attention to Function 3. Partner A, explain Function 3. (A table of values is given.) Partner B, what is the rate of change of the function? (6) Record. Partner A, explain how you determined the rate of change. (Find the difference in y-values of two points and divide by the difference in x-values of the same two points. Then, check to be sure that the rate applies to all of the other points. (Example: y: 7 1 = 6 and x: 0-1 = 1. Ratio: 6 1 = 6) Record. Step 4: Direct students attention to Function 4. Partner B, explain Function 4. (It is a line which represents the graph of a function.) Partner A, what is the rate of change of the function? (1) Record. Partner B, explain how you determined the rate of change. (Identify two points on the graph and find the change in y-values and change
Mathematics Success Grade 8 T543 in x-values from the first to second point. Then, divide the change in y-values by the change in x-values. Change in y-values: 1; Change in x-values: 1. Ratio: 1 1 = 1.) Record. Step 5: Have students look at the four functions and discuss which function has the greatest rate of change. Partner A, which function has the greatest rate of change? (Function 3) Record. Partner B, explain your thinking and defend your answer. (Function 3 has the greatest rate of change of the four functions. The rate of change, or slope, for Function 3 is 6.) Record. CP, IP, WG: Have students turn to S271 in their books. Students will complete Questions 6 10. Students will identify the rate of change and explain the strategy for finding the rate of change. After students have completed this page, review the answers as a whole group. {Verbal Description, Graphic Organizer, Graph, Table, Algebraic Formula} Linear and Non-linear Functions (M, GP, WG, CP, IP) S272, S273, S274 (Answers on T554, T555, T556.) M, GP, CP, WG: Have students turn to S272 in their books. Be sure students know their designation as Partner A or B. Students will analyze functions to determine if they are linear or non-linear. {Graph, Table, Verbal Description, Graphic Organizer, Algebraic Formula} MODELING Linear and Non-linear Functions Step 1: Have student pairs analyze and discuss the four given functions. After looking at the functions with their partner have students share possible similarities with the whole group. Partner A, describe the exponent of the x-value for Function 1 and identify the rate of change. (The input value, x, is raised to the first power and has a rate of change that is 2.) Record in the graphic organizer. Partner B, describe Function 2 and explain how to find the rate of change. (It is a function table. When I divide the difference in y-values of any two points by the difference in x-values of the same two points, the ratio is the same for all points in the table.) Record. 16-7 5-2 = 9 3 = 3
T544 Mathematics Success Grade 8 28-16 9-5 = 12 4 = 3 34-28 11-9 = 6 2 = 3 Partner A, describe Function 3 and explain the rate of change. (The graph of a line that is decreasing at a constant rate and it is easy to see that it is a straight line.) Record in the graphic organizer. Partner B, describe Function 4 and explain the rate of change. (It is a verbal description. This function can be written as y = - 7x + 4) Record. What is the exponent with the variable? (The exponent with the variable of x is a 1.) Record. Step 2: Have students discuss Question 5. Based on the functions above, list the characteristics of the functions we described. Partner A, describe the rate of change of the function. (The function had a constant rate of change.) Record. Partner B, describe the variable used in the function examples. (The function had a variable whose exponent was a 1.) Record. Partner A, describe the graph of the function. (The graph of the function was a straight line.) Record. Partner B, explain what type of line is formed for each of these functions and defend your answer. (If each of the functions were graphed they would form a straight line. This means the functions are linear.) Record. Partner A, how is the rate of change different for Problem 3? (It is a negative value and the line goes down from left to right.) Record. Partner B, explain what you think this means about rate of change. (It can be a positive or negative value.) Step 3: Have student turn to page S273 in their books. Partner A, describe the function in Problem 7. (The input value, x, is raised to the second power.) Record in the graphic organizer. Partner B, describe the function in Problem 8. (It is a function table. When I divide the difference in y-values of any two points by the difference in x-values of the same two points, the ratio is different between each set of points in the table.) Record. 25-1 5-1 = 24 4 = 6 64-25 8-5 = 39 3 = 13 81-64 9-8 = 17 1 = 17
Mathematics Success Grade 8 T545 Partner A, describe the function in Problem 9. (The graph is curved and does not show a constant increase or decrease. It shows different rates of change throughout. This graph is not a straight line.) Record. Partner B, describe the function in Problem 10. (It is a verbal description. This function can be written as 4x 2 + 5.) Record. What is the exponent with the variable? (The exponent with the variable of x is 2.) Record. Step 4: Have students discuss Question 11. Based on the functions above, list the characteristics of the functions we described. Partner A, describe the rate of change of the function. (The function did not have a constant rate of change.) Record. Partner B, describe the variable used in the function examples. (The function had a variable whose exponent was a 2.) Record. Partner A, describe the graph of the function. (The graph of the function was a curve. It was not a straight line.) Record. Partner B, explain what the graph of each function in Questions 7-10 would look like. (They would not form a straight line.) Record. Partner A, what term could be used to describe a function that is not represented by a straight line. (non-linear) Record. CP, IP, WG: Have students work in pairs to complete S274. This graphic organizer asks students to identify whether the functions are either linear or non-linear. If the function is linear, the students should explain how they know that the function is a straight line. If the function is non-linear, explain how they know it is not a straight line. Review answers as a whole group. {Verbal Description, Graphic Organizer, Graph, Table, Algebraic Formula} SOLVE Problem (WG, CP, IP) S275 (Answers on T557.) Have students turn to S275 in their books. This SOLVE problem focuses on students identifying the function and then determining if the function is linear or non-linear. Have students complete the SOLVE problems in student pairs and then review the answers as a whole group. {SOLVE, Graphic Organizer, Verbal Description, Algebraic Formula} If time permits (CP, IP) S276 (Answers on T558.) Have students complete the activity on S276 to work with linear and non-linear functions.