MATH GRADE 8 UNIT LINEAR RELATIONSHIPS
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CONTENTS LESSON 1: PRINT JOB... LESSON : MODELING RUNNING SPEEDS... 5 LESSON 3: INVESTIGATING GRAPHS... 13 LESSON : TRANSLATING GRAPHS... 3 LESSON 5: LINEAR EQUATIONS... 31 LESSON 6: LINEAR RELATIONSHIPS PROJECT... 1 LESSON 7: EQUATIONS, GRAPHS, AND TABLES... LESSON 8: NEGATIVE SLOPE... 50 LESSON 9: PROJECT WORK DAY... 57 LESSON 10: THE EFFECT OF SCALE... 58 LESSON 11: PUTTING IT TOGETHER... 7 LESSON 1: PUTTING IT TOGETHER... 73 Note: Some of these problems are designed to be delivered electronicall. Copright 01 Pearson Education, Inc. 3
LESSON 1: PRINT JOB Write our wonderings about Linear Relationships. Write a goal stating what ou plan to accomplish in this unit. Based on our previous work, write three things ou will do differentl during this unit to increase our success. Copright 01 Pearson Education, Inc.
LESSON : MODELING RUNNING SPEEDS Use the graph to answer Eercises 1 5. The graph represents four runners competing in the 100-m dash. 100 A B C D Distance (m) 80 60 0 0 5 10 15 0 Time (sec) 1. Who ran the 100-m dash the fastest? A Runner A B Runner B C Runner C D Runner D. At what time does Runner D cross the 80-m mark? A 8 sec. B 11 sec. C 13 sec. D 16 sec. Copright 01 Pearson Education, Inc. 5
LESSON : MODELING RUNNING SPEEDS 3. What is the rate (in distance over time) for Runner C? A 5 m/sec B 5.5 m/sec C 6 m/sec D 6.5 m/sec. Which is the correct equation representing the graph of Runner A? A = 5 B = 10 C = 15 D = 0 5. a. Compare the times of each runner at 0 m, 0 m, 60 m, and 80 m. You ma want to create a chart to organize our data. b. Create an equation for each runner s graph. Copright 01 Pearson Education, Inc. 6
LESSON : MODELING RUNNING SPEEDS 6. Use the figure to create the equation for a line that goes through as man hoops as possible. First write the equation for our line. Then graph the line to check how man hoops it goes through. Tr to go through at least three hoops in one straight line. 10 8 6 5 10 15 0 6 7. Was our best shot in Eercise 6 a proportional relationship? Eplain wh or wh not. Copright 01 Pearson Education, Inc. 7
LESSON : MODELING RUNNING SPEEDS Challenge Problem 8. Use the graph to find the answers. 8 6 (5, 6) (, 0) 10 8 6 6 8 (0, ) 6 (, 8) 8 10 a. What is the equation for this graph? Eplain how ou determined it. b. Draw the graph of = c. What is the significance of the point (0, )? Copright 01 Pearson Education, Inc. 8
LESSON : MODELING RUNNING SPEEDS ANSWERS 1. A Runner A Runner A ran the fastest because the slope of Runner A s graphs is the steepest.. D 16 sec. The graph for Runner D intersects the 80-m mark at 16 sec. 3. D 6.5 m/sec. 100 m The rate for Runner C is, which is 6.5 m/sec. 16 sec. B = 10 The slope of the line for Runner A is 100 Runner A is = 10. 10 = 10. Therefore, the equation for 5. a. Runner A Runner B Runner C Runner D 0 m sec..8 sec. 3. sec. sec. 0 m sec. 5.6 sec. 6. sec. 8 sec. 60 m 6 sec. 8. sec. 9.6 sec. 1 sec. 80 m 8 sec. 11. sec. 1.8 sec. 16 sec. b. Runner A: = 10 Runner B: = 50 7 Runner C: = 6.5 Runner D: = 5 Copright 01 Pearson Education, Inc. 9
LESSON : MODELING RUNNING SPEEDS 6. Answers will var. Possible answer: One equation that hits three hoops is = 1 5 : 10 8 6 = 1 5 5 10 15 0 6 7. Answers will var. The eplanation should be based on whether the equation starts at the origin. Possible answer: In Eercise 6, the best straight-line shot is a proportional relationship, because it goes through the origin and has a constant slope. Copright 01 Pearson Education, Inc. 10
LESSON : MODELING RUNNING SPEEDS Challenge Problem 8. a. The equation for this line =. The graph crosses the -ais at so the 0 ( ) -intercept is. The slope is the change in over the change in or or. 0 Thus the equation = m + b with a slope of and -intercept of is =. b. = is shown as the green line on the graph: 8 6 10 5 5 6 8 10 c. Possible answers: (0, ) is the -intercept of both lines; it is also where the intersect. Copright 01 Pearson Education, Inc. 11
LESSON : MODELING RUNNING SPEEDS Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 a. 5 b. 6 7 Challenge Problem Eercise 8 a. 8 b. 8 c. I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 1
LESSON 3: INVESTIGATING GRAPHS 1. What is the slope of this line? 10 8 6 5 A Slope = 1 B Slope = C Slope = 3 D Slope = Copright 01 Pearson Education, Inc. 13
LESSON 3: INVESTIGATING GRAPHS. Which graph shows a proportional relationship? A B 3 1 3 1 3 1 1 3 1 3 1 1 1 3 3 3 C D 3 3 1 1 3 1 1 3 1 3 3 1 1 3 1 3 3. Pedra sas, In this proportional relationship, when is 3, is 9. What is the slope of Pedra s proportional relationship? A Slope = 1 3 B Slope = 1 3 C Slope = 3 D Slope = 3 Copright 01 Pearson Education, Inc. 1
LESSON 3: INVESTIGATING GRAPHS. Proportional relationships alwas contain what point? A (1, 1) B (0, 0) C (1, 0) D (0, 1) 5. Use the Graphing tool to create four proportional relationships with the slopes given. A Slope = 1 B Slope = C Slope = 3 5 D Slope = 3 6. Use the table to find the answers. 6 3 0 6 0 6 1 16 0 a. Draw the graph that corresponds with the points in the table. b. Eplain whether the graph is a proportional relationship. c. What is the slope of the graph? d. Where does the line intercept the -ais? 7. a. Draw a linear graph that intersects the -ais at (0, ) and has a slope of 3. b. Write an equation that represents this linear graph. Copright 01 Pearson Education, Inc. 15
LESSON 3: INVESTIGATING GRAPHS Challenge Problem 8. Analze this graph and use it to find the answers. 10 Water Temperatute ( C) 100 80 60 0 0 1 3 5 Heating Time (min.) a. What is the -intercept of this line? b. What is the slope? (Epress the slope in temperature/time.) c. Create an equation that represents this graph. Copright 01 Pearson Education, Inc. 16
LESSON 3: INVESTIGATING GRAPHS ANSWERS 1. B Slope = Slope = rise/run = 1 =. A A graph that shows a proportional relationship must go through the point (0, 0). 3 1 3 1 1 1 3 3 3. C Slope = 3 Slope = rise/run = 9 3 = 3 Copright 01 Pearson Education, Inc. 17
LESSON 3: INVESTIGATING GRAPHS. B (0, 0) Proportional relationships alwas contain the point (0, 0). b. = 6 5 3 c. = 5 3 1 a. = 1 10 9 8 7 6 5 3 1 1 3 5 6 7 8 9 10 3 d. = 3 5 6 a. = 1 b. = c. = 3 5 d. = 3 Copright 01 Pearson Education, Inc. 18
LESSON 3: INVESTIGATING GRAPHS 5. a. 5 0 15 10 5 5 10 b. The graph is a linear graph, but it is not a proportional relationship, because it does not include the origin point (0, 0). c. Slope = d. -intercept = 1 Copright 01 Pearson Education, Inc. 19
LESSON 3: INVESTIGATING GRAPHS 6. a. 10 8 6 5 5 b. = 3 + Challenge Problem 7. a. -intercept = (0, 50) b. Slope = c. = 10 + 50 10, or an increase in 10 for ever heating minute 1min. Copright 01 Pearson Education, Inc. 0
LESSON 3: INVESTIGATING GRAPHS Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 a. 5 b. 5 c. 5 d. 6 a. 6 b. 6 c. 6 d. 7 a. 7 b. Copright 01 Pearson Education, Inc. 1
LESSON 3: INVESTIGATING GRAPHS Challenge Problem Eercise 8 a. 8 b. 8 c. I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc.
LESSON : TRANSLATING GRAPHS 1. When a proportional relationship line is translated, which attribute alwas remains unchanged? A -intercept B -intercept C Slope D Its equation. After a proportional relationship line is translated, does the new line still represent a proportional relationship? Wh or wh not? Copright 01 Pearson Education, Inc. 3
LESSON : TRANSLATING GRAPHS 3. This graph shows the proportional relationship line = 1 being translated to a new line, represented with the equation = 1 +. Which description(s) is accurate about this translation? (There ma be more than one correct answer. Choose all that appl.) 10 8 6 1 = + = 1 10 5 5 10 6 A The line was translated units upward. B The line was translated units downward. C The line was translated 8 units to the left. D The line was translated 8 units to the right. E The line maintains the same slope. Copright 01 Pearson Education, Inc.
LESSON : TRANSLATING GRAPHS. This graph shows the proportional relationship line = 3 being translated to a new line. Which is the correct equation for the new line? 6 10 5 5 10 = 3? 6 8 A = 3 3 B = 3 + 3 C = 3 6 D = 3 + 6 5. Use the Graphing tool to find the answers. a. Translate the line = 1 3 downward b units. What is the equation for the new line? b. Translate the line = + upward b 3 units. What is the equation for the new line? c. Translate our new line from part b to the right units. What is the equation of the third line? Copright 01 Pearson Education, Inc. 5
LESSON : TRANSLATING GRAPHS 6. The following graph started as a proportional relationship. How must it have been translated to get to its current state? 8 6 6 6 8 Challenge Problem 7. a. Draw five different lines that could result from = 1 + b, where b could be an real number. b. How are the lines ou created related? Copright 01 Pearson Education, Inc. 6
LESSON : TRANSLATING GRAPHS ANSWERS 1. C Slope The slope remains the same when a line is translated.. No, the resulting line will not be a proportional relationship because it no longer contains the point (0, 0). 3. A The line was translated units upward. C The line was translated 8 units to the left. E The line maintains the same slope. The translated line is 8 units to the left of the original line; the translated line is also units above the original line. The slope of the translated line remains the same as the slope of the original line.. C = 3 6 The new line is 6 units below the original line, so the equation is = 3 6. 5. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. Copright 01 Pearson Education, Inc. 7
LESSON : TRANSLATING GRAPHS EXAMPLE OF A -POINT RESPONSE: points a. 6 5 5 6 1 The equation for the new line is = ( ) 3 b. The equation for the new line = + 7. c. The equation for the third line = ( ) + 7 or = + 15. 8 Up 3 and the Right 6 Up 3 5 5 10 Copright 01 Pearson Education, Inc. 8
LESSON : TRANSLATING GRAPHS 6. Answers will var. Possible answer: Since a proportional relationship must include the origin point, ou know that this graph must have been translated up units, since its current -intercept is at (0, ). Challenge Problem 7. a. Possible graph: 3 1 1? 3 b. Answers will var. Possible answers: All of the lines are parallel. The are all translations of the same line. From = 1, ou can go up or down, left or right, an amount and the resulting equation will fit. The equation = 1 + b represents all the possible vertical translations of = 1. Copright 01 Pearson Education, Inc. 9
LESSON : TRANSLATING GRAPHS Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 a. 5 b. 5 c. 5 d. 6 Challenge Problem Eercise 7 a. 7 b. I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 30
LESSON 5: LINEAR EQUATIONS 1. Which equation correctl represents this line? 10 8 6 5 5 A = 5 + B = 5 + C = + 5 D = + 5 Copright 01 Pearson Education, Inc. 31
LESSON 5: LINEAR EQUATIONS. Which equation correctl represents this line? 6 5 5 6 A = 5 3 B = 3 5 C = 3 5 D = 3 5 + 3 Copright 01 Pearson Education, Inc. 3
LESSON 5: LINEAR EQUATIONS 3. Which line correctl matches this equation: = 1 +? 8 6 5 5 6 8 A The blue line B The green line C The purple line D The red line. What properties do ou know about a line that has an equation in slope-intercept form with a negative m-value and a positive b-value? (There ma be more than one correct answer. Choose all that appl.) A The slope is positive (upward to the right). B The slope is negative (downward to the right). C The -intercept is positive. D The -intercept is negative. E The line goes through the origin point (0, 0). Copright 01 Pearson Education, Inc. 33
LESSON 5: LINEAR EQUATIONS For Eercises 5 7, graph each equation. Then find the slope and -intercept for each. 5. = 1 3 + 6. = + 1 7. = 3 Challenge Problem 8. There is a relation between two quantities. The relation is linear, but it is not proportional. The slope is the same as a proportional relationship that has a ratio of (on the -ais) to 5 (on the -ais). The relation fits the formula = m + 3. Draw the graph of this relation, based on the description. Copright 01 Pearson Education, Inc. 3
LESSON 5: LINEAR EQUATIONS ANSWERS 1. D = + 5 The -intercept is 5, and the slope is positive.. B = 3 5 The -intercept is, and the slope is 3 5. 3. B The blue line The intercept is, and the slope is negative.. B The slope is negative (downward to the right). C The -intercept is positive. In slope-intercept form ( = m + b), m represents the slope, and b represents the -intercept. Copright 01 Pearson Education, Inc. 35
LESSON 5: LINEAR EQUATIONS 5. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. EXAMPLE OF A -POINT RESPONSE: points 6 1 = + 3 15 10 5 5 Slope = 1 3 -intercept = Copright 01 Pearson Education, Inc. 36
LESSON 5: LINEAR EQUATIONS 6. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. EXAMPLE OF A -POINT RESPONSE: points 6 = + 1 5 5 Slope = -intercept = 1 Copright 01 Pearson Education, Inc. 37
LESSON 5: LINEAR EQUATIONS 7. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. EXAMPLE OF A -POINT RESPONSE: points 3 = 10 5 5 Slope = 3 -intercept = Copright 01 Pearson Education, Inc. 38
LESSON 5: LINEAR EQUATIONS Challenge Problem 8. Interpret the ratio to determine the slope. A on the -ais and a 5 on the -ais indicate a slope of 5. So the equation of the mster relation is = 5 + 3. 6 5 5 Copright 01 Pearson Education, Inc. 39
LESSON 5: LINEAR EQUATIONS Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 6 7 Challenge Problem Eercise 8 I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 0
LESSON 6: LINEAR RELATIONSHIPS PROJECT Continue to work on our project. Check that ou have addressed all the rubric criteria for our project. Complete an eercises that ou have not finished from this unit. Copright 01 Pearson Education, Inc. 1
LESSON 7: EQUATIONS, GRAPHS, AND TABLES For Eercises 1, match the equations to a graph and a table. Equations 1. = + 3. = 1 3. = 3 5. = 3 + 1 Copright 01 Pearson Education, Inc.
LESSON 7: EQUATIONS, GRAPHS, AND TABLES Graphs 6 6 6 6 Copright 01 Pearson Education, Inc. 3
LESSON 7: EQUATIONS, GRAPHS, AND TABLES Tables I II III IV 1 0 1.5 1 0.5 1 0 1 3..6 1. 0.8 1 0 1 1 0.5 0 0.5 1 1 0 1 1 1 3 5 7 5. Use this table to create a graph and an equation that represent the linear relationship. 0 7 1 5 6. Marshall is tracking the height of a tomato plant in his garden. It starts at 5 cm tall After each week, it grows another cm. Create a table, graph, and equation that all represent this scenario. Copright 01 Pearson Education, Inc.
LESSON 7: EQUATIONS, GRAPHS, AND TABLES 7. Use this graph to create a table and an equation that represent the same linear relationship. 8 6 10 8 6 6 8 6 8 10 Challenge Problem 8. a. On one coordinate plane, draw the graphs corresponding to these two equations: = 3 = 8(6 ) b. Create a table of values for each equation in part a. Be sure to include the - and -intercepts in each table. c. At what point do the two graphs intersect? Copright 01 Pearson Education, Inc. 5
LESSON 7: EQUATIONS, GRAPHS, AND TABLES ANSWERS 1. Graph C, Table IV -intercept = 3, slope = ; points (0, 3) and (1, 5). Graph B, Table III -intercept = 0, slope = 1 ; points (0, 0) and (, 1) 3. Graph D, Table II -intercept =, slope = 3 5 ; points (5, 1) and (0, ). Graph A, Table I -intercept = 1, slope = 3 ; points (0, 1) and (, ) 5. 6 3 = 1 10 5 5 6 8 The -intercept is 1. The slope can be calculated using an two points. For eample, ( 1) 3 using the points (, ) and (0, 1), the slope is =. 0 Copright 01 Pearson Education, Inc. 6
LESSON 7: EQUATIONS, GRAPHS, AND TABLES 6. Weeks 0 1 3 5 Height (cm) 5 7 9 11 13 15 16 1 Height (cm) 1 10 8 = + 5 6 Weeks 6 8 7. Possible table 0 6 8 10 3 0.5.5 7 9.5 = ( 5 ) 3 Copright 01 Pearson Education, Inc. 7
LESSON 7: EQUATIONS, GRAPHS, AND TABLES Challenge Problem 8. a. = 3 0 30 = 8(6 ) 0 10 10 5 5 10 10 b. = 3 : 0 6 8 10 3 16 8 0 8 Includes the -intercept at (0, 3) and the -intercept at (8, 0). = 8(6 ): 0 6 8 10 8 3 16 0 16 3 Includes the -intercept at (0, 8) and the -intercept at (6, 0). c. The intersection point is at (, 16). Copright 01 Pearson Education, Inc. 8
LESSON 7: EQUATIONS, GRAPHS, AND TABLES Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 6 7 Challenge Problem Eercise 8 a. 8 b. 8 c. I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 9
LESSON 8: NEGATIVE SLOPE Eercises 1 refer to the lines on this graph. 0 18 16 1 Height (cm) 1 10 8 6 5 10 15 1. What is the slope and -intercept of the orange line?. What is the slope and -intercept of the purple line? 3. What is the slope and -intercept of the green line?. What is the slope and -intercept of the blue line? Copright 01 Pearson Education, Inc. 50
LESSON 8: NEGATIVE SLOPE 5. How can ou tell when a slope is negative? Use a right triangle diagram on a line to show what a negative slope looks like. 6. Without graphing, determine which of these equations will have a negative slope. Eplain how ou can find the answers just b looking at the equation. a. = 3 + 5 b. = c. = 6 3 7. a. Draw the graph of this formula for maimum heart rate: = 00 1 age. Use a coordinate plane with age on the -ais and heart rate on the -ais. b. What are the coordinates of the -intercept? What is the slope epressed in bpm/age? (Remember, bpm is beats per minute.) c. A woman is 0 ears old. What is her maimum heart rate using this formula? Challenge Problem 8. The maimum heart rate for a good athlete under the age of 30 can be determined using a combination of scientific results: Ma HR = 17 age Subtract 3 beats when training for running. Subtract 3 beats when training for rowing. Subtract 5 beats when training for biccling. Calculate our ma heart rate for running, rowing, and biking. Copright 01 Pearson Education, Inc. 51
LESSON 8: NEGATIVE SLOPE ANSWERS 1. Slope = 5, -intercept = 0 Two points on the red line are (, 0) and (0, 0). The slope can be calculated as: 0 0 0 = = 5 0. Slope = 3, -intercept = 15 Two points on the purple line are (5, 0) and (0, 15). The slope can be calculated as: (0 15)/(5 0) = 15 5 = 3 3. Slope = 1, -intercept = 8 Two points on the green line are (16, 0) and (0, 8). The slope can be calculated as: 0 8 8 1 = = 16 0 16. Slope = 1 3, -intercept = 6 Two points on the blue line are (18, 0) and (0, 6). The slope can be calculated as: 0 6 6 1 = = 18 0 18 3 Copright 01 Pearson Education, Inc. 5
LESSON 8: NEGATIVE SLOPE 5. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. EXAMPLE OF A -POINT RESPONSE: points A line with negative slope alwas goes downward as increases to the right. Here is a good eample of a line with slope 3 5. 1 6 1 3 = 3 5 3 5 Copright 01 Pearson Education, Inc. 53
LESSON 8: NEGATIVE SLOPE 6. Equations a and b have negative slopes, but equation c does not. Possible eplanation: You can tell when a slope will be negative based on the m-value of the equation. If the m-value is negative, then the graph will have a negative slope. 7. a. 00 180 1 = 00 160 10 Heart Rate (bpm) 10 100 80 60 0 0 0 0 60 80 Age b. -intercept = (0, 00); slope = 1 bpm/age c. = 00 1 (0) = 00 0 = 180 The 0-ear-old woman s maimum heart rate is 180 bpm. Copright 01 Pearson Education, Inc. 5
LESSON 8: NEGATIVE SLOPE Challenge Problem 8. Answers will var based on age. For a 13-ear-old student who considers herself a good athlete: Running: 17 13 3 = 01 bpm Rowing: 17 13 3 = 01 bpm Biccling: 17 13 5 = 199 bpm Copright 01 Pearson Education, Inc. 55
LESSON 8: NEGATIVE SLOPE Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 6 a. 6 b. 6 c. 7 a. 7 b. 7 c. Challenge Problem Eercise 8 I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 56
LESSON 9: PROJECT WORK DAY Continue to work on our project. Check that ou have addressed all the rubric criteria for our project. Complete an eercises that ou have not finished from this unit. Copright 01 Pearson Education, Inc. 57
LESSON 10: THE EFFECT OF SCALE 1. What is the slope of this line? 3 1 10 1 A Slope = 1 5 B Slope = 1 5 C Slope = D Slope = 10 Copright 01 Pearson Education, Inc. 58
LESSON 10: THE EFFECT OF SCALE. What is the slope of this line? 5 0 15 10 5 1 3 5 5 A Slope = 1 0 B Slope = 1 10 C Slope = 1 5 D Slope = 1 Copright 01 Pearson Education, Inc. 59
LESSON 10: THE EFFECT OF SCALE 3. What is the slope-intercept equation for this line? 5 0 15 10 5 6 8 5 A = 6 B = + 3 C = 6 + 3 D = 3 Copright 01 Pearson Education, Inc. 60
LESSON 10: THE EFFECT OF SCALE. What is the slope-intercept equation for this line? 1 0.8 0.6 0. 0. 10 5 5 10 0. A = 0.1 + 0.5 B = + 0.5 C = D = 0.5 Copright 01 Pearson Education, Inc. 61
LESSON 10: THE EFFECT OF SCALE 5. Look at the two graphs. 1 1 10 8 6 5 5 8 6 Copright 01 Pearson Education, Inc. 6
LESSON 10: THE EFFECT OF SCALE a. Calculate the slope of each line. b. Write a slope-intercept equation for each line. c. Redraw both graphs on a standard coordinate plane (with both aes scaled from 10 to 10) d. Eplain the similarities and differences between the two graphs. 6. The line in this graph seems to make an angle of about 5 degrees with the -ais, which usuall indicates a slope of ±1. 6 5 3 1 10 9 8 7 6 5 3 1 1 3 5 6 7 8 9 3 5 6 7 Find the formula for this graph. What is the slope? Copright 01 Pearson Education, Inc. 63
LESSON 10: THE EFFECT OF SCALE 7. This graph consists of five linear parts. Each part of the graph has a duration of 6 sec. There are two different slopes in the graph. 0 15 Value 10 5 5 10 15 0 5 30 Time a. What is the slope for the increasing parts? b. What is the slope for the decreasing parts? c. Give a formula for each part of the graph. Challenge Problem 8. a. Draw the points (0, 0) and (30, 60) in a coordinate plane. Draw the straight line through these points. b. In the same coordinate sstem, draw the points (0, 50) and (30, 60). Draw the straight line through these points. c. Draw the graph of = 1 3 + 10. d. What, if an, is the relation between the three graphs? Copright 01 Pearson Education, Inc. 6
LESSON 10: THE EFFECT OF SCALE ANSWERS 1. B Slope = 1 5 Two points on the line are (0, ) and (10, 0). The -intercept is. The slope is calculated as: 0 = 10 0 10 = 1 5. D Slope = 1 Two points on the line are (5, 5) and (0, 0). The -intercept is 0. The slope is 5 0 calculated as: = 5 5 0 5 =1 3. C = 6 + 3 Two points on the line are ( 1, 0) and (0, 3). The -intercept is 3. The slope is 3 0 0 0.5 = 3 0.5 =6 calculated as: ( ). A = 0.1 + 0.5 Two points on the line are (0, 0.5) and (5, 0). The -intercept is 0.5. The slope is 0 0.5 ( ) calculated as: = 0.5 = 0.1 5 0 5 5. RUBRIC points Response shows a sound approach, a correct answer, and good communication of the process b which the answer was obtained. 3 points Response shows a sound approach, but errors along the wa ma result in an incorrect response. Work is clearl shown, but some of the detail of the steps taken ma be incomplete. points Response has an approach that is flawed but carried through appropriatel, with work shown to document what was done. 1 point Response begins to take an inappropriate approach and does not follow through well, with work shown being potentiall sketch. Or the correct answer is shown, but no communication is given about how the solution was obtained. Copright 01 Pearson Education, Inc. 65
LESSON 10: THE EFFECT OF SCALE EXAMPLE OF A -POINT RESPONSE: points a. Both lines have a slope of. b. The slope-intercept form of the equation for both lines is = + 3. c. 10 8 6 5 5 6 8 10 d. Both graphs represent the same equation, the therefore have the same slope and -intercept and contain all the same points. The different scaling of the aes make the two graphs appear to have different slopes, though. Copright 01 Pearson Education, Inc. 66
LESSON 10: THE EFFECT OF SCALE 6. You could estimate the slope as being closer to 3 rather than 1. The equation could be something like = 3 + 1. 7. a. The slope of the increasing parts is 1. You can use the endpoints to determine this. For eample, the first section goes from (0, 0) to (6, 6). b. The slope of the decreasing parts is 1. You can again use the endpoints to 6 determine this. For eample, the second section goes from (6, 6) to (1, 5). c. The formulas will all include either the positive slope of 1 or the negative slope of 1 6. (0, 0) to (6, 6): = (6, 6) to (1, 5): = 1 6 + 7 (1, 5) to (18, 11): = 7 (18, 11) to (, 10): = 1 6 + 1 (, 10) to (30, 16): = 1 Copright 01 Pearson Education, Inc. 67
LESSON 10: THE EFFECT OF SCALE Challenge Problem 8. Answers will var. Possible answer: Analze the equations for all three lines: 70 60 (30, 60) (0, 50) 50 0 (0, 0) 30 0 10 1 = + 10 3 60 0 0 0 6 8 The line for part a can be written as = 3 + 0. The line for part b can be written as = 1 3 + 50. The line for part c was given as = 1 3 + 10 Copright 01 Pearson Education, Inc. 68
LESSON 10: THE EFFECT OF SCALE The third equation represents the difference between the first two equations: 1 1 + 50 + 0 = + 3 3 3 10 The easiest place to see this on the graph is where the first two equations intersect (the difference between them is 0); the given equation has value 0. You can check an -value for all three equations and the result will be the same. The difference between the first two equations will be the value of the given equation. Copright 01 Pearson Education, Inc. 69
LESSON 10: THE EFFECT OF SALE Self Assessment After ou use the answer ke to check our answers, use the chart below to self-assess our work. For each eercise, place a check mark in the column that best describes how ou did on that eercise. Eercise Number Yes! I got it. I was confused, but now I get it. I need help! 1 3 5 a. 5 b. 5 c. 6 7 a. 7 b. 7 c. Copright 01 Pearson Education, Inc. 70
LESSON 10: THE EFFECT OF SALE Challenge Problem Eercise 8 a. 8 b. 8 c. 8 d. I gave it a tr, but I m not sure I did it right. I did it, and m answer makes sense. Copright 01 Pearson Education, Inc. 71
LESSON 11: PUTTING IT TOGETHER Read through our work on the Self Check task and think about our other work in this lesson. Write what ou have learned. What would ou do differentl if ou were starting the Self Check task now? Record our ideas. Keep track of an strategies ou have learned. Complete an eercises that ou have not finished from this unit. Copright 01 Pearson Education, Inc. 7
LESSON 1: PUTTING IT TOGETHER Read through our work on the Self Check task and think about our other work in this lesson. Write what ou have learned. What would ou do differentl if ou were starting the Self Check task now? Record our ideas. Keep track of an strategies ou have learned. Complete an eercises that ou have not finished from this unit. Copright 01 Pearson Education, Inc. 73