2.3 Up and Down the Staircase Again: PROBLEM Exploring Slope Investigation 2 Linear Models and Equations 31 In any problem that calls for a linear model that is an equation, the goal is to find the values of m and b for the linear equation y = mx + b that fits the data pattern well. Recall from Moving Straight Ahead that the y-intercept is (0, b) or b. Also, for linear relationships, as x increases by 1 unit, y changes by a constant amount. This is called the rate of change or slope, which is represented by m in the equation. The rate of change also describes the steepness of a line. To measure the steepness of a linear graph, it helps to imagine a staircase that lies underneath the line. What is the slope of the line that represents the steepness of the staircase? Another way to think about slope is to look at the ratio of the rise to run in the staircase. SAMPLE
+1 +5 +5 +5 +5 +5 +1 +1 +1 +1 32 Thinking with Mathematical Models Steepness of the Staircase Steepness of the Staircase x y 0 5 1 10 2 15 3 20 4 25 5 30 6 35 0 2 31 5 4 5 6 7 10 15 20 25 30 1 5 3 horizontal change y-intercept 15 vertical change (0, b) y = mx + b y x The change in x from one point to another is also called the horizontal change or run change. The change in y from one point to another is called the vertical change or rise change. The ratio of rise change to run change is the slope of the line which is also the coefficient m in the equation, y = mx + b. slope = vertical change _____________ horizontal change = ___ rise run = m The equation y = mx + b is also called the slope-intercept form of a linear equation. INITIAL CHALLENGE The following are representations for nine linear functions. Linear Function 1 –4 –2 0 2 4 2 4 –2 –4 y x Linear Function 2 –4 –2 0 2 4 2 4 –2 –4 y x Linear Function 3 –4 –2 0 2 4 2 4 –2 –4 y x 2.3 SAMPLE
Investigation 2 Linear Models and Equations 33 Linear Function 4 x −2 −1 0 1 2 y −1 1 3 5 7 Linear Function 5 x −6 −2 2 6 10 y −4 −2 0 2 4 Linear Function 6 x −1 0 1 2 3 y 4 1 −2 −5 −8 Linear Function 7 The line passes through the points (6, 1) and (2, −1). Linear Function 8 The line passes through (0, 3) and (3, 3). Linear Function 9 y = 1 − 3x For each function do the following: • Find the slope and y-intercept. • Write the equation in the form y = mx + b. WHAT IF . . . ? Situation A. Student Claims from Mr. Cai’s Class Mr. Cai asks students to look for patterns in the representations in the Initial Challenge. They make some claims about linear functions. Study each claim. Are they correct? Explain. Jackie’s Claim To find the slope of a line, it makes a difference which two points you use to find the slope. The slope is different if I pick two different points. Jackie’s Linear Function –4 –2 0 2 4 2 4 –2 –4 y x With points (–2, –3) and (2, 3), the vertical change is 6, and the horizontal change is 4. With points (–2, –3) and (0, 0), the vertical change is 3, and the horizontal change is 2. 2.3 SAMPLE
34 Thinking with Mathematical Models Malcolm’s Claim I found the constant rate of change by looking at the change in y-values in a table. For example, in this table, the constant rate of change is 6. Malcom’s Linear Function x y 0 0 2 6 4 12 6 18 Situation B. Connecting Rates and Slopes Juan and Stella were studying a table of values that represents the relationship between the variables x and y and recorded their observations. Are they correct? Explain your reasons. Relationships Between x and y x −1 0 1 2 3 y 3 5 7 9 11 Yvonne’s Claim Both a horizontal line and a vertical line are linear functions. 0 y x 0 y x Gus’s Claim For each unit change in the independent variable there is a constant change in the dependent variable. Lisa’s Claim The slope is the same as the constant rate of change m between the variables x and y. Kara’s Claim The line with equation y = 2x passes through the points (0, 0) and (1, 2). The line with equation y = −3x passes through the points (0, 0) and (1, −3). In general, lines with equations of the form y = mx always pass through the points (0, 0) and (1, m). And m is the constant of proportionality. 2.3 SAMPLE
Investigation 2 Linear Models and Equations 35 NOW WHAT DO YOU KNOW? What information do you need to write an equation for a linear function? How is the constant rate of change for a linear function the same as the ratio of rise/run between two points of the graph of the function? Juan’s Comment I noticed that as x increases by 1, y increases by 2. So this is a linear function. But I don’t think it is possible to use the table to find the ratio of rise to run, which is the slope. Stella’s Comment The relationship is a proportional relationship, and the slope of the graph is the unit rate, or the constant of proportionality. 2.3 SAMPLE
36 Thinking with Mathematical Models We have used linear functions to model linear situations and make predictions. Our understanding of linear relationships or functions can be used to explore some ideas about groups of lines. For example: Suppose the slope of a line is 3. Sketch a line with this slope. Can you sketch a different line with this slope? Explain. In Problem 2.4, we will use slope to explore some patterns among linear relationships. INITIAL CHALLENGE Ms. Chen’s class explores the graphs of some linear equations and found patterns. For each group of equations, answer the questions that follow. Hank’s Group of Equations Pattern Example 1 y = 3x y = 5 + 3x y = 10 + 3x y = 3x − 5 Example 2 y = −2x y = −2x + 4 y = 8 − 2x y = −4 − 2x Gia’s Group of Equations Pattern Example 1 y = 2x y = −_1 2 x Example 2 y = 4x y = −0.25x Example 3 y = −3x + 5 y = _1 3 x − 1 Chris’s Group of Equations Pattern Example 1 y = 2x + 1 y = 2(x + 1) − 1 Example 2 y = 5 − 2x y = 3 − 2(x − 1) Example 3 y = 2(x − 1) y = 4x − 2x − 2 • What features do the equations in each group have in common? • Graph the equations on the same coordinate axes. What patterns do you notice about the graphs? How could you predict this from the equations? • For each group, describe another example that has the same pattern. PROBLEM 2.4 Exploring Patterns with Lines SAMPLE
Investigation 2 Linear Models and Equations 37 Maddie’s Claim The equations for the costs C1 , C2, and C3 represent the same relationship. Mitch’s Claim The graphs of C1 and C2 are parallel. NOW WHAT DO YOU KNOW? How can you predict if two lines are parallel or perpendicular from their equations? WHAT IF . . . ? Situation A. Jose Makes a Claim Jose claims that the following two lines have the same pattern as one of those in the Initial Challenge. Which pattern or pair of lines do these lines belong to? Explain why. Line 1 contains the points (1, 4) and (3, 12). Line 2 contains the points (1, 6) and (3, 14). Situation B. Boat Rentals Again Several competing canoe rentals use expressions to represent the rental cost C in dollars for t minutes. C1 = 2.5 + 0.25t C2 = 0.25(10 + t) C3 = 0.25(10 + 10t) C4 = 0.5(10 + 0.5t) 1. Which plan(s) is the least expensive per minute? Explain why. 2. Describe the shape of each graph. What information about rental costs does the slope represent? 3. Maddie and Mitch each made a claim about the costs. Do their claims make sense? Explain why. 2.4 SAMPLE
38 Thinking with Mathematical Models In this unit, we are exploring linear and nonlinear relationships or functions and how they are used to model real-life phenomena and solve problems. At the end of this investigation, ask yourself: What are the characteristics of linear functions? How might you use what you know about linear functions to model a situation? MR Mathematical Reflection SAMPLE
Investigation 2 Linear Models and Equations 39 INVESTIGATION 2 APPLICATIONS 1. Assume that the relationships in ACE 1 are linear. a. Kaya buys a $20.00 ride card for the local fair. She is charged $1.25 for each ride. What equation gives the value v in dollars left on her card after she takes r rides? b. A typical American baby weighs about 8 pounds at birth and gains about 1.5 pounds per month for the first year of life. What equation relates weight w in pounds to age a in months? c. Dakota lives 1,500 meters from school. She leaves for school, walking at a speed of 60 meters per minute. Write an equation for her distance d in meters from school after she walks for t minutes. d. A car can average 140 miles on 5 gallons of gasoline. Write an equation for the distance d in miles the car can travel on g gallons of gas. 2. Tell whether each table represents a linear relationship. Explain your reasoning. a. x 2 4 6 8 10 12 14 y 0 1 2 3 4 5 6 b. x 1 2 3 4 5 6 7 y 0 3 8 15 24 35 48 c. x 1 4 6 7 10 12 16 y 2 −1 −3 −4 −7 −9 −13 APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) SAMPLE
40 Thinking with Mathematical Models 3. The Business Club has decided to get caps. They look at two different stores. Caps Galore’s ad lists a $20.00 design creation charge and then $3.50 per cap. Hats for Success’s ad lists that each cap is $7.50. Which company should they use? 4. Solve each equation for x using tables, graphs, guess and check, or symbols. a. 3x + 4 = 10 b. 6x + 3 = 4x + 11 c. −3x + 5 = 7 d. 4x − _1 2 = 8 5. Here is a graph of three lines. –4 –2 0 2 4 2 4 –2 –4 y A C B x a. Complete the table. Line Constant Rate of Change y-Intercept A B C b. Match each line with its equation. y = 2 + x y = −4 + 2x y = 3 − x ACE SAMPLE
Investigation 2 Linear Models and Equations 41 6. Find the slope, y-intercept, and equation of each line. a. 2 4 6 8 10 20 4 6 8 10 x y (6, 6) (0, 3) b. 4 8 12 16 20 40 8 12 16 20 x y (6, 6) (2, 18) c. 2 4 6 8 10 20 4 6 8 10 x y (2, 9) (8, 9) d. 4 8 12 16 20 20 4 6 8 10 x y (6, 3) (6, 10) ACE SAMPLE
42 Thinking with Mathematical Models 7. For each equation, do the following: • Make a table of x- and y-values for the equation. • Sketch a graph of the equation. • Find the slope of the line. • Make up a problem that can be represented by the equation. a. y = x b. y = 2x − 2 c. y = −0.5x + 2 8. Write an equation for each line in the following graph. –4 –2 0 2 4 2 4 –2 –4 y x L1 L2 L3 9. Find an equation for the line that satisfies the conditions. a. slope 4.2; y-intercept (0, 3.4) b. slope _ 2 3 ; y-intercept (0, 5) c. slope 2; passing through (4, 12) d. passing through (0, 15) and (5, 3) e. passing through (−2, 2) and (5, −4) f. parallel to the line with equation y = 15 − 2x and passing through (3, 0) 10. Use the line in the graph to answer each question. a. Find the equation of a line that is parallel to this line. b. Find the equation of a line that is perpendicular to this line. –4 –2 0 2 4 2 4 –2 –4 y x ACE SAMPLE
Investigation 2 Linear Models and Equations 43 11. a. Graph a line with slope 3. i. Find two points on your line. ii. Write an equation for the line. b. On the same set of axes, graph a line with slope − _1 3 . i. Find two points on your line. ii. Write an equation for the line. c. Compare the two graphs you made in ACE 11a–b. 12. Use the line in the graph to answer each question. –4 –2 0 2 4 2 4 –2 –4 y x a. Find the equation of a line that is parallel to this line. b. Find the equation of a line that is perpendicular to this line. 13. a. Find the slope of each line. Then write an equation for the line. i. –4 –3 –2 –1 0 1 2 3 4 2 4 6 –2 –4 –6 y x ACE SAMPLE
44 Thinking with Mathematical Models ii. –1 0 1 4 8 12 –4 –8 –12 y x iii. –12 –8 –4 0 4 8 12 100 200 300 –100 –200 –300 y x b. Compare the slopes of the three lines. c. How are the three graphs similar? How are they different? CONNECTIONS 14. Complete each table. Tell whether the relationship is linear, and explain your reasoning. a. y = −3x − 8 x −5 −2 1 4 y b. y = 4(x − 7) + 6 x −3 0 3 6 y ACE SAMPLE
Investigation 2 Linear Models and Equations 45 c. y = x(3x + 2) x −3 0 3 6 y d. y = 4 − 3x x −3 0 3 10 y 15. Madeline sets the scale factor on a copy machine at 150%. She then uses the machine to copy a polygon. Write an equation that relates the perimeter of the polygon after the enlargement, a, to the perimeter before the enlargement, b. 16. The figures below are similar. (They have the same shape but are different sizes.) 2.16 in. 1.28 in. 1.08 in. 1 in. 0.64 in. A B x a. Find the value of x. b. What is the scale factor from Triangle A to Triangle B? c. What is the scale factor from Triangle B to Triangle A? d. How are the scale factors in ACE 16b–c related? 17. One of the most popular items at a farmers market is sweet corn. This table shows relationships among the price of the corn, the supply of corn (how much corn the market has), and the demand for the corn (how much corn people want to buy). Sweet Corn Supply and Demand Price per Dozen ($) 1.00 1.50 2.00 2.50 3.00 3.50 Demand (dozens) 200 175 140 120 80 60 Supply (dozens) 40 75 125 175 210 260 a. Write a linear equation that models the relationship between demand d and price p. b. Write a linear equation that models the relationship between supply s and price p. c. Use graphs to estimate the price for which the supply equals the demand. Then find the price by solving symbolically. ACE SAMPLE
46 Thinking with Mathematical Models 18. You can express the slope of a line in different ways. The slope of the following line is __6 10 , or 0.6. You can also say the slope is 60% because the rise is 60% of the run. 2 4 6 20 4 6 8 10 x y These numbers represent slopes of other lines: ___ −4 −2 60% __ 4 4 1.5 150% 200% a. Which numbers represent the same slope? b. Which number(s) represent the greatest slope? c. Which number(s) represent the least slope? 19. Read the following stories, and look at the graphs. a. Match each story with a graph. Tell how you would label the axes of the graph. Explain how each part of the story is represented in the graph. y x Graph A y x Graph B y x Graph C y x Graph D • Story 1: Brittany flew a kite. The kite stayed in the air for a few minutes. As the kite came down, it got tangled in the branches of a bush. • Story 2: Ella puts some money in the bank. She leaves it there to earn interest for several years. Then one day, she withdraws half the money in the account. • Story 3: Gerry has a big pile of gravel to spread on his driveway. On the first day, he moves half the gravel from the pile to his driveway. The next day, he is tired and moves only half of what is left. The third day, he again moves half of what is left in the pile. He continues in this way until the pile has almost disappeared. b. One of the graphs does not match a story. Make up your own story for that graph. ACE SAMPLE
Investigation 2 Linear Models and Equations 47 20. Suppose the slopes of two lines are the negative reciprocal of each other. Here is an example: y = 2x and y = −_1 2 x What must be true about the two lines? Is your conjecture true if the y-intercept of either equation is not zero? Explain. 21. Write equations for four lines that intersect to form the sides of a parallelogram. Explain what must be true about such lines. 22. Write equations for three lines that intersect to form a right triangle. Explain what must be true about such lines. EXTENSIONS 23. Recall that Custom Steel Products builds trusses from steel pieces. Here is a 7-foot truss. 7-foot truss made from 27 rods a. Which of these formulas represent the relationship between truss length L in feet and number of pieces r? r = 3L r = L + (L − 1) + 2L r = 4(L − 1) + 3 r = 4L − 1 b. How might you have reasoned to come up with each formula? 24. Multiple Choice Recall that Custom Steel Products uses steel pieces to make staircase frames. Here are staircase frames with 1, 2, and 3 steps. 1 step made from 4 rods 2 steps made from 10 rods 3 steps made from 18 rods ACE SAMPLE
48 Thinking with Mathematical Models Which of these formulas represent the relationship between the number of steps n and number of pieces r? A. r = n2 + 3n B. r = n(n + 3) C. r = n2 + 3 D. r = (n + 3)n 25. Custom Steel Products (CSP) builds cubes out of square steel plates measuring 1 foot on a side. Below is a 1-foot cube. 1 ft. 1 ft. 1 ft. a. How many square plates are needed to make a 1-foot cube? b. Multiple Choice Suppose CSP wants to triple the dimensions of the cube. How many times greater than the original surface area will the surface area of this larger cube be? A. 2 B. 3 C. 4 D. 9 c. Multiple Choice Suppose CSP triples the dimensions of the original cube. How many times the volume of the original cube is the volume of the new cube? A. 8 B. 9 C. 27 D. 81 26. A bridge painting company uses the formula C = 5,000 + 150L to estimate painting costs. C is the cost in dollars, and L is the length of the bridge in feet. To make a profit, the company increases a cost estimate by 20% to arrive at a bid price. For example, if the cost estimate is $10,000, the bid price will be $12,000. ACE SAMPLE
Investigation 2 Linear Models and Equations 49 a. Find bid prices for bridges 100 feet, 200 feet, and 400 feet long. b. Write a formula relating the final bid price to bridge length. c. Use your formula to find bid prices for bridges 150 feet, 300 feet, and 450 feet long. d. How would your formula change if the markup for profit were 15% instead of 20%? 27. At Dominique’s Auto Detailing, car washes cost $5 for any time up to 10 minutes plus $0.40 per minute after that. The managers at Dominique’s are trying to agree on a formula for calculating the cost c in dollars for a t-minute car wash. Dominique’s Auto Detailing a. Sid thinks c = 0.4t + 5 is correct. Is he right? b. Tina proposes the formula c = 0.4(t − 10) + 5. Is she right? c. Jamal says Tina’s formula c = 0.4(t − 10) + 5 can be simplified to c = 0.4t + 1. Is he right? ACE SAMPLE
50 In Investigation 1, we explored the relationship of strength, number of layers, and length of a bridge. We found that the relationship between strength and number of layers was approximately linear. We also found that the relationship between strength and length was not linear. To understand linear functions, it is important to recognize nonlinear situations. Number of Layers Breaking Weight (pennies) 20 40 20 Bridge Thickness Experiment 4 y x Length (in.) Breaking Weight (pennies) 20 40 60 40 Bridge Length Experiment 8 12 y x In this investigation, we will explore other linear and nonlinear functions. Inverse Variation: Linear or Nonlinear? INVESTIGATION 3 SAMPLE
Investigation 3 Inverse Variation: Linear or Nonlinear? 51 In the Covering and Surrounding unit, to study the relationship between area and perimeter, we looked at rectangular plots of lands that had a fixed area. In this problem, we will revisit a similar situation and explore whether the relationship between the length and width of rectangles with fixed area is linear. INITIAL CHALLENGE Roseville is building a new development. The city planners decide that the lots should be rectangular with an area of 24,000 square feet. The planners invite the Business Club to help find a quick way to check lot sizes to see which rectangular shapes to use. They suggest investigating rectangles that have a fixed area of 24 square feet. Then they will look for patterns in the data to make predictions about the rectangles that have an area of 24,000 square feet. To decide on which rectangle to use, help the club by doing the following: • Look at all the rectangles that have whole-number dimensions and an area of 24 square feet. • Create a table and graph for the length and width of rectangles that have an area of 24 square feet. • Write an equation that represents the relationship between the length l and width w of rectangles with area 24 square feet. Is this a linear function? Explain. • How might this information be useful to Roseville planners? WHAT IF . . . ? Situation A. More Rectangles Before making decisions, the Business Club checks their data with rectangles that have an area of 32 square feet. 1. Make a table and a graph of values for width w and length l of a rectangle with an area of 32 square feet. 2. Write an equation for the relationship between the width w and the length l. 3.1 PROBLEM Rectangles with Fixed Area SAMPLE
52 Thinking with Mathematical Models 3.1 3. Compare tables, graphs, and equations for the functions relating width and length of rectangles with area 24 square feet and area 32 square feet. 4. What can you say about the perimeters of the rectangles? 5. Use these results and those from the Initial Challenge to describe the shape of rectangular lots with an area of 24,000 square feet. Which would you choose? Explain why. Situation B. Niral’s Claims Niral’s Claims The relationship between length and width for rectangles with a fixed area, like 24 square feet, is not a linear function. But if I look at the area of rectangles with widths of 15 feet, then the relationship between the area and length for rectangles is a linear function. 1. What evidence can you use to show his claims are or are not correct? 2. If we fix the width to be 4 feet, is the relationship between the area and length linear? Explain. NOW WHAT DO YOU KNOW? What examples using the variables of area, length, and width show one variable related to another is a nonlinear relationship? Is a linear relationship? Explain. SAMPLE
Investigation 3 Inverse Variation: Linear or Nonlinear? 53 To understand a concept like linear functions, we will continue to look at nonlinear situations. The relationship between length l and width w for rectangles with a fixed area A is a function, but it is not a linear function. As l increases, w decreases, but not at a constant rate. The graph is a curve showing how l and w relate to each other. From our work with Problem 3.1 and the Covering and Surrounding unit, we know that the area A of a rectangle is equal to length l times width w or A = lw If an area A is fixed, then the relationship is not linear. If l or w is fixed, is the relationship is linear? The relationship between l and w in Problem 3.1 is an example of an important nonlinear relationship called an inverse variation. The word inverse suggests that as one variable increases in value, the other variable decreases in value. However, the meaning of inverse variation is more specific than that. Two nonzero variables x and y are related by an inverse variation if y = _ k x or xy = k where k is a constant other than 0. The value of k is determined by the specific relationship. How are the equations y = _ k xand xy = k related? For the same x-value, will the two equations give different y-values? Inverse variation occurs in many situations. For example, consider the table and graph below. They show the (bridge length, breaking weight) data collected by a group of students. 3.2 PROBLEM Distance, Speed, and Time SAMPLE
54 Thinking with Mathematical Models Bridge Breaking Experiment Data Table Bridge Length (in.) Breaking Weight (pennies) 4 50 6 27 8 22 10 16 Graph Bridge Length (in.) 10 20 30 40 50 60 70 80 90 50 0 2 4 6 8 y x Breaking Weight (pennies) 10 Some students tried to find an equation that fits the inverse data and wrote 160 = bl, where b is the breaking weight in pennies and l is the length in inches. Does this equation make sense? Why? Is there an equation that fits the data better? Explain. In the following problem, we will investigate how the length of a trip depends on speed. INITIAL CHALLENGE The Business Club is in Los Angeles, California. They are planning to attend the Young Entrepreneurs conference in Mexicali, Mexico. They need information on how long it would take to drive to the conference. They record some times in a table. Travel Times for Different Speeds Average Speed (miles per hour) 30 40 50 60 70 Travel Time (hours) 10 7.5 6 5 4.3 • Describe the change in travel time as the average speed increases. • If the distance is 300 miles, what equation relates travel time t to average speed s? • How is the pattern relating travel time to average speed shown in a graph of (s, t) data? • Is the relationship between travel time and average speed a function? A linear function? Explain why or why not. 3.2 SAMPLE
Investigation 3 Inverse Variation: Linear or Nonlinear? 55 WHAT IF . . . ? Situation A. Driving 50 Miles per Hour to Mexicali Suppose Mr. Cordova, who is the club’s advisor, decides to aim for an average speed of 50 miles per hour for the trip to Mexicali. 1. Make a table and a graph to show how the distance traveled changes as time passes from when they leave Los Angeles to when they reach Mexicali. 2. Write an equation relating distance traveled d to time t. 3. Is the relationship between distance and time a linear function? Explain why or why not. Situation B. Mr. Cordova’s Sister 1. Mr. Cordova’s sister also lives in Los Angeles and often drives to Mexicali. She claims it takes 5 hours to make the drive. What is her average speed if she makes the drive in 5 hours? 2. Write an equation relating distance traveled d to speed s. 3. Is the relationship between the variables d and s a linear function? Explain. NOW WHAT DO YOU KNOW? What examples using distance, speed, and time show one variable related to another with an inverse variation relationship? A linear relationship? Explain why. 3.2 SAMPLE
PROBLEM 3.3 Planning a Fundraising Event: Finding Individual Costs 56 Thinking with Mathematical Models In this problem, we use a new situation to continue to explore whether relationships are linear or nonlinear. INITIAL CHALLENGE The Business Club is planning a fundraising event at a local nature center. It costs $750 to rent the center facilities. They want a way to determine what to charge per person to be sure to cover the rental cost of $750. Investigate the relationship between cost per person c in dollars and number of people n who buy tickets. • Describe a way to represent this relationship. • Use this relationship to help you decide how much money they should charge per person. What did they charge? How many people must buy tickets to make a profit? WHAT IF . . . ? Situation A. Strategies from Club Members The following strategies were used to answer the question in the Initial Challenge. For each strategy, explain how it will or will not help answer the questions. Zane’s Group Our group used a table to determine how much money we would have to charge each person just to cover our rental cost of $750. Cost Per Person for Different Numbers of People Number of People 50 100 150 200 250 300 350 Cost per Person ($) SAMPLE
3.3 Investigation 3 Inverse Variation: Linear or Nonlinear? 57 Molada’s Group We tried to write an equation that relates the cost per person c in dollars to the number of people n who buy tickets. But we could not find an equation, so we made a graph of the data and used it to make predictions. Xander’s Group We made a table. We could not see a constant rate of change, which means it is not linear. To be sure, we made a graph of the data. It looked a bit like our bridge length data. We found an inverse variation equation to fit the data. Situation B. A Fixed Cost The students decide to charge $5 per person for the event. They want to know › the amount of money A in dollars they collect if n people buy tickets; and › the amount of the profit P in dollars they will have after they pay for the rental cost. 1. Write an equation that will help them predict the amount of money they collect if they charge $5 per person. 2. Write an equation that will help them predict the profit if they charge $5 per person. 3. Sketch a graph of each equation. 4. Do the graphs indicate a linear relationship or an inverse variation? Explain. 5. How would you use the graphs to show how many tickets must be sold to make a profit? How would you use the equations to decide how many tickets must be sold to make a profit? Situation C. Linear, Inverse, or Other? A local business donated $500 to use for expenses for the fundraising event. The following table shows the amount of money in dollars in the account over several weeks. Is the relationship between the amount of money in the account and the number of weeks a linear or an inverse function? Or neither? Explain why. Account Balance for Different Weeks Balance in Account ($) 500 400 370 300 250 350 300 100 Number of Weeks 0 1 2 3 4 5 6 7 SAMPLE
58 Thinking with Mathematical Models NOW WHAT DO YOU KNOW? When a problem involves three variables, x, y, and z, where z = xy, when is the relationship linear? When is the relationship an inverse variation? Use examples from the investigation to help you explain. 3.3 SAMPLE
Investigation 3 Inverse Variation: Linear or Nonlinear? 59 In this unit, we are exploring linear and nonlinear relationships or functions and how they are used to model real-life phenomena and solve problems. At the end of this investigation, ask yourself: What are the characteristics of linear functions? How might you use what you know about linear functions to model a situation? Mathematical Reflection MR SAMPLE
INVESTIGATION 3 APPLICATIONS 1. Consider rectangles with an area of 16 square inches. a. Copy and complete the table. Rectangles with Area 16 in.2 Length (in.) 1 2 3 4 5 6 7 8 Width (in.) b. Make a graph of the data. c. Describe the pattern of change in width as length increases. d. Write an equation that shows how the width w depends on the length l. Is the relationship linear? 2. Consider rectangles with an area of 20 square inches. a. Make a table of length and width data for at least five rectangles. b. Make a graph of your data. c. Write an equation that shows how the width w depends on the length l. Is the relationship linear? d. Compare and contrast the graphs in this ACE and those in ACE 1. e. Compare and contrast the equations in this ACE and those in ACE 1. 3. A student collected these data from the bridge length experiment. Bridge Length Experiment Length (in.) 4 6 8 9 10 Breaking Weight (Pennies) 24 16 13 11 9 a. Find an inverse variation equation that models the data. b. Explain how your equation shows that breaking weight decreases as length increases. Is this decrease reasonable for the situation? Explain. APPLICATIONS—CONNECTIONS—EXTENSIONS (ACE) 60 Thinking with Mathematical Models SAMPLE
For ACE 4–7, tell whether the relation between x and y is an inverse variation. If it is, write an equation for the relationship. 4. x 1 2 3 4 5 6 7 8 9 10 y 10 9 8 7 6 5 4 3 2 1 5. x 1 2 3 4 5 6 7 8 9 10 y 48 24 16 12 9.6 8 6.8 6 5.3 4.8 6. x 2 3 5 8 10 15 20 25 30 40 y 50 33 20 12.5 10 6.7 5 4 3.3 2.5 7. x 0 1 2 3 4 5 6 7 8 9 y 100 81 64 49 36 25 16 9 4 1 8. A marathon is a 26.2-mile race. The best marathon runners can complete the race in a bit more than 2 hours. a. Make a table and graph that show how the average running speed for a marathon changes as the time to complete the race increases. Show times from 2 to 8 hours in 1-hour intervals. b. Write an equation for the relationship between time t in hours and average running speed s in miles per hour for a marathon. c. Tell how the average running speed changes as the time increases from 2 hours to 3 hours, from 3 hours to 4 hours, and from 4 hours to 5 hours. d. How do the answers for ACE 8c show that the relationship between average running speed and time is not linear? 9. The route for one day of a charity bike ride covers 50 miles. Individual participants ride this distance at different average speeds. a. Make a table and a graph that show how the riding time changes as the average speed increases. Show speeds from 4 to 20 miles per hour in intervals of 4 miles per hour. b. Write an equation for the relationship between the riding time t in hours and average speed s in miles per hour. c. Tell how the riding time changes as the average speed increases from 4 to 8 miles per hour, from 8 to 12 miles per hours, and from 12 to 16 miles per hour. d. How do the answers for ACE 9c show that the relationship between average speed and time is not linear? ACE Investigation 3 Inverse Variation: Linear or Nonlinear? 61 SAMPLE
ACE CONNECTIONS 10. Here are some possible descriptions of a line. For each equation, list all of the properties that describe the graph of that equation. Descriptions of a Line • slope positive, 0, or negative • y-intercept positive, 0, or negative • crossing the x-axis to to right of the origin • passing through the origin at (0, 0) • crossing the x-axis to the left of the origin • never crossing the x-axis a. y = x b. y = 2x + 1 c. y = −5 d. y = 4 − 3x e. y = −3 − x 11. Write equations and sketch the graphs of lines with the following properties. a. slope of 3.5, y-intercept at (0, 4) b. slope _ 3 2 , passing through (−2, 0) c. passing through the points (2, 7) and (6, 15) d. slope −3, passing through the point (−2.5, 4.5) 12. The following net folds to make a rectangular prism. 5 cm 5 cm 5 cm 10 cm 10 cm 10 cm 10 cm a. What is the volume of the prism? b. Suppose the dimensions of the shaded face of the prism are doubled. The other dimensions are adjusted so the volume remains the same. What are the new dimensions of the prism? c. Which prism has the smaller surface area: the original prism or the prism from ACE 12b? Explain. 62 Thinking with Mathematical Models SAMPLE
ACE 13. Suppose the town of Roseville is giving away rectangular lots with a perimeter of 500 feet rather than with an area of 24,000 square feet. a. Copy and complete this table. Rectangles with a Perimeter of 500 ft. Length (ft.) 50 100 150 200 225 Width (ft.) b. Make a graph of the (length, width) data. Draw a line or curve that models the data pattern. c. Describe the pattern of change in width as length increases. d. Write an equation for the relationship between length and width. Explain why it is or is not a linear function. 14. Suppose a car travels at a speed of 60 miles per hour. The function d = 60t relates time t in hours and distance d in miles. This function is an example of direct variation. A relationship between variables x and y is a direct variation if it can be expressed as y = kx, where k is a constant. a. Describe two functions in this unit that are direct variations. Give the rule for each function as an equation. b. For each function from ACE 14a, find the ratio of the dependent variable to the independent variable. How is the ratio related to k in the general function? c. Suppose the relationship between x and y is a direct variation. How do the y-values change as the x-values increase? How does this pattern of change appear in a graph of the relationship? d. Compare direct variation and inverse variation. Be sure to discuss the graphs and equations of each. For ACE 15–17, tell which store offers the better buy. Explain your choice. 15. The Super Market Tomatoes are 8 for $4.60. Tomatoes are 6 for $4.00. DeAndre's Groceries 16. DeAndre's Groceries Onions are 4 for $1.75. The Super Market Onions are 5 for $2.00. Investigation 3 Inverse Variation: Linear or Nonlinear? 63 SAMPLE
ACE 17. DeAndre's Groceries Apples are 6 for $3.00. The Super Market Apples are 5 for $2.89. EXTENSIONS 18. The drama club members at Henson Middle School are planning their spring show. They decide to charge $4.50 per ticket. They estimate their expenses for the show at $150.00. a. Write a function for the relationship between the number of tickets sold and the club’s total profit in dollars. b. Make a table to show how the profit changes as the ticket sales increase from 0 to 500 in intervals of 50. c. Make a graph of the (tickets sold, total profit) data. d. Add a column (or row) to your table to show the per-ticket profit for each number of tickets sold. For example, for 200 tickets, the total profit is $750.00, so the per-ticket profit is $750.00 ÷ 200, or $3.75. e. Make a graph of the (tickets sold, per-ticket profit) data. f. How are the patterns of change for the (tickets sold, total profit) data and (tickets sold, per-ticket profit) data similar? How are they different? How are the similarities and differences shown in the tables and graphs of each function? For ACE 19–21 find the value of c for which the two ordered pairs satisfy the same inverse variation. Then write an equation for the relationship. 19. (3, 16), (12, c) 20. (3, 9), (4, c) 21. (3, 4), (4, c) 64 Thinking with Mathematical Models SAMPLE
ACE 22. Multiple Choice The acceleration of a falling object is related to the object’s mass and the force of gravity acting on it. For a fixed force F, the relationship between mass m and acceleration a is an inverse variation. Which equation describes the relationship of F, m, and a? A. F = ma B. m = Fa C. __ m F = a D. __ m a = F 23. Multiple Choice Suppose the time t in the equation d = rt is held constant. What happens to the distance d as the rate r increases? A. d decreases. B. d increases. C. d stays constant. D. There is not enough information. 24. Multiple Choice Suppose the distance d in the equation d = rt is held constant. What happens to the time t as the rate r increases? A. t decreases. B. t increases. C. t stays constant. D. There is not enough information. Investigation 3 Inverse Variation: Linear or Nonlinear? 65 SAMPLE
66 In the Data About Us unit, we looked at data sets that involved one variable, called univariate data. Some of the issues we studied about these data sets were variability and outliers. In this unit, we continue to explore sets of data. In Investigation 1, we examined the data we collected from two experiments. Each experiment collected data by linking a pair of variables. This data is called bivariate data. Experiment 1: bridge strength and thickness of the bridge Experiment 2: bridge strength and length of the bridge The data collected for each set of quantitative variables (variables that have measures attached to them) varied among the groups. Statisticians study large data sets. One of the things that they are interested in is the accuracy of the predictions made from the data. › How well does the model fit the data? That is, can it be used to make reliable predictions? Making reliable predictions depends on having carefully gathered data sets and finding a model that fits the data well. Fitting a model to a data set is challenging when the data come from experiments or surveys, which naturally include a lot of variability. The first problem in this investigation looks at the issue of choosing a model that fits well and interpreting what the fit of our model tells us about the situation. Variability in Numerical and Categorical Data INVESTIGATION 4 SAMPLE
Organizing and displaying the data from experiments such as the tests of bridge strength helps us see patterns and make predictions. For linear data, or data that is close to linear, we can usually find a graph and an equation to express the approximate relationship between the variables. We can see that the points do not lie exactly on a line, but we can draw a line that is a good match for the data pattern. Drawing such a line gives us a model for the data. When statisticians represent the bivariate data in a coordinate graph, it is called a scatterplot. Scatterplots are used to analyze if there is a relationship between the variables. INITIAL CHALLENGE The Business Club revisited the bridge experiments. The scatterplots that follow come from members of the Business Club who were doing the experiment to test bridge weight and thickness in Investigation 1. Their data is represented in the table on the right. Business Club Bridge Data Bridge Thickness (layers) Breaking Weight (pennies) 1 12 2 18 3 32 4 40 5 48 6 64 PROBLEM Lines of Best Fit 4.1 Investigation 4 Variability in Numerical and Categorical Data 67 SAMPLE
68 Thinking with Mathematical Models 4.1 Recall that each student drew a different line as a model to fit the same data set. Luke’s Model Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Maria’s Model Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Billie’s Model Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) Chris’s Model Line Number of Layers 10 20 30 40 50 60 70 0 1 2 3 4 5 6 7 y x Breaking Weight (pennies) • Write an equation for each line. • What information do the slope and y-intercept provide for the experiment? • Use each equation to make predictions for 3 layers, for 5 layers, and for 10 layers. How accurate do you think each of these predictions is? • Which line do you think best fits the data? Why? WHAT IF . . . ? Situation A. Judging Accuracy with Residuals To test the accuracy of each model, the mathematics teacher Ms. Gomez suggests that they calculate the differences between the actual data and what the model predicts. This error is called a residual. 1. For each linear model in the Initial Challenge, find the residuals using a table. SAMPLE
Investigation 4 Variability in Numerical and Categorical Data 69 4.1 2. Do the residuals suggest that one of the models is better than the others? Explain your reasons. Maria’s Linear Model Number of Layers 1 2 3 4 5 6 Breaking Weight (pennies) Actual 12 18 32 40 48 64 Predicted by Linear Model Residual (ACTUAL – PREDICTED) Luke’s Linear Model Number of Layers 1 2 3 4 5 6 Breaking Weight (pennies) Actual 12 18 32 40 48 64 Predicted by Linear Model Residual (ACTUAL – PREDICTED) Billie’s Linear Model Number of Layers 1 2 3 4 5 6 Breaking Weight (pennies) Actual 12 18 32 40 48 64 Predicted by Linear Model Residual (ACTUAL – PREDICTED) Chris’s Linear Model Number of Layers 1 2 3 4 5 6 Breaking Weight (pennies) Actual 12 18 32 40 48 64 Predicted by Linear Model Residual (ACTUAL – PREDICTED) SAMPLE
4.1 70 Thinking with Mathematical Models Situation B. Exploring Scatterplots on Bridge Strength and Bridge Thickness The members of the Business Club did several trials testing bridge thickness and decided to combine their data to make one scatterplot. The scatterplot shows clusters of points in a cloud of data. The pattern still looks linear, but figuring out which line is the best model for all the data is complicated because the data produced by the three groups shows variability. Business Club Bridge Thickness by Different Bridge Strengths Number of Layers 1 2 3 4 5 6 Group 1 Bridge Strength (pennies) 6 13 15 21 22 28 Group 2 Bridge Strength (pennies) 4 5 17 21 26 28 Group 3 Bridge Strength (pennies) 5 12 14 22 25 32 Number of Layers 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 0 1 2 3 4 5 6 y x Breaking Strength (pennies) 1. Describe any variability in the data that is produced by the three different groups. What might cause any outliers? 2. Draw a line that looks like a good model for all the data, write its equation, and find the residuals. 3. What are the advantages and disadvantages of having a cloud of data to consider instead of the data from one experiment? 4. Does having a cloud of data make you more or less convinced that the relationship between Breaking Strength and Number of Layers is linear? SAMPLE
4.1 Investigation 4 Variability in Numerical and Categorical Data 71 Situation C. Deciding If There Is a Linear Relationship When statisticians are convinced that there is a linear relationship between two variables, they say there is a strong association. If the relationship is increasing, they say this is a positive association; if the relationship is decreasing, they say it is a negative association. If no linear model fits well, they say there might not be a linear relationship, or that the linear association is weak. If the linear model does not fit at all, they might think there is a relationship but not linear, or they may say there is no relationship at all, no association. Review the following four different data sets. Data Set A Bridges Data Set B Roller Coasters Data Set C Amusement Parks Data Set D Practice Time 2 31 1 0 4 65 2 3 4 5 6 7 8 9 10 11 12 y x 7 2 310 1 4 65 2 3 4 5 6 7 8 9 10 11 12 y x 7 2 310 1 4 65 7 2 3 4 5 6 7 8 9 10 11 12 y x 8 2 310 1 4 65 2 3 4 5 6 7 8 9 10 11 12 y x 7 8 9 10 1. Choose one of these statements to describe the situations you see in the graphs. (One statement may fit more than one situation.) ° The variables are probably linearly related. ° The variables are related, but the relationship is probably not linear. ° The variables are probably not related at all. 2. For the data set(s) that you thought showed a linear association, draw a linear model that seems to fit well. Describe the association. 3. Will your linear model(s) give accurate predictions? Explain. 4. For the relationship(s) that you thought were not linear, draw a nonlinear model that seems to fit well. Will your model give good predictions? SAMPLE
4.1 72 Thinking with Mathematical Models NOW WHAT DO YOU KNOW? How can you find an equation for a linear function that is a good model for a set of data? Describe how finding the residuals for the line can help measure the accuracy of the model. Did You Know? Due to the increase in data generation and collection across industries and because businesses are now realizing the value in making decisions based on large data sets, there has been a spike in demand for statisticians. According to the U.S. Bureau of Labor Statistics, the job outlook for the industry is positive. Overall employment for mathematicians and statisticians is expected to grow 30 percent from 2016 to 2032—nearly five times as fast as growth for all occupations. The U.S. Bureau of Labor Statistics reports on the fastest-growing occupations and median pay. Learn more on their website. You might find some information that surprises you. Source: U.S. Bureau of Labor Statistics, “Fastest Growing Occupations” and “Mathematicians and Statisticians,” Occupational Outlook Handbook. Updated September 6, 2023. SAMPLE
Investigation 4 Variability in Numerical and Categorical Data 73 Did You Know? Early roller coasters had wooden frames. Now, most roller coasters have steel frames, even though wood is still popular. A recent Roller Coaster Census Report counted 183 wood-frame coasters in the world, with 116 in North America. How can people compare wood- and steel-frame roller coasters? Wood and steel are types of frames, not numbers. Type of roller coaster is what statisticians call a categorical variable that has values wood and steel. Categorical data are variables that do not have numerical measures. They are associated with qualitative characteristics, such as color or the number of people who prefer T-shirts to those who prefer sweatshirts to represent their school. Since one of the variables in this situation does not have a measure, we cannot use equations or graphs to show the relationship between the two variables. In this problem, we will use two-way tables to investigate if there is an association between the age of the rider and the type of roller coaster preferred. Two-way tables display the number of times an event occurs, also called the relative frequencies, for categorical variables. Wood or Steel? Relationships in PROBLEM Categorical Data 4.2 SAMPLE
74 Thinking with Mathematical Models INITIAL CHALLENGE To plan a new amusement park, a team of coaster designers asked customers, “Do you prefer wood or steel frames in roller coasters?” The table shows the preferences by age group. Roller Coaster Customer Preference by Age Group Prefer Wood Perfer Steel Age ≤ 40 years 45 60 Age > 40 years 15 20 After visiting the amusement park, members of the Business Club make the following claims using the survey data about the roller coaster survey data by age of rider. Amelia’s Claim Younger riders are three times as likely as older riders to prefer wood-frame coasters. Julian’s Claim Younger riders are three times as likely as older riders to prefer steel-frame coasters. Mateo’s Claim The number of riders who prefer wood-frame coasters is about three quarters of the number who prefer steel-frame coasters. Chris’s Claim Younger riders are more likely than older riders to prefer steel-frame coasters. Gus’s Claim Older riders are more likely than younger riders to prefer wood-frame coasters. Nola’s Claim Considering people over 40, for every 15 who prefer wood, there are 20 that prefer steel. • Is each claim true or false? Explain. • Make a recommendation about the type of coaster that should be installed in the new park. 4.2 SAMPLE
Investigation 4 Variability in Numerical and Categorical Data 75 WHAT IF . . . ? Situation A. Making Predictions Suppose that a park installed one of each type of roller coaster. One day, there were 210 riders over the age of 40 and 420 riders age 40 or under. Use the survey data from the Initial Challenge. Roller Coaster Rider Preference by Age Group Prefer Wood Perfer Steel Total Age ≤ 40 years 420 Age > 40 years 210 Total 1. How many riders would you expect on the wood-frame coaster? On the steel-frame coaster? 2. How would you expect those riders to be distributed by age and coaster type in the table? 3. If only one roller coaster type could be installed in the park, which would you recommend? Explain your choice. Situation B. Jasmine Makes a Connection to Proportionality Examine Jasmine’s reasoning. Jasmine’s Connection I remember that I used percentages to make comparisons with “for every” comparisons in the Comparing and Scaling unit. I wonder if I can use percentages to make predictions with the steel and wood roller coaster data. I made the following table to enter the percentages of all riders who prefer wood and the percentage of all riders who prefer steel. Prefer Wood Perfer Steel Total Percentage Age ≤ 40 years 420 ___ 420 630 = 67% Age > 40 years 210 ___ 210 630 = 33% Jasmine Uses Percentages on Roller Coaster Data 4.2 SAMPLE
76 Thinking with Mathematical Models 1. Does her reasoning make sense? Explain why. 2. Describe other ways you could use proportional reasoning to make predictions with the data in the table. NOW WHAT DO YOU KNOW? How does a two-way table show similarities and differences among groups? How can you use a two-way table to make predictions? How is this way of making predictions similar to or different from the way we made predictions in Investigations 1, 2, and 3? 4.2 SAMPLE
Investigation 4 Variability in Numerical and Categorical Data 77 In the last problem, we studied two-way tables of categorical data and made predictions. In this problem, we get a chance to set up a two-way table and to decide if there is an association between activities and the location to run them. INITIAL CHALLENGE The teachers were thinking of doing a team-building activity for the beginning of the year. They asked the Business Club to organize the event. To help plan, they asked students if they would rather do challenging game activities or art activities and if they would rather be outside or inside for the event. Here are the results. Game Activity and Outside Game Activity and Inside Art Activity and Outside Art Activity and Inside 32 100 48 60 • Make a two-way table to display the data on students and participation in afterschool art or game activity. • Use your table to decide if each comparison is true or false. Justify your answers. Comparison 1 Students who would rather do game activities are more likely to want to be inside rather than be outside. Comparison 2 Students who would rather be outside are more likely to want the game activities than to want the art activities. PROBLEM School Team-Building Activity: Setting Up a Two-Way Table 4.3 SAMPLE
78 Thinking with Mathematical Models Comparison 3 Students who would rather do game activities are three times as likely to want to do the activities inside as those who want to do them outside. WHAT IF . . . ? Situation A. Using Percentages or Ratios Since the numbers of student answers in each category are not the same, Jasmine likes to work with ratios and percentages. This will let her know the relative frequency of a response. To test Jasmine’s claim, answer the following questions using a ratio, decimal, or percent. 1. What portion of the students want to be inside for the activity? 2. What portion of the students want to do the game activity? 3. How can you express the number of students who want to do art and be inside? 4. Nony said that about half of the students want to do the game activity and about half want to do the art activity. Do you agree? 5. Sari said that about of the half of the students want to do the activity inside and about half want to do the activity outside. Do you agree? NOW WHAT DO YOU KNOW? How does a two-way table model data and help you decide if two groups are the same or not relative to a characteristic? How does it tell you if there is an association between two variables? 4.3 Comparison 4 Students who would rather be inside are less likely to want to do art activities than those who want to do game activities. SAMPLE
Investigation 4 Variability in Numerical and Categorical Data 79 In this unit, we have seen how real situations can be modeled by tables, graphs, and equations. The challenge is always to decide if there is a relationship between the variables and, if so, what kind of relationship it is. If the underlying relationship is linear, we have learned now to find the equation of a linear model and how to measure how well our model fits the data. If our best attempt at a linear model does not fit well, what does it tell us about the association between the variables? We have also used two-way tables with categorical data to decide if there is an association between variables. If there is no association between the variables, how would our two-way table show this? When mathematicians and statisticians study data sets, they are looking for relationships. If they have enough evidence, they may claim there is a strong linear association, positive or negative, or a weak linear association, positive or negative; or they may decide there is no evidence of a relationship and claim there is no association. INITIAL CHALLENGE The Business Club collected data about various events from the events that occurred during the year. Some events represent categorical or numerical data. For each event, • Decide if the data contains categorical or numerical data. • If it is numerical data, can it be modeled by a linear function or some other kind of function? • Write a claim about the data. Explain why your claim is true. • How might your claim be useful in making predictions about the situation? PROBLEM Linear, Bivariate, and Categorical Data 4.4 SAMPLE
80 Thinking with Mathematical Models Students Who Prefer Dancing or Singing The following data shows the number of middle and high school students who prefer dancing before an audience of 1,000 or singing before an audience of 1,000. Middle and High School Preferences for Dancing and Singing Student Category Dancing Singing Total Middle School 40 80 120 High School 20 60 80 Total 60 140 200 Roller Coaster Riders and Ages The scatterplot shows the number of roller coaster riders and their ages on a given day. The pink dots represent wood-frame roller coasters. The blue dots represent steel-frame coasters. Riders and Their Ages Age of Riders 10 20 30 40 50 60 70 80 90 0 10 20 30 40 50 60 70 80 y x Number of Riders Older and Faster Lapeer Elementary is a school with students who are 5 to 14 years old. One field day, all students were timed in a 100-meter race. The table shows data for some of the students. Race Time for Different Ages Student Age (years) 5 5 6 8 8 8 9 9 10 10 10 11 11 12 13 13 14 Race Time (seconds) 25 22 23 18 16 17 15 16 17 20 14 15 13 14 17 12 13 4.4 SAMPLE