Lesson Mathematical Linear Models

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1 STATWAY STUDENT HANDOUT STUDENT NAME DATE INTRODUCTION Jean needs to buy some meat for her housing co-operative. She can go to the Fresh-Plus store to buy it for $3.50 per pound. Or she can go to the warehouse store, which is further away, to buy it for $3 per pound. In terms of her time and car expenses, a trip to the nearby store will cost $5 and a trip to the warehouse store will cost $9. Write a mathematical model for the cost of buying meat at each of these two stores. Use your models to determine how much meat she will need to buy to make it cheaper overall to go to the warehouse store. This scenario is not exactly like those we saw in Module 3, where we were given some data and investigated it to determine whether there was a pattern and how to best summarize any pattern we saw. Here the patterns are described and our task is to write the patterns in mathematical symbols and use those to efficiently answer questions. We can make a table. Let x = number of pounds of meat purchased and y = cost of purchasing the meat at Fresh-Plus. 1 Complete the next 2 values in the table. X y (Fresh-Plus cost) x = 1 y = $5 +$3.50 $8.50 x = 2 y = $5 + 2($3.50) $12 x = 3 y = $5 + 3($3.50) $15.50 x = 4 x = 5 2 Look at the pattern in the formula. How is the cost changing each time one more pound is added to the purchase?

2 STATWAY STUDENT HANDOUT 2 Now, continue to let x = number of pounds of meat purchased, but now let w = cost of purchasing the meat at the warehouse store. 3 Make a table as above, leading to a formula for finding the cost of buying x pounds of meat from the warehouse store. X x = 1 x = 2 x = 3 x = 4 x = 5 w (Warehouse store cost) 4 Graph the formulas you found for Fresh-Plus and the warehouse store on the same set of axes. 5 Do your graphs cross? 6 If not, do you suppose that if you continued computing for larger values of x, they would eventually cross? If so, extend the graphs. 7 What does it mean to be the point where the two lines cross? 8 Can you now answer the question about when it would be better, financially, to make the trip to the warehouse store to buy meat?

3 STATWAY STUDENT HANDOUT 3 Finding the Formula From Two Points 9 We are told that temperature Celsius and temperature Fahrenheit are linearly related and that when it is 10 C, then it is 50 F, and when it is 40 C then it is 104 F. We want to find the formula to predict temperature F from temperature C. This takes four steps: (1) Decide how to label the independent variable and the dependent variable. (2) Find the slope. (3) Find the y-intercept. (4) Write the formula for the line, which is the mathematical model. (1) Since we want to find a formula for F, then want F to be the y-variable and then C must be the x-variable. (2) The slope is defined to be the amount y changes when x changes by 1. We find that by taking a ratio with the values we have. m change in change in y x (104 50) degrees F m (40 10) degrees C 54degrees F 30 degrees C 1.8degrees F 1 degrees C m 1.8degrees F per degree C (3) Find the y-intercept. The equation of a line is written as y mx b, where m is the slope and b is the y-intercept. Then every point on the line ( xy, ) must fit this equation, so we can plug in any point for the x and y in the equation and it must be

4 STATWAY STUDENT HANDOUT 4 true. That means that we can plug in either point with the slope, m, that we just computed, and the only unknown value will be b. So we can find b. y mx b ( ) ( ) ( ) b The two points I am given in this problem are (10,50) and (40,104). I choose to use the first of these, but it would be equally good to use the second. y mx b (50) (1.8) (10) b b b b (4) So the formula for the line, which is the mathematical model, is y mx b y 1.8x 32

5 STATWAY STUDENT HANDOUT 5 YOU NEED TO KNOW When computing the slope, we must be sure that we have the right order for the x s and y s. If we get that mixed up, we will not correctly identify whether the slope is positive or negative. A more precise way of computing the slope, to emphasize the need to keep the order straight, is to use this algebraic notation. Also, once we have seen the units, it isn t necessary to write them at every step as we did in the first part of this problem, when we computed the slope. More precise algebraic notation: If the points ( x1, y1) and ( x2, y2) are on a line, the slope of the line is y2 y1 m. x x 2 1 Solution: Since x = degrees C and y = degrees F, then our two given points are (10,50) and (40,104). It doesn t matter which we call the first point and which we call the second point. Just to illustrate that, I will change the order from the obvious order. I will use ( x1, y1) (40,104) and ( x2, y2) (10,50), so the formula gives me y y m x x m m 1.8 Notice that we have the same value for the slope that we found before. Practice: It is a good idea to always check your work. In problems like this, it is easy to check by calculating the y-intercept twice. Use one point to calculate the y-intercept. Then use the second point to calculate the y-intercept. If you do not get the same value, you have done something wrong somewhere. The most likely place to make a mistake is in the sign of the slope. That s not the only place, but a different answer here suggests that you should look back and check your slope calculation very carefully. If that is clearly correct, then check each of your y-intercept calculations.

6 STATWAY STUDENT HANDOUT 6 TRY THESE Find the formula for the line from a graph, use the formula for the line to predict y for a given x, and use the graph to estimate a value of x, given y. For the following graph, 10 Choose two points on the line, then use steps 2, 3, and 4 from question 0 to find the formula for the line. 11 Use the formula to predict y when x = Use the graph to estimate x if y = 6. Then use the formula to check whether your estimate is close to correct.

7 STATWAY STUDENT HANDOUT 7 TAKE IT HOME 1 For the example about the exact linear relationship between temperature Celsius and temperature Fahrenheit, it is known that the freezing point of water is 0 C and 32 F. It is also known that the boiling point of water at sea level is 100 C and 212 F. A Find the find the formula for the line to predict temperature F from temperature C. (This is the same problem as in class, but using two different points.) B Did you get the same formula for the line as in the example? If not, what did you do to try to decide whether / where you made a mistake? C If the temperature is 25 Celsius, what is the temperature F?

8 STATWAY STUDENT HANDOUT 8 2 For the following graph, A find / estimate the formula for the line. B Use the formula to find y when x = 0 C Use the graph to estimate x when y = 10 and then use the formula to check your estimate.

9 STATWAY STUDENT HANDOUT This lesson is part of STATWAY, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. Version 1.0, A Pathway Through Statistics, Statway was created by the Charles A. Dana Center at the University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version 1.5 and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is an open-resource research and development community that seeks to harvest the wisdom of its diverse participants in systematic and disciplined inquiries to improve developmental mathematics instruction. For more information on the Statway Networked Improvement Community, please visit carnegiefoundation.org. For the most recent version of instructional materials, visit Statway.org/kernel STATWAY and the Carnegie Foundation logo are trademarks of the Carnegie Foundation for the Advancement of Teaching. A Pathway Through College Statistics may be used as provided in the CC BY license, but neither the Statway trademark nor the Carnegie Foundation logo may be used without the prior written consent of the Carnegie Foundation.