# Experience is a hard teacher because she gives the test first, the lesson afterward. Vernon Law DRAFT

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2 112 Unit 2 Rela/onships and Reasoning 2. List other factors that are likely to impact your grade in this course. Because these factors can cause a change in your course grade, and not the other way around, the course grade (which can vary) is referred to as a dependent variable in this situation. Other factors, like the amount of time you spend studying, are called independent variables. Independent variables cause changes in dependent variables. 3. Consider the following relationships, where one quantity or event causes another to change. Identify the independent and dependent variable in each case, and don t forget to think about WHY it makes sense that these things are variables. a. The age of a tree and the height of a tree c. The number of practice sessions and the quality of a musical performance d. Your score on a placement test and the math courses you ve taken previously e. Your blood pressure and the amount of time you spend exercising each week f. The value of a share of Apple stock and what year it is g. The number of songs a band sells on itunes and the amount of money spent on marketing h. The cost of a cab ride and the number of miles you re driven i. The number of customers that want to buy a certain product and the price of that product

3 ! Lesson 2-4 Ins and Outs Group 1. A few key terms used in this section are listed below. Discuss each term with your group and note how the definitions are similar, and how they re different. Expression A combination of variables and constants using mathematical operations and grouping symbols Examples: 4x + 2y, 11 t 2, x2 20, ( 4n 6) ( 2 + n) Equation A statement that two quantities are equal built using expressions and an equal sign (=) Examples: 3x + 2 = 7, y = z 2 z 6, 100e 0.02t = 500 Formula An expression or equation with multiple variables that is used to calculate some quantity of interest Examples: πr 2, P = 2l + 2w, A = P + Prt In many cases, relationships like the ones we thought about in the Class portion of this lesson can be described mathematically using a formula. For example, it might cost \$2 for a cab ride plus \$2.50 for each mile; in that case we could write the formula C = m where C represents the variable cost of the cab ride, and m represents the variable number of miles. It s the number of miles that affects the cost, so m is the independent variable and C is the dependent variable. We might say that the number of miles is the input, while the cost is the output. Think of it like a machine: you input a number of miles, and the machine gives you back the cost. 2. If we wanted to know the cost of a 7 mile cab ride, we could replace the variable m, which represents miles; the result would be C = (7). Finding the value of the calculation on the right will tell us the cost. We call this evaluating the expression (or formula) for cost. Find the cost of the 7-mile cab ride. 3. Evaluate the cost formula for m = 4, then attach units to your answer and write a sentence describing what it tells us. Include information about each variable. Math Note There s no reason you HAVE to use a le>er to represent a variable quan/ty: You could use a smiley face, a zodiac sign, a picture of your mom, whatever. But we ll use le>ers because we don t know what your mom looks like.

4 114 Unit 2 Rela/onships and Reasoning The formula A = P( 1+ r) t gives the value A of an account where P dollars have been deposited for t years at a compound interest rate r (written in decimal form). 4. Evaluate this formula for P = \$10,000, t = 40, and r = Write a short paragraph describing the significance of the calculation in Question 4. What does it tell us? Include information about each of the variables. Next, we re going to look at how the types of relationships we re studying can be applied to a small business. This spreadsheet describes the costs associated with running a snow cone business, as well as revenue and profit. 6. If 300 snow cones are sold, how many ounces of syrup are needed?

5 ! Lesson 2-4 Ins and Outs A formula was typed into cell D5 to obtain the result you used to answer Question 6. What formula would give the correct result? (The formula should use cell A6.) 8. Write a verbal description of the relationship between the number of snow cones sold and the number of ounces of syrup that will be needed. 9. Use your answer to Question 8 to write an expression with variable x that represents the number of ounces of syrup used. The variable x should represent the number of snow cones sold. 10. Using the letter S to represent the amount of syrup used, write a formula that describes the ounces of syrup needed in terms of the number of snow cones sold. 11. Does the number of snow cones sold depend on the amount of syrup used, or does the amount of syrup used depend on the number of snow cones sold? 12. In your formula from Question 10, which is the independent variable, and which is the dependent variable?

6 116 Unit 2 Rela/onships and Reasoning 13. If x represents the number of snow cones sold, what quantity described in the spreadsheet would be represented by 5x? 14. If 70 snow cones are sold, the value of 5x is. Write a sentence or two explaining exactly what that result tells us about the snow cone business. 15. Write a unit rate that describes the combined costs of all supplies needed per snow cone sold. 16. Notice that the fixed costs are \$245 for any number of snow cones sold. That s what makes them fixed costs: the aren t affected by the sales. The formula for finding total costs in terms of the number of snow cones sold (x) is C = 0.12x Explain how we got that formula.

8 118 Unit 2 Rela/onships and Reasoning 2-4 Applica;ons Name Use the spreadsheet on Page 114 to answer the following questions. 1. Find the total cost from selling 400 snow cones. 2. Find the revenue from selling 400 snow cones. 3. A formula that uses cells J7 and K7 was typed in cell L7. What is that formula? (Focus on the dollar amounts in cells J7 and K7.) 4. Write a verbal explanation of the relationship between the profit made from selling a certain number of snow cones and the revenue and total costs from selling that number of snow cones. 5. If we use P to represent profit, C to represent total costs, and R to represent revenue, write a formula that describes the profit in terms of C and R. 6. In Question 16 of the 2-4 group activity, you discussed how to represent total costs with the expression 0.12x Using the same line of reasoning, find an expression that describes the revenue from selling x snow cones. (Recall that we used a unit rate to find the expression for total costs.) 7. Is 5 (7 3) equal to 5 7 3, or ? Perform each calculation to decide, then try to explain why it worked out that way. 8. Subtract the expression 0.12x from the one you wrote in Question 6. Use what you discovered in Question 7 to perform the subtraction. Then write a description of what exactly this new expression represents.

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