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


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1 ! Lesson 24 Ins and Outs 111 Lesson 24 Ins and Outs Learning Objecves 1. Distinguish between inputs (independent variables) and outputs (dependent variables). 2. Evaluate expressions and formulas. 3. Write and interpret expressions. 24 Class Experience is a hard teacher because she gives the test first, the lesson afterward. Vernon Law The holy grail of energy research: a process that outputs more energy than you put in. Sadly, one of the basic laws of physics is that this is impossible: you can t create energy, just convert it from one form to another. Still, there s a worthwhile metaphor here: in many instances, the output (result you get) is determined by what you input. Don t practice and stay out all night before a big game, your output will probably stink. Put in a ton of work and effort, and you re likely to do your best. In this lesson, we ll study input and output from a mathematical sense, and hopefully give you a MUCH better idea of just what in the world a variable really is. First, let s talk about variables. Answer this question honestly and to the best of your ability without any outside help: 1. What is a variable? If you re like most people, your answer was probably something like this: A variable is a letter instead of a number. I am very sorry to inform you that, and I say this in the most caring possible way, your answer stinks. Think about what the word variable should mean in plain English: able to vary. And THAT is the key to understanding variables: A variable is a quantity that is able to change, or vary. Contrast this with a constant, which can t vary. The number 12 is a constant, because no matter what, it still has the same value. The number of hours you spend studying, on the other hand IS a variable, because it can vary depending on how much effort you choose to expend. But what about the whole letter thing? Isn t x a variable? The short answer: No. We use letters to REPRESENT variables, since that distinguishes them from numbers, which are constant. But a variable is NOT a letter: it s a quantity that can vary which is usually represented by a letter. Now we re ready to move on! 1. Which sentence makes more sense? Explain your reasoning. a. The amount of time you spend studying depends on the grade you earn in a course. b. The grade you earn in a course depends on the amount of time you spend studying.
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 24 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 7mile 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 24 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.
7 ! Lesson 24 Ins and Outs Por8olio Name Check each box when you ve completed the task. Remember that your instructor will want you to turn in the portfolio pages you create. Technology 1. The spreadsheet on Page 114 is in the online resources for Lesson 24. Make a copy of the spreadsheet, then use the copy down feature to extend the existing pattern down to show up to 2,100 snow cones sold. This should be in row 23 if all goes well. 2. Now create a scroll bar that changes the values in cell A23. (We first learned how to make scroll bars in the online resources for Lesson 110.) The settings should be Current = 2,000, minimum = 0, maximum = 2,000. Adjust the scroll bar to the point where the profit is as close to zero dollars as possible. Record the value on the scroll bar, and describe what that tells you about the business. Skills Include any written work from the assignment on ALEKS along with any notes or questions about this lesson s content. Applications 1. Complete the applications problems. Reflections Type a short answer to each question. 1. What is a variable? Why is it wrong to answer a letter? 2. When two variables are related, what does it mean to say that one is the independent variable and the other is the dependent variable? 3. Describe the difference between an equation and an expression. 4. How do the terms input and output apply to the relationships we discussed in this lesson? 5. What questions do you have about this lesson?
8 118 Unit 2 Rela/onships and Reasoning 24 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 24 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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