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Math Foundation Unit Grade 7 Proportional Relationships This book belongs to:
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Table of Contents A Note to Students.......................................................... v 1 Proportional Relationships............................................. 1 2 Constant Ratios........................................................ 8 3 The Graph of a Proportional Relationship.............................. 13 4 The Constant of Proportionality....................................... 19 5 Is It Proportional?..................................................... 26 6 Progress Check 1: Recycling Cans...................................... 36 7 Solving Problems..................................................... 41 8 Formulas and Graphs................................................. 47 9 Relationships......................................................... 52 10 Progress Check 2: Dog Food........................................... 59 11 Connecting Percent to Proportional Relationships..................... 63 12 Percent Increase and Decrease......................................... 68 13 Relating Percent Increase to Percent Decrease......................... 72 14 Connecting Percent Increase and Decrease............................ 83 15 Correcting Mistakes................................................... 91 16 Progress Check 3: Clothes Sale........................................ 95 Talking Mathematics....................................................... 99 A Complete Solution to a Word Problem.................................. 101 What to Do If You Get Stuck............................................... 103 Procedural Help Proportional Relationships........................................... 105 Constant of Proportionality.......................................... 106 Exploring Proportionality............................................ 107 Image Credits............................................................. 109 Proportional Relationships iii
Relationships LESSON 9 setting the direction You can write two formulas for any proportional relationship. Example Lisa said, Our car gets 26 miles per gallon. Her statement tells you that Lisa s car travels 26 miles for each gallon of gasoline it uses. It also tells you that the car uses gas at the rate of 1 gallon for 26 miles. Distance (miles) d Gasoline (gallons) g 26 1 52 2 78 3 104 4 The table shows the relationship between two quantities: distance traveled (in miles), d, and volume of gasoline used (in gallons), g. You already know that if you calculate the constant of proportionality, you can find a formula that represents the number of miles traveled in terms of the number of gallons of gas used: k 1 = d g = 26 1 = 26 The formula for d in terms of g: d = 26 g Use the formula given in the example above to calculate the number of miles Lisa can travel using 35 gallons of gas. 52 Math Foundation Unit Grade 7
Relationships The second formula represents the number of gallons of gas used in terms of the number of miles traveled. Example The constant of proportionality for the number of gallons of gas used to the number of miles traveled: k 2 = g d = 1 26 The formula for g in terms of d: g = 1 26 d The two constants of proportionality (k 1 and k 2 ) have reciprocal values. k 1 = d g = 26 1 = 26 k 2 = g d = 1 26 Both formulas accurately represent the relationship between gallons of gas used and miles traveled. Which formula you use depends on the question you would like to answer. These relationships can be graphed as follows: 30 25 d (1, 26) 30 25 g Distance (miles) 20 15 10 Gasoline (gallons) 20 15 10 5 5 0 g 0 (26, 1) d 0 5 10 15 20 25 30 0 5 10 15 20 25 30 Gasoline (gallons) Distance (miles) Proportional Relationships 53
Lesson 9 work time 1. Suppose you make a stack of identical books. Here is a diagram showing the height of the stack and the thickness (3.6 centimeters) of a single book. a. Create a table that gives the height of the stack for stacks of 1, 2, 5, and 10 books. Let h represent the height of the stack in centimeters and n represent the number of books in the stack. 3.6 cm h b. Find the constant of proportionality and write a formula for the height of the stack in terms of the number of books. : c. Use your formula to find the height of a stack of 13 of these books. 54 Math Foundation Unit Grade 7
Relationships d. Find the constant of proportionality, explain what it represents, and write a formula for the number of books in terms of the height. (Your formula will start with n.) e. Use your formula from Part d to find the number of books when the height is 54 centimeters. 2. The number of coins in a stack of identical coins is proportional to the height of the stack. To find the number of coins in a stack, measure the height of the stack instead of counting the coins. 1 in Suppose a stack of 8 coins is exactly 1 inch high. a. Write a formula for finding the number of coins Choose your own letters to using the height of the stack. stand for the two quantities. b. Use your formula from Part a to find the number of coins when the height is 12.5 inches. Proportional Relationships 55
Clarify Talking Mathematics Are you saying that? Can you say more about? What do you mean by? I have a question about that because Critique I notice that How do you know that? I agree with that because Connect I disagree with that because An example of that is What you said is like I see a connection to The words that I use to say that are My work is similar to yours in that My work is different from yours in that I used a different strategy to solve the problem. My strategy was Compare Use these suggestions to help you talk to other students about their work. Is that the same as? Is that different from? Is it always true that? Is it ever true that? Can you apply what you know about that to? Proportional Relationships 99
A Complete Solution to a Word Problem includes all of the following A written estimate All work that you do An equation A diagram, number line, table, or other representation The answer to the question in a complete sentence Proportional Relationships 101
What to Do If You Get Stuck Look at past work times Look at the charts that are posted Model the problem using counters or other materials Sketch a diagram or other representation Change the numbers to make the problem simpler Write what you do know Write down questions to ask later Check other resources Proportional Relationships 103