Sampler
Math Foundation Unit Grade 7: Intro Using Ratios to Solve Problems
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Table of Contents A Note to Students............................................. v 1 Ratio Tables............................................... 1 2 Representing Ratios......................................... 6 3 Solving Proportion Problems................................. 12 4 Rate.................................................... 18 5 Unit Conversion........................................... 23 Talking Mathematics.......................................... 29 A Complete Solution to a Word Problem........................... 31 What to Do If You Get Stuck.................................... 33 Using Ratios to Solve Problems iii
LESSON 3 Solving Proportion Problems setting the direction Here is a summary of the properties of equality: Properties of Equality Reflexive Property of Equality a = a. Symmetric Property of Equality If a = b, then b = a. Transitive Property of Equality If a = b and b = c, then a = c. Addition Property of Equality If a = b, then a + c = b + c. Subtraction Property of Equality If a = b, then a c = b c. Multiplication Property of Equality If a = b, then a c = b c. Division Property of Equality If a = b and c 0, then a c = b c. Substitution Property of Equality If a = b, then b may be substituted for a in any expression containing a. work time Work with a partner to solve problems 1 and 2 using these three solution methods: Ratio table Tape diagram Solving an equation For each proportion problem, one student should use a ratio table, and the other student should use a tape diagram. Then, partners should work together to solve the problem using an equation. Identify which properties of equality you use to solve the equation. 12 Math Foundation Unit Grade 7: Intro
Solving Proportion Problems Example Dwayne and Chen collect model airplanes. Dwayne has two and a half times as many as Chen. Dwayne has 15 models. How many models does Chen have? Using a Ratio Table The ratio of Dwayne s models to Chen s is 2.5 : 1. The simplest whole number ratio equal to this is 5 : 2. So, if Dwayne has 15 models, Chen has 6. Using a Tape Diagram The simplest whole number ratio is 5 : 2. Since Dwayne has 15 models, then each unit square must represent 3 models. This means that Chen has 6 models. Dwayne 2.5 5 15 Chen 1 2 6 15 models Dwayne 6 models 3 models Chen Solve by Writing and Solving an Equation A proportion is an equation of two ratios. Dwayne has 15 models. Let x represent the number of models that Chen has. Express as fractions Using the multiplication property of equality, multiply by x. Simplify Using the multiplication property of equality, multiply by 2. Simplify Solve and use the symmetric property of equality. 15 : x = 5 : 2 15 x = 5 2 15 x x = 5 2 x 15 = 5 2 x 5 2 x 30 = 5x x = 6 Chen has 6 models. Using Ratios to Solve Problems 13
Lesson 3 1. Lakeside City Hospital has a rule stating that the ratio of the nurses to patients must be as close as possible to 1 : 12. The hospital is expanding and will have space for 672 patients. How many nurses will the hospital need? 14 Math Foundation Unit Grade 7: Intro
Solving Proportion Problems 2. Dwayne s little brother Darell, and two of his friends, Carla and Anthony, deliver newspapers and earn tips. The total dollars in tips for Darell to Carla to Anthony are in a ratio of 2 : 5 : 11. Suppose Darell earns $20.80 in tips. How much do Carla and Anthony earn? Using Ratios to Solve Problems 15
Lesson 3 preparing for the closing 3. Explain the similarities and differences among the three methods of solving problems with proportional ratios. Which method do you prefer? Explain why. closing Notes and questions: show me Show me a tape diagram, a ratio table, and an equation of how many calculators are needed when each calculator can be shared by 3 students and 21 students are in class. 16 Math Foundation Unit Grade 7: Intro
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? 29
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 31
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 33