Comparing and Scaling: Ratios, Rates, Percents & Proportions Name: Per: Investigation 1: Ratios and Proportions and Investigation 2: Comparing and Scaling Rates Standards: 7.RP.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. 7.RP.2a: Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane (observing whether the graph is a straight line through the origin). 7.RP.2c: Represent proportional relationships by equations. 7.RP.2d: Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 7.RP.3: Use proportional relationships to solve multistep ratio and percent problems. Date Learning Target/s Classwork (Check Off Completed/ Corrected Items) Thursday, Dec. 3 Friday, Dec. 4 (Half Day) Monday, Dec. 7 Tuesday, Dec. 8 Wednesday, Dec. 9 I can write part-to-part and part-to-whole ratios to compare mixtures. I can review my mistakes on my Stretching and Shrinking test. I can write part-to-part and part-to-whole ratios to compare mixtures. I can solve proportions by scaling ratios. I can solve proportions by scaling ratios. Pg. 2-3 CS 1.2: Comparing Ratios Coversheet for Stretching and Shrinking Test (separate sheet) Pg. 6-7 CS 1.3: Scaling Ratios Pg. 9-10 CS 1.4: Scaling to Solve Proportions Exit Ticket Pg. 13-14 Inv. 1 Additional Practice Puzzle Homework (Check Off Completed/ Corrected Items) Pg. 4 CS 1.2 Self-Assess Your Understanding of the Learning Target/s Zaption Pg. 5 CS 1.3 Zaption Pg. 8 CS 1.4 Zaption Pg. 11-12 Inv. 1 Additional Practice Zaption Make sure Pg. 2-14 are completed and corrected Get packet signed Thursday, I can solve proportions Check Up Dec. 10 by scaling ratios. Retake for S&S (if needed) MathXL I completed 30 minutes of MathXL between Thursday, Dec. 3 and Thursday, Dec. 10. MathXL Exit Ticket #1 I can solve proportions by scaling ratios. Score: Parent/Guardian Signature: Due: 1
CS 1.2: Mixing Juice Comparing Ratios Every year, the Grade 7 students at Langston Hughes School go on an outdoor education camping trip. During the week-long trip everyone pitches in to help with cooking and cleanup. This year, Arvin and Mariah were in charge of making orange juice for the campers. They planned to make the juice by mixing water and frozen orange juice concentrate. To find the mix that would taste best, they decided to test some mixes. Think and Ink: Explain your thinking. Mix A 2 cups concentrate 3 cups water Mix B 5 cups concentrate 9 cups water Mix C 1 cup concentrate 2 cups water Which mix will make juice that is the most orangey? Which mix will make juice that is the least orangey? Mix D 3 cups concentrate 5 cups water Pair and Share: Discuss your ideas with your group. Work with your group to create a poster that includes the following: How did you compare the mixes? Which mix will make juice that is the most orangey? Which mix will make the juice that is the least orange? Be prepared to share your poster in a class gallery walk. After gallery walk: Did you change your answer? Why or why not? Which mix will make juice that is the most orangey? Which mix will make juice that is the least orangey? 2
A. For each recipe, find the information and then compare the mixes: How much concentrate is How much water is needed to needed to make 1 cup of juice? make 1 cup of juice? Mix A Rate the mixes in order from the most orangey (1) to the least orangey (4). Mix B Mix C Mix D B. Isabelle and Doug used fractions to express their reasoning. Isabelle: 5/9 of Mix B is concentrate Doug: 5/14 of Mix B is concentrate Do you agree with either of them? Explain. Mix A 2 cups concentrate 3 cups water Mix B 5 cups concentrate 9 cups water Mix C C. Max thinks that Mix A and Mix C are the same. Max says, They are both the most orangey since the difference between the number of cups of water and the number of cups of concentrate is 1. Is Max s thinking correct? Explain. 1 cup concentrate 2 cups water Mix D 3 cups concentrate 5 cups water D. Assume that each camper will get ½ cup of juice. Answer the questions below for each recipe: How many batches are needed to make juice for 240 campers? Mix A How much concentrate is needed to make juice for 240 campers? How much water is needed to make juice for 240 campers? Mix B Mix C 3
Homework CS 1.2 Zaption: Appley Juice Complete the problems and correct them with the Zaption. A. Compare these four mixes for apple juice. 1. Which mix would make the most appley juice? Explain your reasoning. 2. Suppose you make a single batch of each mix. What fraction of each batch is concentrate? Mix W Mix X Mix Y Mix Z 3. Rewrite your answers to the previous question as percents. Mix W Mix X Mix Y Mix Z 4. Suppose you make only 1 cup of Mix W. How much water and how much concentrate do you need? 1 cup of Mix W = cup of Water and cup of Concentrate 4
Homework CS 1.3 Zaption: Time to Concentrate Scaling Ratios In Problem 1.2, you may have used the ratios below to determine which recipe was the most orangey. Below are two ratios describing Mix A: Mix A 2 cups concentrate 3 cups water Mix B 5 cups concentrate 9 cups water Mix C The first ratio is a -to- ratio. It compares one (the water) of the whole (the juice) to the other (the concentrate). The second ratio is a -to- ratio. It compares one (the concentrate) to the (the juice). 1 cup concentrate 2 cups water Mix D 3 cups concentrate 5 cups water Write ratios for the mixes: Mix A Part-to-Part Ratio Concentrate to Water Part-to-Whole Ratio Concentrate to Juice Part-to-Whole Ratio Water to Juice 2:3 2/3 2 to 3 2:5 2/5 2 to 5 3:5 3/5 3 to 5 Mix B Mix C Mix D Scaling ratios was one of the comparison strategies Sam used in Problem 1.2. Practice scaling ratios: Mix A Part-to-Part Ratio Concentrate to Water 2 3 = 4 = 6 = = = 12 12 15 Mix C Part-to-Part Ratio Concentrate to Water 1 = 2 = 6 = 8 = 10 = 6 Mix B 5 = 18 = 27 = 20 = 45 = 30 Mix D 5 = 6 = 9 = 20 = 25 = 18 How can you use these ratios to compare Mix A and Mix B? 5
CS 1.3: Time to Concentrate Scaling Ratios A. A typical can of orange juice concentrate holds 12 fluid ounces. The standard recipe is: Mix one can of concentrate with three cans of cold water. How large of a pitcher will you need to hold the juice made from a typical can? Show or explain how you arrived at your answer. B. A typical can of lemonade concentrate holds 12 fluid ounces. The standard recipe is: Mix one can of concentrate with 4 ⅓ cans of cold water. How large of a pitcher will you need to hold the lemonade from a typical can? Show or explain how you arrived at your answer. The pitchers hold ½ gallon, 60 ounces, and 1 gallon. Which container should you use for the lemonade from one can? Explain your reasoning. Note: 1 gallon = 128 ounces. 6
C. Solve these mixing problems: 1. Cece is making orange juice using one 16-ounce can of concentrate. She is using the standard ratio of one can of concentrate to three cans of cold water. How large of a pitcher will she need? 2. Olivia has a one-gallon pitcher to fill with orange juice. She uses the standard ratio of one can of concentrate to three cans of cold water. How much concentrate does she need? 3. August has some leftover cans of lemonade concentrate in his freezer. He uses 1 ½ ten-ounce cans of concentrate and the standard ratio of one can of concentrate to 4 ⅓ cans of cold water. How large of a pitcher does he need? D. Otis likes to use equivalent ratios. For Olivia s problem in Question C, he wrote ratios in fraction form: 1 4 = x 128 1. What do the numbers 1, 4, and 128 mean in each ratio? What does x mean in this equation? 2. How can Otis find the correct value of x? (think about what you know about scale factors ) 7
Homework CS 1.4 Zaption: Keeping Things in Proportion Scaling to Solve Proportions Questions What is a proportion? Notes In Problem 1.3, you used ratios and scaling to solve problems. When you write two equivalent ratios in fraction form and set them equal to each other, you form a. Otis s strategy for solving a problem involving a ratio of orange concentrate to juice was to write this proportion: 1 4 = x 128 What are some other ways Otis might have written the proportion? How can you solve the proportion by scaling up? Suppose that among American doctors men outnumber women by a ratio of 12 to 5. If about 600,000 American doctors are men, how can you figure out how many are women? = = = Otis solved the proportion by scaling up. He wrote: 1 32 4 32 = x 128 There are four ways to write this as a proportion: Does one of the proportions seem easier to solve than the others? Using what you know about equivalent ratios, you can find the number of women doctors from any one of these proportions. Finding the missing value in a proportion is called solving the proportion. How many women doctors are there? 8
CS 1.4: Keeping Things in Proportion Scaling to Solve Proportions For each question, set up a proportion that shows the relationship between known and unknown quantities. Then use equivalent fractions, ratios, and scaling to solve each proportion. A. Imani gives vitamins to her dogs. The recommended dosage is 1 teaspoon per day for adult dogs weighing 10 pounds. She needs to give vitamins to Bruiser, who weighs 80 pounds, and Dust Ball, who weighs 7 pounds. What is the correct dosage for each dog? Proportion: B. Jogging 5 miles burns about 500 calories. How many miles does Tyler need to jog to burn off the 1,200 calorie lunch he ate? Proportion: C. Tyler jogs about 8 miles in 2 hours. How long will it take him to jog 12 miles? Proportion: D. The triangles in this picture are similar. Find the height of the tree. Proportion: 9
E. Solve these proportions for the variable x. Use the reasoning you applied in the earlier problems. 1. 8 5 = 32 x 6. x : 6 = 10 : 150 2. 7 12 = x 9 7. x = 20 15 30 3. 25: x = 5: 7 8. 0.1 48 = x 960 4. x 3 = 8 9 9. x = 3.5 900 15 5. x 5 = 120 3 10. 18 = 4.5 x 1 F. Nic was working on the proportion at right: 1. He could not see a way to scale 10 to make 6. Instead, he scaled both sides of the proportion. His work is shown at right. How could Nic complete his solution? 2. Kevin thinks Nic s idea is great, but he used 30 as a common denominator. Show what Kevin s version of the proportion would look like. Does Kevin s scaled up proportion give the same answer as Nic s? Explain your reasoning. 10
Homework CS Inv. 1 Zaption: Additional Practice Complete and then correct with the Zaption. For each problem: Set up and solve with a proportion. A. Jared and Pedro walk 1 mile in about 15 minutes. They can keep up this pace for several hours. Set up and solve each problem with a proportion. a. About how far can they walk in 90 minutes? b. About how far can they walk in 65 minutes? B. Swimming ¼ of a mile uses about the same number of calories as running 1 mile. Set up and solve each problem with a proportion. a. Gilda ran a 26-mile marathon. About how far would her sister have to swim to use the same number of calories Gilda used during the marathon? b. Juan swims 5 miles a day. About how many miles would he have to run to use the same number of calories used during his swim? C. After testing many samples, an electric company determined that approximately 2 of every 1,000 light bulbs on the market are defective. Americans buy more than 1 billion light bulbs every year. Estimate how many of these bulbs are defective. Set up and solve the problem with a proportion. Assignment continues on next page 11
D. The organizers of an environmental conference order buttons for the participants. They pay $18 for 12 dozen buttons. Set up and solve each problem with a proportion. a. How much do 4 dozen buttons cost? b. How much do 50 dozen buttons cost? c. How many dozens of buttons can the organizers buy for $27? d. How many dozens of buttons can the organizers buy for $63? E. Denzel makes 10 of his first 15 shots in a basketball free-throw contest. His success rate stays about the same for his next 100 free throws. Write and solve a proportion for each part. Round your answer to the nearest whole number. a. About how many baskets do you expect Denzel to make in his next 60 attempts? b. About how many free throws do you expect him to make in his next 80 attempts? c. About how many attempts do you expect Denzel to take to make 30 free throws? d. About how many attempts do you expect him to take to make 45 free throws? 12
Investigation 1 Additional Practice Puzzle: Setting Up and Solving Proportions with Scaling Set up and solve a proportion for each problem: Puzzle continues on the next page 13
Set up and solve a proportion for each problem: Challenge Problem: A fruit bar is 5 inches long. The bar will be split into two pieces. For each situation, find the lengths of the two pieces. Hint: Draw a diagram of the bar each time. a. One piece is 3/10 of the whole bar. b. One piece is 60% of the whole bar. c. One piece is 1 inch longer than the other. 14