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Comparing and Scaling: Ratios, Rates, Percents & Proportions Name: _______________________ Per: _____ 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 Friday, Dec. 11 I can compare ratios in a real-world situation. Monday, Dec. 14 I can find the unit rate and the constant of proportionality from a graph, table, and equation. Tuesday, Dec. 15 I can determine whether or not two quantities are Wednesday, in a proportional relationship. Dec. 16 Classwork (Check Off Completed/ Corrected Items) Pg. 2-3: CS 2.1 – Sharing Pizza Homework (Check Off Completed/ Corrected Items) Pg. 4: CS 2.1 Zaption Self-Assess Your Understanding of the Learning Target/s Pg. 5-7: CS 2.2 – Scaling Rates Pg. 8: CS 2.2 Zaption Pg. 9-10: CS 2.3 – Unit Rates and Constant of Proportionality Exit Ticket Pg. 12-13: CS 2.3 Part 2 Pg. 11: CS 2.3 Zaption Pg. 14-15: CS 2.3 Part 2 Zaption Thursday, Dec. 17 I can explain what a point on a graph of a proportional relationship means. Check Up 2 Retake Check Up 1 (if needed) Pg. 16 – Puzzle Get Packet Signed Friday, Dec. 18 I can play a game to Ratio Rumble! explore equivalent ratios. I can solve proportions by scaling. (Investigation 1) I can find unit rates with rate tables. Enjoy your Winter Break! See you in 2016! Check Up 1 Exit Ticket 2 Score: Score: /18 / Parent/Guardian Signature: _______________________________________ Due: ______________________________ 1 CS 2.1: Sharing Pizza – Comparison Strategies The dining room at Camp Chargers has two sizes of tables. A large table seats ten people and a small table seats eight people. When the campers come for dinner one night, there are four pizzas on each large table and three pizzas on each small table. A. The campers at each table share the pizzas equally. Does a person sitting at the small table get the same amount of pizza as a person sitting at a large table? Solve the problem as many ways as you can! Explain your reasoning. 2 B. Selena wonders whether a person at a small table or a person at a large table gets more pizza. She uses two ratios, 8:3 and 10:4. She says: 1. Do you agree or disagree with Selena’s reasoning? Explain. 2. Tony disagrees with Selena, he says: Do you agree with Tony’s reasoning? Explain. C. There are 160 campers. a. If everyone sits at small tables, how many pizzas should the camp director order? b. If everyone sits at large tables, how many pizzas should the camp director order? c. The camp director also has extra-large tables that seat 25. How many pizzas should be placed on each of these tables? Explain. d. How many pizzas should he order if everyone sits at extra-large tables? 3 Homework: CS 2.1 Zaption – Sharing Pizza Complete the problems on your own and then correct with the Zaption. 1. Guests at a pizza party are seated at three tables. The small table has 5 seats and 2 pizzas. The medium table has 7 seats and 3 pizzas. The large table has 12 seats and 5 pizzas. The pizzas at each table are shared equally. At which table does a guest get the most pizza? Show your thinking. 2. Find the values that make each sentence correct. 3. For each diagram, write three statements comparing the areas of the shaded and unshaded regions. Fraction: Fraction: Percent: Percent: Ratio: Ratio: 4. 5. 4 CS 2.2: Comparing Pizza Prices – Scaling Rates Juliana is in charge of ordering pizzas for a camp dinner. She wonders whether to order the pizzas from Royal Pizza or Howdy’s Pizza. What is the ratio of pizzas to price ($)for each business? Howdy’s Pizza: Royal Pizza: Each of these ratios is a _____________, a comparison of two quantities measured in different units. Each pizzeria allows customers to use the same pricing rate for fewer or more pizzas than the listed number. You can only use the ads to find the cost for any number of pizzas you want to purchase. One way to find the costs is to build a _____________ __________________. This is a table that shows the prices for different numbers of pizzas. As the campers plan their pizza dinner, they need to calculate the costs for many different numbers of pizzas. A. Complete the table with the prices for each of the numbers of pizzas shown for Howdy’s Pizza: Number of Pizzas Price of 1 2 3 4 5 10 15 20 100 150 200 $120 1. What is the price for one pizza from Howdy’s Pizza? This is called a unit rate. A unit rate is a rate in which the second quantity is 1 unit. In this case we are talking about the price per pizza (one pizza). 2. What is the price for 8 pizzas from Howdy’s Pizza? Show how you found your answer. 3. What is the price for 27 pizzas from Howdy’s Pizza? Show how you found your answer. 4. What is the price for 53 pizzas from Howdy’s Pizza? Show how you found your answer. 5 B. Complete the table with the prices for each of the numbers of pizzas shown for Royal Pizza: Number of Pizzas Price 1 2 3 4 5 10 15 20 100 150 200 $195 5. What is the price for one pizza from Royal Pizza? Again, this is called a unit rate. 6. What is the price for 8 pizzas from Royal Pizza? Show how you found your answer. 7. What is the price for 27 pizzas from Royal Pizza? Show how you found your answer. 8. What is the price for 53 pizzas from Royal Pizza? Show how you found your answer. C. If you know the price of one pizza (the unit rate), how can you find the price of additional numbers of pizzas? One strategy would be to write an equation: 1. For each pizza place, write an equation for the total price (P) for any number of pizzas (n). Hint: Price = Price Per Pizza × Number of Pizzas Price = Unit Rate × Number of Pizzas Howdy’s Pizza Equation: ____________ = _____________________ Royal Pizza Equation: ____________ = _____________________ 2. How does having equations help you solve problems such as those in Question A? 3. Use your equations to find the price of 115 pizzas from each pizza place. Show your work. Howdy’s Pizza Equation: Royal Pizza Equation: 6 D. On the coordinate plane below, plot the data for both pizza places (use different colors for the two places). 1. Give your graph a title and label the axes. 2. Create a key so that it is clear which graph goes with which pizza place. 3. Think about how to scale your graph on each axis before you start plotting points. 4. Look at the tables that you created on pages 5 and 6 so that you can find the coordinates to plot. Number of pizzas will be your x-value, price will be your y-value (Number of pizzas, Price) (x,y) 5. What do you notice about the graphs of the pizza places? 6. Which pizza place is more expensive? How can you tell based on the graph? 7 Homework – CS 2.2 Zaption – Comparing and Scaling Rates Complete the problems on your own and then correct with the Zaption. 1. Maralah can drive her car 580 miles at a steady speed using 20 gallons of gasoline. a. Complete the rate table. b. What are the two unit rates? c. Graph the points on the coordinate grids. d. Write two equations that relate miles and gallons of gasoline (use m for miles and g for gallons of gasoline). e. Use one equation to find how many miles she can travel using 14 gallons of gasoline. 8 CS 2.3: Finding Costs – Unit Rate and Constant of Proportionality The unit rate of one pizza at Howdy’s is $13. The equation P = 13n relates the price of the pizza and the number of pizzas. This equation represents a ____________________________ ____________________________________, because you multiply one variable by a constant number to get the value of the other variable. The constant multiplier is called the ________________________ _____ _____________________________________. If a delivery charge of $5 is added to the cost, the relationship is no longer proportional. P = 13n + 5 is not a proportional relationship. How can you recognize a proportional relationship from a table, graph, or equation? A. FreshFoods has oranges on sale at 10 for $2. This is a ratio or rate. 1. What is the cost per orange? This is a unit rate, because we are talking about the price of one orange. 2. How many oranges can you buy for $1? This is also a unit rate, because we are talking about how many oranges you can get for $1. 3. Complete the rate table: Number of Oranges, n Cost, C 10 $2 1 20 11 $1 $2.60 4. How many oranges can you buy for $5? 5. How much do 25 oranges cost? 6. The equation n = 5C relates cost, C, to number of oranges, n. a. What does this equation tell you about the relationship between the number of oranges and the cost of the oranges? b. What is another equation relating these same two variables? What information does this other equation give you? c. Identify the two unit rates in the equations. What information do the unit rates give you? d. How does the constant of proportionality relate to the unit rate? 9 7. Graph the two equations for the oranges problem on separate coordinate grids. a. Think about how to scale your graph on each axis before you start plotting points. b. Look at the tables that you created on page 9 so that you can find coordinates to plot. Cost of Oranges Equation: __________________ (oranges, cost) (x,y) Cost of Oranges Equation: n = 5C (cost, oranges) (x,y) B. Noralie’s car uses 20 gallons of gas to go 600 miles. Rate Table 0 0 1 30 Graph 2 60 4 120 10 10.5 Find the unit rate (miles per gallon) by using the graph: Find the missing values in the table. Find the unit rate (miles per gallon) by using the table: How can you tell by looking the table that this is a proportional relationship? What does (1,30) mean? Equation Find the equation for this situation: Use the graph to find how many miles she can drive with 4 gallons of gas. How can you tell by looking the equation that this is a proportional relationship? How can you tell by looking the graph that this is a proportional relationship? 10 Homework – CS 2.3 Zaption – Unit Rates and Constant of Proportionality Complete the problems on your own and then correct with the Zaption. 1. Joel can drive his car 450 miles at a steady speed using 15 gallons of gasoline. a. Complete the rate table. b. What are the two unit rates? f. Graph the points on the coordinate grids. g. Write two equations that relate miles and gallons of gasoline (use m for miles and g for gallons of gasoline). h. Is this a proportional relationship? If so, how can you tell from the table, graph, and equation? 11 CS 2.3: Finding Costs – Unit Rate and Constant of Proportionality (Part 2) C. Gus wants to determine which store has better prices for groceries. At More for Your Money grocery story, pasta is 7 boxes for $6. At FreshFoods grocery store, pasta is 6 boxes for $5. i. Create a rate table for each grocery store and write two unit rates (include units): More for Your Money Grocery Store Cost, C $6 $1 Boxes, n 7 1 Set up and solve proportions to find the two unit rates: FreshFoods Store Cost, C $5 Boxes, n 6 $1 1 Set up and solve proportions to find the two unit rates: ii. Write two equations for each store that relates the number of boxes to the cost. FreshFoods Store: Equation: Equation: More for Your Grocery Store Equation: Equation: 12 iii. Graph the relationships of cost and boxes at the two grocery stores on separate coordinate grids. More for Your Money Grocery Store Equation: FreshFoods Store Equation: More for Your Money Grocery Store Equation: FreshFoods Store Equation: iv. At which store should Gus buy pasta? Explain your reasoning. 13 Homework: CS 2.3 Part 2 Zaption Complete the problems on your own and then correct with the Zaption. A. Several students wonder which is a better buy: a 40-pack of pencil-top erasers for $2.82 or a 2-pack of pencil-top erasers for $0.12. Which of these methods are correct? Which method do you prefer? Explain. B. For each situation, find a unit rate and write an equation relating the two quantities. (Challenge: Find two unit rates and write two equations…) a. 3 dozen apples for $4.50 b. 30 bottles of water for $4.80 c. 24 ounces of mozzarella cheese for $2.88 14 C. Which of these items is the better buy? Show your thinking. (Hint: Find a unit rate…) a. An 8-pack of glue sticks for $3.99 or 1 glue stick for $0.54 b. A 12-pack of tape for $2.50 or 1 roll of tape for $0.19 c. A 100-pack of pencils for $4.88 or 1 pencil for $0.05 d. A 50-pack of paper clips for $0.89 or a 25-pack of paper clips for $0.45 D. The dairy uses 50 pounds of milk to make 5 pounds of cheddar cheese. a. Complete the rate table: Pounds of Cheese Pounds of Milk 5 10 15 20 25 30 50 50 b. Graph the relationship between pounds of milk and pounds of cheddar cheese. Decide which variable should go on each axis. c. Write an equation relating pounds of milk, m, to pounds of cheddar cheese, c. d. What is the constant of proportionality in your equation? e. Explain one advantage of each method (the graph, the table, and the equation) to express the relationship between milk and cheese production. 15 16