Ratios and Proportional Relationships: Lessons 16


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1 Unit 71 Lesson 16 Ratios and Proportional Relationships: Lessons 16 Name Date Classwork Book Math 7: Mr. Sanford Lessons 16: Proportional Relationship
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3 Lesson 11 Lesson 1: An Experience in Relationships as Measuring Rate Example 1: How fast is our class? Record the results from the paper passing exercise in the table below. Trial Number of Papers Passed Time (in seconds) Ratio of Number of Papers Passed to Time Rate Unit Rate Key Terms from Grade 6 Ratios and Unit Rates A ratio is an ordered pair of nonnegative numbers, which are not both zero. The ratio is denoted A: B to indicate the order of the numbers: the number A is first and the number B is second. Two ratios A: B and C: D are equivalent ratios if there is a positive number, c, such that C = ca and D = cb. A ratio of two quantities, such as 5 miles per 2 hours, can be written as another quantity called a rate. The numerical part of the rate is called the unit rate and is simply the value of the ratio, in this case 2.5. This means that in 1 hour the car travels 2.5 miles. The unit for the rate is miles/hour, read miles per hour. Example 2: Our Class by Gender Class 1 Number of boys Number of girls Ratio of boys to girls Class 2 Whole 7 th grade Create a pair of equivalent ratios by making a comparison of quantities discussed in this Example. Grade 7 Lesson 11: An Experience in Relations as Measuring Rate 3
4 Lesson 11 Exercise 1: Which is the better buy? ValueMart is advertising a BacktoSchool sale on pencils. A pack of 30 sells for $7.97 whereas a 12pack of the same brand cost for $4.77. Which is the better buy? How do you know? Exercise 2: Problem Solving Each year Lizzie s school purchases student agenda books, which are sold in the school store. This year, the school purchased 350 books at a cost of $1, If the school would like to make a profit of $1,500 to help pay for field trips and school activities, what is the least amount they can charge for each agenda book? Explain how you found your answer. Grade 7 Lesson 11: An Experience in Relations as Measuring Rate 4
5 Lesson 11 Lesson 2: Proportional Relationships Example 1: Pay by the Ounce Frozen Yogurt! A new selfserve frozen yogurt store opened this summer that sells its yogurt at a price based upon the total weight of the yogurt and its toppings in a dish. Each member of Isabelle s family weighed their dish and this is what they found. Weight (ounces) Cost ($) Cost = Weight Example 2: A Cooking Cheat Sheet! In the back of a recipe book, a diagram provides easy conversions to use while cooking. 0 ½ 1 1 ½ 2 Cups Ounces Ounces = Cups Exercise 1 During Jose s physical education class today, students visited activity stations. Next to each station was a chart depicting how many Calories (on average) would be burned by completing the activity. Calories burned while Jumping Rope Time (minutes) Calories Burned a. Is the number of calories burned proportional to time? How do you know? NOTE: Proportional means the relationship is made up of equivalent ratios Grade 7 Lesson 12: Proportional Relationships 5
6 Lesson 11 b. If Jose jumped rope for 6.5 minutes, how many calories would he expect to burn? Example 3: Summer Job Alex spent the summer helping out at his family s business. He was hoping to earn enough money to buy a new $220 gaming system by the end of the summer. Halfway through the summer, after working for 4 weeks, he had earned $112. Alex made the same amount of money each week that he worked. Alex wonders, If I continue to work and earn money at this rate, will I have enough money to buy the gaming system by the end of the summer? To check his assumption, he decided to make a table. He entered his total money earned at the end of week 4. Week Total Earnings $112 a. Work with a partner to complete the table and answer Alex s question. b. Are Alex s total earnings proportional to the number of weeks he worked? How do you know? c. Try to write a multiplication problem using a number of weeks and the total earnings. Lesson Summary: Measures in one quantity are proportional to measures of a second quantity if the relationship is made up of equivalent ratios. We can tell if a relationship is proportional by comparing the unit rates to see if they are the same. If each set of values can be divided to get the same unit rate we can say that each measure multiplied by the unit rate gives you the other measure. In the example given below, the distance is proportional to time since each measure of distance, y, can be calculated by multiplying each corresponding time, t, by the same value, 10. This table illustrates a proportional relationship between time, t, and distance, y. Time (hrs), t Distance (km), y Grade 7 Lesson 12: Proportional Relationships 6
7 Lesson 13 Lesson 3: Identifying Proportional Relationships in Tables Opening Exercise You have been hired by your neighbors to babysit their children on Friday night. You are paid $8 per hour. Complete the table relating your pay to the number of hours you worked. Hours Worked 1 Pay Based on the table above, is pay proportional to hours worked? How do you know? ½ Examples 1 4 For Examples 1 3, determine if y is proportional to x. Justify your answer. 1. The table below represents the amount of snow fall in 5 counties (in inches) to hours of a recent winter storm. x Time (hrs) y Snowfall (In) Unit Rate (in/hr) Grade 7 Lesson 13: Identifying Proportional Relationships in Tables 7
8 Lesson The table below shows the relationship between cost of renting a movie to the number of days on rent. x Number of Days y Cost Unit Rate ($/day) The table below shows the relationship between the amount of candy (pounds) bought and the total cost. x Pounds y Cost Unit Rate ($/lb) Randy is planning to drive from New Jersey to Florida. Randy recorded the distance traveled and the total number of gallons used every time he stopped for gas. Complete the table assuming miles driven is proportional to gallons consumed. Gallons Consumed Miles Driven In real life are miles driven proportional to gallons consumed? Grade 7 Lesson 13: Identifying Proportional Relationships in Tables 8
9 Lesson 14 Lesson 4: Identifying Proportional Relationships in Tables Example: Which Team Will Win the Race? You have decided to run in a long distance race. There are two teams that you can join. Team A runs at a constant rate of 2.5 miles per hour. Team B runs 4 miles the first hour and then 2 miles per hour after that. Task: Create a table for each team showing the distances that would be run for times of 1, 2, 3, 4,,5 and 6 hours. Team A Team B Time (hrs) Distance (miles) Unit Rate (m/hr) Time (hrs) Distance (miles) Unit Rate (m/hr) a. For which team is distance proportional to time? Explain your reasoning. b. Explain how you know distance for the other team is not proportional to time. c. If the race were 2.5 miles long, which team would win? Explain. If the race were 3.5 miles long, which team would win? Explain. Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 9
10 Lesson 14 If the race were 4.5 miles long, which team would win? Explain. d. For what length race would it be better to be on Team B than Team A? Explain. e. Using this relationship, if the members on the team ran for 10 hours, how far would each member run on each team? f. Will there always be a winning team, no matter what the length of the course? Why or why not? g. If the race is 12 miles long, which team should you choose to be on if you wish to win? Why would you choose this team? Challenge: How much sooner would you finish on that team compared to the other team? Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 10
11 Lesson 15 Lesson 5: Identifying Proportional Relationships in Graphs Opening Exercise Isaiah sold candy bars to help raise money for his scouting troop. The table compares the amount of candy he sold to the money he received. Is the amount of candy bars sold proportional to the money Isaiah received? How do you know? x Candy Bars Sold y Money Received ($) Example 1: From a Table to Graph Complete the table below if Isaiah earned $3.00 for every 2 candy bars. x Candy Bars Sold y Money Received ($) 2 3 Important Note: Characteristics of graphs of proportional relationships: Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 11
12 Lesson 15 Example 2 x y Similarities with Example 1: Differences from Example 1: Example 3 x y Similarities with Example 1: Differences from Example 1: Lesson Summary: When two proportional quantities are graphed on a coordinate plane, the points lie on a straight line that passes through the origin. Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 12
13 Lesson 16 Lesson 6: Identifying Proportional Relationships in Graphs and Tables Today s activity is an extension of Lesson 5. You will be working in groups to table, graph and identify whether or not the two quantities are proportional to each other. Classwork Poster Layout Use for notes Problem Table Graph Proportional or not? Explain. Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 13
14 Lesson 16 Gallery Walk: Take notes and answer the following questions: Were there any differences found in groups that had the same problem. Did you notice any common mistakes? How could they be fixed? Was there any group that stood out by representing their problems and findings exceptionally clearly? Problem 1: Problem 2: Problem 3: Problem 4: Grade 7 Lesson 15: Identifying Proportional Relationships in Graphs 14
15 Extra Notes Page
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