Identifying Proportional and Non-Proportional Relationships

Review Key Vocabulary Proportional – when two quantities simplify to the same ratio. Constant – a quantity having a value that does not change or vary. Constant of Proportionality - a constant value of the ratio of two proportional quantities.

Exercise: Babysitting You have been hired by your neighbors to babysit their children on Friday night. You are paid $8 per hour. Complete the table relating your pay to the number of hours you worked. Hours Worked Pay ($) 1 2 3 4 4.5 5 6 6.5

Discussion Describe the relationship between the amount of money earned and the number of hours worked in this example. How can multiplication and division be used to show the earnings are proportional to number of hours worked?

Discussion Explain how you completed the table. How did you determine the pay for 4 ½ hours? How could you use the information to determine the pay for a week in which you worked 20 hours? How many other ways can the answer be determined? If the quantities in the table were graphed, would the point (0, 0) be on the graph? What would mean in the context of the problem.

TEACHER MODEL: Example 1 Determine if y is proportional to x. Justify your answer. The table below represents the amount of snow fall in 5 countries (in inches) to hours of a recent winter storm. x Time (hrs) y Snowfall (in) 2 10 6 12 8 16 2.5 5 7 14

CLASS MODEL: Example 2 Determine if y is proportional to x. Justify your answer. 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 6 2 9 3 24 8 1

PARTNER MODEL: Example 3 Determine if y is proportional to x. Justify your answer. The table below shows the relationship between the amount of candy (pounds) bought and the total cost. x Pounds y Cost 5 10 4 8 6 12 16 20

Discussion When looking at ratios that describe two quantities that are proportional in the same order, do the ratios always have to be equivalent? For example, if the quantities in the table were graphed, would point (0,0) be on the graph? Describe what the point (0,0) would represent in each table. Do the x and y values need to go up at a constant rate? In other words, when the x and y values both go up at constant rate, does this always indicate the relationship is proportional?

PARTNER MODEL: Example 4 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. Assume miles driven is proportional to Gallons Consumed in order to complete the table. Gallons Consumed 2 4 8 10 12 Miles Driven 54 189 216

Discussion Why is it important for you to know that miles are proportional to the gallons used? Describe the approach you used to complete the table. What is the value of the constant? Explain how the constant was determined. Explain how to use multiplication and division to complete the table.

Review Key Vocabulary Proportional – when two quantities simplify to the same ratio. Constant – a quantity having a value that does not change or vary. Constant of Proportionality - a constant value of the ratio of two proportional quantities.

Warm-Up: Price of Roses The table below shows the price for the number of roses indicated. Number of Roses 3 6 7 12 15 Price (Dollars) 9 18 27 36 45 Is the price proportional to the number of roses? How do you know? Find the cost of purchasing 30 roses.

Lesson Summary One quantity is proportional to a second if a constant (number) exists such that each measure in the first quantity multiplied by this constant gives the corresponding measure in the second quantity. Steps to determine if two quantities in a table are proportional to each other: 1. For each given measure of Quantity A and Quantity B, find the value of 𝐵 𝐴 . 2. If the value of 𝐵 𝐴 is the same for each pair of numbers, then the quantities are proportional to each other.