Lanier Middle School “An International Baccalaureate Middle Years Programme School” MYP 7th – LESSON 2.3 Proportional Relationships and Graphs Daily Objective: How can you use graphs to represents and analyze proportional relationships? Comparing and Ordering Integers In the testing of a new kind of cooling system, the thermostat recorded that the temperature of the cooled space changed  2.5 C each minute for 15 consecutive minutes. How much did the temperature of the cooled space change in all? Operations with Decimals Amber’s baby-sitting fee is $4.50 per hour. Last Saturday she worked 4 1 2 hours for Mr. and Mrs. Marlee. How much did she charge the Marlee family? Locate the following numbers on the number line: 0.5, 2/5, 1/2, 45%, 12/25 Concept and Skill Development Guided Practice Most showerheads that were manufactured before 1994 use 5 gallons of water per minute. Is this relationship between the number of gallons of water and the number of minutes a proportional relationship? A Complete the table. B Based on the table, is this a proportional relationship? Explain. D Plot the ordered pairs. C Write the data in the table as ordered pairs (time, water used). (1,5), (2, ___), (3,___), (___,35), (10,___) E Draw Conclusions If you continued the table to include 23 minutes, would the point (23,125) be on this graph? Why or why not? E If the showerhead is used for 0 minutes, how many gallons of water will be used? What ordered pair represents this situation? What is this location called? In addition to using a table to determine if a relationship is proportional, you also can use a graph. A relationship is a proportional relationship if its graph is a straight line through the origin. A house cleaning company charges $45 per hour. Is the relationship a proportional relationship? Explain. Step 1 Make a table. Time (hr) 1 2 3 5 8 Lanier Middle School “An International Baccalaureate Middle Years Programme School” Total Cost ($) Is the graph a straight line that goes through the origin? Step 2 Write the data in the table as ordered pairs (time, cost). (1,___), (2,___), (3,___), (____,____), (___,____) Is this a proportional relationship? Explain. Step 3 Graph the ordered pairs. Your Turn: 1 Jared rents bowling shoes for $6 and pays $5 per bowling game. Graph the data. Is the relationship a proportional relationship? Explain. Games Total Cost ($) 1 2 3 4 Recall that you can describe a proportional relationship with the equation y  kx . The constant of proportionality k tells you how steep the graph of the relationship is. The greater the absolute value of k , the steeper the line. The graph shows the relationship between time in minutes and the number of miles Damon Runs. Write an equation for this relationship. Step 1 Chose a point on the graph and tell what it represents. The point (25, 2.5) represents the – Lanier Middle School “An International Baccalaureate Middle Years Programme School” Because dis tan ce  time Step 2 What is the constant of proportionality? The constant of proportionality is Step 3 Write an equation in the form y  kx y Reflect 2 Communicate Mathematical Ideas What does the point (0,0) on the graph represent? 4 The graph shows the relationship between the distance a bicyclist travels and the time in hours. 3 What if? Esther runs faster than Damon. Suppose you drew a graph representing the relationship between time in minutes and distance run for Esther. How would the graph compare to the one for Damon? A What does the point (4,60) represent? B What is the constant of proportionality? C Write an equation in the form y  kx for this relationship. Lanier Middle School “An International Baccalaureate Middle Years Programme School” Guided Practice Complete each table. Tell whether the relationship is a proportional relationship. Explain why or why not. Tell whether the relationship is a proportional relationship. Explain why or why not. Write an equation of the form y  kx for the relationship shown in each graph. For Exercises 8 – 12, the graph shows the relationship between time and distance run by two horses. 8. Explain the meaning of the point (0,0)? 9. How long does it take each horse to run a mile? Lanier Middle School “An International Baccalaureate Middle Years Programme School” 10. Multiple Representations. Write an equation for the relationship between time and distance for a horse. 11. Draw Conclusions. At the given rates, how far would each horse run in 12 minutes? 12. Analyze Relationships. Draw a line on the graph representing a horse that runs faster than horses A and B. 13. A bullet train can travel at 170 miles per hour. Will a graph representing distance in miles compared to time in hours show a proportional relationship? Explain. 14. Critical Thinking. When would it be more useful to represent a proportional relationship with a graph rather than an equation? The graph shows the relationship between distance and time as Glenda swims. 16. How far did Glenda swim in 4 seconds? 17. Communicate Mathematical Ideas. Is this a proportional relationship? Explain your reasoning. 18. Multiple Representations. Write an equation that shows the relationship between time and distance. Lanier Middle School “An International Baccalaureate Middle Years Programme School” The tables show the distance traveled by three cars. Challenge Problems If you know that a relationship is proportional and are given one ordered pair other than (0,0), how can you find another pair? Closure How does a graph show a proportional relationship? Homework