1. Representing Proportional Relationships 8.EE.5 - GRAPH PROPORTIONAL RELATIONSHIPS, INTERPRETING THE UNIT RATE AS THE SLOPE OF THE GRAPH. COMPARE TWO DIFFERENT PROPORTIONAL RELATIONSHIPS REPRESENTED IN DIFFERENT WAYS. FOR EXAMPLE, COMPARE A DISTANCE-TIME GRAPH TO A DISTANCE-TIME EQUATION TO DETERMINE WHICH OF TWO MOVING OBJECTS HAS GREATER SPEED.

2. LESSON OBJECTIVES  GRAPH PROPORTIONAL RELATIONSHIPS  INTERPRET THE UNIT RATE OF A PROPORTIONAL RELATIONSHIP AS THE SLOPE OF ITS GRAPH  UNDERSTAND THAT THE Y-INTERCEPT IS ALWAYS 0 (zero) FOR PROPORTIONAL RELATIONSHIPS  COMPARE TWO DIFFERENT PROPORTIONAL RELATIONSHIPS REPRESENTED IN DIFFERENT WAYS

3. Let’s Review!  Proportional Relationship – the relationship among a group of ratios that are equivalent.  Constant of Proportionality – what the unit rate is called in a proportional relationship

4. Something to think about….. Quantity (ex: number of pens, number of servings, number of anything) is always represented by the variable “x” Unit (ex: cost, distance, measurement) is represented by the variable “y”

5. What is a proportional relationship? Suppose you and some friends plan to go to a movie and the tickets cost $8 each. You will pay $8 for 1 ticket, $16 for 2 tickets, $24 for 3 tickets, $32 for 4 tickets, and so on. The ratios of the total cost of the tickets to the number of tickets are all equivalent. A group of ratios that are equivalent are in a proportional relationship. When ratios are equivalent, they all have the same unit rate. In a proportional relationship, the unit rate is called the constant of proportionality. Does the data represent a proportional relationship?

6. What is a proportional relationship? The unit rate is 8. If we plug 8 into the equation y = mx, our equation is y = 8x.

7. How can you use a graph to tell if a relationship is proportional?

8. The points on the graph are on a straight line for both sets of data, but only the data for the cost of the movie tickets goes through the origin. Only the total cost of the movie tickets compared to the number of tickets is a proportional relationship.

9. Linear Equations and Proportional Relationships A proportional relationship is a linear function with slope m and that goes through the origin (0,0). *Plot the following points on your graph and connect with a straight line: (0,0), (2,2), (4,4), (6, 6), (8,8)

10. Proportional Relationships In this equation, m is the slope of the line and is also called the unit rate. As the x-coordinate gets larger, the y-coordinate also gets larger at a proportional rate. * m represents the slope and tells us how to move between points.

11. Proportional Relationships Let’s start by finding the unit rate (also called slope). Change in Y = Cost Change in X # of Servings Unit Rate (slope) = __________ Now, write the equation for this proportional relationship using y = mx. y = ___x

12. Example One: Graphing a proportional relationship equation. Y = 2/3x Rise = 2 Run = 3 1.Begin at the origin (0,0). 2.Move with m.

13. Example Two: Let’s create a table to represent the data on the graph. Writing a proportional relationship equation. 1.Find the slope using rise over run. 2.Substitute into y = mx The equation of the line is _________________ m = ________

14. Examples Continued…

15. 1 yard = 3 feet