Proportional Relationships (Graphs) Lesson 4.2.2 Proportional Relationships (Graphs) NOTE TO TEACHER… LOOK FOR SPECIAL INSTURCTIONS AND ANSWERS IN THE NOTE SECTION OF EACH SLIDE. 7.RP.2a, 7.RP.2b

Prior Knowledge We have been learning about proportional relationships. Proportions are the comparison of two equal ratios. In our Jet Ski table from yesterday… We determined that the data in the table was proportional by writing each 𝒙,𝒚 pair as a ratio of 𝒚 𝒙 and then dividing to get equal unit rates for every row. Answer to the questions: Constant of Proportionality is the fancy word for Unit Rate. We use the letter k to represent the Constant of Proportionality. Who can tell me what the fancy word is for Unit Rate in proportional situations? What letter do we use?

Remember, on a graph… The independent variable appears on the x-axis. The dependent variable appears on the y-axis.

Today Before we get started… We are going to learn how to determine proportional relationships from a Graph. We are also going to learn how to find the Constant of Proportionality from a Graph. Before we get started… Let’s do an Exploration to see if you can determine the characteristics of a proportional graph!

Exploration Circle the word that describes each table (Proportional or Non-Proportional). Then, graph each 𝑥,𝑦 pair on the coordinate plane. Connect your points with a straight line. Extend the straight line out until it touches the y-axis! 𝒌= 𝒚 𝒙 𝒌= 𝒚 𝒙 See lesson plan for details on how to do this explore. Proportional Non-Proportional Proportional Non-Proportional

Exploration TABLE A TABLE B TABLE C TABLE D Allow several students to give their opinion. Refer to the characteristics on the t-chart if necessary. Ask class if they agree or disagree with opinions. Based on what you noticed about the proportional and non-proportional graphs on the previous slide… Can you pick out the 2 proportional graphs shown here? What makes you think that your 2 choices are correct?

Section 1 What makes a Graph Proportional?

Proportional Graphs The graph of a proportional relationship is a Straight-Line that passes through the origin (0, 0). PROPORTIONAL NOT PROPORTIONAL NOT PROPORTIONAL

Is the graph proportional or non-proportional? Why? YOU TRY! Is the graph proportional or non-proportional? Why? 1) Proportional – It is a straight line that passes through the origin (0, 0)

Is the graph proportional or non-proportional? Why? YOU TRY! Is the graph proportional or non-proportional? Why? 2) NOT Proportional – It is not a straight line.

Is the graph proportional or non-proportional? Why? YOU TRY! Is the graph proportional or non-proportional? Why? 3) NOT Proportional – It does not pass through the origin (0, 0)

Is the graph proportional or non-proportional? Why? YOU TRY! Is the graph proportional or non-proportional? Why? 4) Proportional – It is a straight line that passes through the origin (0, 0)

Section 2 How do you find the Constant of Proportionality from a Graph?

The graph shows the total cost in dollars, y, of purchasing x amount of concert tickets. Is the relationship between number of tickets purchased and total cost proportional? Explain your reasoning. How much do you think it costs to buy 1 concert ticket? Explain your reasoning? Where can we place a dot on the line that would represent the cost of purchasing 1 concert ticket. The relationship is proportional because it is a line that goes through the origin. 1 ticket costs $21. Allow students to try to explain why they think that. Someone may want to come to the board to illustrate their reasoning. Student should place a dot at (1, 21). (1, 21) represents the cost of purchasing 1 ticket. Be sure students realize that the first number is always the x and the second number is always the y. What x, y pair could be written that shows the cost of purchasing 1 ticket? ( , )

The rate for 1 of something is called the Unit Rate. On the previous slide, the cost for 1 concert ticket was $21. This value is also called the Constant of Proportionality. Let’s look at the steps for finding the Constant of Proportionality from a graph…

Finding the Constant of Proportionality from a Graph Step 1: Look for the quantity of 𝟏 on the x-axis. Step 2: Travel up the grid until you reach the graphed line. Step 3: Turn left to the y-axis to find the constant of proportionality (or unit rate). 𝒌 = 𝟐𝟏 This means that each concert ticket is $𝟐𝟏.

Find the Constant of Proportionality. YOU TRY! 1) Find the Constant of Proportionality. Write an equation in 𝒚=𝒌𝒙 format that represents the graph. Answer: k = $4 This is the cost of 1 pound of Salad. The equation is 𝑦=4𝑥

Finding the Constant of Proportionality from a Graph Notice that our previous technique would not work on this graph because the graphed line does not pass though a lattice point at the quantity of 1. We need different steps for this one! Step 1: Place a dot on ANY lattice point on the graphed line. Step 2: Use the 𝒙,𝒚 pair associated with the lattice point. Step 3: Find the Constant of Proportionality with the formula 𝑘= 𝑦 𝑥 This means that the pool is filled with 5 gallons of water in 1 minute. 𝑘= 𝑦 𝑥 = 10 2 = 𝟓

Finding the Constant of Proportionality from a Graph Notice… Using the lattice point at (4, 20) would result in the same constant of proportionality. 𝒌= 𝒚 𝒙 = 𝟐𝟎 𝟒 = 𝟓

Find the Constant of Proportionality. YOU TRY! 2) Find the Constant of Proportionality. Write an equation in 𝒚=𝒌𝒙 format that represents the graph. Answer: k = 𝑦 𝑥 = 100 5 =20 It means that it cost 20 Rand for 1 kg in weight. The equation is y = 20x. Show how using different lattice points would give the same Constant of Proportionality. (R) Rand is the money for South Africa.

Find the Constant of Proportionality. YOU TRY! 3) Find the Constant of Proportionality. Write an equation in 𝒚=𝒌𝒙 format that represents the graph. Answer: k = 𝑦 𝑥 = 160 5 =32 It means that $32 in 1 month. The equation is y = 32x.

END OF POWERPOINT