Proportional Relationships and Tables Ex: 1a Complete the table. Write all ratios in simplest form. Is the relationship between perimeter and side length proportional? Side length 1 2 3 5 Perimeter 4 16 24 Ratio = perimeter side length 𝟖 𝟐 = 𝟒 𝟏 4 6 8 12 20 𝟒 𝟏 𝟏𝟐 𝟑 = 𝟒 𝟏 𝟏𝟔 𝟒 = 𝟒 𝟏 𝟐𝟎 𝟓 = 𝟒 𝟏 𝟐𝟒 𝟔 = 𝟒 𝟏 𝟒 𝟏 =𝟒 What is the unit rate in this table?

Proportional Relationships and Tables Ex: 1b The amount of time Jarod spends studying and his test scores have a proportional relationship. Complete the table. Hours studying 2 3 4 Test Scores 46 69 92 Ratio= test scores hrs studying 𝟒𝟔 𝟐 = 𝟐𝟑 𝟏 𝟔𝟗 𝟑 = 𝟐𝟑 𝟏 𝟗𝟐 𝟒 = 𝟐𝟑 𝟏 𝟐𝟑 𝟏 = 23 What is the unit rate in this table?

Proportional Relationships and Tables Ex: 2a Determine whether the table shows a proportional relationship using 𝒚 𝒙 . x y 1 2 4 3 6 8 2 1 𝒚 𝒙 = ÷ 2 = 𝟐 𝟏 𝒚 𝒙 = 4 2 ÷ 2 6 3 ÷ 3 = 𝟐 𝟏 Yes, the table is proportional (equivalent). 𝒚 𝒙 = ÷ 3 𝒚 𝒙 = 8 4 ÷ 4 𝟐 𝟏 =𝟐 = 𝟐 𝟏 What is the unit rate in this table? ÷ 4

Proportional Relationships and Tables Ex: 2a Determine whether the table shows a proportional relationship using 𝒚 𝒙 . 𝒚 𝒙 = 4 8 ÷ 4 = 𝟏 𝟐 x y 8 4 6 3 ½ ¼ 11 5 ½ Yes, the table is proportional (equivalent). ÷ 4 3 6 𝒚 𝒙 = ÷ 3 = 𝟏 𝟐 ÷ 3 1 4 1 2 = 1 4 1 2 = 1 4 2 1 = 2 4 ÷ 2 = 𝟏 𝟐 𝒚 𝒙 = ÷ x ÷ 2 5 1 2 11 𝒚 𝒙 = = 5 1 2 11 = 11 2 11 1 = 11 2 1 11 = 11 22 ÷ 11 = 𝟏 𝟐 ÷ ÷ x ÷ 11 𝟏 𝟐 What is the unit rate in this table?

Proportional Relationships and Tables Ex: 2c Is there a proportional relationship between the number of copies and the cost? How do you know? x y No proportional relationship. As the number of copies increase the price per copy decreases. # of copies Cost ($) 150 18 225 27 550 55 1,050 84 What is the unit rate in this table? 𝒄𝒐𝒔𝒕 𝒄𝒐𝒑𝒊𝒆𝒔 = $𝟏𝟖.𝟎𝟎 𝟏𝟓𝟎 = $𝟎.𝟏𝟐 𝟏 There is no unit rate for this table. 𝒄𝒐𝒔𝒕 𝒄𝒐𝒑𝒊𝒆𝒔 = $𝟐𝟕.𝟎𝟎 𝟐𝟐𝟓 = $𝟎.𝟏𝟐 𝟏 𝒄𝒐𝒔𝒕 𝒄𝒐𝒑𝒊𝒆𝒔 = $𝟓𝟓.𝟎𝟎 𝟓𝟓𝟎 = $𝟎.𝟏𝟎 𝟏 𝒄𝒐𝒔𝒕 𝒄𝒐𝒑𝒊𝒆𝒔 = $𝟖𝟒.𝟎𝟎 𝟏𝟎𝟓𝟎 = $𝟎.𝟎𝟖 𝟏

Proportional Relationships and Tables REVIEW Determine whether the table shows a proportional relationship between x and y. x y 5 25 6 30 7 35 8 40 𝟓 𝟏 𝒚 𝒙 = 𝟐𝟓 𝟓 = 𝟑𝟎 𝟔 = 𝟓 𝟏 𝒚 𝒙 = 𝒚 𝒙 = 𝟑𝟓 𝟕 = 𝟓 𝟏 Yes, this table shows a proportional relationship between x and y . 𝒚 𝒙 = 𝟓 𝟏 = 5 𝟒𝟎 𝟖 = 𝟓 𝟏 What is the unit rate in this table?

2-2 Proportional Relationships and Graphs Ex: 1a In order to determine if a graph shows a proportional relationship, the graph must make a line, the line must pass through t (0,0) and every point for 𝑦 𝑥 must be . straight origin equivalent . 𝒚 𝒙 = 1 1 =1 𝒚 𝒙 = 3 3 =1 . y . 2 2 =1 𝒚 𝒙 = 4 4 =1 𝒚 𝒙 = . . x

2-2 Proportional Relationships and Graphs Ex: 1b Which represents a proportional relationship? How do you know? Yes, this graph shows a proportional relationship between x and y .

. . . y = 6x y = 6x x y y = 6(0) y = 6(1) y = 0 y = 6 1 6 y = 6x 2 12 2-2 Proportional Relationships and Equations Ex: 2a Does the equation y=6x show a proportional relationship between x and y? Explain In order to graph we need at least 3 points. . y = 6x y = 6x x y y = 6(0) y = 6(1) y = 0 y = 6 1 6 . y = 6x 2 12 y = 6(2) y = 12 . Yes, this graph shows a proportional relationship between x and y .

. . . y = 2x+1 y = 2x+1 x y y = 2(0)+1 y = 2(1)+1 1 y = 3 y = 1 1 3 2-2 Proportional Relationships and Equations Ex: 2b Does the equation y=2x + 1 show a proportional relationship between x and y? Explain In order to graph we need at least 3 points. y = 2x+1 y = 2x+1 x y y = 2(0)+1 y = 2(1)+1 1 . y = 3 y = 1 1 3 . y = 2x+1 2 5 . y = 2(2)+1 y = 5 This graph does not show a proportional relationship.

2-2 Proportional Relationships and Equations Ex: 3 Distance Run by a Cheetah The graph shows a proportional relationship between time and the distance run by a cheetah. a) What does the point (1,90) represent? The cheetah travels 90 feet in 1 second, which is also the unit rate. b) What does the point (0, 0) represent? 450 0 feet traveled in 0 seconds by the cheetah. 360 Distance (feet) c) How far did the cheetah run in 3 sec? 270 The cheetah travels 270 feet in 3 seconds. 180 d) Assuming the cheetah continues to run at the same speed, how long would it take the cheetah to run 900 ft? 90 The unit rate 𝟗𝟎 𝒇𝒕 𝟏 𝒔 = 𝟗𝟎𝟎 𝒇𝒕 𝒔 x 10 0 1 2 3 4 5 6 10 x 10 Time (seconds)

2-3 Constant of Proportionality Ex: 1a The weight of the stack depends on the number of books in the stack. Identify the constant of proportionality for this situation. Then use the constant of proportionality to find the weight of 11 books. Constant of proportionality 𝒚 𝒙 = 𝐰𝐞𝐢𝐠𝐡𝐭 𝐨𝐟 𝐭𝐡𝐞 𝐬𝐭𝐚𝐜𝐤 𝐨𝐟 𝐛𝐨𝐨𝐤𝐬 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐛𝐨𝐨𝐤𝐬 15.75 lb. 𝒚 𝒙 = 𝟏𝟓.𝟕𝟓 𝐥𝐛 𝟗 𝐛𝐨𝐨𝐤𝐬 ÷9 1.75 x11 19.25 = 𝐥𝐛 𝟏 𝐛𝐨𝐨𝐤 = 𝐥𝐛 𝟏𝟏 𝐛𝐨𝐨𝐤𝐬 ÷9 x11

2-3 Constant of Proportionality Ex: 1b The weight of 3 eggs is shown. Identify the constant of proportionality of total weight to number of eggs. The weight of 3 eggs is 120 g. Constant of proportionality 40 𝒚 𝒙 = = 𝟏𝟐𝟎 𝐠 𝟑 𝐞𝐠𝐠𝐬 ÷3 𝐭𝐨𝐭𝐚𝐥 𝐰𝐞𝐢𝐠𝐡𝐭 𝐧𝐮𝐦𝐛𝐞𝐫 𝐨𝐟𝐞𝐠𝐠𝐬 = 𝐠 𝟏 𝐞𝐠𝐠 ÷3

2-3 Constant of Proportionality Ex: 2a You have a recipe that calls for 2 cups of flour to make 3 dozen cookies. Your friend has a cookie recipe that calls for 3 cups of flour to make 60 cookies. Are the constants of proportionality the same for the two recipes? Are the recipes for the same cookie? How do you know? You 𝟑𝟔 𝐜𝐨𝐨𝐤𝐢𝐞𝐬 𝟐 𝐜𝐮𝐩𝐬 𝐟𝐥𝐨𝐮𝐫 = 𝟏𝟖 𝐜𝐨𝐨𝐤𝐢𝐞𝐬 𝟏 𝐜𝐮𝐩 𝐟𝐥𝐨𝐮𝐫 c.o.p = Friend 𝟔𝟎 𝐜𝐨𝐨𝐤𝐢𝐞𝐬 𝟑 𝐜𝐮𝐩𝐬 𝐟𝐥𝐨𝐮𝐫 = 𝟐𝟎 𝐜𝐨𝐨𝐤𝐢𝐞𝐬 𝟏 𝐜𝐮𝐩 𝐟𝐥𝐨𝐮𝐫 c.o.p = The constant of proportionalities are not the same. This means the recipes are different.

2-3 Constant of Proportionality Ex: 3 Does the table show a proportional relationship? If so, what is the constant of proportionality. x 5 6 7 8 y 90 108 126 144 The table shows a proportional relationship. The c.o.p. is 18. Constant of proportionality 𝒚 𝒙 = 𝟗𝟎 𝟓 = 𝟏𝟖 𝟏 =18 𝟏𝟖 𝟏 =18 𝒚 𝒙 = 𝟏𝟐𝟔 𝟕 = 𝟏𝟒𝟒 𝟖 = 𝟏𝟖 𝟏 =18 𝒚 𝒙 = 𝟏𝟎𝟖 𝟔 = 𝟏𝟖 𝟏 =18 𝒚 𝒙 =

The point (1,80) represents 80 wingbeats in 1 second. 2-3 Constant of Proportionality Part 4 Male Hummingbird Wing Beats The graph shows the number of times a male hummingbird beats its wings based on time. a) What is the constant of proportionality for this situation? y x = 80 1 Number of wing beats 320 b) What does the point (1, 80) represent? 240 160 The point (1,80) represents 80 wingbeats in 1 second. 80 0 1 2 3 4 5 Time (seconds)

2-4 Proportional Relationships and Equations Equation Form y = mx constant of proportionality (unit rate) The number in front of x is the ___________________________________.

What is the constant of proportionality shown in the equation? 2-4 Proportional Relationships and Equations Ex: 1 Your friend uses the equation y = 8.5 x to calculate the total cost y in dollars for x movie tickets. What is the constant of proportionality shown in the equation? What does the constant of proportionality represent in this situation? How much will 13 movie tickets cost?

2-4 Proportional Relationships and Equations Ex: 1 Your friend uses the equation y = 8.5 x to calculate the total cost y in dollars for x movie tickets. What is the constant of proportionality shown in the equation? $8.50 per ticket y = 8.5 x

2-4 Proportional Relationships and Equations Ex: 1 Your friend uses the equation y = 8.5 x to calculate the total cost y in dollars for x movie tickets. b. What does the constant of proportionality represent in this situation? The constant of proportionality represents the unit cost or the price per movie ticket.

2-4 Proportional Relationships and Equations Ex: 1 Your friend uses the equation y = 8.5 x to calculate the total cost y in dollars for x movie tickets. c. How much will 13 movie tickets cost? y = 8.5 x Substitute 13 for x. = 8.5 (13) = $110.50

y = 8x 8 = mi 1 hr 2-4 Proportional Relationships and Equations Ex: 2a You ride a bike 9.6 miles in 1.2 hours at a steady rate. What equation represents the proportional relationship between the x hours you bike and the distance y in miles that you travel? 1st: Find the constant of proportionality (unit rate) 8 9.6 mi 1.2 hr ÷ 1.2 = mi 1 hr ÷ 1.2 2nd : Write your equation y=mx y = 8x

y = 7x 7 = mi 1 hr 2-4 Proportional Relationships and Equations Ex: 2b You ride a bike 12.25 miles in 1.75 hours at a steady rate. What equation represents the proportional relationship between the x hours you bike and the distance y in miles that you travel? 1st: Find the constant of proportionality (unit rate) 7 12.25 mi 1.75 hr ÷ 1.75 = mi 1 hr ÷ 1.75 2nd : Write your equation y=mx y = 7x

Topic 2 Review 1 Determine whether the equation y = 3x shows a proportional relationship. Explain. x y 1 3 y = 3x y = 3x y = 3x 2 6 y = 3(0) y = 3(1) y = 3(2) y = 0 y = 3 y = 6 𝒚 𝒙 = 𝟑 𝟏 Yes the equation is proportional because the ratios are equivalent. 𝒚 𝒙 = 𝟔 𝟐 = 𝟑 𝟏

Topic 2 Review 2 The equation below describes a proportional relationship between x and y. What is the constant of proportionality? y = 6x 6 is the constant of proportionality

Topic 2 Review 3 You bike 12 miles in 1.5 hours at a steady rate. What equation represents the proportional relationship between the x hours you bike and the distance y in miles that you travel? 8 𝟏𝟐 𝐦𝐢 𝟏.𝟓 𝐡𝐫 y = ___x 1.5 12 8

Topic 2 Review 4 Suppose the relationship between x and y is proportional. When x is 17 and y is 51. Identify the constant of proportionality. 𝑦 𝑥 = 51 17 = 3