1. Proportions & Unit Rate 7 th Grade Math

2. 1. Melanie and Valerie went shopping last week. Melanie bought twice as many shirts as Valerie. If Valerie bought x number of shirts, which expression shows the total number of shirts that Melanie and Valerie bought? a) x ²b) x +2 c) x + 2x d) x + x 2.What is the value of the expression (x-y)(x+y) when x = 12 and y = -12? 3.Greg can ride at the rate of 24 miles to Gainesville in two hours on his bike. If he rides at the same rate, how long will it take him to ride his bike the 36 miles to the soccer field? (Set up a proportion.) 0 3

3. CC.7.RP.2b Identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. Standard

4. How do I identify the constant of proportionality in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships? Essential Question

5. Ratio is … A comparison of two numbers. There are twelve girls and nine boys in a class at MTSU. Write the ratio of boys to girls in fraction form. 9 12

6. A Proportion is… Two ratios that are equivalent to each other. $4 for 3 folders = $8 for 6 folders 4 8 3 6 =

7. Do these ratios form a proportion? ? Cross multiply and see if the products are equal. 2  7 = 143  4 = 12 14≠ 12 They do not form a proportion.

8. Find the missing part. 5w = 120 5 5 w = 24 Cross multiply and solve the equation.

9. Real World Application Miles Minutes Cross multiply and solve the equation.

10. Unit Rate is… A ratio that has a denominator of one… showing the value of one unit. Examples: price per pound pages read in one hour pages read in one hour cups that can fit on one shelf cups that can fit on one shelf price per gallon price per gallon

11. Real World Application If a person walks ½ mile in ¼ hour, what is the unit rate? (miles per hour) 2 miles per hour

12. How do you determine the better buy using unit rates? The CHEAPER of two unit rates…(the lowest amount). Example: 2 cokes for $1.00 4 cokes for $1.75 4 cokes for $1.75 is cheaper so it is the better buy. ($.50 each) ($.44 each)

13. Write the unit rate: $30 for six books ____ for one book $5

14. X independent Y dependent Record your information in a table. Books Price 1 5 2 10 315 420 525 6 30 The independent variable does not depend on anything. The dependent variable depends on the independent variable in order to find its value.

15. Graph It! independent dependent Books Price 1 2 3 4 5 1 2 3 4 5 6 8 10 7 9 A straight line through the origin shows a direct proportion on a graph… As x is increasing, y is also increasing.

16. Proportionality The graph of a proportional relationship is a straight line that passes through the origin. Proportional quantities can be described by the equation y = kx, where k is a constant ratio... also known as constant of proportionality.

17. X independent Y dependent Using the form y = kx, write an equation for the information in the chart. Books Price 1 5 210 315 420 525 630 What did you do to x to get y? 5x = y

18. X independent Y dependent Books Price 1 5 210 315 420 525 630 5x = y If y = kx then doesn’t k = y/x? To find the constant of proportionality, you can divide the y coordinate by the x coordinate. 5/1 = 5 10/2=5 15/3=5 20/4=5 25/5=5 30/6=5 The constant of proportionality is 5.

19. Constant of Proportionality Do you remember where the 5 came from? The unit rate…it is the number that you used to multiply by each x value to get each y value. Unit rate is also known as the constant of proportionality because it is the number that remains constant in each proportional situation.

20. Write the unit rate: Read 80 pages in 2 hours _______ pages in one hour 40

21. X independent Y dependent Record your information in the table. Hours Pages 1 40 280 3120 Is this a direct proportion? Yes…as x increases, y increases. What is the constant of proportionality? 40 Which is the dependent variable and which is the independent variable? (hours and pages)

22. Graph It! independent dependent Hours Pages 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 How do you determine if the graph is a direct proportion? A straight line through the origin shows a direct proportion.

23. Write the unit rate: A plane travels 2250 km in 3 hours. What is the average speed? (km per hour) 750 km/hr What is the constant of proportionality? They are asking for unit rate… 750

24. Graph It! independent dependent Hours Km 250 500 750 1000 1250 1500 1750 2000 2250 2500 1 2 3 4 5 Is this a direct proportion? Yes…A straight line through the origin shows a direct proportion.

25. k = y/x k = 20/2 k = 10

26. k = 2 Write the equation for the graph above. y = 2x

27. Write an equation for the information in the table. Xy 110 220 330 440 550 y = 10x What is the constant of proportionality?10

28. k = 5 Write the equation for the graph above. y = 5x

29. k = 7.5 Write the equation for the graph above. y = 7.5x

30. Y = 12/x No. The equation is not in the form y = kx.

31. k = 12.5

32. Closing How do you find the constant of proportionality from a: Ratio? Table? Equation? Graph? How do you determine if the proportional relationship shown is a direct proportion in a: Table?Graph?