Proportions & Unit Rate

7th GradeMath

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) x² b) 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

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

Standard

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

Essential Question

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.

912

A Proportion is…

Two ratios that are equivalent to each other.

$4 for 3 folders = $8 for 6 folders

4 83 6

=

Do these ratios form a proportion?

23=

47

?

Cross multiply and see if the products are equal.

2 7 = 14 3 4 = 12

14 ≠ 12They do not form a

proportion.

Find the missing part.

5w = 120 5 5

w = 24

Cross multiply and solve the equation.

Real World Application

Set up a proportion and solve.

Remember, when setting up a proportion to label your numerators and label your denominators.

4m = 540÷4 ÷4m = 135

Miles

Minutes

Cross multiply and solve the equation.

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 cups that can fit on one

shelf price per gallon

Real World Application

If a person walks ½ mile in ¼ hour, what is the unit rate? (miles per hour)

2 miles per hour

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)

Write the unit rate:

$30 for six books

____ for one book$5

Xindependent

Ydependent

Record your information in a table.

Books Price

1 5

2 10

3 15

4 20

5 25

6 30

The independent variable does not depend on anything.

The dependent variable depends on the independent variable in order to find its value.

Graph It!

independentdepend

ent

BooksPri

ce1 2 3 4 5

123456

8

10

7

9A straight line through the origin shows a direct proportion on a graph…

As x is increasing, y is also increasing.

Proportionality

The graph of a proportional relationship is a straightline that passes through the origin.

Proportional quantities can be describedby the equation y = kx, where k is a constant ratio... also known as constant of proportionality.

Xindependent

Ydependent

Using the form y = kx, write an equation for the information in the

chart.

Books Price

1 5

2 10

3 15

4 20

5 25

6 30

What did you do to x to get y?

5x = y

Xindependent

Ydependent

Books Price

1 5

2 10

3 15

4 20

5 25

6 30

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 = 510/2=515/3=520/4=525/5=530/6=5

The constant of proportionality is 5.

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.

Write the unit rate:

Read 80 pages in 2 hours_______ pages in one hour40

Xindependent

Ydependent

Record your information in the table.

Hours Pages

1 40

2 80

3 120

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)

Graph It!

independent

depend

ent

HoursPa

ges

102030405060708090

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.

Write the unit rate:

A plane travels 2250 km in 3 hours. What is the average speed? (km per hour)

750 km/hrWhat is the constant of proportionality?

They are asking for unit rate…

750

Graph It!

independent

depend

ent

Hours

Km

250

500

750

1000125015001750200022502500

1 2 3 4 5

Is this a direct proportion?

Yes…A straight line through the origin shows a direct proportion.

k = y/xk = 20/2k = 10

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

Write an equation for the information in the table.

X y

1 10

2 20

3 30

4 40

5 50

y = 10x

What is the constant of proportionality? 10

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

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

Y = 12/x

No. The equation is not in the form y = kx.

k = 12.5

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?