constant of proportionality

Constant of Proportionality

7th GradeChapter 2

 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

Remember: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 4 mile in 2 hours, what is the unit rate? (miles per hour)

=

2 miles per hour

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 multiply 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

X Y

Record your information in the table.

Hours Pages

1 40

2 80

3 120

Is this a direct proportion?Yes…as x increases by the same rate, by y increases by the same rate.

What is the constant of proportionality?k= =

40

Which is the dependent variable?

independent dependent

Which is the independent variable?

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

Find the first point on the graph and name it using the form (x,y).

(2, 20)

x

y

Plug those into the formula to find the constant of proportionality.

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?