Unit 7: Direct Variation/Proportion

#19 Direct Variation/Proportion Notes #20 Frayer Diagram #21 Direct Variation/Proportion HW Sheet

Your Georgia Performance Standard:

M6A2. Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from

mathematical rules, tables, and graphs.d. Describe proportional relationships mathematically using y = kx, where k is the constant of proportionality.

IT / NOT IT

8 minutes One person in your group is the recorder Everyone will look at the IT / NOT IT and

decide what you think makes it fit each of the categories….

If you can’t participate quietly and appropriately have no fear, I’ll have you sit down and you can view the IT / NOT IT examples from your desk.

Direct Proportion also known as

Direct Variation

Consider this:On Friday chicken biscuits are sold before school.

1. Complete the chart to the right.

2. What number do you think should go in the “?” column? How do you know?

# of biscuits

times?

=Total cost

1

2

5

$2.50, because every biscuit costs $2.50.

Direct Proportion Defined:

The relationship between 2 quantities where one quantity changes based on what happens to the other quantity.

Direct Proportion

Look at the BISCUIT table. Does the cost per biscuit change? What affects the total cost?

The number of biscuits determines the total cost.

The number of biscuits is “x” and the total cost is “y”.

What equation could we write to represent the total cost of the biscuits?

y = 2.50 x

no – it is always $2.50

the # of biscuits

Direct Proportion: Equations

The cost of the biscuits is the same no matter how many you purchase.

Each time the number of biscuits changes, it changes by the same amount….$2.50 per biscuit. It is the constant of proportionality. (k)

y = 2.50 x

All equations in a direct proportion are in the form y =

k x

Constant of Proportionality (AKA: unit rate)

IS a DIRECT VARIATION NOT!

y = 3x y = 3x +1

y = x y = x-1

y = ½ x or y=x/2 y = x + 3

y = 1000000x y = 3/x

What is the constant of proportionality for each of these equations?

Direct Proportion : Equations

Direct Proportion : Graphs

Look at the graph at the bottom of your notes Take 5 minutes to work with your shoulder

partner to graph the biscuit table What do you notice about your graph?

Direct Variation and its graphDirect Variation and its graph

Observations:

1.the graph will always be a straight line

2.The graph will always go through…

the ORIGIN!!!!!

Special point known as the “ORIGIN”

(0,0)

Tell if the following graph is a Direct Variation or not.

No Yes

No No

No Yes

Yes No

Tell if the following graph is a Direct Variation or not.

Let’s look again at theFrayer Diagram

1. Look at your Frayer Diagram sheet that we give you (it’s pink).

2. You have 3 minutes - complete the Graph and Equation blocks.

3. I’ll use the sticks to call on you to tell us how you completed it. If you have made any errors in your Frayer, be sure to correct them as we go through it together. This will be an excellent study tool!

Homework tonight:

The first side of your Homework Sheet.

Direct Proportion: Tables

Spend 5 minutes completing the X-Men Rock! Table.

How did you know what to put for y values?

direct proportion : the table

Notice…

• As x increases in value, y increases by the same factor ….

This “same factor” is the constant of proportionality. You can find the constant of proportionality k by using k = y/x

People(x)

Total Cost(y)

3 $30

5 $50

9 $90

Is it a Direct Proportion?

x y 6 12 7 14 8 16

Note, x increases: 6 , 7 , 8

and y increases: 12, 14, 16

•Is it a direct proportion? •Find the constant of proportionality for each row using k =y/x.

 k = 12/6 or 2

The constant of proportionality in the table above is 2.If all the values are the same it is a Direct Proportion

The equation would be y=2x

yk

x

x y 20 10 18 9 14 7 8 4

Yes!

k = 10/20 or ½

k= 9/18 or ½

k= 7/14 or ½

k= 4/8 or ½

Equation?

y = 1/2 x

Is this a direct variation? If yes, give the constant of variation (k) and the equation.

x y 15 5 3 26 1 75 2 150

No!

The k values are different!

Is this a direct variation? If yes, give the constant of variation (k) and the equation.

k

Direct Proportion: Word Problems

A car uses 8 gallons of gasoline to travel 280 miles. If the gas used and miles driven are proportional, how far will the car go on 10 gallons of gas? 25 gallons of gas? How many gallons are needed to drive 420 miles?

1. Spend 5 minutes with your shoulder partner completing the table from the word problem.

2. What is the constant of proportionality?

3. What equation would represent the scenario?

4. What would the graph look like?

xgallons used

ymiles driven

8 280

10 350

25 875

12 420

k = y/x280/8= 35

10 x 35 = 350

25 x 35 = 875

35x= 420x=12

2. constant of proportionality k = 353. equation: y = 35x4. a straight line through the origin

Frayer Diagram

Based on what we know so far, we’ll create a rough draft. I’ll write on the board, you write on your Assignment #20 page.