direct and inverse variation student instructional module use the buttons below to move through this...

Direct and Inverse VariationStudent Instructional Module

Use the buttons below to move through this module

Algebra A: 4-7 & 4-8

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Review your notes Return to the Question

First decide if your problem is Direct or Inverse Variation

• If it is Direct Variation Look for the phrases:– “ varies directly as”– “is directly

proportional to x”

• If it is Inverse Variation Look for the phrases:– “ varies inversely as”– “is indirectly


Now you Try!

• Read the following problem and determine if it is Direct variation or Inverse variation. Select the correct term below.

Assume that y varies directly as x. If y=3 when x = 15, find y when x = -25.

Direct Variation Inverse Variation

Try Again!

• Remember the problem states:


• Click the button below to review your notes. Then try again!

Direct Variation vs. Inverse Variation

(Return to the question)





Correct !

• The problem:


• Is a Direct Variation problem because you see the phrase “y varies directly as x.”

Try this one!


Assume that y varies inversely as x. If y=6 when x = 12, find x when x = 9.


Try Again!

• Remember the problem states:Assume that y varies inversely as x. If y=6

when x = 12, find x when x = 9.


Correct !

• The problem:

Assume that y varies inversely as x. If y=6 when x = 12, find x when x = 9.

.

• Is a Inverse Variation problem because it says “y varies inversely as x.”

Story Problems also use Direct Variation & Inverse Variation• In Direct Variation

story problems:– as one thing increases

the other increases.

• In Inverse Variation story problems:– as one thing increases

the other decreases.– Ex: Fulcrum Problems

An Example of Inverse Variation: Fulcrum Problems (see-saw)

• Did you ever notice that the heavier person needs to sit closer to the middle in order to balance on a seesaw?

• When weights are placed on a lever it works the same way.

Fulcrum

W1 W2

lever

D1 D2

W1D1 = W2D2

An Example of Inverse Variation: Fulcrum Problems

• As the distance from the fulcrum increases the weight decreases.

• That means it is an Inverse Variation Problem.

Source: http://www.tooter4kids.com

Example of a Direct Variation Problem

• If 4 pounds of peanuts cost $7.50, how much will 2.5 pounds cost.

• This is a Direct Variation problem because:– as the weight of the

peanuts increases the cost also increases.

Now you Try!

• Read the following problem and determine if it is an example of Direct Variation or Inverse Variation. Select the correct term below.

• Charles Law says that the volume of gas is directly proportional to its temperature. If the volume of gas is 2.5 cubic feet at 150 degrees absolute temperature, what is the volume of that same gas at 200 degrees absolute temperature.


Try Again!

• The problems stated:Charles Law says that the volume of gas is directly proportional to its temperature. If the volume of gas is 2.5 cubic feet at 150 degrees absolute temperature, what is the volume of that same gas at 200 degrees absolute temperature.





• Also look for:– Story problems where

as one thing increases the other increases.

• If it is Inverse Variation Look for the phrases:– “ varies inversly as”– “is indirectly



as one thing increases the other decreases.

– Ex: Fulcrum Problems


Correct!

• The problems stated:Charles Law says that the volume of gas is directly proportional to its temperature. If the volume of gas is 2.5 cubic feet at 150 degrees absolute temperature, what is the volume of that same gas at 200 degrees absolute temperature.

• Since it stated that gas was directly proportional to temperature it is a direct variation problem.

Try this one!


• Grant, who weights 150 pounds, is seated 8 feet from the fulcrum of a seesaw. Mariel is seated 10 feet from the fulcrum. If the see saw is balanced, how much does Mariel weigh?


Try Again!

• The problems stated:Grant, who weights 150 pounds, is seated 8 feet from the fulcrum of a seesaw. Mariel is seated 10 feet from the fulcrum. If the see saw is balanced, how much does Mariel weigh?


Correct!

• The problems stated:• Grant, who weights 150 pounds, is seated 8 feet

from the fulcrum of a seesaw. Mariel is seated 10 feet from the fulcrum. If the see saw is balanced, how much does Mariel weigh?

• Since, on a balanced seesaw, weight increases as distance from the fulcrum decreases; this is inverse variation.

Give this a shot


• The frequency of a vibrating string is inversely proportional to its length. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8-inch string.


Try Again!

• The problems stated:The frequency of a vibrating string is inversely proportional to its length. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8-inch string.


Correct!

• The problems stated:The frequency of a vibrating string is inversely proportional to its length. A violin string 10 inches long vibrates at a frequency of 512 cycles per second. Find the frequency of an 8-inch string.

• Since it stated that frequency was inversely proportional to length it is an inverse variation problem.

Congratulations

• You can figure out if a problem is an example of Direct Variation or Inverse Variation.

• Now we can teach you how to solve these problems!

Learn the Equations

• Once you have decided whether an example is a direct or inverse variation problem you must use one of the following equations:

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Identifying “Partners”

• In variation problems the variables have “partners”, or numbers that belong together. We use subscripts to identify that these numbers are partners.

Ex: Solve this problem assuming y varies directly as x:

If y=6 when x=8, find y when x=12.

Partners (1) Partners (2)

Plugging the Information In

Then plug the information in to the correct equation.

• Direct Variation – Partners are across from each other.• Inverse Variation – Partners are on the same side

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

For Example:

Ex: Solve this problem assuming y varies directly as x:

This sentence tells us it is a direct variation problem

If y=6 when x=8, find y when x=12.

Partners (1) Partners (2)Plug them into the Direct Variation equationY1 = X1 6 = 8Y2 X2 y 12

Now you Try!

• Set up the following problem:

Assuming y varies directly as x:

If x= -5 when y = 6, find y when x= -8

A. (-5)(6) = (-8)y

B. (-5)y = (-8)(6)

C. 6 = -5 y-8

D. y = -5

6 -8

Not Quite Right …

• The problem said:



• Check your equationVariation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

You’re not there yet…

• Check out how you plug in your partners:



Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Correct!

Assuming y varies directly as x:You chose the direct variation equation!

You correctly identified and plugged in the partners.

If x= -5 when y = 6, find y when x= -86 = -5y -8

Try this one!

• Set up the following problem:

Assume that y varies inversely as x

If y= 15 when x = 21, find x when y= 27

A. (21)(15) = (x)27

B. (x)15 = (21)(27)

C. 15 = 21 27x

D. 15 = x

27 21

You’re not there yet…

• Check out how you plug in your partners:



Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Not Quite Right …

• The problem said:



• Check your equationVariation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Correct!


You chose the inverse variation equation!

You correctly identified and plugged in the partners.


(21)(15) = (x)27

Congratulations!

• So far you can:– Figure out if a problem is an example of Direct

Variation or Inverse Variation.– Remember the correct equation– Look for the partners– And plug them in.

• Now it is time to learn how to solve the problem!

Solving Equations

• Before we go on, it is time to review how to solve equations.

• Watch the video on the next slide.

• Then we will practice solving direct and inverse variation equations.

Solving Inverse Variation Equations

• If you have an inverse variation problem like:Assume that y varies inversely as x


(21)(18) = (x)27 which means 378 = 27x

• Once you have set it up you must get x by itself by dividing both sides by 27

378 = 27x

27 27

then x = 18

Solving Direct Variation Equations

• Direct Variation Equations look a little different.

• After you have plugged into a direct variation equation it will still be in fraction form like this example:


6 = -5

y -8

Cross Multiplying

• You can cross multiply to solve these equationsIf x= -5 when y = 6, find y when x= -8

6 = -5y -86(-8) =(-5)y

• Then you solve the equation the same way we did the inverse variation problems.

-48 = -5y-5 -5

9.6 = y

Lets Review

• Decide if a problem is Direct or Inverse Variation

• Remember the Correct Equation

• Look for the Partners

• Plug them into the Equation

• Solve!

Try a whole problem!

Assume that y varies directly as x.

If y=3 when x = 15, find y when x = -25.

A. -5

B. -125

C. - 3 5

D. 5

Oops!

• Did you set up the partners correctly?

• Assume that y varies directly as x. If y=3 when x = 15, find y when x = -25.

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Not there yet!

Remember:

• A negative X a negative = A positive

• A negative X a positive = A negative

No Quite Right

• Did you use the right Equation?

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

That is correct!

• You chose direct variation

• 3 = y

15 -25

• You correctly plugged in the pairs

• And cross multiplied to get y= -5

Try this one!

• Grant, who weights 150 pounds, is seated 8 feet from the fulcrum of a seesaw. Mariel is seated 10 feet from the fulcrum. If the see saw is balanced, how much does Mariel weigh?

A. 187.5lbs

B. 120lbs

C. 1.875lbs

D. 0.53lbs

Not Quite

• Did you choose the correct Equation?

• Did you choose the right partners?

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Does this answer make sense?

• Do you think she weighs such a small amount?

• Maybe you should double check how you plugged the pairs in to the equation.

Variation Equations


Y1 = X1 X1Y1 = X2Y2

Y2 X2

Hooray!

• You can do Direct and Indirect Variation Problems!!!!

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