Transcript Direct and Inverse Variation Student Instructional Module

Algebra A: 4-7 & 4-8
Direct and Inverse Variation
Student Instructional Module
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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
proportional to x”
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:
Assume that y varies directly as x. If y=3
when x = 15, find y when x = -25.
• Click the button below to review your
notes. Then try again!
Direct Variation vs. 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
proportional to x”
(Return to the question)
Correct !
• The problem:
Assume that y varies directly as x. If y=3
when x = 15, find y when x = -25.
• Is a Direct Variation problem because you
see the phrase “y varies directly as x.”
Try this one!
• Read the following problem and determine
if it is Direct variation or Inverse variation.
Select the correct term below.
Assume that y varies inversely as x. If y=6
when x = 12, find x when x = 9.
Direct Variation
Inverse Variation
Try Again!
• Remember the problem states:
Assume that y varies inversely as x. If y=6
when x = 12, find x when x = 9.
• Click the button below to review your
notes. Then try again!
Direct Variation vs. 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
proportional to x”
(Return to the question)
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.
D1
D2
W1
W2
lever
Fulcrum
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.
Direct Variation
Inverse Variation
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.
• Click the button below to review your
notes. Then try again!
Direct Variation vs. Inverse Variation
• If it is Direct Variation
Look for the phrases:
– “ varies directly as”
– “is directly
proportional to x”
• 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
proportional to x”
• Also look for:
– Story problems where
as one thing increases
the other decreases.
– Ex: Fulcrum Problems
(Return to the question)
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!
• Read the following problem and determine if it is
Direct variation or Inverse variation. Select the
correct term below.
• 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?
Direct Variation
Inverse Variation
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?
• Click the button below to review your
notes. Then try again!
Direct Variation vs. Inverse Variation
• If it is Direct Variation
Look for the phrases:
– “ varies directly as”
– “is directly
proportional to x”
• 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
proportional to x”
• Also look for:
– Story problems where
as one thing increases
the other decreases.
– Ex: Fulcrum Problems
(Return to the question)
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
• Read the following problem and determine if it is
Direct variation or Inverse variation. Select the
correct term below.
• 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.
Direct Variation
Inverse Variation
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.
• Click the button below to review your
notes. Then try again!
Direct Variation vs. Inverse Variation
• If it is Direct Variation
Look for the phrases:
– “ varies directly as”
– “is directly
proportional to x”
• 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
proportional to x”
• Also look for:
– Story problems where
as one thing increases
the other decreases.
– Ex: Fulcrum Problems
(Return to the question)
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
Direct Variation
Inverse Variation
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
Direct Variation
Inverse Variation
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 equation
Y1
=
X1
6
=
8
Y2
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:
Assuming y varies directly as x:
If x= -5 when y = 6, find y when x= -8
• Check your equation
Variation Equations
Direct Variation
Inverse Variation
Y1
=
X1
X1Y1 = X2Y2
Y2
X2
You’re not there yet…
• Check out how you plug in your partners:
Assuming y varies directly as x:
If x= -5 when y = 6, find y when x= -8
Variation Equations
Direct Variation
Inverse Variation
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= -8
6
=
-5
y
-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
27
x
D. 15
=
x
27
21
You’re not there yet…
• Check out how you plug in your partners:
Assume that y varies inversely as x
If y= 15 when x = 21, find x when y= 27
Variation Equations
Direct Variation
Inverse Variation
Y1
=
X1
X1Y1 = X2Y2
Y2
X2
Not Quite Right …
• The problem said:
Assume that y varies inversely as x
If y= 15 when x = 21, find x when y= 27
• Check your equation
Variation Equations
Direct Variation
Inverse Variation
Y1
=
X1
X1Y1 = X2Y2
Y2
X2
Correct!
Assume that y varies inversely as x
You chose the inverse variation equation!
You correctly identified and plugged in the
partners.
If y= 15 when x = 21, find x when y= 27
(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
If y= 18 when x = 21, find x when y= 27
(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
then
378
27
x
=
=
27x
27
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:
If x= -5 when y = 6, find y when x= -8
6
=
-5
y
-8
Cross Multiplying
• You can cross multiply to solve these equations
If x= -5 when y = 6, find y when x= -8
6
=
-5
y
-8
6(-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
Direct Variation
Inverse Variation
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
Direct Variation
Inverse Variation
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.
B.
C.
D.
187.5lbs
120lbs
1.875lbs
0.53lbs
Not Quite
• Did you choose the correct Equation?
• Did you choose the right partners?
Variation Equations
Direct Variation
Inverse Variation
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
Direct Variation
Inverse Variation
Y1
=
X1
X1Y1 = X2Y2
Y2
X2
Hooray!
• You can do Direct and Indirect Variation
Problems!!!!