Transcript Fun with Fractions!

Fun with Fractions!
Kaitlyn Murray
EDCI 270
Spring 2012
Students
Teachers
Teachers
Have the students go through this application with a pencil
and paper to try out all the problems to get the right
answers!
•1. Target Audience
•
This power point presentation is directed towards upper
elementary students that are currently learning all about
fractions. These students can be first discovering
fractions, or students that already have an
understanding, but are building new ideas on top of
fractions, such as simple mathematic procedures.
•2. Learning Environment
•
This power point presentation is intended to use in the
classroom and as homework. First the students will learn how
to go through the presentation for help in the classroom,
where the teacher is available to help to enable the students
to understand the material at hand. Then, the students can
use this presentation for homework to fully understand the
material for the specific night’s homework.
•3. Objective/Purpose
1.
2.
3.
Given a fraction, students will be able to compare other fractions in
relation to size with the use of strip diagrams.
Given a fraction, students will describe where the fraction falls on
the number line between numbers zero and one.
Given an addition/subtraction, or multiply/divide fraction problem,
students will be able to do the mathematical process through strip
diagrams and equations with high accuracy.
Students
This is Fraction Jackson!
He’s here to help you through
the lesson!
•Students: How to use this application:
Click this to go to the HOME page
• Click this to find out what this word means
•
Link
•
Click these buttons to go the next or the previous
slide
•First of all……
•
You need to understand
what a fraction is, how to
compare it to the whole
unit, and do simple
mathematic processes.
•
LET’S GET STARTED!!
• You
and your four friends
order a pizza. Before the
pizza arrives, you want to
know how much of the
pizza each of you and
your four friends will get.
• Before solving this
problem, you need to
know more about
FRACTIONS.
START
•Understanding the meaning of A/B
1.
2.
A
-------B
3.
•
The unit, or the whole, is clearly
in mind.
(What=1?)
The denominator (B) tells how
many pieces of equal size the unit
is cut into (or thought of being
cut into)
The numerator (A) tells how
many such pieces are being
considered.
There can be different ways
that the fraction can look
like. Click here to find out!
•What a fraction looks like on the
computer!
•
This is what a fraction looks
like when we write them
down:
•
AND this is what a fraction
looks like on the computer:
•Draw strip diagram!!
•Now you try! -- ⅔-First draw the whole unit – one long strip
• Then, draw how many sections the whole is cut up
into (the denominator)
• Finally, shade in the number of pieces that are
being considered.
• Look back at the previous slide for examples!!
•
Now that you know what a
fraction is, it is important to
know Equivalent Fractions
Lowest Terms
and
all in
Greatest Common Factor
terms of fractions to be able
to do the mathematical
processes!
•Greatest Common Factor (GCF)
The greatest common factor of two (or more) whole numbers is
the largest number that is a factor of the two (or more)
numbers.
For example: 2, 3, and 6 are all common factors of 12 and 18,
and 6 is the GREATEST COMMON FACTOR of 12 and 18.
(sometimes called the greatest common divisor)
•Common Factors and Lowest Terms
A factor of a number divides the number, leaving no
remainder. (2 is a factor of 4)
A common factor is one that each of the numbers will
share and have in common.
12/30 = (6x2)/(15x2) = 6/1; 6/15 = (2x3)/5x3) = 2/5
Because 2 and 5 have no common factors (except 1),
we say that 2/5 is the SIMPLEST FORM of 12/30.
Alternatively, we say 12/30 in LOWEST TERMS is 2/5
Click here for
examples!
•EXAMPLES:
•Equivalent Fractions
2/3 and 100/150 look different but they are equal
EX: 2/3 and 4/6 Strip Diagrams!!!
•Now you try!
1.
What is the GCF of 45 and 20.
a)
b)
c)
2.
9
5
4
What is 18/24 ‘s equivalent fraction?
a)
b)
c)
¼
½
¾
•YES!!
1. The GCF of 45 and 20 IS 5!
•
Good job!!!
•UH-OH!!
• Remember
that the GCF is the largest factor that
both the numbers share.
• HINT** write out the factors for both numbers
and circle the greatest common factor
•YES!!
2. The equivalent fraction to 18/24 is ¾!!!
•
Good Job!!
•UH-OH!!!
• Remember
to draw out the strip diagrams to
compare how much is shaded in with each one
to find out which one is the equivalent fraction.
• OR find the lowest terms of the fraction to find
the equivalent one!
•Fraction Jackson!
Now that you know more
about fractions in lowest
terms, and their common
factors, it is important to
relate them to their
decimal values and what
category of numbers
fractions fall into. Let’s Go!
•Relating Fractions and Decimals:
Terminating Decimals
A fraction in simplest form can be presented with
terminating decimals when the denominator
has only 2’s and 5’s as factors, because we can
always find an equivalent fraction with a
denominator that is a power of 10.
EXAMPLE: ½ = 0.5
•Relating Fractions and Decimals:
Nonterminating Decimals
A fraction in simplest form can be presented with
a nonterminating repeating decimal if the
denominator has factors other than 2’s and 5’s.
Try with your calculator what are
nonterminating, repeating decimals!
EXAMPLE: 1/3 = 0.333333333333333333333333
•How to convert Fractions to decimals
The simple method is to divide. A fraction bar is
a division symbol, so divide the numerator by
the denominator to get the decimal value of
that fraction
•How to convert Decimals to Fractions
Take the decimal, for example, .25, and use that
number as the numerator. However many digits are
after the decimal, is how many zeros that need to
be in the denominator in base 10; in this case it is
100. Now simplify the fraction into the lowest
terms to get the answer of ¼.
•Numbers!
•
Real Numbers
•
Rational Numbers
•
Whole Numbers
•
Irrational Numbers
•Real Numbers
The set of rational numbers together with the set
of irrational numbers.
EXAMPLES: 2, -3, ½
•Rational Numbers
Any fraction or its equivalent terminating or
repeating decimal names
EXAMPLES: 1, 7/9, 1/3, 9
•Whole Numbers
Any of the numbers 0, 1, 2, 3, 4…… etc
•Irrational Numbers
Nonterminating, nonrepeating, decimals cannot be
represented as factors with the whole numbers as
the numerator and as the denominators
EXAMPLES: ∏(3.14), √2
•Decimals to Fractions Practice
Which fraction is closer to the decimal??
.29
• 1/3
• 1/4
.78
• 3/4
• 7/10
.05
•0
• 1/2
•YES!
.29 is close to and very similar to 1/3
Good Job!
•UH-OH!!
¼ is very close to .29, but think about how to
convert fractions to decimals….
HINT** Set the decimal as the numerator and the
denominator is a base 10 number….
•YES!!
.78 is closer to ¾!
Good Job!!
•UH-OH!!
.78 is close to 7/10, but remember to think about
changing decimals to fractions….
•YES!!
0.05 is closer to 0!
Good Job!!
•UH-OH!!!
Although .05 may look like 0.5, it is not.
Remember to keep in mind how to convert
decimals to fractions….
•Fraction Jackson!
Now that we know what
the fraction is equivalent
to in decimal form, we
can now estimate the
fractional values and
compare them to one
another to know which
fraction is the largest in a
set of numbers!
•Estimating Fractional Values
•
Look at the denominators for the fraction sizes
•
•
A larger denominator means a smaller piece of the whole
A smaller denominator means a larger piece of the whole
You would rather have a larger slice of pizza than a smaller
one when sharing with you and your four friends….
•Fractions Close to Zero
2/11, 11/108, 3/36
*Look at the size of the denominator, imagine the pieces and
how many are in the entire whole and how many are being
considered. The reason these fractions are close to zero is
because there aren’t many pieces being considered in
comparison to the whole.
•Fractions Close to ½
17/35, 4/7, 25/50
*Look at the denominator and figure out what is
exactly half of it to determine if the fraction is close
to ½
•Fractions Close to 1
9/8, 11/12, 99/102
*Look at the denominator and determine how many
pieces there are in relation to the numerator.
•Practice with Estimating Fractions
Look at each fraction and determine if the fraction is
close to 0, ½, or 1.
A.
B.
C.
D.
11/9 Close to 0, Close to ½, Close to 1
18/36 Close to 0, Close to ½, Close to 1
7/12 Close to 0, Close to ½ , Close to 1
1/25 Close to 0, Close to ½, Close to 1
•YES!!!
11/9 is close to 1
9 goes into 11 one full time. It is an improper
fraction
•UH-OH!!!
Remember to look at the denominator and
determine how many pieces the whole is cut up
into and consider how many pieces you want
from the numerator!
•YES!!!
18/36 is equal to ½!!
•YES!!!
7/12 is just over ½, so it is closer to ½!
•YES!!
1/25 is close to zero!!
It is one small piece over a large denominator
•Fraction Jackson!!
•
Now you know what a
fraction is, and how to
compare it to other
sizes to find the
common factors, you
can now determine the
answer to computation
with fractions!! You can
do it!!
•Computing with Fractions: Adding
When adding fractions, it is important to find the
common denominator by finding the common
factors of the denominators of both the
fractions in the problem.
It is also good to draw strip diagrams.
•Draw strip diagram!!
•Adding ½ + 1/3
½+1/3
-Must find a common multiple the
denominators share.
½*3/3+1/3*2/2 -In this case, the common
multiple is 6. Multiply each
fraction to have the common
denominator!
3/6+2/6
- Add only the numerators!!
=5/6
•Computing with Fractions: Subtracting!
Subtracting is very similar to addition in finding a
common denominator.
¾-1/3 -Must find the common denominator
9/12-4/12 -Subtract!
=5/12
•
When using strip diagrams, shade in the whole unit with was the first fraction
is and then X out what is being subtracted after there is a common
denominator shown on the diagram. The boxes without an X make up the
answer. Try it with this problem!
•Now you try!
Add or subtract the following, using strip diagrams if
you need them!
• ¾-16/64
•
•
Check your answer!
3/5+1/2
•
Check your answer!
•¾-16/64
• 64
is the common denominator!!
• When reducing the final fraction, the answer
is….
½!
•3/5+1/2
10 is the common denominator of 2 and 5.
• When adding, you should have gotten an improper
fraction.
• When the fraction is reduced, the answer is…
=1 1/10
•
•Computing with Fractions:
Multiplying!
THE RULE: multiply the numerators to obtain the
numerator of the product and multiply the denominators
to find the denominator of the product.
*To make multiplication of fractions meaningful, one must
understand the referent units for the fractions and what
the product actually stands for.
•Referent Unit
The referent unit is the number that starts on the left.
For example, 1/3 of ½ of 1 whole
1/3 is the referent unit
1/3 is multiplied by ½ of the 1 whole unit
•Multiplying Fractions by Whole Numbers
When multiplying fractions by whole numbers, the denominator of the
whole number is 1. Then you simply multiply across and simplify the
fraction.
Example: 3 * 1/5
3/1 * 1/5
3*1/1*5
= 3/5!
•
•Multiplying A Fraction by another
Fraction
The process is still the same when multiplying fractions.
Instead of one whole number and a fraction, there are
now two fractions.
Example:
3/5 * 3*4
Multiply across – 9/20
ANSWER: 9/20!
•
•Computing with Fraction:
Dividing
Dividing by a fraction gives the same result as multiplying
by the reciprocal (flip the numerator and the
denominator – ½ -> 2/1) of the fraction symbolically,
when the divisor is a fraction, N ÷ c/d = N * d/c, or if N
itself is a fraction, a/b ÷ c/d = a/b * d/c
(c/d cannot equal 0)
•a/b ÷ c/d = (a*d)/(b*c)
Dividend ÷ Divisor = Quotient
•Dividing by Unit Fractions
a/b ÷ 1/c = a/b ÷ c/1 = ac/b
Examples:
11 ÷ 1/3 = 11 * 3/1 = 11*3 / 1 = 33
½ ÷ 1/3 = ½ * 3/1 = (3*1)/(1*2) = 3/2 = 1 ½
•Now You Try!
Multiply or Divide the following fraction problems:
•
•
2/3 * 5/2
•
Check your answer!
•
Check your answer!
2/3 ÷ 1/6
•2/3 * 5/2
When multiplying across, you should get 10/6.
• When you simplify the improper fraction into a
mixed number, the answer is…
•
=1 2/3
•2/3 ÷ 1/6
•
When going through the processes of dividing, the
reciprocal is 6/1. After the multiplication across, the
answer is…
=4
•Video!!
•
Now watch this video to recap how to draw strip diagrams, how to add,
subtract, multiply, and divide using them.
•Now It’s Time for the Quiz!!
I know! Quizzes aren’t cool, but
see how you do! You can do
it! You and your friends
splitting the pizza need you
to do well!!
Make sure you click the answer
to check your work to see
how well you do!
Once you start the quiz, you
cannot go “home” or go
onto the next question until
you get the answer correct.
You must finish it.
•1. Quiz
A
---------B
What is the indicator of how
many pieces of the whole
are being considered in
the fraction?
A?
B?
•A!
Right!
The numerator tells us how many pieces are
being considered in the fraction of the whole!
•B….
OH-NO!!!
Remember what each part of the fraction is called
and what each part stands for of the whole of the
fraction.
•2. Quiz
What is the lowest term for
the fraction to the left??
25
----------625
A.
B.
¼
1/25
•A. ¼
UH-OH!!!
You have the right thinking in that when you see 25,
you think of fourths, but think of what the common
factor in both the numerator and the denominator
to find the lowest tem.
•B. 1/25
Good Job!!!
You knew to look at the common factors of the
numerator and the denominator to simplify the
fraction into its lowest term!
•3. Quiz
14
---------27
Is the fraction to the left
equal to a terminating or a
repeating decimal?
A.
B.
Repeating decimal
Terminating decimal
•A. Repeating Decimal
YES!!
You knew that the multiples of the denominator do
not include 2 or 5, so that it had to be a repeating
decimal.
•B. Terminating Decimal
UH-OH!!
Remember that terminating decimals have only
2’s or 5’s as factors. Look again at the fraction..
•4. Quiz
What kind of number is
shown to the right?
-31/2
A.
B.
Rational Number
Real Number
•A. Rational Numbers
YES!!
Rational Numbers are any fractions or its
equivalent terminating or repeating decimal
names.
•B. Real Number
YES!!
Real Numbers are the sets of rational numbers
together with the set of irrational numbers.
•4. Quiz
-3 ½ is a Real AND a Rational Number!!
You might have been questioning your answer, but
you are right in the thinking that -3 ½ fits both of
the descriptions of rational and real numbers!
GOOD JOB!!!
•5. Quiz
What fraction is the
decimal closest to?
0.29
A.
B.
1/3
¼
•A. 1/3
YES!!
That’s right!! 0.29 is close to 0.3333 which is the
same as the fraction 1/3
•B. 1/4
UH-OH!!
Although 0.29 seems like it would be close to 0.25
which is ¼, but think of a smaller fraction, that has
a larger decimal value…
•6. Quiz
Is this fraction closer to ½ or
1?
67
------------100
A.
B.
½
1
•A. ½
UH-OH!!
Remember to think of what exactly ½ is of the
denominator and think of how far away the numerator is
from the midpoint of the denominator. It is close to ½,
but not as close to 1 is.
•B. 1
YES!!!
67
---------100
is closer to 1 than ½!!
GOOD JOB!!
•7. Quiz
5
--------8
Add these two fractions together.
A.
B.
1
----------3
6/11
11/12
•A. 6/11
UH-OH!!!
Remember that in adding or subtracting fractions,
there first must be a common denominator
between the two original fractions.
•B. 11/12
YES!!
You knew to find the common denominator of 24 and add
the numerators equally 22/24, but when the fraction is
simplified, the answer is 11/12!
GOOD JOB!!
•8. Quiz
3
-----4
Subtract these two fractions.
A.
B.
1
-------3
5/12
2/1
•A. 5/12
YES!!
You knew to find the common denominator of 12
and subtract across to get the answer of 5/12!!
GOOD JOB!!!
•B. 2/1
UH-OH!!
2/1 is the same as the whole number 2. When subtracting
fractions, you cannot get a whole number. Try to remember to
find the common denominator of the two fractions. Whatever
you do to the denominator, you must do to the numerator.
•9. Quiz
2
-------3
Multiply these fractions together.
A.
B.
5
---------2
10/6
60/36
•A. 10/6
YES!!!
When multiplying fractions, you knew not to find a common
denominator, and just simply multiply across the numerators
to the get the numerator of the product, and multiply across
the denominators to get the denominator of the product.
GOOD JOB!!!
•B. 60/36
UH-OH!!!
When multiplying fractions, there is no need to find
a common denominator. Just simply multiply across
the numerators to get the numerator of the
product and multiply across the denominators to
get the denominator of the product.
•10. Quiz
6 1/3
÷
1/3
Divide these fractions. Use the
first fraction as the dividend
and the second fraction as
the divisor to get the
quotient.
A.
39/2
B.
19/1
•A. 39/2
YES!!
You knew to change the mix number into an improper
fraction and multiply that by the reciprocal of the divisor
(the second fraction)
GOOD JOB!!
•B. 19/1
UH-OH!!
Remember that you need to change the divisor to its
reciprocal, then multiply. Don’t just change the mix
number into an improper fraction and divide by
1/3.
•Fraction Jackson!!
Congratulations!!!! You did
it!! Now you can finally do
the arithmetic to find out
how much pizza you and
your four friends will get.
•Conclusion!
You and your four
friends order a pizza.
Before the pizza
arrives, you want to
know how much of
the pizza each of you
and your friends will
get.
•
•
•
Remember that the pizza is
the whole unit.
There are five people (which
makes the five the
denominator)
One person will get 1/5 of
the pizza then, to be able to
share it with you and your
four friends!