Transcript Modeling Multiplication of Fractions

Modeling Multiplication of
Fractions
MCC4.NF.4; MCC5.NF.4; MCC5.NF.5;
MCC5.NF.6
Deanna Cross – Hutto Middle School
FRACTION BY A WHOLE NUMBER
Multiplying on a Number Line
• Fraction by a WHOLE number
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the
paper. How much paper did she use? How
much paper did she have left over?
Suggestions on how to solve?
Number Line
Number Lines
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
Start and end with an arrow
Divided into equal (equivalent) increments
Can start and end at any number
Are there any numbers that can be “renamed” or
written as an equivalent fraction?
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the paper.
How much paper did she use? How much paper
did she have left over?
• Isabel has used only ¾ of the paper. What if
she had used ½ of the paper? How much
would she have used?
• You have to multiply 8 x ¾.
8x¾
• First – Model what you have on a number line
– “She had 8 feet of wrapping paper”
0
1
2
3
4
5
6
7
8
• Now, she is multiplying by ¾ . What is the
denominator?
8x¾
• Now, divide the total amount (8) into 4 pieces. (8 ÷ 4 =
2 – so each piece is equal to 2)
0
1
2
3
4
5
6
7
• Shade in 3 of the four pieces.
• Look to see if this lines up with a number on your
number line.
8
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the paper.
How much paper did she use? How much paper
did she have left over?
• So, 8 x ¾ = 6.
• Why is the answer smaller than 8?
• Because whenever you multiply a whole
number by a fraction, you will get a smaller
answer.
4x½
• Will your answer be bigger or smaller than 4?
• First – show 4 on the number line.
0
1
2
3
4
4x½
• Now, look at your denominator – 2
• Divide your bar into two EQUAL pieces.
0
1
2
3
4
• Shade in 1 of the two pieces.
• Does this line up with a number on the number
line?
3x
1
3
• Will your answer be bigger or smaller than 3?
• First – show 3 on the number line.
0
1
2
3
4
3x
1
3
• Now, look at your denominator – 3
• Divide your bar into three EQUAL pieces.
0
1
2
3
4
• Shade in 1 of the three pieces.
• Does this line up with a number on the number
line?
6x
2
4
• Will your answer be bigger or smaller than 6?
• First – show 6 on the number line.
0
1
2
3
4
5
6
6x
2
4
• Now, look at your denominator – 4
• Divide your bar into four EQUAL pieces.
HINT: Divide 6 by 4 and determine the decimal portion to
divide this piece into
0
1
2
3
4
5
6
• Shade in 2 of the four pieces.
• Does this line up with a number on the number line?
Practice
• Optional Practice problems
1)
2)
3)
4)
5)
8 x 14
9 x 23
12 x 34
10 x 53
3
4x8
Multiplying with an AREA MODEL
• Fraction by a WHOLE number
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the
paper. How much paper did she use? How
much paper did she have left over?
Area models
• Reminder of area – length x width = area
• Area is the amount INSIDE a rectangular
shape.
• To determine area, you multiply TWO
numbers – the length and the width.
Area models
• Multiply the length and the width
2
5
• 2 x 5 = 10 – AREA = 10
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the paper.
How much paper did she use? How much paper
did she have left over?
• Isabel has used only ¾ of the paper.
• You have to multiply 8 x ¾.
• Suggestions to solve using area model?
8x¾
• Draw a rectangle.
• Divide the rectangle into smaller rectangles to
represent your WHOLE number.
8
8x¾
• Next, along the vertical side, divide your
rectangle into the number of pieces
representing your denominator  4
4
8x¾
• Now, shade in 3 rows of the 4 you just
created.
8
4
8x¾
• Hard part – This model started out with 8
wholes. How much would 1 box be worth?
THINK…
8
4
8x¾
• This is 1 whole…
• So how much would 1 box be worth?
• 1 box equals ¼
8
4
8x¾
• Now, count how many ¼’s you have shaded
green.
24
• 24 boxes =
4
8
4
24
4
•Can we leave like this, or is there another way to
write this improper fraction?
8x¾
24
•
= 24 ÷ 4 = 6
4
• Proof: If you divided 8 dollars up among 4
people, how much would each get?
8
4
8x¾
24
• = 24 ÷ 4 = 6
4
• Proof: If you divided 8 dollars up among 4
people, how much would each get?
– TWO
• Now, how much would 3 people get?
– SIX
• So, ¾ of 8 = 6
4x½
• This is one you already know the answer to –
if you have ½ of 4 you have 2. Let’s prove that
with an area model.
4x½
• First, draw a rectangle divided into your whole
number – 4
4
4x½
• Next, divide your rectangle into the number of
pieces for your denominator along the vertical
edge.
4
2
4x½
• Shade in the number represented by the
numerator…
4
2
4x½
• Now THINK – how much is ONE square worth?
What is your WHOLE?
4
2
4x½
• One square = ½
4
• There are 4 “halves” – or
2
4
2
4x½
4
• =4÷2=2
2
• So – if you have half of 4 you have 2.
4
2
3 x 1/3
• Try to draw this model on your own – you
already know what 1/3 of 3 would be…
3
3
3 x 1/3
• Now, think about what each square
represents…
3
So, each square =
1/3, there are 3
thirds… 3
3
3
3 x 1/3 = 1
• Draw the model.
6
4
6 x 2/4
6 x 2/4
• What does each square represent?
6
1
4
How many fourths?
12
4
12
 12  4  3
4
Practice
• Optional Practice problems
1)
2)
3)
4)
5)
8 x 34
6 x 23
12 x 14
5 x 53
5
4x8
Multiplying with TAPE DIAGRAM
• Fraction by a WHOLE number
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the
paper. How much paper did she use? How
much paper did she have left over?
Tape diagrams
• Tape diagrams are like adding strips of paper
together to determine lengths. For example,
if I had 3 chocolate cupcakes and someone
gave me 2 more, I would have five.
3 chocolate
2 more
5 chocolate cupcakes
Multiplying with Tape Diagrams
• Fraction by Whole numbers are easy with tape
diagrams…it is like repeated addition.
Isabel had 8 feet of wrapping paper to wrap
Christmas gifts with. She Used 3/4 of the paper.
How much paper did she use? How much paper
did she have left over?
• Isabel has used only ¾ of the paper.
• You have to multiply 8 x ¾.
• Suggestions to solve using tape diagram
model?
8 x 3/4
• Think, how many 3/4ths do you need?
• 8
• Make a tape model to represent 3/4. Copy this
eight times.
8 x 3/4
+
+
+
+
+
+
+
• Add up how many
fourth’s you
have…
3 3 3 3 3 3 3 3 24
       
4 4 4 4 4 4 4 4 4
Can you leave the
fraction as it is?
8 x 3/4
24
4
How do you change an improper
fraction to a mixed number?
24
 24  4  6
4
4x½
• Draw a diagram to represent ½.
• Repeat this 4 times.
4x½
• Add up each piece…
1 1 1 1 4
   
2 2 2 2 2
+
+
+
4
 42  2
2
Ahhh…there is a
large number on
top of a small
number!
3 x 1/3
• Model
• Add
• Reduce
+
+
1 1 1 3
  
3 3 3 3
How else can you write
a number over itself?
3
1
3
6 x 2/4
• Model
• Add
• Reduce
+
+
+
+
+
2 2 2 2 2 2 12
     
4 4 4 4 4 4 4
Can you simplify this fraction?
12
 12  4  3
4
Practice
• Optional Practice problems
1)
2)
3)
4)
5)
12 x 34
9 x 23
5 x 14
6 x 53
7
4x8
Algorithm?
• Now…let’s look at our practice problems and
try to determine an algorithm to solve
multiplication of a whole by a fraction.
3 24
8 
6
4 4
1 4
4   2
2 2
1 3
3   1
3 3
2 12
6   3
4 4
Is there a
pattern? What
is being done
each time?
Algorithm
a
( ) x q = (a x q) ÷ b
b
HUH??? LETTERS???
Each letter is a variable. It represents or takes
the place of a number. Let’s look at an
example of what these letters mean.
Algorithm
a
( ) x q = (a x q)÷b
b
a = your numerator
b = your denominator
3
 8  (3  8)  4
4
q = your whole number
Practice
• Optional Practice problems
1)
2)
3)
4)
5)
7
9
2
3
1
4
x5
x 18
x 32
3
x
10
5
7
x4
8
FRACTION BY A FRACTION
•
3
4
2
3
of a class are boys. Of those boys, are
wearing tennis shoes. What fraction shows
how many boys are wearing tennis shoes?
• Suggestions on how to solve this?
Multiplying on a Number Line
• First, draw a line graph to represent the
amount of boys (3/4).
0
1
4
2
4
3
4
4
4
5
4
Multiplying on a Number Line
• Next, divide this bar into the denominator of
the first fraction (3). Shade in the numerator
(2).
0
1
4
2
4
3
4
4
4
5
4
Multiplying on a Number Line
• Finally, see if this matches any of your points
on the number line.
Can 2/4 be
written any
other way?
0
1
4
2
4
3
4
4
4
5
4
Multiplying on a Number Line
2
4
These are both even numbers, so the
fraction can be reduced (or simplified) by
dividing the numerator and denominator
by 2.
22 1

42 2
• Model the second fraction (factor).
0
1
8
2
8
3
8
4
8
5
8
6
8
7
8
8
8
• Divide this amount into two equal sections (how
can you divide 7 in half?)
0
1
8
2
8
3
8
4
8
5
8
6
8
7
8
• When you divide 7 by 2, it does not produce an
even number. Instead, you get 3.5 - Model this
amount (three sections and half of a section).
8
8
• This does NOT fall at an exact mark on the
number line, which means more numbers
must be added to the number line.
0
1
8
2
8
3
8
4
8
5
8
What could
fall 6between
7
3/8
8 and8
4/8???
8
8
0
1
8
2
8
3
8
4
8
5
8
6
8
7
8
We need a number half way in between these
two fractions, which means we need two
TIMES as many increments (or lines) on the
number line. What is 2 x 8?
8
8
0
2
16
4
16
6
16
8
16
10
16
12
16
14
16
16
16
0
1
8
2
8
3
8
4
8
5
8
6
8
7
8
8
8
Let’s make equivalent fractions with 16 as a
denominator by multiplying all by 2/2.
0
2
16
4
16
6
16
8
16
10
16
12
16
14
16
16
16
0
1
8
2
8
3
8
4
8
5
8
6
8
7
8
8
8
Now, what could fall between 6/16 and 8/16?
7/16
• Model the second fraction (factor).
• Divide into 3 sections and shade 2.
0
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
10
10
10
• Check to see if this lines up with a number on the
number line.
• Divide the numerator and denominator by 2.
0
1
10
2
10
3
10
4
10
5
10
42 2

10  2 5
Can this fraction be
reduced (simplified)
or written in any
other way?
6
10
7
10
8
10
9
10
10
10
Practice
• Optional Practice problems
1)
2)
3)
4)
5)
3 4

4 12
2 6

3 7
1 3

2 8
3 8

4 13
2 2

4 8
Multiplying with an AREA MODEL
• Fraction by a Fraction
•
3
4
2
3
of a class are boys. Of those boys, are
wearing tennis shoes. What fraction shows
how many boys are wearing tennis shoes?
• Suggestions on how to solve this using an area
model?
2
3
2 3
of
OR 
3
4
3 4
• Draw a rectangular model showing 3
4
3
horizontally.
4
2
3
2 3
of
OR 
3
4
3 4
2
• Next, model vertically.
3
3
4
2
3
2
3
2 3
of
OR 
3
4
3 4
• To determine your answer, count the boxes
you shaded twice.
– SIX
2
3
2 3
of
OR 
3
4
3 4
• Next, count the total number of boxes.
– TWELVE
2
3
2 3
of
OR 
3
4
3 4
• So, your numerator = 6
• Your denominator = 12
Can you reduce
or simplify
this?
6
12
6 2 3 3 1
   
12 2 6 3 2
6 3 2 2 1
   
12 3 4 2 2
6 6 1
 
12 6 2
• Draw a rectangular model showing
horizontally. 7
8
7
8
• Next, model
1
2
1
2
vertically.
7
8
• To determine your answer, count the boxes
you shaded twice.
– SEVEN
• Next, count the total number of boxes.
– SIXTEEN
• So, your numerator = 7
• Your denominator = 16
7
16
Can you reduce
or simplify
this?
• Draw a rectangular model
6
horizontally.
10
6
showing
10
• Next, model
2
3
2
3
vertically.
6
10
• To determine your answer, count the boxes
you shaded twice. 6
– TWELVE
2
3
10
• Next, count the total number of boxes.
– THIRTY
2
3
6
10
• So, your numerator = 12
• Your denominator = 30
12
30
Can you reduce
or simplify
this?
12 2 6 3 2
12
3
4
2
2
   
   
30 2 15 3 5
30 3 10 2 5
12 6 2
 
30 6 5
Algorithm?
• Now…let’s look at our practice problems and
try to determine an algorithm to solve
multiplication of a whole by a fraction.
2 3 6
 
3 4 12
1 7 7
 
2 8 16
2 6 12
 
3 10 30
Is there a
pattern? What
is being done
each time?
Algorithm
a c a  c ac
 

b d b  d bd
HUH??? LETTERS???
Each letter is a variable. It represents or takes
the place of a number. Let’s look at an
example of what these letters mean.
Algorithm
a c a  c ac
 

b d b  d bd
2 3 23 6 6 1
 
  
3 4 3  4 12 6 2
a = your numerator of your
first fraction
b = your denominator of
your first fraction
c = your numerator of your
second fraction
d = your denominator of
your second fraction