modeling multiplication of fractions
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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 LineNumber Lines
Start and end with an arrowDivided into equal (equivalent) incrementsCan start and end at any number
Are there any numbers that can be “renamed” or written as an equivalent fraction?
10
10
10
9
10
8
10
7
10
6
10
5
10
4
10
3
10
2
10
1
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.
8 x ¾
• First – Model what you have on a number line – “She had 8 feet of wrapping paper”
• Now, she is multiplying by ¾ . What is the denominator?
0 1 2 3 4 5 6 7 8
8 x ¾
• Now, divide the total amount (8) into 4 pieces. (8 ÷ 4 = 2 – so each piece is equal to 2)
• Shade in 3 of the four pieces.• Look to see if this lines up with a number on your number
line.
0 1 2 3 4 5 6 7 8
• 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.
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?
4 x ½
• Will your answer be bigger or smaller than 4?• First – show 4 on the number line.
0 1 2 3 4
4 x ½
• Now, look at your denominator – 2• Divide your bar into two EQUAL pieces.
• Shade in 1 of the two pieces.• Does this line up with a number on the number
line?
0 1 2 3 4
3 x
• Will your answer be bigger or smaller than 3?• First – show 3 on the number line.
0 1 2 3 4
3
1
• Now, look at your denominator – 3• Divide your bar into three EQUAL pieces.
• Shade in 1 of the three pieces.• Does this line up with a number on the number line?
3 x 3
1
0 1 2 3 4
6 x
• Will your answer be bigger or smaller than 6?• First – show 6 on the number line.
0 1 2 3 4 5 6
4
2
• 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
• Shade in 2 of the four pieces.• Does this line up with a number on the number line?
6 x 4
2
0 1 2 3 4 5 6
Practice
• Optional Practice problems
1) 8 x 2) 9 x 3) 12 x 4) 10 x 5) 4 x
4
1
3
2
4
3
5
3
8
3
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 x 5 = 10 – AREA = 10
2
5
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?
8 x ¾ • Draw a rectangle.• Divide the rectangle into smaller rectangles to
represent your WHOLE number. 8
8 x ¾ • Next, along the vertical side, divide your
rectangle into the number of pieces representing your denominator 4
4
8 x ¾ • Now, shade in 3 rows of the 4 you just created.
4
8
8 x ¾ • Hard part – This model started out with 8
wholes. How much would 1 box be worth? THINK…
4
8
8 x ¾ • This is 1 whole…• So how much would 1 box be worth?• 1 box equals ¼
4
8
8 x ¾ • Now, count how many ¼’s you have shaded
green.• 24 boxes =
4
84
24
•Can we leave like this, or is there another way to write this improper fraction?
4
24
8 x ¾ • = 24 ÷ 4 = 6• Proof: If you divided 8 dollars up among 4
people, how much would each get?
4
8
4
24
8 x ¾
• = 24 ÷ 4 = 6• 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
4
24
4 x ½
• 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.
4 x ½
• First, draw a rectangle divided into your whole number – 4
4
4 x ½
• Next, divide your rectangle into the number of pieces for your denominator along the vertical edge. 4
2
4 x ½
• Shade in the number represented by the numerator…
4
2
4 x ½
• Now THINK – how much is ONE square worth? What is your WHOLE?
4
2
4 x ½
• One square = ½ • There are 4 “halves” – or
4
2
2
4
4 x ½
• = 4 ÷ 2 = 2• So – if you have half of 4 you have 2.
4
2
2
4
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
3
So, each square = 1/3, there are 3 thirds…
3
3
3 x 1/3 = 1
6 x 2/4 • Draw the model.
6
4
6 x 2/4 • What does each square represent?
6
4
4
1
How many fourths?
12
34124
12
Practice
• Optional Practice problems
1) 8 x 2) 6 x 3) 12 x 4) 5 x 5) 4 x
4
3
3
2
4
1
5
3
8
5
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…+
++++++
4
24
4
3
4
3
4
3
4
3
4
3
4
3
4
3
4
3
Can you leave the fraction as it is?
8 x 3/4
4
24 How do you change an improper fraction to a mixed number?
64244
24
4 x ½
• Draw a diagram to represent ½.• Repeat this 4 times.
4 x ½
• Add up each piece…
+++
2
1
2
1
2
1
2
1
2
4
Ahhh…there is a large number on
top of a small number!
2242
4
3 x 1/3
• Model • Add• Reduce
++
3
3
3
1
3
1
3
1
How else can you write a number over itself?
13
3
6 x 2/4
• Model• Add• Reduce
+++++
4
12
4
2
4
2
4
2
4
2
4
2
4
2
Can you simplify this fraction?
34124
12
Practice
• Optional Practice problems
1) 12 x 2) 9 x 3) 5 x 4) 6 x 5) 4 x
4
3
3
2
4
1
5
3
8
7
Algorithm?• Now…let’s look at our practice problems and
try to determine an algorithm to solve multiplication of a whole by a fraction.
34
12
4
26
13
3
3
13
22
4
2
14
64
24
4
38
Is there a
pattern? What is being done
each time?
Algorithm
( ) x q = (a x q) ÷ 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.
b
a
Algorithm
( ) x q = (a x q)÷bb
a
4)83(84
3
a = your numerator
b = your denominator
q = your whole number
Practice
• Optional Practice problems
1) x 52) x 183) x 324) x 10 5) x 4
9
7
3
2
4
1
5
3
8
7
FRACTION BY A FRACTION
• 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?
4
3
3
2
Multiplying on a Number Line
• First, draw a line graph to represent the amount of boys (3/4).
0 4
1
4
2
4
3
4
4
4
5
Multiplying on a Number Line
• Next, divide this bar into the denominator of the first fraction (3). Shade in the numerator (2).
0 4
1
4
2
4
3
4
4
4
5
Multiplying on a Number Line
• Finally, see if this matches any of your points on the number line.
0 4
1
4
2
4
3
4
4
4
5
Can 2/4 be written any other way?
Multiplying on a Number Line
4
2 These are both even numbers, so the fraction can be reduced (or simplified) by dividing the numerator and denominator by 2.
2
1
24
22
• Model the second fraction (factor).
8
108
2
8
4
8
3
8
5
8
6
8
8
8
7
• Divide this amount into two equal sections (how can you divide 7 in half?)
• 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
108
2
8
4
8
3
8
5
8
6
8
8
8
7
• This does NOT fall at an exact mark on the number line, which means more numbers must be added to the number line.
8
108
2
8
4
8
3
8
5
8
6
8
8
8
7
What could fall between
3/8 and 4/8???
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
108
2
8
4
8
3
8
5
8
6
8
8
8
7
Let’s make equivalent fractions with 16 as a denominator by multiplying all by 2/2.
8
108
2
8
4
8
3
8
5
8
6
8
8
8
7
16
2016
4
16
8
16
6
16
10
16
12
16
16
16
14
Now, what could fall between 6/16 and 8/16?7/16
8
108
2
8
4
8
3
8
5
8
6
8
8
8
7
16
2016
4
16
8
16
6
16
10
16
12
16
16
16
14
• Model the second fraction (factor).• Divide into 3 sections and shade 2.
10
1010
2
10
4
10
3
10
5
10
6
10
8
10
7
10
9
10
10
• Check to see if this lines up with a number on the number line.
• Divide the numerator and denominator by 2.
10
1010
2
10
4
10
3
10
5
10
6
10
8
10
7
10
9
10
10
Can this fraction be reduced (simplified)
or written in any other way?
5
2
210
24
Practice
• Optional Practice problems
1) 2) 3) 4) 5)
12
4
4
3
7
6
3
2
8
3
2
1
13
8
4
3
8
2
4
2
Multiplying with an AREA MODEL
• Fraction by a Fraction
• 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?
4
3
3
2
• Draw a rectangular model showing horizontally. 4
3
4
3
3
2 OR
4
3
3
2of
4
3
• Next, model vertically.3
2
4
3
3
2 OR
4
3
3
2of
4
3
3
2
• To determine your answer, count the boxes you shaded twice.– SIX
4
3
3
2 OR
4
3
3
2of
• Next, count the total number of boxes.– TWELVE
4
3
3
2 OR
4
3
3
2of
• So, your numerator = 6• Your denominator = 12
4
3
3
2 OR
4
3
3
2of
12
6
2
1
3
3
6
3
2
2
12
6
2
1
2
2
4
2
3
3
12
6
Can you reduce or simplify
this?
2
1
6
6
12
6
• Draw a rectangular model showing horizontally.
8
7
8
7
• Next, model vertically.2
1
8
7
2
1
• 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
16
7
Can you reduce or simplify
this?
• Draw a rectangular model showing horizontally.
10
6
10
6
10
6• Next, model vertically.3
2
3
2
10
6
3
2
• To determine your answer, count the boxes you shaded twice.– TWELVE
10
6
3
2
• Next, count the total number of boxes.– THIRTY
• So, your numerator = 12• Your denominator = 30
30
12
Can you reduce or simplify
this?
5
2
3
3
15
6
2
2
30
12
5
2
6
6
30
12
5
2
2
2
10
4
3
3
30
12
Algorithm?• Now…let’s look at our practice problems and
try to determine an algorithm to solve multiplication of a whole by a fraction.
30
12
10
6
3
216
7
8
7
2
112
6
4
3
3
2
Is there a
pattern? What is being done
each time?
Algorithm
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.
bd
ac
db
ca
d
c
b
a
Algorithm
2
1
6
6
12
6
43
32
4
3
3
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
bd
ac
db
ca
d
c
b
a
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