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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

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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

multiplying on a number line
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 Line

Number Lines

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?

slide5

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.
slide6
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

slide7
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

slide8

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.
slide9

0 1 2 3 4

4 x ½
  • Will your answer be bigger or smaller than 4?
  • First – show 4 on the number line.
slide10

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?
slide11

0 1 2 3 4

3 x
  • Will your answer be bigger or smaller than 3?
  • First – show 3 on the number line.
slide12

0 1 2 3 4

3 x

  • 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?
slide13

0 1 2 3 4 5 6

6 x
  • Will your answer be bigger or smaller than 6?
  • First – show 6 on the number line.
slide14

0 1 2 3 4 5 6

6 x

  • 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?
practice
Practice
  • Optional Practice problems
  • 8 x
  • 9 x
  • 12 x
  • 10 x
  • 4 x
multiplying with an area model
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
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 models1
Area models
  • Multiply the length and the width
  • 2 x 5 = 10 – AREA = 10

2

5

slide19

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?
slide20
8 x ¾
  • Draw a rectangle.
  • Divide the rectangle into smaller rectangles to represent your WHOLE number.

8

slide21
8 x ¾
  • Next, along the vertical side, divide your rectangle into the number of pieces representing your denominator  4

4

slide22
8 x ¾
  • Now, shade in 3 rows of the 4 you just created.

8

4

slide23
8 x ¾
  • Hard part – This model started out with 8 wholes. How much would 1 box be worth? THINK…

8

4

slide24
8 x ¾
  • This is 1 whole…
  • So how much would 1 box be worth?
  • 1 box equals ¼

8

4

slide25
8 x ¾
  • Now, count how many ¼’s you have shaded green.
  • 24 boxes =

8

4

  • Can we leave like this, or is there another way to write this improper fraction?
slide26
8 x ¾
  • = 24 ÷ 4 = 6
  • Proof: If you divided 8 dollars up among 4 people, how much would each get?

8

4

slide27
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
slide28
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.
slide29
4 x ½
  • First, draw a rectangle divided into your whole number – 4

4

slide30
4 x ½
  • Next, divide your rectangle into the number of pieces for your denominator along the vertical edge.

4

2

slide31
4 x ½
  • Shade in the number represented by the numerator…

4

2

slide32
4 x ½
  • Now THINK – how much is ONE square worth? What is your WHOLE?

4

2

slide33
4 x ½
  • One square = ½
  • There are 4 “halves” – or

4

2

slide34
4 x ½
  • = 4 ÷ 2 = 2
  • So – if you have half of 4 you have 2.

4

2

3 x 1 3
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 31
3 x 1/3
  • Now, think about what each square represents…

3

So, each square = 1/3, there are 3 thirds…

3

3 x 1/3 = 1

6 x 2 4
6 x 2/4
  • Draw the model.

6

4

6 x 2 41
6 x 2/4
  • What does each square represent?

6

How many fourths?

12

4

practice1
Practice
  • Optional Practice problems
  • 8 x
  • 6 x
  • 12 x
  • 5 x
  • 4 x
multiplying with tape diagram
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
  • 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
Multiplying with Tape Diagrams
  • Fraction by Whole numbers are easy with tape diagrams…it is like repeated addition.
slide43

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
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 41
8 x 3/4

+

+

+

+

+

+

+

  • Add up how many fourth’s you have…

Can you leave the fraction as it is?

8 x 3 42
8 x 3/4

How do you change an improper fraction to a mixed number?

slide47
4 x ½
  • Draw a diagram to represent ½.
  • Repeat this 4 times.
slide48
4 x ½

Ahhh…there is a large number on top of a small number!

  • Add up each piece…

+

+

+

3 x 1 32
3 x 1/3

How else can you write a number over itself?

  • Model
  • Add
  • Reduce

+

+

6 x 2 42
6 x 2/4

+

+

+

+

+

  • Model
  • Add
  • Reduce

Can you simplify this fraction?

practice2
Practice
  • Optional Practice problems
  • 12 x
  • 9 x
  • 5 x
  • 6 x
  • 4 x
algorithm
Algorithm?
  • Now…let’s look at our practice problems and try to determine an algorithm to solve multiplication of a whole by a fraction.

Is there a pattern? What is being done each time?

algorithm1
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.

algorithm2
Algorithm

a = your numerator

b = your denominator

q = your whole number

( ) x q = (a x q)÷b

practice3
Practice
  • Optional Practice problems
  • x 5
  • x 18
  • x 32
  • x 10
  • x 4
slide57

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 line1
Multiplying on a Number Line
  • First, draw a line graph to represent the amount of boys (3/4).

0

multiplying on a number line2
Multiplying on a Number Line
  • Next, divide this bar into the denominator of the first fraction (3). Shade in the numerator (2).

0

multiplying on a number line3
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

multiplying on a number line4
Multiplying on a Number Line

These are both even numbers, so the fraction can be reduced (or simplified) by dividing the numerator and denominator by 2.

slide63

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).

0

slide64

This does NOT fall at an exact mark on the number line, which means more numbers must be added to the number line.

What could fall between 3/8 and 4/8???

0

slide65

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?

0

slide68

Model the second fraction (factor).

  • Divide into 3 sections and shade 2.

0

  • Check to see if this lines up with a number on the number line.
slide69

Divide the numerator and denominator by 2.

Can this fraction be reduced (simplified) or written in any other way?

0

practice4
Practice
  • Optional Practice problems
multiplying with an area model1
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?
slide76

Can you reduce or simplify this?

  • So, your numerator = 6
  • Your denominator = 12
slide81

Can you reduce or simplify this?

  • So, your numerator = 7
  • Your denominator = 16
slide86

Can you reduce or simplify this?

  • So, your numerator = 12
  • Your denominator = 30
algorithm3
Algorithm?
  • Now…let’s look at our practice problems and try to determine an algorithm to solve multiplication of a whole by a fraction.

Is there a pattern? What is being done each time?

algorithm4
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.

algorithm5
Algorithm

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

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