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

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Modeling Multiplication of Fractions

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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


Fraction by a whole number

Fraction by a Whole Number


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?


Modeling multiplication of fractions

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.


Modeling multiplication of fractions

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


Modeling multiplication of fractions

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


Modeling multiplication of fractions

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.


Modeling multiplication of fractions

0 1 2 3 4

4 x ½

  • Will your answer be bigger or smaller than 4?

  • First – show 4 on the number line.


Modeling multiplication of fractions

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?


Modeling multiplication of fractions

0 1 2 3 4

3 x

  • Will your answer be bigger or smaller than 3?

  • First – show 3 on the number line.


Modeling multiplication of fractions

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?


Modeling multiplication of fractions

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.


Modeling multiplication of fractions

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


Modeling multiplication of fractions

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?


Modeling multiplication of fractions

8 x ¾

  • Draw a rectangle.

  • Divide the rectangle into smaller rectangles to represent your WHOLE number.

    8


Modeling multiplication of fractions

8 x ¾

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

4


Modeling multiplication of fractions

8 x ¾

  • Now, shade in 3 rows of the 4 you just created.

8

4


Modeling multiplication of fractions

8 x ¾

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

8

4


Modeling multiplication of fractions

8 x ¾

  • This is 1 whole…

  • So how much would 1 box be worth?

  • 1 box equals ¼

8

4


Modeling multiplication of fractions

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?


Modeling multiplication of fractions

8 x ¾

  • = 24 ÷ 4 = 6

  • Proof: If you divided 8 dollars up among 4 people, how much would each get?

8

4


Modeling multiplication of fractions

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


Modeling multiplication of fractions

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.


Modeling multiplication of fractions

4 x ½

  • First, draw a rectangle divided into your whole number – 4

4


Modeling multiplication of fractions

4 x ½

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

4

2


Modeling multiplication of fractions

4 x ½

  • Shade in the number represented by the numerator…

4

2


Modeling multiplication of fractions

4 x ½

  • Now THINK – how much is ONE square worth? What is your WHOLE?

4

2


Modeling multiplication of fractions

4 x ½

  • One square = ½

  • There are 4 “halves” – or

4

2


Modeling multiplication of fractions

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.


Modeling multiplication of fractions

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?


Modeling multiplication of fractions

4 x ½

  • Draw a diagram to represent ½.

  • Repeat this 4 times.


Modeling multiplication of fractions

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


Fraction by a fraction

Fraction by a fraction


Modeling multiplication of fractions

  • 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.


Modeling multiplication of fractions

  • Model the second fraction (factor).

0


Modeling multiplication of fractions

  • 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


Modeling multiplication of fractions

  • 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


Modeling multiplication of fractions

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


Modeling multiplication of fractions

Let’s make equivalent fractions with 16 as a denominator by multiplying all by 2/2.

0

0


Modeling multiplication of fractions

Now, what could fall between 6/16 and 8/16?

7/16

0

0


Modeling multiplication of fractions

  • 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.


Modeling multiplication of fractions

  • 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?


Modeling multiplication of fractions

  • Draw a rectangular model showing horizontally.


Modeling multiplication of fractions

  • Next, model vertically.


Modeling multiplication of fractions

  • To determine your answer, count the boxes you shaded twice.

    • SIX


Modeling multiplication of fractions

  • Next, count the total number of boxes.

    • TWELVE


Modeling multiplication of fractions

Can you reduce or simplify this?

  • So, your numerator = 6

  • Your denominator = 12


Modeling multiplication of fractions

  • Draw a rectangular model showing horizontally.


Modeling multiplication of fractions

  • Next, model vertically.


Modeling multiplication of fractions

  • To determine your answer, count the boxes you shaded twice.

    • SEVEN


Modeling multiplication of fractions

  • Next, count the total number of boxes.

    • SIXTEEN


Modeling multiplication of fractions

Can you reduce or simplify this?

  • So, your numerator = 7

  • Your denominator = 16


Modeling multiplication of fractions

  • Draw a rectangular model showing horizontally.


Modeling multiplication of fractions

  • Next, model vertically.


Modeling multiplication of fractions

  • To determine your answer, count the boxes you shaded twice.

    • TWELVE


Modeling multiplication of fractions

  • Next, count the total number of boxes.

    • THIRTY


Modeling multiplication of fractions

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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