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Modeling Multiplication of FractionsPowerPoint Presentation

Modeling Multiplication of Fractions

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### 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 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?

MCC4.NF.4; MCC5.NF.4; MCC5.NF.5; MCC5.NF.6

Deanna Cross – Hutto Middle School

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

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.

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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

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.

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

- 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?

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 3 4 5 6

- Optional Practice problems
- 8 x
- 9 x
- 12 x
- 10 x
- 4 x

Multiplying with an AREA MODEL 3 4 5 6

- Fraction by a WHOLE number

Area models 3 4 5 6

- 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.

- Isabel has used only ¾ of the paper.
- You have to multiply 8 x ¾.
- Suggestions to solve using area model?

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Draw a rectangle.
- Divide the rectangle into smaller rectangles to represent your WHOLE number.
8

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

4

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

8

4

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

8

4

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- This is 1 whole…
- So how much would 1 box be worth?
- 1 box equals ¼

8

4

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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?

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- = 24 ÷ 4 = 6
- Proof: If you divided 8 dollars up among 4 people, how much would each get?

8

4

8 x ¾ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- = 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 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

4

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

4

2

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Shade in the number represented by the numerator…

4

2

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

4

2

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- One square = ½
- There are 4 “halves” – or

4

2

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- = 4 ÷ 2 = 2
- So – if you have half of 4 you have 2.

4

2

3 x 1/3 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Try to draw this model on your own – you already know what 1/3 of 3 would be…

3

3

3 x 1/3 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Draw the model.

6

4

6 x 2/4 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- What does each square represent?

6

How many fourths?

12

4

Practice with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Optional Practice problems
- 8 x
- 6 x
- 12 x
- 5 x
- 4 x

Multiplying with TAPE DIAGRAM with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Fraction by a WHOLE number

Tape diagrams with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Fraction by Whole numbers are easy with tape diagrams…it is like repeated addition.

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

+

+

+

+

+

+

+

- Add up how many fourth’s you have…

Can you leave the fraction as it is?

8 x 3/4 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Draw a diagram to represent ½.
- Repeat this 4 times.

4 x ½ with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

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

- Add up each piece…

+

+

+

3 x 1/3 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

How else can you write a number over itself?

- Model
- Add
- Reduce

+

+

6 x 2/4 with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

+

+

+

+

+

- Model
- Add
- Reduce

Can you simplify this fraction?

Practice with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Optional Practice problems
- 12 x
- 9 x
- 5 x
- 6 x
- 4 x

Algorithm? with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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?

Algorithm with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

( ) 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.

Algorithm with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

a = your numerator

b = your denominator

q = your whole number

( ) x q = (a x q)÷b

Practice with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- Optional Practice problems
- x 5
- x 18
- x 32
- x 10
- x 4

Fraction by a fraction with. She Used 3/4 of the paper. How much paper did she use? How much paper did she have left over?

- 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 tennis shoes. What fraction shows how many boys are wearing tennis shoes?

- First, draw a line graph to represent the amount of boys (3/4).

0

Multiplying on a Number Line tennis shoes. What fraction shows how many boys are wearing tennis shoes?

- Next, divide this bar into the denominator of the first fraction (3). Shade in the numerator (2).

0

Multiplying on a Number Line tennis shoes. What fraction shows how many boys are wearing tennis shoes?

- 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 Line tennis shoes. What fraction shows how many boys are wearing tennis shoes?

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

- Model the second fraction (factor). tennis shoes. What fraction shows how many boys are wearing tennis shoes?

0

- 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

- 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

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

- Model the second fraction (factor). by multiplying all by 2/2.
- Divide into 3 sections and shade 2.

0

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

- Divide the numerator and denominator by 2. by multiplying all by 2/2.

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

0

Practice by multiplying all by 2/2.

- Optional Practice problems

Multiplying with an AREA MODEL by multiplying all by 2/2.

- 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?

- Draw a rectangular model showing horizontally. by multiplying all by 2/2.

- Next, model vertically. by multiplying all by 2/2.

- To determine your answer, count the boxes you shaded twice. by multiplying all by 2/2.
- SIX

- Next, count the total number of boxes. by multiplying all by 2/2.
- TWELVE

Can you reduce or simplify this? by multiplying all by 2/2.

- So, your numerator = 6
- Your denominator = 12

- Draw a rectangular model showing horizontally. by multiplying all by 2/2.

- Next, model vertically. by multiplying all by 2/2.

- To determine your answer, count the boxes you shaded twice. by multiplying all by 2/2.
- SEVEN

- Next, count the total number of boxes. by multiplying all by 2/2.
- SIXTEEN

Can you reduce or simplify this? by multiplying all by 2/2.

- So, your numerator = 7
- Your denominator = 16

- Draw a rectangular model showing horizontally. by multiplying all by 2/2.

- Next, model vertically. by multiplying all by 2/2.

- To determine your answer, count the boxes you shaded twice. by multiplying all by 2/2.
- TWELVE

- Next, count the total number of boxes by multiplying all by 2/2..
- THIRTY

Can you reduce or simplify this? by multiplying all by 2/2.

- So, your numerator = 12
- Your denominator = 30

Algorithm? by multiplying all by 2/2.

- 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?

Algorithm by multiplying all by 2/2.

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 by multiplying all by 2/2.

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

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