Multiplication and Division of Fractions

In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics well only when they construct their own mathematical understanding

Everybody Counts National Research Council, 1989

Operating With Fractions

 Meaning of the denominator (number of equal-sized pieces into which the whole has been cut);  Meaning of the numerator (how many pieces are being considered);  The more pieces a whole is divided into, the smaller the size of the pieces;  Fractions aren ’t just between zero and one, they live between all the numbers on the number line;  Understand the meanings for operations for whole numbers.

A Context for Fraction Multiplication

 Nadine is baking brownies. In her family, some people like their brownies frosted without walnuts, others like them frosted with walnuts, and some just like them plain. So Nadine frosts 3/4 of her batch of brownies and puts walnuts on 2/3 of the frosted part.

How much of her batch of brownies has both frosting and walnuts?

Multiplication of Fractions

Consider: 2  3 3 4  How do you think a child might solve each of these?

 Do both representations mean exactly the same thing to children?  What kinds of reasoning and/or models might they use to make sense of each of these problems?

 Which one best represents Nadine’s brownie problem?

Models for Reasoning About Multiplication

 Fraction of a fraction  Linear/measurement  Area/measurement models  Cross Shading

We will think of multiplying fractions as finding a

fraction of another fraction

.

2 3

How much is of ?

3 4 We use a fraction square to represent the fraction .

4 Then, we shade of 3 3 4 3 4 We can see that it is the same as .

12 2  3 3 4  1 2

The Linear Model with multiplication utilizes the number line and partitions the fractions 3 4 2 3

How much is of ?

3 4 0 1 4 2 4 3 4 4 4 1 3

of

3 4 2 3

of

3 4 1 2 2  3 3 4  1 2 3 3

of

3 4

We can also use the linear model with shapes and partition accordingly Break into Identify ¾ of the circle 2 3

How much is of ?

3 4 3 pieces Take 2 pieces Answer is ½ 2  3 3 4  1 2

In the third method, we will think of multiplying fractions as

multiplying a length times a length to get an area.

Length is 3 4 2 3

How much is of ?

3 4 2 3 Area Width is Number of square units Is 6 out of 12 This area is X 3 3 4 = 6 12 2  3 3 4  1 2

Modeling multiplication of fractions using the

length times length equals area

approach requires that the children understand how to find the area of a rectangle.

A great advantage to this approach is that the area model is consistently used for multiplication of whole numbers and decimals. Its use for fractions, then is merely an extension of previous experience.

In the fourth method, we will represent both fractions on the same square.

2 3 2 3

How much is of ?

3 4 is 2 3 3 4 is 3 4 2  3 3 4  1 2

Modeling multiplication of fractions using the

cross shading

approach does produce correct answers. However, many elementary students may not grasp the “because it is shaded in both directions” overlapping concept. This may require some additional explanations

Classroom Problem



Eric and his mom are making cupcakes. Each cupcake gets 1/4 of a cup of frosting. They are making 20 cupcakes. How much frosting do they need?

Sample children’s strategies

1/4 of a cup 1 cup 2 cups “…so 5 cups altogether.” 3 cups 4 cups 5 cups

Another student strategy

1/4 of a cup So, 5, 6, 7, 8 - that’s 2 cups.

9, 10, 11, 12 - that’s 3 cups.

13, 14, 15, 16 - that’s 4 cups.

4 of these is 1 cup… 17, 18, 19, 20 - that’s 5 cups.

…so 5 cups altogether.

Another student strategy

1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 1/4 + 1/4 + 1/4 + 1/4 = 1 5 cups Q: What’s a number sentence for this problem? A: 20 x 1/4 = 5 (there are others)

Other Contexts for Multiplication of Fractions

 Finding part of a part (a reason why multiplication doesn’t always make things “bigger”)  Pizza (pepperoni on ⅓ of ½ pizza)  Recipes ( 1 ¾ cups of sugar is used but we want to make ½ a batch)  Ribbon (you have ⅜ yd , ⅓ of the ribbon is used to make a bow)

Division With Fractions

Division with Fractions

 Sharing meaning for division: 1  1 3 • One shared by one-third of a group? • How many in the whole group?

• How does this work?

Division With Fractions

 Repeated subtraction / measurement meaning 1  1 3 • • • • How many times can one-third be subtracted from one?

How many one-thirds are contained in one?

How does this work?

How might you deal with anything that’s left?

Division of Fractions examples

    How many quarters are in a dollar?

Ground beef cost 2.80 for ½ pound. What is the price per pound?

Maggie can walk the 2 ½ miles to school in 3/4 of an hour. How long would it take to walk 4 miles?

Barb had ¾ of a pizza left over from her party. She wants to store it in plastic containers. Each container holds ⅓ of a pizza. How many containers will she use? How many will be completely full? How full will the last container be?

Division of Fractions examples

 1 1

You have 1 cups of sugar. It takes

2 3

cup to make 1 batch of cookies.

 How many batches of cookies can you make?

 How many cups of sugar are left?

 How many batches of cookies could be made with the sugar that’s left?

“How many one eighths are in three fourths?”

3 4  1 8  ?

Our pizza is cut into 8 pieces. If three fourths of a pizza is left, how many slices remain? Recall: a slice represents one eighth of the pizza

Pizza

How many one eighths are in three fourths?

3 4  1 8  ?

To find this we must first find 3/4 of the pizza.

We then cut each fourth into halves to make eighths. We can see there are 6 eighths in three fourths.

3 4  1 8  6

Pizza

Now only half of the pizza is left. How many slices remain?

How many one eighths are in one half?

1 2  1 8  ?

Using a fraction manipulative, we show one half of a circle.

To find how many one eighths are in one half, we cover the one half with eighths and count how many we use.

We find there are 4. There are four one eighths in one half.

1 2  1 8  4

3 1 : 4 4 = 3 1 1 : 2 6 = 3 1 1 : 3 12 = 4 2 2 : 3 3 = 1

Now that you know how to divide fractions, let’s try some together .

3 1 : 3 3 = 3 1 1 : 2 12 = 6