40 Multiplication and Division of Fractions

You will need:

Exploration Activity

You are sharing a rectangular garden with your best friend. You each get half of the garden.

1. 1. Make a diagram of the garden showing your half and your friend’s half.
   2. You are going to plant tomatoes in [latex]\frac{3}{5}[/latex] of your half. Add this to your diagram.
   3. Your friend is only going to plant flowers. Consider your diagram. What fraction of the whole garden will be tomatoes?

In the Exploration Activity you visualized [latex]\frac{1}{2}\times\frac{3}{5}[/latex] using the rectangle model of multiplication. In general, we can use the dot model for a fraction, the repeated addition model of multiplication or the rectangle interpretation of multiplication.

Multiplying a Fraction and a Whole Number

If one of our two numbers is a whole number, we can use the dot model for fractions or the repeated addition approach to multiplication.

The dot model only works if we can evenly partition our whole number into the proper amount of equal groups. You would not want to use the dot model for something like [latex]5\times\frac{4}{11}[/latex] because you cannot evenly divide 5 dots into elevenths. In this case, the better approach is to use repeated addition.

Students typically start with logic similar to the dot model because it matches our concept of what a fraction is. The repeated addition model requires students to understand the repeated addition model of multiplication and know how to add fractions.

Multiplying two Fractions

If we do not have a whole number, we need another way to visualize multiplication of fractions. The easiest way is to use rectangles. Recall that [latex]5\times 8[/latex] can be represented by the area of a 5 by 8 rectangle.

(picture)

We can use the same concept for fractions. We simply need to keep track of what makes a unit so that we can interpret our answer.

Let’s revisit our situation from the Exploration Activity, [latex]\frac{1}{2}\times\frac{3}{5}[/latex]. Our unit is the entire garden.

First we split that unit into halves. Since we want one-half, shade one of the two halves.

Next we consider the unit split into fifths. Since we did the halves horizontally, we make the fifths vertical. Since we want three-fifths, shade three of the five fifths.

If we lay the two fractions on top of each other, their product is the overlap of the two shaded areas.

Our overlap is three pieces and the unit now consists of ten pieces. Thus our answer is [latex]\frac{3}{10}[/latex].

When you solved the exploration activity, your picture probably looked something like the following.

Does this look familiar to our overlap picture? When you multiply two fractions you are dividing each dimension of the rectangle into each fraction. It just helps to draw the lines all the way through so that we can determine our denominator.

Algorithm for Multiplying Fractions

Models work well to visualize what is happening with smaller numbers, but they are not reasonable for products like [latex]\frac{56}{101}\times\frac{7}{43}[/latex]. The “multiply across” algorithm, is an approach that works for all fraction combinations. This algorithm was covered in the Pre Unit Review Multiply and Divide Fractions. But now you have a stronger understanding of what multiplication of fractions looks like, so let’s review this algorithm.

Reconsider the rectangles you drew in Exercise 3. The number of pieces in the overlap was always the product of the numerators and the number of pieces in the entire rectangle was always the product of the denominators. That is exactly what our algorithm says!

Applications of Fraction Multiplication

Now that you can visualize a product of fractions, use that skill in the following word problems. For each one, draw a picture using one of our models for fractions. You may check your answer using the “multiply across” algorithm.

Exploration Activity

You need a lot of quarters for the laundromat.

a. If you convert 10 dollars into quarters, how many quarters will you have?

H5P

b. Represent [latex]10\div 2[/latex] on the number-line using the repeated subtraction model of division.

Solution

c. Similar to what you did in part (b), represent [latex]10\div\frac{1}{4}[/latex] on the number-line using the repeated subtraction model of division.

Solution

d. A quarter is [latex]\frac{1}{4}[/latex] of a dollar. Explain how your process in part (a) relates to your process in part (c).

Solution

Dividing by fractions can feel counterintuitive. Recall the classroom scenario from the beginning of Understanding Division. The students were confused about the difference between dividing BY half and dividing IN half. Let’s use our two models of division, partitioning into equal subsets and repeated subtraction, to visualize what is going on when we divide by a fraction.

Fraction Division by Partitioning into Equal Subsets

If we have a fraction divided by a whole number, we can use the partitioning into equal subsets model of division. The steps are the same as with whole numbers:

• start with the first number
• divide that number up into equal subsets.

Your answer will be how many things are in one subset. The only difference from whole numbers is we need to keep track of how many pieces make up a unit in order to determine our denominator.

This model can be extended to situations where the divisor is a nice fraction. For example, [latex]\frac{6}{7}\div\frac{1}{2}[/latex] can be thought of as

• starting with [latex]\frac{6}{7}[/latex] and
• making [latex]\frac{1}{2}[/latex] a subset/group out of that [latex]\frac{6}{7}[/latex].

If [latex]\frac{6}{7}[/latex] is half a group, then [latex]\frac{12}{7}[/latex] is a whole group. Thus, [latex]\frac{6}{7}\div\frac{1}{2}=\frac{12}{7}[/latex]. This approach quickly becomes difficult if you are trying to solve something like [latex]\frac{53}{12}\div\frac{5}{12}[/latex]. If [latex]\frac{53}{12}[/latex] is [latex]\frac{5}{12}[/latex] of a subset, how much is a full subset? We need another approach.

Fraction Division by Repeated Subtraction

Recall the exploratory activity with the quarters. You split the $10 into quarters by finding how many quarters make up $10. That is exactly how the repeated subtraction model works. If we want to solve [latex]\frac{9}{2}\div\frac{3}{4}[/latex] we start by drawing out [latex]\frac{9}{2}[/latex].

picture

Then, we count how many [latex]\frac{3}{4}[/latex] we can get out of [latex]\frac{9}{2}[/latex].

picture

In this case, we can make six groups of size [latex]\frac{3}{4}[/latex]. This means [latex]\frac{9}{2}\div\frac{3}{4}=6[/latex].

We could also have represented the problem on a number-line.

picture

Look back at your answer to Exercise 8d. Did you say the answer was [latex]5\frac{3}[12}[/latex]? That is not correct! It may be [latex]5[/latex] with a remainder of [latex]\frac{3}{12}[/latex], but that is not the same as saying there are [latex]5\frac{3}[12}[/latex] groups. Let’s look at how to handle remainder in terms of groups.

(Example with remainder)

(Revisit 5 and 3/12 problem)

(Exercises with remainder)

Algorithm for Dividing Fractions

One method that does work for all fraction division problems is the “flip and multiply algorithm.” This algorithm was covered in the Pre Unit Review Multiply and Divide Fractions. But now you have a stronger understanding of what division of fractions really means according to our models, so let’s review this algorithm.

Applications of Fraction Division

(word problems)

Negative Fractions

(comment on mult/div for negative fractions)