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Making Sense of Division

Meanings of Fraction Division

Algorithm for Fraction Division

Fractions in Lengths, Areas, and Volumes

This week, your student will be thinking about the meanings of division to prepare to learn about division of fraction. Suppose we have 10 liters of water to divide into equal-size groups. We can think of the division 10÷2 in two ways, or as the answer to two questions:

Here are two diagrams to show the two interpretations of 10÷2:

In both cases, the answer to the question is 5, but it could either mean “there are 5 bottles with 2 liters in each” or “there are 5 liters in each of the 2 bottles.”

Here is a task to try with your student:

Solution:

Earlier, students learned that a division such as 10÷2=? can be interpreted as “how many groups of 2 are in 10?” or “how much is in each group if there are 10 in 2 groups?” They also saw that the relationship between 10, 2 and the unknown number ("?") can also be expressed with multiplication:

This week, they use these ideas to divide fractions. For example, 6÷112=?can be thought of as “how many groups of 112 are in 6?” Expressing the question as a multiplication and drawing a diagram can help us find the answer.

From the diagram we can count that there are 4 groups of 112 in 6.

We can also think of 6÷112=? as “how much is in each group if there are 112equal groups in 6?” A diagram can also be useful here.

From the diagram we can see that if there are three 12-groups in 6. This means there is 2 in each 12 group, or 4 in 1 group.

In both cases 6÷112=4, but the 4 can mean different things depending on how the division is interpreted.

Many people have learned that to divide a fraction, we “invert and multiply.” This week, your student will learn why this works by studying a series of division statements and diagrams such as these:

Because there are 3 thirds in 1, there are (2⋅3) or 6 thirds in 2. So dividing 2 by 13 has the same outcome as multiplying 2 by 3.

We already know that there are (2⋅3) or 6 thirds in 2. To find how many 23s are in 2, we need to combine every 2 of the thirds into a group. Doing this results in half as many groups. So 2÷23=(2⋅3)÷2, which equals 3.

Again, we know that there are (2⋅3) thirds in 2. To find how many 43s are in 2, we need to combine every 4 of the thirds into a group. Doing this results in one fourth as many groups. So 2÷43=(2⋅3)÷4, which equals 112.

Notice that each division problem above can be answered by multiplying 2 by the denominator of the divisor and then dividing it by the numerator. So 2÷abcan be solved with 2⋅b÷a, which can also be written as 2⋅ba. In other words, dividing 2 by ab has the same outcome as multiplying 2 by ba. The fraction in the divisor is “inverted” and then multiplied.

Over the next few days, your student will be solving problems that require multiplying and dividing fractions. Some of these problems will be about comparison. For example:

If Priya ran for 56 hour and Clare ran for 32 hours, what fraction of Clare’s running time was Priya’s running time?

We can draw a diagram and write a multiplication equation to make sense of the situation.

Other problems your students will solve are related to geometry—lengths, areas, and volumes. For examples:

We know that the area of a rectangle can be found by multiplying its length and width (?⋅212=1114), so dividing 1114÷212 (or 454÷52) will give us the length of the room. 454÷52=454⋅25=92. The room is 412meters long.

What is the volume of a box (a rectangular prism) that is 312 feet by 10 feet by 14 foot?

We can find the volume by multiplying the edge lengths. 312⋅10⋅14=72⋅10⋅14, which equals 708. So the volume is 708 or 868cubic feet.

IM 6–8 Math was originally developed by Open Up Resources and authored by Illustrative Mathematics, and is copyright 2017-2019 by Open Up Resources. It is licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). OUR's 6–8 Math Curriculum is available at https://openupresources.org/math-curriculum/.