Mathematics Support

Math 6 | Unit 1 2 3 4 5 6 7 8 9


Making Sense of Division

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:

  • “How many bottles can we fill with 10 liters if each bottle has 2 liters?”
  • “How many liters are in each bottle if we divide 10 liters into 2 bottles?”

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 liters in each of the 2 bottles.”

Here is a task to try with your student:

  1. Write two different questions we can ask about 15÷6.
  2. Estimate the answer: Is it less than 1, equal to 1, or greater than 1? Explain your estimate.
  3. Find the answer to one of the questions you wrote. It might help to draw a picture.

Solution:

  1. Questions vary. Sample questions:
    • A ribbon that is 15 inches long is divided into 6 equal sections. How long (in inches) is each section?
    • A ribbon that is 15 inches is divided into 6-inch sections. How many sections are there?
     
  2. Greater than 1. Sample explanations:
    • 12÷6 is 2, so 15÷6 must be greater than 2.
    • If we divide 15 into 15 groups (15÷15), we get 1. So if we divide 15 into 6, which is a smaller number of groups, the amount in each group must be greater than 1.
     
  3. 212. Sample diagram:

Meanings of Fraction Division

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:

2?=10
?2=10

 

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.

?112=6

 

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.

Here is a task to try with your student:

  1. How many groups of 23 are in 5?
    1. Write a division equation to represent the question. Use a “?” to represent the unknown amount.
    2. Find the answer. Explain or show your reasoning.
     
  2. A sack of flour weighs 4 pounds. A grocer is distributing the flour into equal-size bags.
    1. Write a question that 4÷25=? could represent in this situation.
    2. Find the answer. Explain or show your reasoning.
     

Solution:

    1. 5÷23=?
    2. 712. Sample reasoning: There are 3 thirds in 1, so there are 15 thirds in 5. That means there are half as many two-thirds, or 152 two-thirds, in 5.
     
    1. 4 pounds of flour are divided equally into bags of 25-pound each. How many bags will there be?
    2. 10 bags. Sample reasoning: Break every 1 pound into fifths and then count how many groups of 25 there are.
     

Algorithm for Fraction Division

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:

  • 2÷13=? can be viewed as “how many 13s are in 2?”

    Because there are 3 thirds in 1, there are (23) or 6 thirds in 2. So dividing 2 by 13 has the same outcome as multiplying 2 by 3.

  • 2÷23=? can be viewed as “how many 23s are in 2?”

    We already know that there are (23) 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=(23)÷2, which equals 3.

  • 2÷43=? can be viewed as “how many 43s are in 2?”

    Again, we know that there are (23) 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=(23)÷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 2b÷a, which can also be written as 2ba. 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.

Here is a task to try with your student:

  1. Find each quotient. Show your reasoning.
    1. 3÷17
    2. 3÷37
    3. 3÷67
    4. 37÷67
     
  2. Which has a greater value: 910÷9100 or 125÷625? Explain or show your reasoning.

Solution:

    1. 21. Sample reasoning: 3÷17=371=21
    2. 7. Sample reasoning: 3÷37=373=7
    3. 312. Sample reasoning: 3÷17=376=72. The fraction 67 is two times 37, so there are half as many 67s in 3 as there are 37s.
    4. 12. Sample reasoning: 37÷67=3776=36
     
  1. The have the same value. Both equal 10. 910÷9100=9101009=10 and 125÷625=125256=10.

Fractions in Lengths, Areas, and Volumes

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.

    (fraction)(Clare’s time)=(Priya’s time)
    ?32=56
    We can find the unknown by dividing. 56÷32=5623, which equals 1018. So Priya’s running time was 1018 or 59 of Clare’s.

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

  • What is the length of a rectangular room if its width is 212 meters and its area is 1114 square meters?

    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=45425=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. 3121014=721014, which equals 708. So the volume is 708 or 868cubic feet.

Here is a task to try with your student:

  1. In the first example about Priya and Clare’s running times, how many times as long as Priya’s running time was Clare’s running time? Show your reasoning.
  2. The area of a rectangle is 203 square feet. What is its width if its length is 43feet? Show your reasoning.

Solution:

  1. 95. Sample reasoning: We can write ?56=32 to represent the question “how many times of Priya’s running time was Clare’s running time?” and then solve by dividing. 32÷56=3265=1810. Clare’s running time was 1810or 95 as long as Priya’s.
  2. 5 feet. Sample reasoning: 203÷43=20334=204=5

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/.


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