22.3: Using Diagrams to Find the Number of Groups
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Lesson
Let's draw tape diagrams to think about division with fractions.
Exercise 22.3.1: How Many of These in That?
- We can think of the division expression 10÷212 as the question: “How many groups of 212 are in 10?” Complete the tape diagram to represent this question. Then find the answer.

- Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then find the answer.

Exercise 22.3.2: Representing Groups of Fractions with Tape Diagrams
To make sense of the question “How many 23s are in 1?,” Andre wrote equations and drew a tape diagram.
?⋅23=1
1÷23=?

- In an earlier task, we used pattern blocks to help us solve the equation 1÷23=?. Explain how Andre’s tape diagram can also help us solve the equation.
- Write a multiplication equation and a division equation for each question. Then, draw a tape diagram and find the answer.
- How many 34s are in 1?

- How many 23s are in 3?

- How many 32s are in 5?

Exercise 22.3.3: Finding Number of Groups
- Write a multiplication equation or a division equation for each question. Then, find the answer and explain or show your reasoning.
- How many 38-inch thick books make a stack that is 6 inches tall?
- How many groups of 12 pound are in 234 pounds?
- Write a question that can be represented by the division equation 5÷112=?. Then, find the answer and explain or show your reasoning.
Summary
A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for 25 kilogram of flour per batch. How many batches did she make?
We can think of the question as: “How many groups of 25 kilogram make 2 kilograms?” and represent that question with the equations:
?⋅25=2
2÷25=?
To help us make sense of the question, we can draw a tape diagram. This diagram shows 2 whole kilograms, with each kilogram partitioned into fifths.

We can see there are 5 groups of 25 in 2. Multiplying 5 and 25 allows us to check this answer: 5⋅25=105 and 105=2, so the answer is correct.
Notice the number of groups that result from 2÷25 is a whole number. Sometimes the number of groups we find from dividing may not be a whole number. Here is an example:
Suppose one serving of rice is 34 cup. How many servings are there in 312 cups?
?⋅34=312
312÷34=?

Looking at the diagram, we can see there are 4 full groups of 34, plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make 23 of a group. So the number of servings (the “?” in each equation) is 423. We can check this by multiplying 423 and 34.
423⋅34=143⋅34, and 143⋅34=144, which is indeed equivalent to 312.
Practice
Exercise 22.3.4
We can think of 3÷14 as the question “How many groups of 14 are in 3?” Draw a tape diagram to represent this question. Then find the answer.
Exercise 22.3.5
Describe how to draw a tape diagram to represent and answer 3÷35=? for a friend who was absent.
Exercise 22.3.6
How many groups of 12 day are in 1 week?
- Write a multiplication equation or a division equation to represent the question.
- Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.
Exercise 22.3.7
Diego said that the answer to the question “How many groups of 56 are in 1?” is 65 or 115. Do you agree with him? Explain or show your reasoning.
Exercise 22.3.8
Select all the equations that can represent the question: “How many groups of 45 are in 1?”
- ?⋅1=45
- 1⋅45=?
- 45÷1=?
- ?⋅45=1
- 1÷45=?
(From Unit 4.2.2)
Exercise 22.3.9
Calculate each percentage mentally.
- What is 10% of 70?
- What is 10% of 110?
- What is 25% of 160?
- What is 25% of 48?
- What is 50% of 90?
- What is 50% of 350?
- What is 75% of 300?
- What is 75% of 48?
(From Unit 3.4.5)