Lesson 6Using Diagrams to Find the Number of Groups

Let’s draw tape diagrams to think about division with fractions.

Learning Targets:

  • I can use a tape diagram to represent equal-sized groups and find the number of groups.

6.1 How Many of These in That?

  1. We can think of the division expression 10 \div 2\frac12 as the answer to the question: “How many groups of 2\frac 12  are in 10?” Complete the tape diagram to represent the question. Then answer the question. 
  2. Complete the tape diagram to represent the question: “How many groups of 2 are in 7?” Then answer the question.

6.2 Representing Groups of Fractions with Tape Diagrams

To make sense of the question “How many \frac 23 s are in 1?,” Andre wrote equations and drew a tape diagram.

{?} \boldcdot \frac 23 = 1

1 \div \frac 23 = {?}

A tape diagram with three equal parts. The first two parts are shaded and are each labeled one third. Above the tape diagram is a bracket labeled 1, and contains all three parts. Below the diagram there is a bracket labeled "1 group of two thirds," and contains the first two parts.
  1. In an earlier task, we used pattern blocks to help us solve the equation 1 \div \frac 23 = {?} . Explain how Andre’s tape diagram can also help us solve the equation.

  2. Write a multiplication equation and a division equation for each of the following questions. Draw a tape diagram to find the solution. Use the grid to help you draw, if needed. 

    1. How many \frac 34 s are in 1?
      A blank grid with a height of 7 units and length of 16 units.
    2. How many \frac23 s are in 3?
      A blank grid with a height of 7 units and length of 16 units.
    3. How many \frac32 s are in 5?
      A blank grid with a height of 7 units and length of 16 units.

6.3 Finding Number of Groups

  1. For each question, draw a diagram to show the relationship of the quantities and to help you answer the question. Then, write a multiplication equation or a division equation for the situation described in the question. Be prepared to share your reasoning.

    1. How many \frac38 -inch thick books make a stack that is 6 inches tall?
    2. How many groups of \frac12 pound are in  2\frac 34 pounds?
  2. Write a question that can be represented by the division equation 5 \div 1\frac12 = {?} . Then answer the question. Show your reasoning.

Lesson 6 Summary

A baker used 2 kilograms of flour to make several batches of a pastry recipe. The recipe called for \frac25 kilogram of flour per batch. How many batches did she make?

We can think of the question as: “How many groups of \frac25 kilogram make 2 kilograms?” and represent that question with the equations:

{?} \boldcdot \frac25=2 2 \div \frac25 = {?}

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 \frac 25 in 2. Multiplying 5 and \frac25 allows us to check this answer: 5 \boldcdot \frac 25 = \frac{10}{5} and \frac {10}{5} = 2 , so the answer is correct. 

Notice the number of groups that result from 2 \div \frac25 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 \frac34 cup. How many servings are there in 3\frac12 cups?

{?}\boldcdot \frac34 = 3\frac12 3\frac12 \div \frac34 = {?}

Looking at the diagram, we can see there are 4 full groups of \frac 34 , plus 2 fourths. If 3 fourths make a whole group, then 2 fourths make \frac 23 of a group. So the number of servings (the “?” in each equation) is 4\frac23 . We can check this by multiplying  4\frac23 and \frac34 .

4\frac23 \boldcdot \frac34 = \frac{14}{3} \boldcdot \frac34 , and \frac{14}{3} \boldcdot \frac34 = \frac{14}{4} , which is indeed equivalent to 3\frac12 .

Lesson 6 Practice Problems

  1. We can think of 3\div \frac14 as the answer to the question “How many groups of \frac14 are in 3?” Draw a tape diagram to represent the question. Then answer the question.

  2. Describe how to draw a tape diagram to represent and answer 3 \div \frac35 = {?} for a friend who was absent.

  3. How many groups of \frac12 days are in 1 week?

    1. Write a multiplication equation or a division equation to represent the question.
    2. Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.
  4. Diego said that the answer to the question “How many groups of \frac56 are in 1?” is \frac 65 or 1\frac15 . Do you agree with his statement? Explain or show your reasoning.
  5. Select all equations that can represent the question: “How many groups of \frac45 are in 1?”

    1. {?} \boldcdot 1=\frac45
    2. 1 \boldcdot \frac45 = {?}
    3. \frac45 \div 1 = {?}
    4. {?} \boldcdot \frac45 =1
    5. 1\div \frac45 = {?}
  6. Calculate each percentage mentally.

    1. What is 10% of 70?
    2. What is 10% of 110?
    3. What is 25% of 160?
    4. What is 25% of 48?
    1. What is 50% of 90?
    2. What is 50% of 350?
    3. What is 75% of 300?
    4. What is 75% of 48?