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Lesson 7What Fraction of a Group?

Let’s think about dividing things into groups when we can’t even make one whole group.

Learning Targets:

  • I can tell when a question is asking for the number of groups and that number is less than 1.
  • I can use diagrams and multiplication and division equations to represent and answer “what fraction of a group?” questions.

7.1 Estimating a Fraction of a Number

  1. Estimate the following quantities:

    1. What is \frac13 of 7?
    2. What is \frac45 of 9\frac23 ?
    3. What is 2\frac47 of 10\frac19 ?
  1. Write a multiplication expression for each question.

7.2 Fractions of Ropes

The segments in the applet represent 4 different lengths of rope. Compare one rope to another, moving the rope by dragging the open circle at one endpoint. You can use the yellow pins to mark off lengths.

open applet in presentation mode
  1. Compare the lengths of rope B, C, and D to the length of rope A and complete each statement. Then, use the measurements shown on the grid to write a multiplication equation and a division equation for each statement.
    1. Rope B is _______ times as long as rope A.

      Multiplication equation: _________________
      Division equation: _________________

    2. Rope C is _______ times as long as rope A.

      Multiplication equation: _________________
      Division equation: _________________

    3. Rope D is _______ times as long as rope A.

      Multiplication equation: _________________
      Division equation: _________________

  2. Each equation can be used to answer a question about rope C and D. What could each question be?

    1. {?} \boldcdot 3=9 and 9 \div 3={?}

    2. {?} \boldcdot 9=3 and 3 \div 9= {?}

7.3 Fractional Batches of Ice Cream

One batch of an ice cream recipe uses 9 cups of milk. A chef makes different amounts of ice cream on different days. Here are the amounts of milk she used:

  • Monday: 12 cups
  • Tuesday: 22 \frac12 cups
  • Thursday: 6 cups
  • Friday: 7 \frac12 cups

  1. How many batches of ice cream did she make on each of the following days? Write a division equation and draw a tape diagram for the question about each day. Then answer the question.

    1. Monday
      A blank grid with a height of 5 units and a length of 24 units.
    2. Tuesday
      A blank grid with a height of 5 units and a length of 24 units.

  2. What fraction of a batch of ice cream did she make on each of the following days? Write a division equation and draw a tape diagram for the question about each day. Then answer the question.

    1. Thursday
      A blank grid with a height of 5 units and a length of 24 units.
    2. Friday
      A blank grid with a height of 5 units and a length of 24 units.

  3. Write a division equation, and draw a tape diagram for each question. Then answer the question.

    1. What fraction of 9 is 3?
      A blank grid with a height of 5 units and a length of 24 units.
    2. What fraction of 5 is \frac 12 ?
      A blank grid with a height of 5 units and a length of 24 units.

Lesson 7 Summary

It is natural to think about groups when we have more than one group, but we can also have a fraction of a group.

To find the amount in a fraction of a group, we can multiply the fraction by the amount in the whole group. If a bag of rice weighs 5 kg,  \frac34 of a bag would weigh ( \frac34 \boldcdot 5) kg.

A diagram showing kilograms and bags

Sometimes we need to find what fraction of a group an amount is. Suppose a full bag of flour weighs 6 kg. A chef used 3 kg of flour. What fraction of a full bag was used? In other words, what fraction of 6 kg is 3 kg?  

This question can be represented by a multiplication equation and a division equation, as well as by a diagram.

{?} \boldcdot 6 = 3 3\div 6 = {?}  

A tape diagram of 6 equal parts. Above the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled "6 kilograms."Below the diagram, a brace from the beginning of the diagram to the end of the diagram is labeled 1 bag. A third brace that contains the first three parts is labeled "three ." Below the diagram, a fourth brace which also contains the first three parts is labeled "question mark bag."
We can see from the diagram that 3 is \frac12 of 6, and we can check this answer by multiplying:  \frac12 \boldcdot 6 = 3 .

In any situation where we want to know what fraction one number is of another number, we can write a division equation to help us find the answer.

For example, “What fraction of 3 is 2\frac14 ?” can be expressed as {?} \boldcdot 3 = 2\frac14 , which can also be written as  2\frac14\div 3 = {?} .

The answer to “What is 2\frac14 \div 3 ?” is also the answer to the original question.

A diagram showing cups and groups

The diagram shows that 3 wholes contain 12 fourths, and 2\frac14 contains 9 fourths, so the answer to this question is \frac{9}{12} , which is equivalent to \frac34 . 

We can use diagrams to help us solve other division problems that require finding a fraction of a group. For example, here is a diagram to help us answer the question: “What fraction of \frac94  is \frac32 ?,” which can be written as  \frac32 \div \frac94 = {?} .

A diagram showing the breakdown of 2.25 into groups

We can see that the quotient is \frac69 , which is equivalent to \frac23 . To check this, let’s multiply.   \frac23 \boldcdot \frac94 = \frac{18}{12} , and \frac{18}{12} is, indeed, equal to \frac32 .

Lesson 7 Practice Problems

  1. A recipe calls for \frac12 lb of flour for 1 batch. How many batches can be made with each of the following amounts?

    1. 1 lb
    2. \frac34 lb
    3. \frac14 lb

  2. Whiskers the cat weighs 2\frac23 kg. Piglio weighs 4 kg. For each question, write a multiplication and a division equation, decide whether the answer is greater or less than 1, and then answer the question.

    1. How many times as heavy as Piglio is Whiskers?
    2. How many times as heavy as Whiskers is Piglio?

  3. Andre is walking from home to a festival that is 1\frac58 kilometers away. He takes a quick rest after walking \frac13 kilometers. In this situation, which question can be represented by the equation: {?} \boldcdot 1\frac58 = \frac13 ?

    1. What fraction of the trip has Andre completed?
    2. How many more kilometers does he have to walk to get to the festival?
    3. What fraction of the trip is left?
    4. How many kilometers is it from home to the festival and back home?

  4. Draw a tape diagram to represent and answer the question: What fraction of 2\frac12 is \frac45 ?

  5. How many groups of \frac34 are in each of the following quantities?

    1. \frac{11}{4}
    2. 6\frac12

  6. Which question can be represented by the equation 4\div \frac27 = {?}

    1. What is 4 groups of \frac 27 ?
    2. How many \frac27 s are in 4?
    3. What is \frac 27 of 4?
    4. How many 4s are in \frac27 ?