Lesson 5How Many Groups? (Part 2)

Let’s use blocks and diagrams to understand more about division with fractions.

Learning Targets:

  • I can find how many groups there are when the number of groups and the amount in each group are not whole numbers.

5.1 Reasoning with Fraction Strips

Write a fraction or whole number as an answer for each question. If you get stuck, use the fraction strips. Be prepared to share your strategy.

  1. How many \frac 12 s are in 2?
  2. How many \frac 15 s are in 3?
  3. How many \frac {1}{8} s are in 1\frac 14 ?
  1. 1 \div \frac {2}{6} = {?}
  2. 2 \div \frac 29 = {?}
  3. 4 \div \frac {2}{10} = {?}

5.2 More Reasoning with Pattern Blocks

Use the pattern blocks in the applet to answer the questions. (If you need help aligning the pieces, you can turn on the grid.)

  1. If the trapezoid represents 1 whole, what do each of the following shapes represent? Be prepared to show or explain your reasoning.
    1. 1 triangle

    2. 1 rhombus

    3. 1 hexagon

  2. Use pattern blocks to represent each multiplication equation. Use the trapezoid to represent 1 whole.

    1. 3 \boldcdot \frac 13=1

    2. 3 \boldcdot \frac 23=2

  3. Diego and Jada were asked “How many rhombuses are in a trapezoid?”

    • Diego says, “ 1\frac 13 . If I put 1 rhombus on a trapezoid, the leftover shape is a triangle, which is \frac 13 of the trapezoid.”
    • Jada says, “I think it’s 1\frac12 . Since we want to find out ‘how many rhombuses,’ we should compare the leftover triangle to a rhombus. A triangle is \frac12 of a rhombus.” 
  4. Select all equations that can be used to answer the question: “How many rhombuses are in a trapezoid?”

    1. \frac 23  \div {?} = 1

    2. {?} \boldcdot \frac 23 = 1

    3. 1 \div \frac 23 = {?}

    4. 1 \boldcdot \frac 23 = {?}

    5. {?}  \div \frac 23 = 1

5.3 Drawing Diagrams to Show Equal-sized Groups

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

  1. The distance around a park is \frac32 miles. Noah rode his bicycle around the park for a total of 3 miles. How many times around the park did he ride?
  2. You need \frac34 yard of ribbon for one gift box. You have 3 yards of ribbon. How many gift boxes do you have ribbon for?
  3. The water hose fills a bucket at \frac13 gallon per minute. How many minutes does it take to fill a 2-gallon bucket?

Are you ready for more?

How many heaping teaspoons are in a heaping tablespoon? How would the answer depend on the shape of the spoons?

Lesson 5 Summary

Suppose one batch of cookies requires \frac23 cup flour. How many batches can be made with 4 cups of flour?

We can think of the question as being: “How many \frac23 s are in 4?” and represent it using multiplication and division equations.

{?} \boldcdot \frac23 = 4 4\div \frac23 = {?}

Let’s use pattern blocks to visualize the situation and say that a hexagon is 1 whole. 

4 hexagons are divided into 3 parallelograms

Since 3 rhombuses make a hexagon, 1 rhombus represents \frac13 and 2 rhombuses represent  \frac 23 . We can see that 6 pairs of rhombuses make 4 hexagons, so there are 6 groups of \frac 23 in 4.

Other kinds of diagrams can also help us reason about equal-sized groups involving fractions. This example shows how we might reason about the same question from above: “How many \frac 23 -cups are in 4 cups?”

A diagram showing the breakdown of cups into groups

We can see each “cup” partitioned into thirds, and that there are 6 groups of \frac23 -cup in 4 cups. In both diagrams, we see that the unknown value (or the “?” in the equations) is 6. So we can now write:

  6 \boldcdot \frac23 = 4 4\div \frac23 = 6

Lesson 5 Practice Problems

  1. Use the tape diagram to represent and find the value of  \frac12\div\frac13 .

    Mark up and label the diagram as needed.

    A tape diagram on a square grid is composed of 6 squares and is partitioned into two equal parts. Each part is partitioned by a vertical dashed line resulting in three equal parts. A brace extends from the beginning of the first part to the end of the first part and is labeled "one half".
  2. What is the value of  \frac12\div\frac13 ? Use pattern blocks to represent and find this value. The yellow hexagon represents 1 whole. Explain or show your reasoning.

    Four pattern blocks: One large yellow hexagon, one blue rhombus, one red trapezoid, and one green triangle.
  3. Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answer the question.

    1. How many \frac12 s are in 7?

    2. How many \frac38 s are in 6?

    3. How many \frac{5}{16} s are in 1\frac78 ?

    an image of a ruler in inches and centimeters
  4. Use the tape diagram to represent and answer the question: How many \frac25 s are in 1\frac12 ?

    Mark up and label the diagram as needed.

    A tape diagram of two equal parts on a square grid. Each part is composed of 5 squares. A brace from the beginning of the diagram to the middle of the eighth square is labeled "one and one half."
  5. Write a multiplication equation and a division equation to represent each question, statement, or diagram. 

    1. There are 12 fourths in 3.  
    2. A tape diagram of 4 equal parts with each part labeled one half. Above the diagram is a brace, labeled 2, that contains all 4 parts.
    1. How many \frac 23 s are in 6?
    2. The number 2 is broken into 5 pieces
  6. At a farmer’s market, two vendors sell fresh milk. One vendor sells 2 liters for $3.80, and another vendor sells 1.5 liters for $2.70. Which is the better deal? Explain your reasoning.

  7. A recipe uses 5 cups of flour for every 2 cups of sugar.

    1. How much sugar is used for 1 cup of flour?
    2. How much flour is used for 1 cup of sugar?
    3. How much flour is used with 7 cups of sugar?
    4. How much sugar is used with 6 cups of flour?