22.1: How Many Groups? (Part 1)

Basic Math (Grade 6) · LibreTexts

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    Lesson

    Let's play with blocks and diagrams to think about division with fractions.

    Exercise \(\PageIndex{1}\): Equal-Sized Groups

    Write a multiplication equation and a division equation for each sentence or diagram.

    1. Eight $5 bills are worth $40.
    2. There are 9 thirds in 3 ones.
    clipboard_ea65cebf78ef1e31405680a145e644376.png
    Figure \(\PageIndex{1}\)

    Exercise \(\PageIndex{2}\): 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 a hexagon represents 1 whole, what fraction do each of the following shapes represent? Be prepared to show or explain your reasoning.
      1. 1 triangle
      2. 1 rhombus
      3. 1 trapezoid
      4. 4 triangles
      5. 3 rhombuses
      6. 2 hexagons
      7. 1 hexagon and 1 trapezoid
    2. Here are Elena’s diagrams for \(2\cdot\frac{1}{2}=1\) and \(6\cdot\frac{1}{3}=2\). Do you think these diagrams represent the equations? Explain or show your reasoning.
    clipboard_ee8fe4f85c47cb4fe2ffa9d56640a4933.png
    Figure \(\PageIndex{2}\): Two diagrams of pattern blocks: The first is of one red hexagon composed of two red trapezoids. The diagram is labeled with the equation 2 times one half equals 1. The second diagram is of two blue hexagons where each hexagon is composed of three blue rhombuses. The second diagram is labeled with the equation 6 times, one third, equals 2.
    1. Use pattern blocks to represent each multiplication equation. Remember that a hexagon represents 1 whole.
      1. \(3\cdot\frac{1}{6}=\frac{1}{2}\)
      2. \(2\cdot\frac{3}{2}=3\)
    2. Answer the questions. If you get stuck, consider using pattern blocks.
      1. How many \(\frac{1}{2}\)s are in \(4\)?
      2. How many \(\frac{2}{3}\)s are in \(2\)?
      3. How many \(\frac{1}{6}\)s are in \(1\frac{1}{2}\)?

    Summary

    Some problems that involve equal-sized groups also involve fractions. Here is an example: “How many \(\frac{1}{6}\) are in \(2\)?” We can express this question with multiplication and division equations.

    \(?\cdot\frac{1}{6}=2\)

    \(2\div\frac{1}{6}=?\)

    Pattern-block diagrams can help us make sense of such problems. Here is a set of pattern blocks.

    clipboard_e2edd3a0eb5c069604b4ead7900e9c493.png
    Figure \(\PageIndex{3}\)

    If the hexagon represents 1 whole, then a triangle must represent \(\frac{1}{6}\), because 6 triangles make 1 hexagon. We can use the triangle to represent the \(\frac{1}{6}\) in the problem.

    clipboard_e07b32b8004ab74a6a129693d37b05e44.png
    Figure \(\PageIndex{4}\)

    Twelve triangles make 2 hexagons, which means there are 12 groups of \(\frac{1}{6}\) in 2.

    If we write the 12 in the place of the “?” in the original equations, we have:

    \(12\cdot\frac{1}{6}=2\)

    \(2\div\frac{1}{6}=12\)

    Practice

    Exercise \(\PageIndex{3}\)

    Consider the problem: A shopper buys cat food in bags of 3 lbs. Her cat eats \(\frac{3}{4}\) lb each week. How many weeks does one bag last?

    1. Draw a diagram to represent the situation and label your diagram so it can be followed by others. Answer the question.
    2. Write a multiplication or division equation to represent the situation.
    3. Multiply your answer in the first question (the number of weeks) by \(\frac{3}{4}\). Did you get 3 as a result? If not, revise your previous work.

    Exercise \(\PageIndex{4}\)

    Use the diagram to answer the question: How many \(\frac{1}{3}\)s are in \(1\frac{2}{3}\)? The hexagon represents 1 whole. Explain or show your reasoning.

    clipboard_e9baa3aaeb8701d3a40b8a5a182c71db7.png
    Figure \(\PageIndex{5}\)

    Exercise \(\PageIndex{5}\)

    Which question can be represented by the equation \(?\cdot\frac{1}{8}=3\)?

    1. How many \(3\)s are in \(\frac{1}{8}\)?
    2. What is \(3\) groups of \(\frac{1}{8}\)?
    3. How many \(\frac{1}{8}\)s are in \(3\)?
    4. What is \(\frac{1}{8}\) of \(3\)?

    Exercise \(\PageIndex{6}\)

    Write two division equations for each multiplication equation.

    1. \(15\cdot\frac{2}{5}=6\)
    2. \(6\cdot\frac{4}{3}=8\)
    3. \(16\cdot\frac{7}{8}=14\)

    Exercise \(\PageIndex{7}\)

    Noah and his friends are going to an amusement park. The total cost of admission for 8 students is $100, and all students share the cost equally. Noah brought $13 for his ticket. Did he bring enough money to get into the park? Explain your reasoning.

    (From Unit 4.1.2)

    Exercise \(\PageIndex{8}\)

    Write a division expression with a quotient that is:

    1. greater than \(8\div 0.001\)
    2. less than \(8\div 0.001\)
    3. between \(8\div 0.001\) and \(8\div\frac{1}{10}\)

    (From Unit 4.1.1)

    Exercise \(\PageIndex{9}\)

    Find each unknown number.

    1. \(12\) is \(150\)% of what number?
    2. \(5\) is \(50\)% of what number?
    3. \(10\)% of what number is \(300\)?
    4. \(5\)% of what number is \(72\)?
    5. \(20\) is \(80\)% of what number?

    (From Unit 3.4.5)

    22.1: How Many Groups? (Part 1) is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.