Lesson 4

How Many Groups? (Part 1)

4.1: Equal-sized Groups (5 minutes)

Warm-up

This warm-up reviews the idea of multiplication as representing equal-sized groups and the relationship between multiplication and division.

There are multiple equations students can write for each of the problems; the equations that connect multiplication and division to equal-sized groups are the important ones to highlight. As students work, identify students whose equations reflect these ideas.

Launch

Give students 2 minutes of quiet think time, followed by a whole-class discussion.

Student Facing

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.
  3. A tape diagram of 5 equal parts. Each part is labeled one fifth. Above the bar is a bracket, labeled 1, that spans the entire length of the bar.

 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may struggle to frame repeated addition as multiplication. To help them see the connection, refer to one of their addition statements and ask questions such as, “How many same-sized groups are being added?” or “What is in each group?”.

Activity Synthesis

Select 1–2 students to share their responses. Record the responses for all to see. Ask students to indicate whether they agree or disagree with each one.

As students present the equations for each problem, connect the pieces in each equation to the idea of equal-sized groups. Ask questions such as:

  • “Which number in the multiplication equation refers to the number of groups?”
  • “Which number in the multiplication equation refers to how much is in each group?”
  • “In this case, what does the division \(3 \div 9 = \frac 13\) mean?”
  • “In this case, what does the division \(1 \div \frac15 = 5\) mean?”

4.2: Reasoning with Pattern Blocks (25 minutes)

Activity

In this activity, students use the relationships between the areas of geometric shapes to reason about division situations that involve fractions. The focus is on the “how many groups?” interpretation of division.

Students start by using pattern blocks to represent multiplication of a whole number and a fraction. For example, if a hexagon represents 1 and six triangles make a hexagon, then each triangle represents \(\frac16\). They can then use six triangles to represent \(6 \boldcdot \frac16 = 1\).  

Later, students use the blocks to reason in the opposite direction, answering questions such as, “How many \(\frac12\)s are in 4?” These kinds of questions serve as a stepping stone to more abstract questions such as, “What is 4 divided by \(\frac 12\)?”

Launch

Arrange students in groups of 3–4. Provide access to pattern blocks. Give students 10–12 minutes to collaborate on the first three questions and 3–4 minutes of quiet think time for the last question.

Remind students of the following:

  • We can think of a fraction such as \(\frac12\) or \(\frac13\) in relation to 1 whole. In this task, the hexagon is 1 whole.
  • We worked with the same shapes earlier in the course. We saw that two triangles make a rhombus, because if we place two triangles (joined along one side with no gap) on top of a rhombus, the triangles would match the rhombus exactly. This means that a triangle is half of a rhombus.

Classrooms with no access to pattern blocks or those using the digital materials can use the provided applet. Physical pattern blocks are still preferred, however.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Once students have determined what fraction of a hexagon each of the shapes represents, pause the class. Invite students to demonstrate and share their strategies for finding each fraction using pattern blocks to justify their reasoning. Create a display that includes an image of each shape labeled with the name and the fraction it represents of a hexagon. Keep this display visible as students move on to the next problems.
Supports accessibility for: Conceptual processing; Memory
Representing, Writing: MLR3 Clarify, Critique, Correct. At the appropriate time, pause the class for a brief discussion of the first question. Display the following incorrect response that reflects a possible common misunderstanding: “The area of the rhombus is 3 because 3 fit inside the hexagon.” Ask students, “Do you agree with the statement? Why or why not?” Invite students to identify the error, correct the statement, and draw a diagram to represent the situation. Improved statements should include fractional language and direct connections to the diagram. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness; Optimize output (for justification)

Student Facing

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 \(2 \boldcdot \frac12 = 1\) and \(6 \boldcdot \frac13 = 2\). Do you think these diagrams represent the equations? Explain or show your reasoning.

    Two diagrams of pattern blocks. 
  3. Use pattern blocks to represent each multiplication equation. Remember that a hexagon represents 1 whole.
    1. \(3 \boldcdot \frac 16=\frac12\)

    2. \(2 \boldcdot \frac 32=3\)
  4. Answer the questions. If you get stuck, consider using pattern blocks.
    1. How many \(\frac 12\)s are in 4?

    2. How many \(\frac23\)s are in 2?

    3. How many \(\frac16\)s are in \(1\frac12\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 3–4. Provide access to pattern blocks. Give students 10–12 minutes to collaborate on the first three questions and 3–4 minutes of quiet think time for the last question.

Remind students of the following:

  • We can think of a fraction such as \(\frac12\) or \(\frac13\) in relation to 1 whole. In this task, the hexagon is 1 whole.
  • We worked with the same shapes earlier in the course. We saw that two triangles make a rhombus, because if we place two triangles (joined along one side with no gap) on top of a rhombus, the triangles would match the rhombus exactly. This means that a triangle is half of a rhombus.

Classrooms with no access to pattern blocks or those using the digital materials can use the provided applet. Physical pattern blocks are still preferred, however.

Representation: Develop Language and Symbols. Display or provide charts with symbols and meanings. Once students have determined what fraction of a hexagon each of the shapes represents, pause the class. Invite students to demonstrate and share their strategies for finding each fraction using pattern blocks to justify their reasoning. Create a display that includes an image of each shape labeled with the name and the fraction it represents of a hexagon. Keep this display visible as students move on to the next problems.
Supports accessibility for: Conceptual processing; Memory
Representing, Writing: MLR3 Clarify, Critique, Correct. At the appropriate time, pause the class for a brief discussion of the first question. Display the following incorrect response that reflects a possible common misunderstanding: “The area of the rhombus is 3 because 3 fit inside the hexagon.” Ask students, “Do you agree with the statement? Why or why not?” Invite students to identify the error, correct the statement, and draw a diagram to represent the situation. Improved statements should include fractional language and direct connections to the diagram. This will help students evaluate and improve on the written mathematical arguments of others.
Design Principle(s): Maximize meta-awareness; Optimize output (for justification)

Student Facing

Your teacher will give you pattern blocks as shown here. Use them to answer the questions.

Four pattern blocks: One large yellow hexagon, one blue rhombus, one red trapezoid, and one green triangle.
  1. If a hexagon represents 1 whole, what fraction does each of the following shapes represent? Be prepared to show or explain your reasoning.

    • 1 triangle

    • 1 rhombus

    • 1 trapezoid
    • 4 triangles

    • 3 rhombuses

    • 2 hexagons
    • 1 hexagon and 1 trapezoid
  2. Here are Elena’s diagrams for \(2 \boldcdot \frac12 = 1\) and \(6 \boldcdot \frac13 = 2\). Do you think these diagrams represent the equations? Explain or show your reasoning.

    Two diagrams of pattern blocks. 
  3. Use pattern blocks to represent each multiplication equation. Remember that a hexagon represents 1 whole.

    1. \(3 \boldcdot \frac 16=\frac12\)

    2. \(2 \boldcdot \frac 32=3\)

  4. Answer the questions. If you get stuck, consider using pattern blocks.

    1. How many \(\frac 12\)s are in 4?
    2. How many \(\frac23\)s are in 2?
    3. How many \(\frac16\)s are in \(1\frac12\)?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Some students may not remember the names of the shapes for these blocks. Consider reviewing the names of these shapes before beginning the activity and having students write them next to the pictures for reference.

Some students may simply look at the blocks and incorrectly guess the size of each block relative to the hexagon. Encourage them to place the blocks on top of the hexagon, to use non-hexagons to compose a hexagon, or to otherwise manipulate the blocks in order to make comparisons.

Activity Synthesis

Select a few students to show their pattern-block arrangements or drawings for \(3 \boldcdot \frac 16=\frac12\) and \(2 \boldcdot \frac 32=3\). After each person shares, poll the class to see if others did it the same way or had alternative solutions.

Select other students to share their responses and reasoning for the last set of questions. If no one reasoned about the questions by using pattern blocks, show how the blocks could be used to answer the questions. For instance:

  • For “how many \(\frac12\)s are in 4?”, we could use 8 trapezoids (each representing \(\frac12\)) to make 4 hexagons.
  • For “how many \(\frac23\)s are in 2?”, we could use 2 rhombuses (each representing \(\frac 23\)) to make 2 hexagons.
  • For “how many \(\frac16\)s are in \(1\frac12\)?”, we could use 9 triangles (each representing \(\frac16\)) to make \(1\frac12\) hexagons.

Highlight that, in each case, we know the size of each group (or each block) and are trying to find out how many groups (or how many blocks) are needed to equal a particular area.

You may choose to use the applet at https://ggbm.at/VmEqZvke in the discussion.

Lesson Synthesis

Lesson Synthesis

In this lesson, we learned that we can reason about division with fractions as we have done in division with whole numbers—by thinking in terms of equal-sized groups. We can use pattern blocks, diagrams, and equations to think about questions such as “how many \(\frac34\)s are in 6?”

  • “How do we know which number represents the size of a group, and which represents a total?” (We can often tell by the context of the problem, or by interpreting the question carefully. For the question “how many \(\frac34\)s are in 6?,” we are interested in groups of \(\frac34\)s and we have a total amount of 6.) 

  • “How do diagrams or pattern blocks help us find the answers to these questions?” (Diagrams often allow us to count or see the number of groups.)

  • “What equations can we write to represent the question ‘how many \(\frac34\)s are in 6’?” (We can start with multiplication: “there are ? groups of \(\frac34\) in 6” can be written as \(? \boldcdot \frac34 = 6\). The division equation \(6 \div \frac34 = \,?\) represents the same question.) 

4.3: Cool-down - Halves, Thirds, and Sixths (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Some problems that involve equal-sized groups also involve fractions. Here is an example: “How many \(\frac16\) are in 2?” We can express this question with multiplication and division equations. \(\displaystyle {?} \boldcdot \frac16 = 2\) \(\displaystyle 2 \div \frac16 = {?}\)

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

Four pattern blocks: One large yellow hexagon, one blue rhombus, one red trapezoid, and one green triangle.

If the hexagon represents 1 whole, then a triangle must represent \(\frac16\), because 6 triangles make 1 hexagon. We can use the triangle to represent the \(\frac 16\) in the problem.

Two figures of pattern blocks. Each figure is 6 triangles in the shape of a hexagon.

Twelve triangles make 2 hexagons, which means there are 12 groups of \(\frac16\) in 2.

If we write the 12 in the place of the “?” in the original equations, we have: \(\displaystyle 12 \boldcdot \frac16 = 2\)

\(\displaystyle 2 \div \frac16 = 12\)