Lesson 4

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.

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

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?”.

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?”

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 \(\frac 12\)?”

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.

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.