Lesson 2

The purpose of this warm-up is to review students' prior understanding of division and elicit the ways in which they interpret a division expression. This review prepares them to explore the meanings of division in the lesson. 

Some students may simply write the value of the expression because they struggle to put into words how they think about the problem. Encourage them to think of a story with a question, in which the expression could be used to answer the question.

Arrange students in groups of 2. Ask students to write a list of all of the ways they think about \(20\div 4\). Explain that they can write what the expression means to them, how they think about it when evaluating the expression, or a situation that matches the expression. 

Give students 1 minute of quiet think time, followed by 1 minute of partner discussion. During discussion, ask students to share their responses and notice what they have in common. 

Invite partners to share the interpretations of \(20 \div 4\) that they had in common. Record and display these responses for all to see. Ask students to notice any themes or trends in the range of responses.

Highlight the two ways students will be thinking about division in this unit:

• Division means partitioning a number or a quantity into equal groups and finding out how many groups can be made.  
• Division means partitioning a number or a quantity into equal groups, and finding out how much is in each group.

This activity prompts students to explore two ways of thinking about division by connecting it to multiplication, thinking about what it means in the context of a situation, and drawing visual representations.

Keep students in groups of 2. Ask students to keep their materials closed. Display the following question for all to see:

A baker has 12 pounds of almonds. She puts them in bags, so that each bag has the same weight.  In terms of pounds and bags of almonds, what could \(12 \div 6\) mean?

Give students a minute of quiet think time and 1–2 minutes to explain their thinking to their partner. Ask a few students who interpreted the expression differently to share their interpretations. If students do not bring up one of the two ways to interpret the 6, ask them about it: Could the 6 represent the number of bags (or the amount in each bag)?

Once students see that the divisor could be interpreted in two ways, ask students to open the materials and give students 4–5 minutes to complete the first question.

Reconvene as a class afterwards. Select a couple of students to explain Clare and Tyler's diagrams and equations. Highlight that, in this context, \(12 \div 6\) could mean 12 pounds of almonds being divided equally into 6 bags, or 12 pounds of almonds being divided so that each bag has 6 pounds. 

Give students quiet time to complete the rest of the activity.

Select a few students to share their diagrams and equations for the problems in the last question. After each explanation, highlight the connections between the expression, the diagram, and the context. Make sure students understand that the division expression \(12 \div 6\) can be interpreted as the answer to the question “6 times what number equals 12?” or the question, “What number times 6 equals 12?" (or "How many 6s are in 12?"). More generally, division can be interpreted as a way to find two values: 

• The size of each group when we know the number of groups and a total amount
• How many groups are in a total amount given the size of one group

Note that students may write either \(\text{___} \boldcdot 6 = 12\) or \(6 \boldcdot  \text{___} = 12\) for each interpretation as long as they understand what each factor represents. Because we tend to say “___ groups of ___” in these materials, we follow that order in writing the multiplication: \(\displaystyle \text{(number of groups)} \boldcdot \text{(size of each group)} = \text{total amount}\)

When discussing \(12 \div 2\), make explicit how its multiplication equations and diagram connect to those of \(12 \div 6\) in the first question. Students may see that the diagrams for \(2 \boldcdot  \text{___} = 12\) and \(\text{___} \boldcdot 6 = 12\) are partitioned the same way. Point out that:

• In \(2 \boldcdot  \text{___} = 12\), the size of each group (each bag) was unknown, but because there are 2 equal groups in 12, we concluded that there were 6 pounds in each group.
• In \(\text{___} \boldcdot 6 = 12\), we know each group (each bag) has 6 pounds of almonds, so there must be 2 groups of 6 in 12 pounds.

This discussion will be helpful in upcoming work, as students use their understanding of representations of division to divide fractions.

This activity allows students to draw diagrams and write equations to represent simple division situations. Some students may draw concrete diagrams; others may draw abstract ones. Any diagrammatic representation is fine as long as it enables students to make sense of the relationship between the number of groups, the size of a group, and a total amount.

The last question is likely more challenging to represent with a diagram. Because the question asks for the number of jars, and because the amount per jar is a fraction, students will not initially know how many jars to draw (unless they know what \(6\frac34 \div \frac34\) is). Suggest that they start with an estimate, and as they reason about the problem, add jars to (or remove jars from) their diagram as needed. 

As students work, monitor for the range of diagrams that students create. Select a few students whose work represent the range of diagrams to share later.

Students in digital classrooms can use an applet to make sense of the problems, but it is still preferable that they create their own diagrams. 

Select previously identified students to share their diagrams, sequenced from the more concrete (e.g., pictures of jars and cups) to the more abstract (e.g, rectangles, tape diagrams). Display the diagrams and equations for all to see. Ask them how they used the diagrams to answer the questions (if at all).

If tape diagrams such as the ones here are not already shown and explained by a student, display them for all to see. Help students make sense of the diagrams and connecting them to multiplication and division by discussing questions such as these:

• “In each diagram, what does the ‘?’ represent?” (The unknown amount)
• “What does the length of the entire tape represent?” (The total amount, which is sometimes known.)
• “What does each rectangular part represent?” (One jar)
• “What does the number in each rectangle represent?” (The amount in each jar)
• “How do the three parts of each multiplication equation relate to the diagram?” (The first factor refers to the number of rectangles. The second factor refers to the amount in each rectangle. The product is the total amount.) 
• “The last diagram doesn't represent all the jars and shows a question mark in the middle of the tape. Why might that be?” (The diagram shows an unknown number of jar, which was the question to be answered.)

Highlight that the last two situations can be described with division: \(7\frac12 \div 5\) and \(6\frac34 \div \frac34\).