Lesson 6

Division as an Unknown Factor

Warm-up: Notice and Wonder: Missing Numbers (10 minutes)

Narrative

The purpose of this warm-up is to elicit the idea that multiplication and division are related, which will be useful as students learn to understand division as an unknown factor problem. While students may notice and wonder many things about these equations, ideas about how multiplication and division are alike and different are the important discussion points.

Students have seen division expressions, but this will be their first time seeing division equations.

Launch

• Groups of 2
• Display the equations.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

Activity

• 1 minute: partner discussion
• Share and record responses.

Student Facing

What do you notice? What do you wonder?

$$3 \times {?} = 12$$

$$12 \div 3 = {?}$$

Activity Synthesis

• “Today, we are going to work more with multiplication and division equations like these.”

Activity 1: Equations about Onions (15 minutes)

Narrative

The purpose of this activity is for students to formalize the relationship between multiplication and division equations. They see that the unknown quantity in a division situation can be represented as a missing factor in a multiplication equation or as a quotient in a division equation. The synthesis should emphasize that both equations are appropriate ways to represent a situation that involves equal groups.

This activity gives students an opportunity to make sense of each quantity and how it relates to the situation (MP2). As students discuss and justify their decisions, they share a mathematical claim and the thinking behind it (MP3).

MLR7 Compare and Connect. Synthesis: Create a visual display of the problem. “What did Lin and Mai’s approaches have in common? How were they different?” As students share their reasoning, annotate the display to illustrate connections. For example, below each equation, write the words total, number of groups, and number in each group depending on the equation and student input.

Launch

• Groups of 2
• “Where are some places that we get food?”
• 30 seconds: quiet think time
• Share responses.
• “This situation is about getting food from a farmer’s market. Farmers markets are places where people in the community come together and sell food they've grown or food they've made.”

Activity

• 3 minutes: independent work time
• Monitor for students who agree with the different equations to pair for discussion.
• 3 minutes: partner discussion
• Monitor for students who can articulate why either student is correct.

Student Facing

A farmer puts 14 onions into 2 bags, with the same number of onions in each bag.

Lin says the situation should be represented by the equation:

$$2 \times \boxed{\phantom{3}} = 14$$

Mai says the situation should be represented by the equation:

$$14 \div 2 = \boxed{\phantom{3}}$$

Whose equation do you agree with? Be ready to explain your reasoning.

Activity Synthesis

• “After discussing your ideas with your partner, who do you agree with? Explain your reasoning.” (They are both correct because the situation can be represented with a multiplication equation with a missing factor or a division equation. Both equations show the number of onions in each box as the missing number.)
• If students don’t see that both Lin and Mai are correct, consider asking, “How could both equations represent this situation?”

Activity 2: At the Farmers' Market (20 minutes)

Narrative

The purpose of this activity is for students to understand how multiplication equations correspond to diagrams and equations they have used to represent division situations. The focus should be on relating the unknown factor to the unknown number of groups or the unknown number of objects in each group. In their explanations, students should make direct connections between the situations, representations, and equations (MP2).

Engagement: Develop Effort and Persistence. Some students may benefit from feedback that emphasizes effort and time on task. For example, give feedback after each row or encourage students to work on the next row if they have difficulty with a specific row on the chart.
Supports accessibility for: Attention, Social-Emotional Functioning

Launch

• Groups of 2
• “We’re going to complete this table. Take a minute to look at what might be missing from the table.”
• 1 minute: quiet think time
• Clarify any questions students have about the situations in the table.

Activity

• “Complete each row. Be prepared to explain your reasoning.”
• 5–7 minutes: independent work time
• 2–3 minutes: partner discussion
• Monitor for different ways that students explain their responses. Listen for strategies that make clear connections between the quantities in the situation and their representations. For example, students may say the 2 plates match with the 2 groups in the drawing or the 2 in the equations represents the 2 groups in the drawing and the 2 plates in the situation.
• If students finish early, or to add movement to the activity, consider asking groups of 4 to create a poster that show their own situation and the corresponding drawing, multiplication equation, and division equation. They can then do a gallery walk.

Student Facing

Complete each row. Be prepared to explain your reasoning.

Student Response

If students don’t record a multiplication equation and division equation for each row, consider asking:
• “How did you decide what type of equation to write?”
• “How could we represent this diagram (or situation) with a multiplication (or division) equation? Where do we see each part of the equation in the diagram (or situation)?”

Activity Synthesis

• Share student responses for each row.
• Consider asking “How does the equation (or drawing) show the numbers or amounts in the situation?”
• “What relationship did you see between the multiplication equations and the division equations?” (The equations used the same numbers to represent the situation. The answer for the division equation was always one of the missing factors for the multiplication equation.)

Lesson Synthesis

Lesson Synthesis

“Today we focused on connecting multiplication and division equations that represent the same situation.”

Display: A farmer puts 14 onions into 2 bags,  with the same number of onions in each bag.

$$\displaystyle 2 \times \boxed{\phantom{3}} = 14$$

$$14 \div 2 = \boxed{\phantom{3}}$$

“The two equations here have the same parts: 2, 14, and an unknown amount. Why are they arranged differently if they represent the same situation?” (In multiplication, the factors are the number of groups and the size of each group. The number on the other side of the equation is the total amount. In division we start with the total and divide by how many groups we have to find the size of the group or we divide by the size of the group to find the number of groups we have, so that is the answer.)

“We call the result in a division equation the quotient. For example, in $$14 \div 2 = \boxed{\phantom{3}}$$, the result is unknown, so we are finding the value of the quotient. In the completed equation $$14 \div 2 = 7$$, we see that the value of the quotient is 7.”