# Lesson 2

Meanings of Division

## 2.1: A Division Expression (5 minutes)

### Warm-up

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.

### Launch

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.

### Student Facing

Here is an expression: $$20\div 4$$.

What are some ways to think about this expression? Describe at least two meanings you think it could have.

### Activity Synthesis

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.

## 2.2: Bags of Almonds (25 minutes)

### Activity

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.

Teacher Notes for IM 6–8 Accelerated
Adjust this activity to 10 minutes.

### Launch

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.

Representation: Internalize Comprehension. Activate or supply background knowledge. Provide students with access to blank tape diagrams. Encourage students to annotate diagrams with details to show how each value is represented—for example number of pounds of almonds in total, number of pounds in one bag, or number of bags of almonds.
Supports accessibility for: Visual-spatial processing; Organization

### Student Facing

A baker has 12 pounds of almonds. She puts them in bags, so that each bag has the same weight.

Clare and Tyler drew diagrams and wrote equations to show how they were thinking about $$12 \div 6$$.

1. How do you think Clare and Tyler thought about $$12 \div 6$$? Explain what each diagram and the parts of each equation could mean about the situation with the bags of almonds. Make sure to include the meaning of the missing number.

Pause here for a class discussion.

2. Explain what each division expression could mean about the situation with the bags of almonds. Then draw a diagram and write a multiplication equation to show how you are thinking about the expression.

1. $$12 \div 4$$

2. $$12 \div 2$$

3. $$12 \div \frac12$$

### Student Facing

#### Are you ready for more?

A loaf of bread is cut into slices.

1. If each slice is $$\frac12$$ of a loaf, how many slices are there?
2. If each slice is $$\frac15$$ of a loaf, how many slices are there?
3. What happens to the number of slices as each slice gets smaller?
4. What would dividing by 0 mean in this situation about slicing bread?

### Activity Synthesis

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.

Speaking: MLR8 Discussion Supports. Use this routine to support whole-class discussion. For each diagram and expression that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they hear to a partner before selecting one or two students to share with the class. Ask the original speaker if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making

## 2.3: Homemade Jams (20 minutes)

### Activity

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.

### Launch

Arrange students in groups of 2. Tell the class that you will read the three story problems, and ask them to be prepared to share at least one thing they notice and one thing they wonder. After reading, give them a minute to share their observation and question with their partner.

Clarify that their job is to draw a diagram and write a multiplication equation to express the relationship in each story and then answer the question. Give students 7–8 minutes of quiet work time, followed by 2–3 minutes to share their work with their partner.

If the applet is used to complete the activity or for class discussion, note that the toolbar includes colored rectangles that represent fractional parts. Encourage students to drop the fractional parts in the work space on the left of the window, and then use the Move tool (the arrow) to drag them into the jars. The blocks do snap, but the grid is very fine so this may be challenging for some students. Troubleshooting tip: if two blocks get stuck together, delete them. Try not to overlap blocks when adding them to the work space.

### Student Facing

Draw a diagram and write a multiplication equation to represent each situation. Then answer the question.

1. Mai had 4 jars. In each jar, she put $$2\frac14$$ cups of homemade blueberry jam. Altogether, how many cups of jam are in the jars?
2. Priya filled 5 jars, using a total of $$7\frac12$$ cups of strawberry jam. How many cups of jam are in each jar?
3. Han had some jars. He put $$\frac34$$ cup of grape jam in each jar, using a total of $$6\frac34$$ cups. How many jars did he fill?

Here is an applet to use if you choose to.

The toolbar includes buttons that represent 1 whole and fractional parts, as shown here. Click a button to choose a quantity, and then click in the work space of the applet window to drop it. When you're done choosing pieces, use the Move tool (the arrow) to drag them into the jars. You can always go back and get more pieces, or delete them with the Trash Can tool.

The jars in this applet are shown as stacked to make it easier to combine the jam and find out how much you have.

Here are the questions again.

1. Mai had 4 jars. In each jar, she put $$2\frac14$$ cups of homemade blueberry jam. Altogether, how many cups of jam are in the jars?
2. Priya filled 5 jars, using a total of $$7\frac12$$ cups of strawberry jam. How many cups of jam are in each jar?
3. Han had some jars. He put $$\frac34$$ cup of grape jam in each jar, using a total of $$6\frac34$$ cups. How many jars did he fill?

### Launch

Arrange students in groups of 2. Tell the class that you will read the three story problems, and ask them to be prepared to share at least one thing they notice and one thing they wonder. After reading, give them a minute to share their observation and question with their partner.

Clarify that their job is to draw a diagram and write a multiplication equation to express the relationship in each story and then answer the question. Give students 7–8 minutes of quiet work time, followed by 2–3 minutes to share their work with their partner.

If the applet is used to complete the activity or for class discussion, note that the toolbar includes colored rectangles that represent fractional parts. Encourage students to drop the fractional parts in the work space on the left of the window, and then use the Move tool (the arrow) to drag them into the jars. The blocks do snap, but the grid is very fine so this may be challenging for some students. Troubleshooting tip: if two blocks get stuck together, delete them. Try not to overlap blocks when adding them to the work space.

### Student Facing

Draw a diagram, and write a multiplication equation to represent each situation. Then answer the question.

1. Mai had 4 jars. In each jar, she put $$2\frac14$$ cups of homemade blueberry jam. Altogether, how many cups of jam are in the jars?
2. Priya filled 5 jars, using a total of $$7\frac12$$ cups of strawberry jam. How many cups of jam are in each jar?
3. Han had some jars. He put $$\frac34$$ cup of grape jam in each jar, using a total of $$6\frac34$$ cups. How many jars did he fill?

### Activity Synthesis

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 $$6\frac34 \div \frac34$$.

Representing: MLR8 Discussion Supports. Use this routine to support whole-class discussion. As students discuss the questions listed in the Activity Synthesis, label the display of the diagrams and equations accordingly. Annotate the display to illustrate connections between equivalent parts of each representation. For example, next to each question mark, write “unknown amount.”
Design Principle(s): Support sense-making

## Lesson Synthesis

### Lesson Synthesis

In this lesson, we explored the relationship between multiplication and division in order to understand the meanings of division. We know that multiplication can represent the number of equal-size groups. For instance, $$3 \boldcdot 5 = 15$$ can mean 3 groups of 5 make 15. Let's review how we can use the same idea of equal-size groups to think about division.

• "How can we interpret $$20 \div 8$$?" (We can think of it as "how many groups of 8 are in 20?" or "how much is in each group if there are 20 in 8 groups?)

• "Suppose we interpret it as 'how many groups of 8 are in 20?'. How might we draw a diagram to show this?" (A bar that represents 20 divided into equal parts of 8.) "What multiplication equation can we write?" ($${?} \boldcdot 8 = 20$$ or $$8 \boldcdot {?} = 20$$, as long as we are clear what each factor represents.)

• "If we think of it as 'how much is in each group if there are 20 in 8 groups?', how would the diagram be different?" (A bar that represents 20 divided into 8 equal parts.) "What multiplication equation can we write?" ($$8 \boldcdot {?} = 20$$ or $${?} \boldcdot 8 = 20$$, as long as we know what each factor represents.)

## Student Lesson Summary

### Student Facing

Suppose 24 bagels are being distributed into boxes. The expression $$24 \div 3$$ could be understood in two ways:

• 24 bagels are distributed equally into 3 boxes, as represented by this diagram:
• 24 bagels are distributed into boxes, 3 bagels in each box, as represented by this diagram:

In both interpretations, the quotient is the same ($$24 \div 3 = 8$$), but it has different meanings in each case. In the first case, the 8 represents the number of bagels in each of the 3 boxes. In the second, it represents the number of boxes that were formed with 3 bagels in each box.

These two ways of seeing division are related to how 3, 8, and 24 are related in a multiplication. Both $$3 \boldcdot 8$$ and $$8 \boldcdot 3$$ equal 24.

• $$3 \boldcdot 8 =24$$ can be read as “3 groups of 8 make 24.”
• $$8 \boldcdot 3 = 24$$ can be read as “8 groups of 3 make 24.”

If 3 and 24 are the only numbers given, the multiplication equations would be: $$\displaystyle 3 \boldcdot {?} =24$$ $$\displaystyle {?} \boldcdot 3 =24$$

In both cases, the division $$24 \div 3$$ can be used to find the value of the “?”  But now we see that it can be interpreted in more than one way, because the “?” can refer to the size of a group (as in “3 groups of what number make 24?”), or to the number of groups (as in “How many groups of 3 make 24?”).

• Next, suppose we have 20 ounces of water to fill 6 equal-sized bottles, and the amount in each bottle is not given. Here we have 6 groups, an unknown amount in each, and a total of 20. We can represent it like this:

This situation can also be expressed using multiplication, but the unknown is a factor, rather than the product: $$\displaystyle 6\, \boldcdot {?} = 20$$

To find the unknown, we cannot simply multiply, but we can think of it as a division problem: $$\displaystyle 20 \div 6 = \,?$$

• Now, suppose we have 40 ounces of water to pour into bottles, 12 ounces in each bottle, but the number of bottles is not given. Here we have an unknown number of groups, 12 in each group, and a total of 40.

Again, we can think of this in terms of multiplication, with a different factor being the unknown: $$\displaystyle ? \boldcdot 12 = 40$$

Likewise, we can use division to find the unknown: $$\displaystyle 40 \div 12 = \,?$$

Whenever we have a multiplication situation, one factor tells us how many groups there are, and the other factor tells us how much is in each group.

Sometimes we want to find the total. Sometimes we want to find how many groups there are. Sometimes we want to find how much is in each group. Anytime we want to find out how many groups there are or how much is in each group, we can represent the situation using division.