# Lesson 8

Relate Quotients to Familiar Products

## Warm-up: Number Talk: Multiplication and Division (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for multiplying and dividing within 100. These understandings help students develop fluency and identify division facts that are related to known products.

When students use the relationship between multiplication and division to find division facts they don’t know, they are looking for and making use of structure (MP7).

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$4 \times 10$$
• $$40 \div 4$$
• $$40 \div 10$$
• $$60 \div 6$$

### Activity Synthesis

• “How do the first 3 expressions show that multiplication and division are related?” (The two division expressions both have one of the factors missing. In the first expression, the 10 is missing. In the second, the 4 is missing.)

## Activity 1: Card Sort: Multiplication (20 minutes)

### Narrative

The purpose of this activity is for students to check-in on their progress towards fluent multiplication within 100. Students work in groups of 2 to sort products into groups they know right away, can find quickly, or don’t know yet. The launch provides time for a class discussion about what it means to know a fact quickly. Students identify five products with which they’d like to be more proficient, share their strategies, and practice finding the products they choose. The cards from this activity will be used in the next activity.

MLR8 Discussion Supports. Synthesis: Display a sentence frame to support whole-class discussion: “The next time I multiply _____ and _____, I will . . . .”
Representation: Internalize Comprehension. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization

### Required Materials

Materials to Copy

• Card Sort: Multiplication
• Card Sort: Multiplication Recording Sheet

### Required Preparation

• Create a set of cards from the blackline master for each group of 2.
• The Multiplication Fact sort cards from this activity will be used again in the next activity.

### Launch

• Groups of 2
• “Today we’re going to revisit the multiplication facts to see how many you’ve learned so far. Remember, though, you have the rest of the year to learn them.”
• “We all know what it means to know a product right away, but what does it mean to know a product quickly?” (We can figure it out in a couple of seconds with a strategy. We can figure it out in less than 5 seconds.)
• Discuss as a class and come to an agreement about what it means to find a product quickly.
• “Does anyone want to add on to what _____ says it means to find a product quickly?”
• “Does anyone have different ideas about what it means to find a product quickly?”
• “Based on this discussion, does anyone want to revise their ideas about what it means to find a product quickly?”
• Give each group one set of pre-cut cards and a sort table.

### Activity

• “Take some time to quiz each other on multiplication facts. As you quiz your partner, use the table to sort the expressions into three groups that show if they know it right away, they can find it quickly, or they don’t know it yet.”
• 7–10 minutes: partner work time
• “Choose 5 multiplication facts that you don’t know yet and write down the expressions. These are the products you will practice finding.”
• 1 minute: independent work time
• “Now, share the products you want to practice with your partner and have them help you think of some strategies you could use to find the products quickly.”
• “After you have some strategies, take some time to practice finding the products you chose.”
• 5–7 minutes partner work time

### Student Facing

Quiz your partner on their multiplication facts. Sort your partner’s facts into one of these columns:

1. know it right away
2. can find it quickly
3. don’t know it yet

Multiplication expressions I’m going to practice:

### Student Response

If students don’t yet have a strategy for one of the facts they’ve chosen to practice, consider asking:
• “What have you tried so far to find this product?”
• “Could you check in with another group to see if they could suggest a strategy for finding this product?”

### Activity Synthesis

• “What were some useful strategies for finding products you didn’t know yet?” (Thinking of a product I already know and using that product to find the one I didn’t know yet. Using products of 2, 5, and 10 to figure out other products.)

## Activity 2: If I Know, Then I Know (15 minutes)

### Narrative

The purpose of this activity is for students to identify division facts that are related to multiplication facts that they know. Students complete “If I know, then I know” statements using their multiplication fact cards from the previous activity. Give students time, if needed, to determine the product before generating the related division equation. Some students may generate 4 related division equations for each product by moving the quotient to the left side of the equal sign. If this comes up, recognize that this is possible, but keep the emphasis on generating two related division facts, one for each of the factors as the unknown number.

When students use the relationship between multiplication and division identify two division facts from a multiplication fact, they look for and make use of structure (MP7).

### Required Preparation

• Each group of 2 needs a set of cards from the previous activity.

### Launch

• Groups of 2
• “Read the first statement of the activity. Talk with your partner about how you could finish the statement.”
• 1 minute: partner discussion
• Share responses.

### Activity

• “Now, you’re going to take turns drawing a card and using the fact you chose to complete an ‘If I know, then I know’ statement with the multiplication fact that you drew and the related division facts. Take some time to figure out the multiplication fact together if you need to. After every turn, record the multiplication equation and related division equations in the table.”
• 7–10 minutes: partner work time

### Student Facing

If I know $$4 \times 5 = 20$$, then I know _____.

1. Set the multiplication fact cards in a stack face down.
2. Take turns drawing a multiplication fact card.
3. Use the multiplication fact on the card to record a multiplication equation in the “If I know . . .” column.
4. Then, record related division equations in the “Then I know . . .” column.

If I know . . . , then I know . . .

### Activity Synthesis

• “How many division equations were you able to come up with for each multiplication equation? Explain your reasoning.” (2, because I could come up with 1 equation where the quotient was 1 factor and 1 equation where it was the other factor. 4, because I could come up with 1 for each of the factors being the quotient, and I could have the quotient on the right or the left of the equal sign.)

## Lesson Synthesis

### Lesson Synthesis

“Today we thought about multiplication facts that we know and worked on some that we don’t know yet. How did this help you with finding division facts?” (We could use a multiplication fact to find related division facts. We realized that if we know a multiplication fact, then there are some division facts that we know too.)