# Lesson 10

Interpret Relationships

## Warm-up: True or False: Multiply and Divide (10 minutes)

### Narrative

The purpose of this True or False is for students to demonstrate understandings they have for the relationship between multiplication and division. Students will use this understanding in the lesson when they recognize multiplicative relationships between patterns.

### Launch

• Display one statement.
• “Give me a signal when you know whether the statement is true and can explain how you know.”
• 1 minute: quiet think time

### Activity

• Share and record answers and strategy.
• Repeat with each problem.

### Student Facing

Decide if each statement is true or false. Be prepared to explain your reasoning.

• $$276 \div 3 = \frac{1}{3} \times 276$$
• $$276 \div 3 = \frac{276}{6}$$
• $$(276 \div 3)\times2=\frac{2}{3}\times 276$$

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “How can the relationship between multiplication and division help you justify your reasoning?” (I used the fact that dividing by 3 is the same as multiplying by $$\frac{1}{3}$$.)

## Activity 1: Mix and Match: 3 Patterns (20 minutes)

### Narrative

The purpose of this activity is for students to practice interpreting relationships between patterns generated from two different rules. Students may need to generate patterns beyond the boxes provided. Encourage them to continue the pattern as needed. Students may describe the patterns and relationships in different, but accurate ways. Encourage them to notice as many relationships as they can and describe the relationships in whatever way makes sense to them. This is the first time students see two patterns where the numbers in one pattern are not multiples of the numbers in the other pattern.

This activity uses MLR7 Compare and Connect. Advances: Representing, Conversing.

• Groups of 2

### Activity

• 5 minutes: independent work time
• 2 minutes: partner discussion

MLR7 Compare and Connect

• “Create a visual display that shows your thinking about the relationships between each set of patterns. Include details such as notes, diagrams, and drawings to help others understand your thinking.”
• 2–5 minutes: independent or group work
• 3–5 minutes: gallery walk

### Student Facing

1. Complete the patterns for each set of rules.
2. What are some relationships between the patterns for each set of rules? Be prepared to explain your thinking.

Set A

Rule 1: Start at 0. Keep adding 3.

Rule 2: Start at 0. Keep adding 6.

Set B

Rule 1: Start at 4. Keep adding 3.

Rule 2: Start at 9. Keep adding 6.

Set C

Rule 1: Start at 0. Keep adding 5.

Rule 2: Start at 0. Keep adding 3.

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• “What is the same and what is different in the way we represented the relationships between the patterns?” (Some of us used numbers and symbols, some of us wrote sentences.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• “What is the same and what is different about Set A and Set B?” (They both add the same amount, but one set starts at 0. Set B has a lot more odd numbers in it. In set A, each number in rule 2 is double the number in rule 1, but in set B, each number in rule 2 is 1 more than double the number in rule 1.)
• “For which pair of rules was it most challenging to notice and describe relationships?” (Set B because there was no multiple that worked to get from one pattern to the other. Set C because there weren’t doubles or halves.)

## Activity 2: Generate Patterns (15 minutes)

### Narrative

The purpose of this activity is for students to interpret more complex relationships in corresponding terms from patterns generated from two different rules. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of the terms in rule 2 is $$\frac{3}{2}$$ times greater than each of the corresponding terms in rule 1 and each of the terms in rule 1 is $$\frac{2}{3}$$ the corresponding term in rule 2. Some students may state these relationships in different ways. The relationships between these patterns build directly on the third pair of rules from the previous activity where the terms for rule 1 were $$\frac{5}{3}$$ the corresponding terms in rule 2.

Representation: Internalize Comprehension. Synthesis: Use multiple examples and non-examples to emphasize the relationship between corresponding terms in patterns.
Supports accessibility for: Attention, Memory, Conceptual Processing

### Launch

• Groups of 2
• “You and your partner will each complete some problems about patterns independently. After you’re done, discuss your work with your partner.”

### Activity

• 5–7 minutes: independent work time
• 3–5 minutes: partner discussion
• “Look back at your work and make any revisions based on what you learned from your discussion.”
• 1–2 minutes: independent work time
• Monitor for students who:
• recognize that each term in pattern 2 is $$1\frac{1}{2}$$ times the corresponding term in pattern 1
• recognize that each term in pattern 1 is $$\frac{2}{3}$$ the corresponding term in pattern 2

### Student Facing

Partner A

1. Generate patterns for the two rules.

2. Compare your patterns. What relationships do you notice?
3. What number will be in pattern 2 when the number in the pattern 1 box is 40?
4. What number will be in pattern 1 when the number in the pattern 2 box is 120?

Partner B

1. Generate patterns for the two rules.

2. Compare your patterns. What relationships do you notice?
3. What number will be in pattern 2 when the number in the pattern 1 box is 30?
4. What number will be in pattern 1 when the number in the pattern 2 box is 60?

### Student Response

For access, consult one of our IM Certified Partners.

### Activity Synthesis

• Ask previously selected students to share their thinking.
• Display or write the numbers in the patterns for partner A.
• “How can we represent the relationship between the numbers in the patterns with multiplication equations?” (Each number in rule 2 is $$\frac{3}{2}$$ the corresponding number in rule 1. Each number in rule 1 is $$\frac{2}{3}$$ the corresponding number in rule 2:  $$6=\frac{3}{2}\times 4$$, $$4= \frac{2}{3}\times 6$$.)
• Display or write the numbers in the patterns for partner B.
• “How can we represent the relationships between the numbers in the patterns with multiplication equations?” ($$3=1\frac{1}{2}\times 2$$ , $$2=\frac{2}{3}\times 3$$.)

## Lesson Synthesis

### Lesson Synthesis

“Today we noticed and explained relationships between patterns. Some of the relationships involved fractions.”

“What relationships did you find between the patterns we studied today?” (Sometimes I could multiply each term in one pattern by the same number to get the corresponding number in the other pattern. Sometimes that number was a fraction.)

Consider asking students to record their response in a math journal and then share their response with a partner.

## Cool-down: Jada’s and Priya’s Patterns (5 minutes)

### Cool-Down

For access, consult one of our IM Certified Partners.