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Interpret Relationships

Grade 5: Unit 7: Lesson 10

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Let’s find relationships between patterns.

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True or False:

Multiply and Divide

Warm-Up

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Mix and Match: 3 Patterns

Activity 1

MLR7: Compare and Connect

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Launch

•Groups of 2

Student Work Time

•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 Responses

a. Set A:

0, 3, 6, 9, 12

0, 6, 12, 18, 24

b. Set B:

4, 7, 10, 13, 16

9, 15, 21, 27, 33

c.

Set C:

0, 5, 10, 15, 20

0, 3, 6, 9, 12

d.

Sample responses:

Set A: Each number with rule 2 is double the corresponding number with rule 1. Each number with rule 1 is half the corresponding number with rule 2.

Set B: Each number with rule 2 is 1 more than double the corresponding number with rule 1. Each number with rule 1 is half of the corresponding number with rule 2 minus ½.

Set C: The difference between the numbers in rule 1 and the corresponding numbers in rule 2 goes up by 2 each time. Each number in rule 1 is the number in rule 2 multiplied by 1 ⅔.

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.

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Launch

•Groups of 2

Student Work Time

•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 Responses

a. Set A:

0, 3, 6, 9, 12

0, 6, 12, 18, 24

b. Set B:

4, 7, 10, 13, 16

9, 15, 21, 27, 33

c.

Set C:

0, 5, 10, 15, 20

0, 3, 6, 9, 12

d.

Sample responses:

Set A: Each number with rule 2 is double the corresponding number with rule 1. Each number with rule 1 is half the corresponding number with rule 2.

Set B: Each number with rule 2 is 1 more than double the corresponding number with rule 1. Each number with rule 1 is half of the corresponding number with rule 2 minus ½.

Set C: The difference between the numbers in rule 1 and the corresponding numbers in rule 2 goes up by 2 each time. Each number in rule 1 is the number in rule 2 multiplied by 1 ⅔.

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.

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Launch

•Groups of 2

Student Work Time

•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 Responses

a. Set A:

0, 3, 6, 9, 12

0, 6, 12, 18, 24

b. Set B:

4, 7, 10, 13, 16

9, 15, 21, 27, 33

c.

Set C:

0, 5, 10, 15, 20

0, 3, 6, 9, 12

d.

Sample responses:

Set A: Each number with rule 2 is double the corresponding number with rule 1. Each number with rule 1 is half the corresponding number with rule 2.

Set B: Each number with rule 2 is 1 more than double the corresponding number with rule 1. Each number with rule 1 is half of the corresponding number with rule 2 minus ½.

Set C: The difference between the numbers in rule 1 and the corresponding numbers in rule 2 goes up by 2 each time. Each number in rule 1 is the number in rule 2 multiplied by 1 ⅔.

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.

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Generate Patterns

Activity 2

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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.”

Student Work Time

•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 ½ times the corresponding term in pattern 1

◦recognize that each term in pattern 1 is ⅔ the corresponding term in pattern 2

Student Responses

Problem 1

a.0, 4, 8, 12, 16, 20, 24, 28, 32, 36

0, 6, 12, 18, 24, 30, 36, 42, 48, 54

b.Sample responses: Some of the same numbers are in both patterns, each number in pattern 2 will be 1 ½ times greater than the corresponding number in pattern 1. Each number in pattern 1 is ⅔ the corresponding number in pattern 2.

c.When 40 is the number for rule 1, 60 is the corresponding number for rule 2.

d.When 120 is the number for rule 2, 80 is the corresponding number for rule 1.

Problem 2

a.0, 2, 4, 6, 8, 10, 12, 14, 16, 18

0, 3, 6, 9, 12, 15, 18, 21, 24, 27

b.Sample responses: Some of the same numbers are in both patterns, each number in pattern 2 will be 1 ½ times greater than the corresponding number in pattern 1. Each number in pattern 1 is ⅔ the corresponding number in pattern 2.

c.When 30 is the number in rule 1, 45 is the corresponding number in rule 2.

d.When 60 is the number in rule 2, 40 is the corresponding number in rule 1.

Activity Synthesis

•Ask previously selected students to share their thinking.

•Display or write the numbers in the patterns for partner A.

Each number in rule 2 is the corresponding number in rule 1. Each number in rule 1 is the corresponding number in rule 2: , .

•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?”

•“How can we represent the relationship between the numbers in the patterns with multiplication equations?”

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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.”

Student Work Time

•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 ½ times the corresponding term in pattern 1

◦recognize that each term in pattern 1 is ⅔ the corresponding term in pattern 2

Student Responses

Problem 1

a.0, 4, 8, 12, 16, 20, 24, 28, 32, 36

0, 6, 12, 18, 24, 30, 36, 42, 48, 54

b.Sample responses: Some of the same numbers are in both patterns, each number in pattern 2 will be 1 ½ times greater than the corresponding number in pattern 1. Each number in pattern 1 is ⅔ the corresponding number in pattern 2.

c.When 40 is the number for rule 1, 60 is the corresponding number for rule 2.

d.When 120 is the number for rule 2, 80 is the corresponding number for rule 1.

Problem 2

a.0, 2, 4, 6, 8, 10, 12, 14, 16, 18

0, 3, 6, 9, 12, 15, 18, 21, 24, 27

b.Sample responses: Some of the same numbers are in both patterns, each number in pattern 2 will be 1 ½ times greater than the corresponding number in pattern 1. Each number in pattern 1 is ⅔ the corresponding number in pattern 2.

c.When 30 is the number in rule 1, 45 is the corresponding number in rule 2.

d.When 60 is the number in rule 2, 40 is the corresponding number in rule 1.

Activity Synthesis

•Ask previously selected students to share their thinking.

•Display or write the numbers in the patterns for partner A.

Each number in rule 2 is the corresponding number in rule 1. Each number in rule 1 is the corresponding number in rule 2: , .

•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?”

•“How can we represent the relationship between the numbers in the patterns with multiplication equations?”

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Lesson Synthesis

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

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Lesson Synthesis

What relationships did you find between the patterns we studied today?

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Jada’s and Priya’s Patterns

Cool-Down

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Cool Down: Jada’s and Priya’s Patterns

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Cool Down: Jada’s and Priya’s Patterns

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Image Citations

https://pixabay.com/photos/abstract-leaf-tree-stump-green-1846401/

Creative Commons BY-NC-SA

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