Lesson 9

Generate Patterns

Warm-up: Choral Count: Two Patterns (10 minutes)

Narrative

The purpose of this Choral Count is to invite students to notice patterns and relationships in two different counts. These understandings help students develop fluency with multiples and will be helpful when students identify relationships between corresponding terms in two patterns in the next several lessons.

Launch

  • Count by 6, starting at 0.
  • Record as students count.
  • Stop counting and recording at 60.

Activity

  • “What patterns do you see?”
  • 1–2 minutes: quiet think time
  • Record responses.
  • “Now count by 12s, starting at 0.”
  • Record as students count.
  • Stop counting and recording at 120.
  • “What patterns do you see?”
  • 1–2 minutes: quiet think time
  • Record responses.

Student Response

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Activity Synthesis

  • “If we continue counting, what numbers will be in both counts?” (All of the numbers on the second list, the ones where we count by 12)

Activity 1: What’s the Pattern? (20 minutes)

Narrative

The purpose of this activity is for students to generate two different patterns, given two different rules, and recognize relationships between corresponding terms (MP7). Students may notice a variety of relationships between the two patterns and may describe them generally (all of the numbers in one pattern are contained in the other pattern) or more specifically (the numbers of the second pattern are every other number from the first pattern). To answer the questions about corresponding terms in the patterns, students may continue the patterns or may use the relationship between the patterns. The number 192 is deliberately chosen to encourage using the relationship between the patterns.

When students find and explain patterns related to the rules and relationships, they look for and express regularity in repeated reasoning (MP8).

This activity uses MLR2 Collect and Display. Advances: Conversing, Reading, Writing. 

Action and Expression: Internalize Executive Functions. Synthesis: Invite students to plan a strategy, including the tools they will use, for identifying apparent relationships between corresponding terms in patterns. If time allows, invite students to share their plan with a partner before they begin.
Supports accessibility for: Conceptual Processing, Memory

Launch

  • Groups of 2

Activity

  • 1–2 minutes: quiet think time
  • 10–12 minutes: partner work time
MLR2 Collect and Display
  • Circulate, listen for and collect the language students use to describe the relationships between Jada's and Priya’s rules. Listen for: double, half of, twice as much, times 2, divided by 2.
  • Record students’ words and phrases on a visual display and update it throughout the lesson. Monitor for students who notice that:
    • all of Priya’s numbers are in Jada’s pattern, but not all of Jada’s numbers are in Priya’s pattern
    • each of Priya’s numbers is double the corresponding number in Jada’s pattern
    • each of Jada’s numbers is half the corresponding number in Priya’s pattern

Student Facing

  1. Jada and Priya create patterns using rules. Use each rule to complete their patterns.

    Jada’s rule: start with 0 and keep adding 4.

    rectangle partitioned vertically into 10 equal size rectangles
    Priya’s rule: start with 0 and keep adding 8.
    rectangle partitioned vertically into 10 equal size rectangles
  2. If Jada and Priya continue their patterns, what number will be in Priya's pattern when Jada's pattern has:

    1. 40
    2. 60
  3. If Jada and Priya continue their patterns, what number will be in Jada’s pattern when Priya’s pattern has 192?
  4. What relationships do you notice between the numbers in Priya's pattern and the numbers in Jada's pattern?

Student Response

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Advancing Student Thinking

If students are not able to identify a relationship between the two different patterns, ask:

  • “What is the same about the numbers in the patterns?”
  • “What is different about the numbers in the patterns?”

Activity Synthesis

  • “Are there any other words or phrases that are important to include on our display?”
  • As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
  • Remind students to borrow language from the display as needed.
  • “Are all of Priya’s numbers in Jada’s pattern? How do you know?” (Yes. There are 2 fours in 8 so each of Priya’s numbers appears twice as far out in Jada’s list.)
  • “What relationships do you see between Jada's pattern and Priya’s pattern?” (Each number in Priya’s pattern is double the corresponding number in Jada’s pattern. Each number in Jada’s pattern is half the corresponding number in Priya’s pattern.)

Activity 2: More Patterns (15 minutes)

Narrative

The purpose of this activity is for students to practice generating two different patterns from two different rules and observe and quantify relationships between corresponding terms. Both sets of rules generate patterns that have the same relationships between corresponding terms. Each of Priya’s terms is 3 times greater than each of Jada’s corresponding terms and each of Jada’s terms is \(\frac{1}{3}\) of Priya’s corresponding terms.

When students find and explain patterns related rules and relationships, they look for and express regularity in repeated reasoning (MP8).

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 minutes: independent work time
  • 3 minutes: partner discussion
  • “Look back at your work and make any revisions based on what you learned from discussing with your partner.”
  • 1–2 minutes: independent work time
  • Monitor for students who:
    • revise their thinking based on partner discussion
    • use multiplication expressions to represent the relationships between the patterns

Student Facing

Partner A

  1. Use each rule to create a pattern.

    Jada’s rule: Start at 0. Keep adding 2.

    rectangle partitioned vertically into 10 equal sized rectangle

    Priya’s rule: Start at 0. Keep adding 6.

    rectangle partitioned vertically into 10 equal sized rectangle
  2. If the patterns continue:

    1. What number will be in Priya’s pattern when Jada’s pattern has 34? Explain or show your reasoning.
    2. What number will be in Jada’s pattern when Priya’s pattern has 120? Explain or show your reasoning.
  3. What relationships do you notice between the numbers in Priya's pattern and the numbers in Jada's pattern?

    Partner B

  4. Use each rule to create a pattern.

    Jada’s rule: Start at 0. Keep adding 3.

    rectangle partitioned vertically into 10 equal sized rectangle
    Priya’s rule: Start at 0. Keep adding 9.
    rectangle partitioned vertically into 10 equal sized rectangle
  5. If the patterns continue,

    1. What number will be in Priya’s pattern when Jada’s pattern has 54? Explain or show your reasoning.
    2. What number will be in Jada’s pattern when Priya’s pattern has 198? Explain or show your reasoning.
  6. What relationships do you notice between the numbers in Priya’s pattern and the numbers in Jada’s pattern?

Student Response

For access, consult one of our IM Certified Partners.

Activity Synthesis

  • Ask previously identified students to share their thinking.
  • Display completed patterns from both sets of rules.
  • “How are the patterns from the pairs of rules the same? How are they different?” (Some of the numbers are in all four patterns. Some numbers aren’t in any pattern like 5 and 11. In both sets of rules, Priya’s numbers are 3 times greater than Jada’s numbers and Jada’s numbers are \(\frac{1}{3}\) Priya’s numbers.)
  • “How would you describe the relationship between Priya's pattern and Jada's pattern?” (Priya's numbers are all 3 times as much as Jada's corresponding numbers.)
  • “What can I multiply the numbers in Priya’s pattern by to get the corresponding numbers in Jada’s pattern?” (\(\frac{1}{3}\), each number in Jada's pattern is \(\frac{1}{3}\) the corresponding number in Priya's pattern.)

Lesson Synthesis

Lesson Synthesis

“Today we generated patterns and recognized relationships between two different patterns.”

Display or rewrite the numbers from the 2 choral counts in the warm-up.

“What relationships do you notice between the patterns in the counts?” (Each number in the second count is twice the matching number in the first count. Each number in the first count is one half the matching number in the second count. All the numbers on the second list are also on the first list. The second list is every other number of the first list.)

“Explain to your partner why these relationships will continue if the patterns continue.” (The second pattern adds 12 each time and that's twice what the first pattern adds each time.)

Cool-down: Patterns and Relationships (5 minutes)

Cool-Down

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