Growing Patterns Lesson 6

  • Corresponding Relationships in Growing Patterns - Two Step Rules

    Lesson 6 of 6
    5th grade

    60-120 minutes


    Pose interesting, but more difficult-to-generalize growing patterns.



    Revisit an adapted version of Mr. Green's tree growing problem from Day 1 of this ARC:

    L dots

    Students access prior knowledge by engaging in a brief discussion about the "what comes next" rule (add two). Engage in a Think/Pair/Share activity where you ask students to think about if there is another relationship going on in this pattern (between Day and Number of Trees). Give students a minute or two to work on this independently, then have them discuss what they are thinking with a partner. After a few minutes share their ideas in whole group discussion. Once the relationship between Day and Number of Trees has been established have students generate rules they think match this growing pattern (SMP4). After a few minutes have them justify the rule they came up with to a partner (SMP8). Reconvene the whole group and ask what they think the rule is for Mr. Green's tree planting. Students may have generated the rule "double it and add 1", but if they haven't, then they should create a table, graph, or visual model.


    Two growing patterns (Happy Counting and Hexagons) are provided for this part of the lesson. These tasks can be sequenced (Smileys then Hexagons), or students can choose one, or you can give half of the class one, and half of the class the other, and they can later pair-share their rule for the relationship.

    Distribute the Happy Counting and/or Hexagon Trails Activity Sheet to students. For Happy Counting, also distribute counters; For Hexagon Trails, distribute hexagon shapes from pattern blocks. Have students work individually or in partners on their task.

    As students are working, circulate and make a note of the different counting strategies that students are using. During the discussion later you will use this evidence of student thinking as examples. (SMP8)

    • “What is the rule for ‘what comes next’?”
    • “What is changing with each new step?”
    • “What is the same with each new step?”
    • “How can you test to see if your rule is correct?”

    After students have completed their Activity Page, regroup students so that they find someone who did the SAME task, but wrote the rule differently. Tell students to each share their rule and to work together to decide if both rules are correct or not, and to be ready to tell how the rule connects to the pattern. Give students enough time for most to familiarize themselves with someone else's strategy.

    If everyone did the same task, repeat this process for the next task. If you had some students doing one, and some doing the other, have students find a partner who solved a DIFFERENT task. On each student's turn, have them explain to their partner how they know their rule works for the pattern they explored.


    As a whole class, discuss the rules for the patterns explored and help students to write the numeric or algebraic expression for the rules.

    For example, if Susan says "First we figured out that each of the hexagons on the end had 5 sides to count. All of the ones in the middle have only 4 sides. See! We have 6 hexagons in the middle and each one has 4 sides!"

    Teacher writes 5 + 4 + 4 + 4 + 4 + 5

    Ask students if there is a shorter way to write this expression. Listen for evidence that students are thinking about using multiplication to simplify the addition [5 + (4 x 4) + 5 or 4 x 4 + 10].

    • As an extension, translate these numeric expressions into algebraic expressions. In this example, that would be 4(n - 2) + 10.
    • Have students describe how these patterns (Mr. Green’s Trees, Happy Counting, and/or Hexagon Trails) are alike and different from the patterns they have done previously (they still grow each time by + 2 or some amount; these three patterns had a constant amount or starting value). This can be done informally by pairing with a partner and discussing, having students write in their math journal, or as an exit slip/ticket out the door activity.



    Revisit any of the patterns explored in Lessons 1-3, and discover the formula for your chosen growing pattern.

    Teacher Reflection

    • How could I modify this lesson (or previous ones) to better help struggling students?
    • What representations were students most comfortable using (tables, graphs, or visual models)? How could I encourage them to explore other representations?
    • What additional guiding questions could I ask to help students move from the representations to using formulas?
    • What were some misconceptions I noticed students making? How could I eliminate these?
    • How did I encourage students to explain and defend their thinking?

    Leave your thoughts in the comments below.

    Related Material

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    Mathematics teaching in the Middleschool

    Other Lessons in This Activity

    Lesson 1 of 6

    Students explore growing patterns using the actual pattern and tables and determine a rule to tell what comes next.

    Lesson 2 of 6

    Students continue to explore growing patterns and rules to determine what comes next. They analyze, describe, and justify their rules for naming patterns. Since students are likely to see growing patterns in a different way compared to their classmates, this is an opportunity to engage them in communicating about mathematics. This lesson requires students to explain correspondences among their verbal descriptions of the patterns, tables, and graphs that will help them eventually build an equation to solve the problem.

    Lesson 3 of 6

    In this lesson, students use the idea of what comes next to determine the relationship between the pattern number and number of objects in the pattern (explicit rule).

    Lesson 4 of 6

    Students explore a toothpick staircase problem to apply their skills of finding the rule to describe the relationship between corresponding terms.

    Lesson 5 of 6

    Continue to explore relationships between terms by exploring a growing pattern that involves several rules.

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  • Essential Question(s)

    • How do we use a formula to determine the nth term in a growing pattern?


    CCSS, Content Standards to specific grade/standard

    • CCSS.Math.Content.5.OA.A.2 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them.
    • CCSS.Math.Content.5.OA.B.3 Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane.

    CCSS, Standards for Mathematical Practices

    • SMP 4 Model with mathematics.
    • SMP 8 Look for and express regularity in repeated reasoning.

    PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS

    • Pose purposeful questions: The teacher's questioning is pivotal in this lesson. By delving deeply into students' counting strategies with probing questions, the teacher is able to uncover the students' different approaches to visualizing and writing the rules for these patterns.
    • Use and connect Mathematical Representations: The teacher's role in focusing students' analysis on the pattern and the table is critical in helping students see what is changing and what is staying the same.