Growing Patterns Lesson 3

  • Identify Relationships and Generate Rules: How Many Beans?

    Lesson 3 of 6
    5th grade

    60 minutes


    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).



    Project the Trinumbers Overhead.

    Revisit the I see, I think, I wonder activity to activate prior knowledge. Connect to previous day's lesson by having students draw a picture of the 5th term in this pattern. Ask students:

    • How is this like what you have been doing?
    • What can we do to figure out how many beans will be needed to build the fifth picture?
    • What pattern do you notice for what comes next?

    Explain that we have been working on figuring out how many objects will be needed for the next term, in this case the next triangle.

    • Is it possible for you to figure out how many beans for the 100th triangle?
    • What about any triangle in the sequence?

    Today we are going to explore these questions!


    Distribute the How Many Beans Student Page. Students complete the table, record the Triangle Pattern, and solve the first three questions.

    If students finish early, have them try to find another strategy for finding the number of beans for the 10th design. Students exchange their work with a partner and discuss similarities and differences in solutions and strategies for generating solutions. Engage in whole class discussion. Focus students' attention on the pattern across their tables and ask them to determine a rule for getting from the pattern number to the number of beans. Have students complete the remainder of the student activity page.

    Note: This same exploration can be done (with growing beans) that generate squares, pentagons, and other polygons.


    These other patterns can be used for differentiation (giving different groups different patterns to explore or having students choose the one they want to explore), and it can be done as a challenge for early finishers or students who need an additional challenge.

    Explore the different polygon growing patterns, as briefly mentioned above. Students can then generalize how the rules themselves are changing with each new polygon rule.



    Close the lesson by having students work with a partner. The partner to the right tells how to find the "what comes next" rule to the person on the left. Trade and have the person on the left tell the person on the right how to find the rule for the relationship between pattern number and number of beans.



    It is important to focus attention on the difference between the pattern going down the table, which is the "what comes next" pattern and the pattern or rule going across the table, which is the "relationship between corresponding terms" (in this case pattern number and number of beans). Have students make an Exit Slip demonstrating their understanding.

    • What is the rule for what comes next? [Add 3] Record it on the board. Ask where they see it in the pattern [from one design to the next] and where they see that pattern in the table [going down].
    • What is the rule for the relationship between pattern number and number of beans? [Times 3] Where do we see that in the design? [The label to the picture] Where do we see it in the table? [Across]
    • Which rule did you use to find out how many beans are needed for the 100th triangle and why?

    Teacher Reflection

    • Where did I see evidence of the representations (pictures or tables) providing support for student reasoning?
    • How did students articulate the differences of the two types of rules?
    • How did students use the rule for what comes next to find the rule for the relationship between the pattern number and the objects needed to build the design?
    • How did I encourage and support students’ explanations and reasoning?
    • How can I support struggling learners with these concepts?

    Leave your thoughts in the comments below.

    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 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.

    Lesson 6 of 6

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

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  • Essential Question(s)

    • How can we use the rule for what comes next to find the rule for the relationship between the pattern number and the objects needed to build the design?


    CCSS, Content Standards to specific grade/standard

    • 5.OA.B.3 Analyze patterns and relationships.
      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 axis.

    CCSS, Standards for Mathematical Practices

    • SMP 7 Look for and make use of structure.

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

    • Implement tasks that promote reasoning and sense making. Students reason about both the "what comes next" rule and the "relationship" rule, as well as reason about how these rules compare to each other.