Growing patterns Lesson 5

  • Analyze Two Patterns and Find Rules: Rows of Houses Design

    Lesson 5 of 6
    5th grade

    60 minutes


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



    Provide students with pattern blocks (triangles) and colored tiles. Let them play and get familiarized with the manipulative.

    Suggest to students to use the blocks to design a house or rocket ship.

    Explain to students that they are going to be exploring a builder's plan to develop townhouses. Using a document camera (or the like), illustrate what one townhouse looks like:

    rocket ship

    To stimulate curiosity and increase student motivation, lead the class in a brief I see... I think... I wonder... routine (see Lesson One for more detailed explanation if needed) Allow students 3-5 min to write their response prior to sharing in class. This can be done in a variety of ways:

    • journals
    • notebooks
    • sticky notes to post on chart paper
    • etc.

    When whole class sharing begins encourage students to state all three of their responses at the same time, i.e. “I see… I think… I wonder…”

    These can be recorded and posted if you choose. Engage the students in a discussion about what two townhouses will look like, and encourage them to build it.

    rocket ships

    Have students discuss with a partner what they see "growing". (SMP7) Encourage them to look for more than one thing. Allow partners to share; engage in a whole class discussion about the three things growing (the number of squares, number of triangles, and number of total pieces). Explain that today they are going to be describing the rule for how each of these quantities are growing.

    Challenge students to create a table to record all of these changing quantities:

    • House Number
    • Number of Squares
    • Number of Triangles
    • Total Number of Pattern Block Pieces


    Place students in groups of 3 and distribute the Exploring Houses Activity Sheet. Explain that they are going to organize their information in a table (activity sheet says ‘some way’). As students are working, ask questions to support student reasoning and connect representations. (SMP7) Examples include:

    • “What do you notice about how the squares and triangles are growing?”
    • “What is the relationship between the house number and the number of squares used to make the house?”
    • “What is the relationship between the house number and the number of triangles needed to make the house?”
    • “What is the relationship between the house number and the total number of pattern blocks needed to make the house?
    • “How would you find the number of total pieces needed for 15 townhouses?”
    • Differentiation: If students are ready to use variables, do the following: Guide students towards thinking of house number as “n” and then deciding the number of total number of pieces (rule is n for number of triangles, 2n for number of squares, and 3n for number of total pieces).

    After students have completed the Exploring Houses Student Page, ask students for the formula for finding how many pattern blocks are needed once they know the number of townhouses to build. Ask how they used this to answer the question for 15 houses.

    Ask students to tell what they think a graph showing the relationship between number of houses and number of pattern block pieces might look like. Have students share their predictions with a partner. Distribute graph paper, or use a graphing interactive, such as Excel.

    Note: This pattern is discrete (as opposed to continuous) and therefore there should not be a line connecting the dots (you cannot have part of a square or triangle, so the in-between answers don't make sense).

    Ask students (still in groups of 3) to each take one thing to graph:

    • Number of squares needed for each house (using an orange marker).
    • Number of triangles needed for each house (using a green marker).
    • Number of total pattern blocks needed for each house (using a black marker).

    Options for differentiation:

    • There are many, many ways to build pattern block patterns in increasingly complex ways (and other manipulatives, too!). The book Building Algebraic Thinking with Progressive Patterns is a collection of many patterns.
    • Ask students to think about whether or not the rule works backwards; i.e. what if the question posed was “How many houses could be built with 40 squares and 20 triangles?”
    • If time allows, explore another pattern block growing pattern that involves two different shapes.


    When the students are finished, ask them to discuss with their group how their graphs are similar and how they are different. (SMP3) As a whole class discuss what the meaning of any point on one of the graphs means, i.e. what two things does each point tell us?



    Create the following exit slip for students:

    • How did you find the next term in the pattern?
    • What rule did you use to generalize the pattern?
    • Where in the picture, table, and graph, do you see the rule of “times 3”?
    • What is the rule telling us? [The relationship between the number of houses and the number of pattern block pieces needed.]

    Teacher Reflection

    • Were students able to analyze and describe the rules for each piece and total pieces?
    • Were students able to graph the points and interpret the meaning of the points on the graph?
    • What other examples of growing patterns could I use in this lesson or for continued practice?
    • Did I encourage students to explain and defend their thinking?

    Leave your thoughts in the comments below.

    Related Material

    By Susan Friel, Sid Rachlin, and Dot Doyle

    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 6 of 6

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

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  • Essential Question(s)

    • How are these growing patterns changing and how is this different than the other patterns we have explored?


    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 plane.

    CCSS, Standards for Mathematical Practices

    • SMP 3 Construct viable arguments and critique the reasoning of others.
    • SMP 7 CLook for and make use of structure.

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

    • Implement tasks that promote reasoning and problem solving: In this lesson, students engage in using varied solution strategies to find the pattern of the sequence of houses and reason about the general rule.
    • Use and connect Mathematical Representations: In this lesson, the students make connections between the visual model, table and graph to deepen their understanding of patterns.
    • Facilitate meaningful mathematical discourse: The students in this lesson discuss, compare, and analyze each other's strategies to build shared understanding of patterns and discovering a general rule for the pattern.