Growing patterns Lesson 5
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:
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:
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.
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:
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:
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:
Options for differentiation:
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:
Leave your thoughts in the comments below.
Students explore growing patterns using the actual pattern and tables and determine a rule to tell what comes next.
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.
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).
Students explore a toothpick staircase problem to apply their skills of finding the rule to describe the relationship between corresponding terms.
Pose interesting, but more difficult-to-generalize growing patterns.
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS