Growing Patterns Lesson 4
Students explore a toothpick staircase problem to apply their skills of finding the rule to describe the relationship between corresponding terms.
Have students reflect on the different geometric growing patterns they have explored in the last few days (Mr. Green's tree planting, garden toothpick patterns, bean triangles, etc.). Ask students to recall what two types of rules have been explored. As students share, ask follow up questions about what strategies they used to find that rule. Explain to students that another name mathematicians often use for these "rules" is formula, which is a set of instructions for creating a desired result.
Explain that today they are going to be exploring a new toothpick pattern -- a more challenging pattern -- and using what they know to eventually determine the rule for the relationship between the shape number and the number of toothpicks needed.
Distribute toothpicks. Project the following image (fig. 1) and ask students to build it. Ask students what the perimeter is. [P = 4 units.]
Repeat for the second image in (fig. 2). [P = 8 units.]
Be sure that students are not counting the toothpicks in the middle when they are finding the perimeter.
A strategy for keeping track is to place a hash mark on each side counted.
Explain to students that they are going to be exploring the perimeter of the staircase as it continues to grow. Encourage them to look for the many patterns that are part of this growing staircase.
Place students with a partner. Distribute the Building with Toothpicks Activity Sheet and coordinate axis graph paper (alternatively, you can ask students to draw coordinate axis on plain graph paper). Engage students in a brief discussion about what strategies they could use to explore this pattern [create a visual model, make a table, create a graph]. Have students select a strategy and complete the page. On Part Two of the sheet, remind students about the rule/formula discussion earlier.
Allow them to add in the word rule above the word formula if they want.
As students are working to generalize the pattern, make sure they see the growth in the shape and in the other representations they are using (e.g., the table). Ask students:
As a class, discuss the rules that students wrote to tell the relationship between the shape number and the number of toothpicks. Invite students to share the ways they saw the toothpicks growing and how that helped them discover the rule (in the picture and in the table).
Challenge students to whisper with their partner to decide whether 100 toothpicks would be enough to build the 30th staircase.
Close by asking students to craft a short letter to an absent friend or a parent and answer the following three questions:
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).
Continue to explore relationships between terms by exploring a growing pattern that involves several rules.
Pose interesting, but more difficult-to-generalize growing patterns.
Hi. I'm not seeing the "Building with Toothpicks" Activity Page. Any ideas? Thanks!
CCSS, Content Standards to specific grade/standard
CCSS, Standards for Mathematical Practices
PtA, highlighted Effective Teaching Practice and/or Guiding Principle CCSS