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.]

Figure 1

Repeat for the second image in (fig. 2). [P = 8 units.]

Figure 2

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.

Explore

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:

Where do you see the growth in the staircases?

Where do you see the growth in the table?

If they notice that the table is increasing by 4 each time and know the rule is then 4n, challenge them to show how that growing pattern occurs in the shapes.

In what ways is the pattern growing?

What rules describes what comes next (how the pattern is growing)?

Where do you see that in the pattern? The table? The graph?

What rule tells the relationship between shape number and number of toothpicks needed?

Synthesize

Reflect

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.

Assessment

Optional

Close by asking students to craft a short letter to an absent friend or a parent and answer the following three questions:

What is a growing pattern?

How do you find a rule that tells how many toothpicks are needed for a shape number?

How is a rule that tells what comes next (e.g., Add 3) different than a rule that tells the corresponding quantity (e.g., 3 times the shape number)?

Teacher Reflection

What did students struggle with when using pictures to find a rule? How could I help them?

Which representations supported student reasoning in determining the rule?

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?

What additional guiding questions could I ask to help students articulate the reason a formula is useful?

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).

How can we use tables, graphs, and visual models to create a rule for a growing pattern?

Standards

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 1 Make sense of problems and persevere in solving them.

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

Support productive struggle in learning mathematics.
Because this pattern grows in a more complex way and students are to connect the rule to the pattern, they are likely to make numerous attempts to see how the pattern is growing. Teacher questioning as students are working can help the students continue to make connections and "see" the rule.