Toothpick Patterns

Project Slide #1, which shows three toothpick “configurations.” Each configuration has a “design number” so we know which one we’re talking about.

As students begin to call out answers, you may want to create a table. Students will start to see patterns emerging. You can ask the following to help them along:

Explain that we now want to extend the pattern to look at larger toothpick designs.

Pass out Handout #1.

Handout #1 Answer Key

Answers and likely responses:

Show Slide #2.

​Separate students into groups. Encourage them to work together to ensure everyone is on board. Give students time to brainstorm about different ways to answer the questions and to share any patterns they may have identified in the previous problem.

Here are the prompts from the slide:

Have students make a poster that answers these questions and explains clearly how they know they are correct.

Hints you may want to offer teams as they proceed.

Look for patterns in the toothpick designs.

Look for patterns in the numbers.

Write a formula to calculate the number of toothpicks for each design. What variable(s) do you need to create the formula?

Suppose you call the total number of toothpicks T. Can you create more than one formula to calculate T? If so, create as many as possible.

The relevant variable is the number of the pattern, which is also the number of squares in the pattern. We often use n for variables like this one.

One (of many) answers is T = 3n + 1. So the tenth design will have 3(10) + 1 or 31 toothpicks; the 100th design will have 301 toothpicks.

This lesson plan provides several more solutions with detailed explanations in the next section.

Have students post their posters around the classroom.

Encourage students to travel around to view the posters created by other groups. Students should be encouraged to pose questions to other groups by attaching a small adhesive notes to posters.

During this time, the teacher should review all the posters and consider which to highlight in Phase 5.

The first question (tenth design) can be answered by drawing and counting (31).

For the second question (100th design), drawing is impractical. But you could extend a table of values, or reason from it (10 needs 31 toothpicks, 20 needs 61, so 30 must be 91, etc…).

For the third—the one with n—students need to generalize. And there are many ways to do this. Some possibilities:

Facilitate a discussion of the different approaches. Select a sequence of posters to use that will help students move from their current thinking (Levels 1–3) up to Levels 4 and 5.

Level 1: Students reason solely from number patterns, and do not connect those patterns to the configurations of toothpicks. They do not generalize or make a formula, even though they can reason to a correct solution for a large number.

Level 2: Students express the “plus 3” well and abstractly, and can see where this approach comes from in the toothpicks, but cannot make a formula that works for any value of n.

Level 3: Students can make a formula that works for all values of n, but don’t connect it to the configurations of toothpicks.

Level 4: Students make a formula that works for all values of n, and can explain how each part of their formula corresponds to part of the toothpick pattern.

Level 5: Students recognize that there are different ways to construct the formula depending on how you view the toothpick configurations, and can show that these expressions are, in fact, equivalent.

Ask students, “How is it that we have different formulas for the same thing?”

Ask students, “Do you think the formulas give the same answer for any number?”

Ask students to use the distributive property to expand all of the expressions, and to combine like terms (i.e., “simplify”) in all of the expressions.

If you introduce this problem in class, pass out Handout #2. Ask students to work in pairs to become familiar with the pattern in the diagram (below).

Handout #2 Answer Key

They might also notice that the number of stones along any one side is always more than the garden size because the stones are outside the garden. But they might not be able to express the problem of “overlap” or the corners clearly.

Once you’re convinced students are well-oriented to the garden context, have them work on the questions on the handout.

Size 10: 46 stones.

Size 100: 406 stones.

Sides and Corners: For garden n, there there are n stones directly to the left and right. That’s 2n. Then there’s one more than that on the top and bottom, so that’s 2(n + 1). Then there are four corners, for 4 more. Total: 2n + 2(n + 1) + 4.

Fenceposts and Rails: On the left and right sides, if you go all the way to the top and bottom, you have n + 2. That’s 2(n + 2). Add the “rails” top and bottom and you get

2(n + 2) + 2(n + 1).

Recursive Approach: The first one has 10, and each one after adds four more stones. So, if S is the number of stones, then S(1) = 10, and S(n + 1) = S(n) + 4. This is correct, but hard to use for size 100! Probe to see how a student using this approach actually gets the answer. Usually students use some other way they just can’t express as a formula—until you help them see how.