Toothpick Patterns: Growing Squares, Growing Triangles, Growing Stairs

We wanted to build off the problem from last meeting, exploring visual patterns, with a focus on different ways of approaching these problems and how can we bring them into the classroom.

Facilitator(s): Solange and Tyler
Date of Meeting: January 13, 2015
Problem: pdf
Further Reading: pdf

Discussion of the Group’s Meeting Norms

Since we are still defining ourselves and our process, we spent some time talking about what norms folks would like to see happen at every meeting.

Eric – Each meeting should have an opening and a closing activity.

Solange – We should emphasize making clear links between the work we do at these meetings and the classroom – questions like, “Who is going to do this activity with students?” and “What happened when we did this this activity/problem with students?”

Tyler – Each problem we do should be prefaced with a rationale – why did we choose this problem, what are we going to focus on. It would also be great to have a debrief at the end of each meeting – “What do we want to do next month?”. Maybe we can have a conversation/survey to see what ares we are all interested in and see what common ground there is. That would help the folks who are organizing the meetings.

Ramon – Having a teacher bring one of their students with them to the meeting

Maggie – More community building activities – maybe break into pairs/partners at each meeting to have someone to check in with 2-3 times before the next meeting

Kevin – Some clarity on our goals, in terms of the problems we choose – it would be nice to alternate between problems that are slightly beyond our abilities, that require struggle and perseverance and other problems that are more directly applicable to the classroom (i.e. we can use them with students)

Cynthia – Ice breakers to build community, organizing groups around talking to new/different folks

Charlie – Going deeper into problems, as opposed to just doing interesting problem after interesting problem.We need to do good problems but also frame the problems in good pedagogy – maybe we can borrow from lesson study and observe each other doing problems from the meetings with students.

Jane – Definitely having a teaching component – participants should commit to trying problems out with students and reporting back at subsequent meetings.

Mark –  I would like these meetings to be documented and posted online – I think a web presence would build continuity between meetings/problems/teaching questions  and allow for many of the norms suggested above.

Meeting Schedule

Eric proposed having a set, regular and ongoing meeting time. Based on the Doodle polls for the first three meetings, it seemed like Tuesdays and Fridays are the two days that work best for people, so we said we would alternate between the second Tuesday and the second Friday of each month, starting with the 2nd Tuesday in February.

A Question of when to open up these meetings

We repeated our eventual goal of having these meetings be open to any and all math teachers who would like to come. At this meeting we talked about how to build up to that.

We decided for our next step in broadening our group, bringing more teachers into the fold, while continuing to build and expand our shared sensibilities, we would each try to bring 1-2 teachers to the next meeting. Ideally they would be folks who are interested in what we do, but who are newer to a problem-solving approach to math instruction. By each of us bringing a teacher, we’ll be able to keep the conversations, questions, problem-solving and teaching attempts going outside of our meeting, hopefully allowing us to go deeper into the problems and how to teach them.

Problem-posing and Problem-solving

Solange and Tyler gave us all an opportunity to work on the problem(s) on our own, before discussing what we were seeing/doing in groups of three.

Strategy Share


Kevin noticed that the second figure had 3 horizontal rows of 2 toothpicks and 3 vertical columns of 2 toothpicks. That is two groups of 6, or 12 toothpicks total. Then he saw that the same was true for Figure 3. There were 4 horizontal rows of 3 toothpicks and 4 vertical columns of 3 toothpicks, or 2 groups of 12 toothpicks, or 24 total toothpicks.

He saw that the same was true for Figure 4. There were 5 horizontal rows of 4 toothpicks and 5 vertical columns of 4 toothpicks, or 2 groups of 20 toothpicks, or 40 total toothpicks.

Given the Nth figure, Kevin realized he would have 2 groups of N(N+1) toothpicks.


Here’s a brief video of Eric explaining his chart (3:31)

2n² + 2n is 4 times the formula for calculating the triangular numbers (n² + n)/2


Jane noticed the 4th figure has 4 rows of 4 horizontal toothpicks (pink), and 4 rows of 4 vertical toothpicks (green) or 2(4)² toothpicks. Then she saw that there were 2 rows of 4 toothpicks remaining (grey), one making up the right side of the square and the other making up the bottom. The same was true for the 5th figure and Jane was able to come up with the same generalization as Eric, 2x² + 2x, in a visual way that is rooted in the picture.


Cynthia saw that the number of toothpicks followed the pattern of 4 times the sequence of triangular numbers. So for example, since the 2nd triangular number is 3, the # of toothpicks in the 2nd figure is 4(3). Since the 3rd triangular number is 6, the # of toothpicks in the 3rd figure would be 4(6).

As with Eric and Jane’s formula, Cynthia’s is 4 times the formula for calculating the triangular numbers – (n² + n)/2



4(n² +n)


 Mark, Charlie, Ramon

After finding the iterative rule for figuring out the number of squares and the number of toothpicks in each figure, we got interested (a.k.a. Mark got fixated) in the idea of visualizing separating each square in a figure and trying to figure out the pattern in how many toothpicks are duplicated (and need to be removed), when the squares are put back together. For example, the 2nd figure is made up of 4 squares. If we separate those 4 squares, it would take 16 toothpicks to construct them. But when we put them back together, we need to remove 4, leaving the 2nd figure with only 12 toothpicks. If we separate the 9 squares in the 3rd figure, it would be 36 toothpicks. But when we put them back together we need to subtract 12 toothpicks, leaving us with 24. When time ran out we were trying to find the relationship between the figure number and the number of toothpicks we had to subtracted when we brought the squares back together.

Figure Number 


Figure Number Squared


# of toothpicks 4(Figure Number Squared)



Difference between 4(Figure Number Squared) and the actual # of toothpicks in the figure 
1 1 4 4 0 1(0)
2 4 12 16 4 2(2)
3 9 24 36 12 3(4)
4 16 40 64 24 4(6)
5 25 60 100 40 5(8)
6 36 84 144 60 6(10)

4x² – ___________

The generalization above should read n² + ((n+1)² -1)

Charlie came up with the idea by looking at the relationship between the number of squares and the number of toothpicks. He noticed in Fig. 1, there were 4 toothpicks and 1 square – a difference of 3. He noticed that figure 2 had 12 toothpicks and 4 squares a difference of 8. He saw that Fig 3 had 24 toothpicks and 9 squares – a difference of 15. Then he realized the the difference between the # of toothpicks and the number of squares for each figure was one less than the number of squares in the next figure. So for example, the difference between the number of toothpicks and the number of squares in the 5th figure would be 35, because that is one less than the square of the next figure (6² = 36). Since we know the 5th figure has 25 squares, we know it would have 25 + (36-1) toothpicks, which it does… 60 toothpicks.

But in testing the generalization as it is written in the photo, I realized it is what was missing from the generalization we were trying to build above when we were trying to separate and then recombine the squares…  4x² – (n² + (n-1)² -1)

Jane’s Growing Triangles

In the growing triangles pattern (Fig 4 is the one drawn above), Jane noticed that the same thing that happens to the vertical toothpicks, happens to the toothpicks on both diagonals – three times in all. That is to say the outside edge of each triangle is the figure number, and then the number of toothpicks goes down by one, all the way to one toothpick. For example – in the 4th figure above, the right side of the triangle has 4 black toothpicks, followed by 3, then 2, then 1. The same is true for the sea foam green and the fern green. This was a pattern she recognized from Gauss and the Handshake problem.

This formula is three times the formula for calculating triangular numbers – (n² + n)/2

Presentation Questions

  • Solange and Tyler asked Kevin to present his solution/approach first? Why is Kevin’s approach a good place to start the whole-group talk?
  • What similarities do you see between the different approaches?
  • What is something you appreciate and/or find interesting about each approach?
  • Can you explain how each of the formulas derived relate to the figures?

Teaching Questions

  • How could we scaffold these growing patterns for students?
    • What problems might we do with students before they tackle these? What would we want them to get from those problems that would help give them some tools for tackling these?
    • How might the Border Problem help students prepare for these growing patterns?
    •  How might Pascal’s Triangle help students prepare for these growing patterns?
    • In Chapter Five of Lessons for Algebraic Thinking, Grades 6-8by Ann Lawrence and Charlie Hennessy, there is a description of an activity called “Go Figure!“. The problem calls for students to do some explorations of square numbers, rectangular numbers and triangular numbers to make recommendations about boxes of candy. The chapter includes teacher notes, student handouts and examples of student thinking. How might that activity scaffold and prepare students for the toothpick patterns?
  • What is the concrete rationale behind Eric’s method?
  • Which is a better problem for students to start with – Growing Squares or Growing Triangles? Which would be harder for students and why?
  • How does the article Problems With Nth-Term Problems (above) address any of the questions above? What new questions does it raise?

The Toothpick Patterns we looked at today are actually part of Problems that Connect Algebraic Thinking to Geometry and Measurement. Check it out.

In attendance: Jane, Maggie, Charlie, Solange, Tyler, Cynthia, Eric, Ramon, Kevin, Mark

Programs represented: Fifth Avenue Committee, Literacy Assistance Center, York College, Borough of Manhattan Community College, CUNY Start, CUNY Adult Language & Literacy Program

Location: The Adult Learning Center at the Borough of Manhattan Community College, 25 Broadway, 8th Floor

Respectfully submitted by Mark

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