Comments on: [3ACTS] Toothpicks

By: Bryan Dickinson

Bryan Dickinson — Mon, 28 Oct 2013 19:56:05 +0000

One addition you could make to the sequel or the whole problem is: what will the length of the base be?

By: Dan Meyer

Dan Meyer — Sun, 06 Oct 2013 00:52:01 +0000

Nice. Thanks for the feedback, Sam.

By: Sam Shah

Sam Shah — Sat, 05 Oct 2013 02:22:36 +0000

I used this as a starting activity before we jumped into sequences and series in my advanced precalculus class today. It was super fun to do. I brought toothpicks for them to use and gave them giant whiteboards, and had them in groups of 3 and 4. Just gads of fun to listen to them work through the puzzle. I barely said anything once they were off working. And I heard some awesome observations.

Their work for over the weekend?

1) For 12,594 toothpicks, what will the final picture look like?
2) How many vertices are there in the final picture?

Thanks for a fun class!

By: Adrian Pumphrey

Adrian Pumphrey — Thu, 15 Aug 2013 19:37:06 +0000

The real magic happened when students shared their ideas at the end about how they solved for the number of triangles and rows. The comment that stuck out to me was one student mentioned that he started to draw out the triangles but then he stopped. As soon as he saw a pattern emerging he could switch to more efficient process of using numbers.

I was able to quite succinctly go on to describe how we use numbers to describe patterns but that algebra gives us the power to describe patterns for any given number, in this case, of toothpicks.

Next week we start the unit on ‘What is a function?’. I hope this has given me enough to build upon.

By: ‘And then I stopped drawing’ | Pumphrey's Math

‘And then I stopped drawing’ | Pumphrey's Math — Thu, 15 Aug 2013 19:35:04 +0000

[…] I introduced the reason for Algebra 2 to my students starting with the Dan Meyer 3Acts Toothpick activity. Some students really loved it and some gave up quite quickly challenging me as a teacher to ask […]

By: Kelly Collins

Kelly Collins — Sat, 19 Jan 2013 15:52:34 +0000

Well I had some of my kiddos play with this one yesterday. After some thinking, they got the number of rows AND could tell me how many toothpicks he had leftover for 250 and 500 toothpicks. Coming up with the formula stumped them but I DID have one group come up with this. t=3[n + (n-1) + (n-2) + (n-3) +… ] Can’t really expect more than that when they haven’t studied series! :) It was a good day

By: Dan Meyer

Dan Meyer — Tue, 15 Jan 2013 21:54:54 +0000

Unless n=12.4196 was the exact solution, I’d anticipate some error from rounding when you subtracted the theoretical answer from the experimental answer.

By: Bilyan

Bilyan — Mon, 14 Jan 2013 18:06:04 +0000

Hi, Dan. I would like to ask you something and I hope that this is the right place. I tried to solve the task by using arithmetic sequence. I noticed that on each level the number of toothpicks increases by 3. The sum of all the toothpicks is 250, d=3, A1=3, An=A1+(n-1)d and using the formula for the sum of arithmetic sequence we end up with a quadratic equation with only 1 solution n=12,4196(which is ok because we don’t have only 12 levels). Furthermore 12,4196-12=0,4196. Multiplied by 36(the value of A12) is equal 15,106(here I tried to get the exact number of the toothpicks left). The actual count is 15. Where do you think my mistake is? In 0,4196*36?

By: Hawke

Hawke — Sun, 13 Jan 2013 01:36:12 +0000

Great stuff as always, Dan.

Regarding the role that finiteness plays, if we view 3-act lessons as an exercise in students posing (or discovering) intriguing questions, this just seems so much more difficult without some sort of constraint. For this particular video, if you were to cut to a graphic of shelves covered with an unnamed number of toothpicks, or some screenshot showing that the supply of toothpicks was unlimited (or unknown), then cut back to you creating triangles and end with an ellipsis, what question would I ask about this task? Those that come to mind are:

1. How does the ratio of rows:toothpicks relate to the number of rows?
2. What is the number of toothpicks it would take before there is no more space in this room to continue the pattern?

You’ll notice that, for #2, I picked my own constraints.

I could be that I’m just not very imaginative, but these are the best I can come up with. To me, these pale in comparison to the questions already generated with the number of toothpicks given.

I think there is a sweet spot where just the right number of constraints are provided: too many and you fall into the textbook trap where it’s all right there; too few and the problem just isn’t all that intriguing.

By: Dan Meyer

Dan Meyer — Fri, 11 Jan 2013 19:25:17 +0000

Thanks for the feedback, Jennifer. We've been having conversations in another thread about the merits of ST Math, which is one of the better tools available, from my limited exposure. Dreambox is good for elementary and DragonBox is an interesting solving equations app for the iPad.