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The Five Triangle Puzzle: Composing and Decomposing

Beginning in first grade and continuing through the grades, the Common Core Math Standards emphasize composing and decomposing shapes. One way to give students experience with composing shapes is through put-together puzzles. Actually, working at put-together puzzles involves lots of composing and decomposing. Sometimes what we compose is a solution; sometimes it’s not.

While the tangram puzzle is an obvious, well known example of such a puzzle, there are others that students will find interesting. One of my favorites is a puzzle made up of five triangles. In this post I want to tell you about this puzzle; I will give you a master so you can make a Five Triangle Puzzle for yourself. I’m then going to leave it for you to solve the puzzle, and finally, I’m going to come back to it in a later post to talk about the solutions. I’m hoping in the mean time to hear from you about how you thought about the puzzle and to hear about your solutions and anything you might have learned by working it.

I first saw this puzzle when a colleague dropped it on my desk, probably 20 years ago. He simply commented that he had found one solution, but was sure there was at least one other one. I was hooked; I couldn’t leave it alone until I’d explored all the possibilities for solutions and found them or demonstrated that they were not possible.

Let me show you the pieces in this short video and then I’ll finish up by telling you about the challenge of the puzzle.

The puzzle consists of 5 triangles. They are right scalene triangles; actually, they are 30-60-90 triangles. The triangles are two sizes. There are two that are smaller, and three larger. You can see that when two of the same size are put together, the form an equilateral triangle. They are related in that the hypotenuse of the smaller triangle is equal to the long leg of the larger triangle. This will be important to know as you try to put the triangles together form other triangles. Read on to see the challenges posed by the puzzle.

The first challenge is to simply put all five of the triangles together to form one large triangle. The second challenge is to compose a second, different triangle using all five triangles. The third challenge is to determine whether there might be yet one more triangle that can be formed using all of the pieces.

I’m hoping you will accept the challenge posed by this puzzle and that you will let us know about your experience.

2 Responses to The Five Triangle Puzzle: Composing and Decomposing

1. […] The Five Triangle Puzzle was the subject of a post back on February 11. I’m hopeful that some of you will have downloaded the pieces and solved the puzzle. The challenge was to put all five pieces together to form a triangle and then to determine if there were other triangles that could be formed using all five pieces. […]