homeschooling,  mathematics

Origami fish and Pythagorean Theorem

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How to make an origami fish My boys were playing with paper the other day, doing some origami and at some point my son asked me to make a fish for him. So I quickly whipped up the origami fish I remembered from my childhood, and passed it to him. Then I realized  the origami fish would be perfect for reviewing the Pythagorean theorem with him so we did that and now I’m writing up a blog post about it.

To make an origami fish fold a square of paper along the diagonals so that you can see the outline of four right-angle triangles each with a hypotenuse the length of the original square. Take two opposing triangles and push them inwards so that their middles touch and you get a shape like the one to the left.

Fold it down tight so that the whole thing is like two triangles attached together in the middle. Then take the loose corner of one triangle, fold it over to make one part of the tail. Take the other side of the same triangle and fold it to make the other side of the tail.

An origami fish with edges marked to show the pythagorean theorem

You should now have a fish like the one in the picture, except of course without the dark lines across it. The dark green line shows one side of the original square of paper. The black lines shows the line from the tip of the fishes nose to the tip of one tail and the same size line from the tip of the nose to the end of one fin.

If you think of each black line being one unit of measurement then the green line is going to be the square root of two. Why? The Pythagorean Theorem is that:

In any right triangle, the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares whose sides are the two legs (the two sides that meet at a right angle).

Black line represents one unit. The hypotenuse, in green, is the square root of two.My little diagram below shows the paper with the green and black lines marked. Choose one of the small sized triangles and look at the edges. Imagine squaring each of the edges. One square whatever-unit-you’re-using would be equal to two of the small triangles.  The square of the hypotenuses is what you have already – four triangles equally two squares.

At some point in a couple of years we’ll go over this again and I’ll assign the children the task of figuring out what of paper is needed to make fish of certain lengths and sizes, and I’ll ask them to figure out the area of the visible parts of the fish. However, right now we left it with reviewing how the Pythagorean theorem is and then moving on to other origami.

My children are starting to invent their own origami figures, and I am delighted with that. When I was a child, I thought of origami as something to memorize specific patterns for. They are learning to see it as a method of play and exploration. My oldest has picked up the terms from the various origami books we’ve borrowed from the library, and he talks of squash folds, mountain folds and valley folds.


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