## Introduction: Teaching Math Through Paper Folding

Years and years ago, when I was still in college studying to become a math teacher, I had a professor who said we could attend the Detroil Area Council of Teachers of Mathematics conference instead of doing one of the projects for class. I do not remember much about that class or what we learned but I remember the conference.

More specifically, I remember one of the sessions that I attended. The session was held in the library at Lake View (I think) High School in St. Clair Shores, Michigan. I remember that I sat in the back of the class. There was a morning session and an afternoon session--I attended both. I remember leaving with tons of ideas and have used many of them over the years. (The second session was called 'scrap paper math'.)

The steps that follow this are from this session of the DACTM conference. I do not remember the instructor's name or I would give him the credit that he so richly deserves.

I have made dozens of these over the years. They make very nice Christmas ornaments. Even made from old math papers they look pretty. Since it is totally hollow inside and can be disassembled without tearing the paper, I have used the 20 pointed star to wrap small gifts.

## Step 1: Getting Started

In the class, the instructor modeled the procedure that he used with his students. I can speak from experience--it works. He was folding paper as he gave instruction but he was also talking and asking questions. He loved asking questions about fractions and the size of different angles. He kept using math vocabulary and encouraging us to use the vocabulary in our answers. The only way to really get comfortable with technical terms is to use them repeatedly.

Start with a square sheet of paper--he used scrap paper. If there is anything that a teacher has a lot of, it is scrap paper. He had brought stacks of old worksheets, news letters, even student work that he had lying around. You do not have to go buy origami paper. All you need is a nice paper cutter--try the art room or the main office. You can use scissors but it takes longer and the cuts are not as straight. I usually mass produce a whole box of squares using whatever paper is available.

A 4 inch square is big enough for most kids when they are starting out. If you go too much smaller, it can be hard for them to get their fingers where they need to be. I usually demonstrate with 8 inch squares--easier for the kids in the back to see what I am doing. The smallest that I have worked with is an inch and a half but it was a bit difficult to work on even with my small fingers. I recommend at least a 2 inch square.

## Step 2: First Folds

Before we start, let me just say that neatness does count on this project. Fold carefully. Crease sharply.

Fold one sheet of paper in half so that you end up with a rectangle. When you unfold the paper, you will see that you have 2 congruent rectangles and that you bisected 2 opposite sides of the square.

Fold the 2 outside edges in toward the original fold. You should have quartered the sheet of paper. Each quarter is a congruent rectangle. You once again bisected a pair of segments (the edge of the paper).

## Step 3: Paper Airplane Folds

These are the hardest folds to make in the whole process, and the hardest to explain. Some students 'know' what you are going to tell them to do so they just don't bother to pay close attention. Once you know who these kids are, position yourself so that you can keep a close eye on them. That way you can stop them before they go too far off track.

I call them paper airplane folds because that is what the instructor called them. And because, who hasn't made a paper airplane. With that said, everyone doesn't make paper airplanes the same way. Keep an eye on the couple of students who don't follow directions well.

Start with the folds running vertically. Fold the top right edge of the square down to line up with the quarter fold. Some students use the original half fold instead of the quarter fold--'cause teenagers hate to be told what to do. Once you get them back on track, they listen better.

Now is an appropriate time to ask about angle measures. You bisected a right angle so you now have a 45 degree angle. The tougher question to ask students is "What fraction of the whole square is this little isosceles right triangle?"

It usually takes a few tries before someone gets it right. Teenagers these days are even worse at fractions than when I was a kid. I blame it on the availability of calculators.

The second airplane fold brings the folded edge to the same quarter fold. This bisectes the 45 degree angle leaving you with a 22 1/2 degree angle. Believe it or not, some teenagers have never considered that there might be something smaller than a whole degree.

## Step 4: Rotate the Piece.

Once everyone has the 2 paper airplane folds in the upper right hand corner, you have the class rotate the piece 180 degrees. Watch to make sure that you catch the student or 2 that flip the thing over.

Repeat the 2 airplane folds in the new upper right hand corner. You should have folded the opposite corners of the square--not adjacent corners.

If you have a student who is left handed, it may be more comfortable for them to do the airplane folds on the upper left corner. The whole project will work but the 'left handed' pieces will not fit together with the 'right handed' ones.

## Step 5: Next Set of Folds

Now, you fold both the left and right quarters in towards the center crease. If you folded neatly, the paper will not fight you. If you went past the quarter crease on your airplane folds, you may have to work to get it to lay flat.

Now your piece is only 1/2 the size of the original square. You need to fold the upper LEFT corner all the way over to lighten up with the right hand edge of the rectangle. What fraction of the whole original square is this isosceles right triangle--tough question for some kids because they are not looking at the whole square of paper at this point.

Open up the paper airplane fold and close it over the isosceles right triangle. This will hold the triangle in place. The shape is now a right trapezoid.

Rotate the whole piece 180 degrees and repeat the fold. Tuck this triangle under the airplane folds.

What shape did you make? Most kids know 'parallelogram' by the time they get to me.

## Step 6: Finishing the Piece

Your piece has 2 long edges and 2 short edges. Lay the piece on the table so that the long edges run horizontally. Flip the piece so that the tucks are face down.

You are going to bisect the long edge by bringing the acute (45 degrees) vertex (corner) over to the obtuse (135 degrees) one. Crease it well--there are a lot of layers of paper. Use the back of your finger nail. Don't worry that it doesn't want to stay folded.

Do the same and bisect the other long edge of the piece.

You now have the square that you need to build your cube.

## Step 7: Make More Pieces.

Have each student begin the whole process over again with a fresh square of paper. You should be ready to remind them of the strips when they get stuck.

By the time they have completed 2 or 3 pieces, some of them will remember the steps--then they can remind each other of the steps.

## Step 8: Assembly

Six pieces will assemble into one cube. I usually have 2 or 3 students pool their pieces to finish the cube.

If you look at the square side of a piece, you will notice that there are 4 pockets--2 that are triangular, 2 that are quadrilaterals. We will only be using the triangles. I colored in the quadrilateral pockets in the picture above. These will be covered by another piece but you will not be using them as pockets.

Take a second piece and insert the acute vertex into one of the triangular pockets. Take a 3rd piece and put one of its acute corners into the pocket of the 2nd piece. Before you grab another piece, tuck the corner of the first piece into the 3rd piece. You should have 3 sides of a cube. Many kids can figure out the rest without help.

Some need a little more instruction. The biggest problem that most kids have is that they allow the acute corners to slip inside the box. When I help them assemble their pieces, my job usually involves holding the acute points outside of the box. Then I have the student tuck it into the triangle pocket.

Sometime, students forget to find the triangle pocket. Nothing will fit right if you use the quadrilateral ones.

If one piece is made 'left handed', it will never fit. All must be made the same 'handedness'.

## Step 9: The Challenge

This is where the instructor stopped teaching us. He started telling us a story.

It had been about 5 years at this point since he had learned what he had just taught us. He went on to tell us that he had made dozens of pieces and had taught people--teens and adults--to assemble the cube. When he was originally taught, his teacher gave him a completed 'stellated icosahedron' and he pulled it out of the paper grocery bag that he had with him. He had been trying for 5 years to reproduce the thing. While he continued to talk, he pulled out all sorts of miss-shaped attempts. One, I remember, looked like like a snake.

He told us that the icosahedron used 30 pieces like the ones we had just made but he couldn't figure out how to assemble them.

I went back to the conference the following year ('cause I wanted to, not to get out of work for a class) and went to the same teacher's class. I got there late so I had to sit in the back (again). While I was learning all the new things he was teaching, I used my spare time to fold 30 pieces and assemble the star.

When I got the last piece in, he stopped class (he knew what I was working on), and told them that he no longer included this project in his seminars. He dropped it because he thought since he could not teach the assembly process, it was a waste of our time.

I told him that not only did I figure it out, but I showed it to my high school students. I taught them what he had taught us. Then I left it as an open challenge. Anyone who could make one by the end of the term got extra credit. 4 of them got the extra credit.

I have included one extra hint if you want it. It is on the next step. You can choose to accept the challenge of creating your own 20 pointed star now or read one last step.

In the 2nd picture above, the cube (center) is made using 6 pieces. The stellated icosahedron (right) is made using 30 pieces. The stellated octahedron (left) is made using 12 pieces.

## Step 10: Hint

There was one more fold that each piece needs if you are assembling all 30 pieces. See the picture above.

The zig zag will help just a tiny bit in the final assembly.

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