The comments on my previous posts where people suggested spirals reminded me of this activity. When the department head group I support was looking at ways to help implement our revised curriculum, we decided to do a lesson study. Two of the lessons they planned are rolled into this one idea. The majority of this activity comes from Deanna Matthews at J. Percy Page, who tells me she got the idea from one of my professors, Florence Glanfield, from the University of Alberta. This post should probably have preceded the last one, but I never think that far ahead.

**Creating a Pythagorean Spiral**

Have the students create a Pythagorean spiral on card stock, by following the steps below.

- Use one piece of card stock to draw the spiral on, and use a different colored piece of card stock as a “ruler”.
- Using the “ruler”, draw a horizontal line one unit long towards the middle, right hand side of the other piece of card stock. The length is arbitrary. I usually tell students to make this longer than an inch, but shorter than 3 inches. If they start too big, it won’t fit, and if they start too small, it’s hard to work with later. Make sure to mark this length of 1 unit on the “ruler”.
- Arrange the “ruler” so that you can create an isosceles right triangle. We are using the corner of the “ruler” paper as a guide to make that right angle.
- Use any straight edge to draw the hypotenuse of this triangle, which will be units long. Notice that I labeled the sides of this triangle on the inside, because we are going to cut it out later.
- Place the “ruler” on the hypotenuse of the 1,1, triangle as shown, to draw another leg that is at a right angle to this hypotenuse. Make sure students position the “ruler” correctly. I’ve seen them trying to guess at a right angle, rather than using the corner of the paper.
- Use a straightedge to draw the hypotenuse of this new triangle. It will be units long.
- Repeat the process. This new hypotenuse is units long. I usually leave it as on the label, because one of the things I’ll use this activity for is to talk about the difference between rational and irrational numbers.
- Extend this pattern as far as you can until you run out of paper. I went to .

**Using the Spiral to Create a Radical Number Line**

- Notice equivalent radicals. Above, we can see that is equivalent to .
- Discuss rational vs. irrational.
- Order radicals without converting or using a calculator. Below, we can see that >