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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.

1. Use one piece of card stock to draw the spiral on, and use a different colored piece of card stock as a “ruler”.
2. 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”.
3. 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.
4. Use any straight edge to draw the hypotenuse of this triangle, which will be $\sqrt{2}$ units long.  Notice that I labeled the sides of this triangle on the inside, because we are going to cut it out later.
5. Place the “ruler” on the hypotenuse of the 1,1, $\sqrt{2}$ 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.
6. Use a straightedge to draw the hypotenuse of this new triangle. It will be $\sqrt{3}$ units long.
7. Repeat the process. This new hypotenuse is $\sqrt{4}$ units long. I usually leave it as $\sqrt{4}$ 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.
8. Extend this pattern as far as you can until you run out of paper.  I went to $\sqrt{12}$.

Using the Spiral to Create a Radical Number Line
Once students complete their spirals, I have groups of three decide which one of them has made the best spiral.  A good check for the accuracy of a spiral is to look at the $\sqrt{9}$ side and compare it to the “ruler”. It should be 3 units long. Whichever student’s is closest to 3 is the one they should use. This is the one we will cut out to create a set of triangles.
We are now going to create a Radical Number Line from these triangles. I like to use a little something I discovered in consulting services called “Sentence Strips”. These strips are foreign to high school teachers like me, but apparently they are used in elementary schools extensively. They are 2 foot long pieces of card stock that come pre-lined on both sides.  Students can create their number lines, and then fold them in half and keep them in their binders.
We will start by putting whole numbers on this number line. We’ll use one of the triangles we cut out and mark the multiples of 1 across the number line. There was some discussion among the group that planned this lesson as to whether we should include negatives or not. My preference is to start on the left side at 0.
Next we will add the multiples of $\frac{1}{2}$ to the number line. Ask students how they could do this without estimating. I folded one of the one unit sides in half, as shown.
At this point, draw their attention to the fact that every other multiple of $\frac{1}{2}$ lands on a whole number. Ask them to try multiples of any fraction they want. They will discover that eventually they hit a whole number. For example, a student who starts with $\frac{2}{9}$ will get $\frac{2}{9}$$\frac{4}{9}$$\frac{6}{9}$$\frac{8}{9}$$\frac{10}{9}$$\frac{12}{9}$$\frac{14}{9}$$\frac{16}{9}$, and finally stop at  $\frac{18}{9}$, which is equivalent to 2. We are steering them towards a definition of rational vs. irrational numbers.
Now we will add radicals, and their multiples to our number line. Starting with $\sqrt{2}$ on the left, we will add $\sqrt{2}$$2 \sqrt{2}$$3 \sqrt{2}$ and so on. Students should notice that these multiples of $\sqrt{2}$ never hit whole numbers (although, this is somewhat dependent on the accuracy of the spiral they created).
Next, add $\sqrt{3}$ and its multiples to the number line.
Next look at the $\sqrt{4}$.  It should hit at 2.  Ask the students why $\sqrt{4}$ is different than the other two we sets we have plotted so far.
Add $\sqrt{5}$ and its multiples to the number line.  Continue to add as many as you think fit without cluttering it up too badly.
Things you can do with this Radical Number Line:
• Notice equivalent radicals.  Above, we can see that $\sqrt{8}$ is equivalent to $2 \sqrt{2}$.
• Discuss rational vs. irrational.
• Order radicals without converting or using a calculator.  Below, we can see that $3 \sqrt{6}$>$4 \sqrt{3}$

### 5 Responses

1. […] John Scammell shows students how to create a Pythagorean spiral on card stock to explore irrational numbers in Radical Number Line. […]

2. I love it and can’t believe I’ve never seen this before! Thanks.

3. […] Math Teachers at Play Article Zero Knowledge LikeBe the first to like this […]

4. how long did it take for you to complete this lesson?

5. It took most of an 85 minute class. If you leave out the construction of the spiral, and just copy one and hand it out, you can save a lot of time.