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Posts Tagged ‘Radicals’

Radical Number Line

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}
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Thanks to the comments on my previous post, I learned that there is a much easier way to make some of the sets I described.  To make the \sqrt{3}set, I originally suggested measuring a 60 degree angle.  Here is a much better way.

Use the \sqrt{2} set that was built in the previous post, and place it on a different color of card stock grid paper.

Create a right triangle with legs 1 and \sqrt{2}.  This makes the hypotenuse \sqrt{3}.

Use the rest of the \sqrt{2} set to make right triangles with legs 2 and \sqrt{8}, 3 and \sqrt{18}, 4 and \sqrt{32}, 5 and \sqrt{50}, 6 and \sqrt{72}, 7 and \sqrt{98}, and 8 and \sqrt{128}.

Cut them out, and you have a much easier \sqrt{3} set to put on your radical ruler.

You could use similar patterns to create other sets.  For example, you can make a \sqrt{6} set by using your \sqrt{5} set.  Make triangles with legs 1 and \sqrt{5} and their multiples, and you have a \sqrt{6} set.

Thanks again for all the feedback on that last post.  You have helped make this activity even better.

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Last year I visited Shannon Sookochoff at Victoria School.  She showed me a neat activity on simplifying radicals on a Geoboard.  As I was thinking about how to introduce adding and subtracting radicals recently, that activity came back to me. I have created an extension of what Shannon Showed me.

Adding and Subtracting Radicals Using a Radical Ruler

Have students create “sets” of triangles.  The set I would start with is the \sqrt{2} set.

Use a 1 cm grid on a piece of colored card stock.  Have students draw right triangles with legs 1 and 1 (makes the hypotenuse \sqrt{2}) and then with legs 2 and 2 (makes the hypotenuse \sqrt{8}) and so on.  Continue all the way to one with legs 8 and 8 (makes the hypotenuse \sqrt{128}) .

Their grid paper will look something like this:

Notice I labeled the hypotenuse inside each triangle.  That’s because we are going to cut them out so we’ll have a set that looks like this:

Next, draw a line on a piece of paper.  This will be our \sqrt{2} number line.  Start by putting the \sqrt{128} triangle on the number line, and marking this point as \sqrt{128}.  Work your way in by putting the next one, which is \sqrt{98}, on the line.  Continue until you get down to the \sqrt{2}.

At this point, students should notice that all the gaps are \sqrt{2} in length.

This will allow them to label the bottom of the number line with \sqrt{2}2 \sqrt{2}3 \sqrt{2}, and so on. Notice that this gives a nice visual showing entire radicals and their equivalent simplified mixed radical.

Now we will extend the \sqrt{2} number line.  The quickest way to do this is to use the \sqrt{128} triangle and place it on \sqrt{128} as shown.

Then just mark the intersections along the hypotenuse.  These are each \sqrt{2} long, so we can extend our number line.  I took mine to 16 \sqrt{2}.

Now we can add radicals pictorially on our number line.  For example, if we wanted to calculate \sqrt{72}+\sqrt{50}, just put the two triangles on the number line as illustrated, and read the mixed radical below.

The answer is 11 \sqrt{2}.

Now you can have the students make other sets.  The easiest to make are ones that don’t require any angle measurements.  I would make a \sqrt{5} set in a different color, by using legs 1 and 2 cm long, and then just extending those by multiples of the original sides (your other legs are 2 and 4, 3 and 6, 4 and 8, 5 and 10, 6 and 12).

Cut them out, and add them to a second number line below the first. Line up the 0’s, so that students could compare and order mixed radicals using these number lines.

Other sets that are easy to make are \sqrt{10} (Legs 1 and 3, and then multiples of those), \sqrt{13} (Legs 2 and 3 and then multiples of those), and \sqrt{17} (Legs 1 and 4 and then multiples of those).  You could make as many of these as you want.

It would be nice to have a \sqrt{3} set, too, but this one will require some angle measurement.  Start with a base of 1, then measure an angle of 60 degrees. Where this meets the vertical will be a height of \sqrt{3}.

Continue this with bases of 2, 3, 4, 5, 6, and 7 to create a \sqrt{3} set. (Sorry about the stray line)

Cut out this set, and add it to your number line. Note that this set is much harder to build accurately.  I measured as carefully as I could, and my gaps are not consistent.

Edit: I have made improvements to the creation of the a \sqrt{3} set based on feedback from the comments section below.  You can see a better way to make the a \sqrt{3} set here.

Now that you’ve built these radical rulers, here are a few things you can do with them.

  1. Simplifying radicals.  This is a quick pictorial representation of what radicals become in simplified form.
  2. Comparing and ordering radicals.  Which is greater: 9 \sqrt{2} or 5 \sqrt{5}?  Typically, students would punch this in a calculator and compare decimals. It works, but has no understanding behind it. With their radical rulers, they can see which is bigger.
  3. Adding and subtracting radicals.  Put the triangles on the number line as shown above.  To illustrate adding only like radicals, ask them to add \sqrt{75} and \sqrt{45}.  They will discover that it can’t be done.  \sqrt{75}, in green, is not like \sqrt{45}, which is yellow.

I’d love some feedback on this one.  It’s not something I have ever tried in a classroom, but I would love it if someone tries it, and lets me know how it worked.

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