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 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 ) and then with legs 2 and 2 (makes the hypotenuse ) and so on. Continue all the way to one with legs 8 and 8 (makes the hypotenuse ) .
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 number line. Start by putting the triangle on the number line, and marking this point as . Work your way in by putting the next one, which is , on the line. Continue until you get down to the .
At this point, students should notice that all the gaps are in length.
This will allow them to label the bottom of the number line with , , , and so on. Notice that this gives a nice visual showing entire radicals and their equivalent simplified mixed radical.
Now we will extend the number line. The quickest way to do this is to use the triangle and place it on as shown.
Then just mark the intersections along the hypotenuse. These are each long, so we can extend our number line. I took mine to .
Now we can add radicals pictorially on our number line. For example, if we wanted to calculate , just put the two triangles on the number line as illustrated, and read the mixed radical below.
The answer is .
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 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 (Legs 1 and 3, and then multiples of those), (Legs 2 and 3 and then multiples of those), and (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 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 .
Continue this with bases of 2, 3, 4, 5, 6, and 7 to create a 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 set based on feedback from the comments section below. You can see a better way to make the a set here.
Now that you’ve built these radical rulers, here are a few things you can do with them.
- Simplifying radicals. This is a quick pictorial representation of what radicals become in simplified form.
- Comparing and ordering radicals. Which is greater: or ? 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.
- Adding and subtracting radicals. Put the triangles on the number line as shown above. To illustrate adding only like radicals, ask them to add and . They will discover that it can’t be done. , in green, is not like , 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.