## Kiss those Math Headaches GOODBYE!

### Invisible Misunderstandings: Square roots of 2 and 3

Would you say that the square root of two is an important number in math? Hmmm … and would you agree that the square root of three, while perhaps not quite so important, is still a quantity whose value students should be able to estimate?

Why not, right? After all, these numbers play key roles in the 30-60-90 and 45-45-90 “special triangles.” And therefore they both appear a lot in geometry, and a great deal in trig. And on top of that, root two, widely believed to be the first irrational number discovered, shows up in a wide range of other math contexts as well.

square root of 2 w/ "parent" triangle

Given all that, here’s an interesting question:  when was the last time your students were asked to demonstrate whether or not they can even roughly estimate the value of either of these important quantities?

If you’re like most educators, odds are that you can’t recall. But I am not blaming you!  For, last I checked, textbooks generally don’t ask students to learn either of these values. [Don’t ask me why; it’s just one of those strange omissions.]

Considering this situation, I recently became curious to see whether or not students have the vaguest idea regarding the value of these numbers.

To use the circumspect language often favored by math people, I discovered that: at least in Santa Fe, New Mexico, for the secondary math students I work with, over the course of the past two weeks … most students HAVEN’T THE FOGGIEST IDEA about the value of these numbers.

Lately, while working with my algebra and geometry students, I’ve been informally “polling” them to see if they can estimate the value of these two radical quantities. I have received the following “interesting” exit-poll replies:

On the value of the square root of 2:

“4. No, 2. Or something like 2. Wait. Is it negative 2?”

“about 1”

“a little less than 2”

“it’s irrational, right, so it doesn’t really have, like, a normal value”

“isn’t it just 2?”

– – – – – – – – – –

On the value of the square root of 3:

“I really don’t know”

“about 3”

“what is it, like 9?”

“isn’t it the same as 1 over 3?”

“basically the same as 1”

– – – – – – –

This all goes to prove the old saying: “You never really know what people are thinking until you ask.”

Scary, isn’t it?

I’m a big believer in understanding as much as possible about a subject that you’re studying. And I believe that it is better for students to have a clear sense of the value of these two numbers as possible.

When I explained the value of the square root of 3 to one student recently, he said, “That’s good to know because it helps you know that it all makes sense.” By “it all,” he was referring to the relative values of side lengths in a 30-60-90 triangle, the one we were looking at.

I’m bringing this whole issue up for two reason:

1)  To point out that there are often large chunks of information that we as educators assume students have, while those chunks are, in fact, missing, and …

2)  To suggest a few ways to help students grasp the value of these two numbers.

Regarding point #1, I plead an unfair advantage. Working as a tutor, I am privy to a certain degree of mathematical-candor that the typical classroom teacher does not have access to. But since I do have this access, I just thought I’d put out the warning, for what it is worth.

Regarding point #2, here’s a suggestion.

Inform students that the square root of two is approximately 1.4, and the square root of 3 is about 1.7. Keep it simple by going this far but no further (into decimal-land).

But do go one step further by helping students grasp the following: a) both numbers are greater than 1 but less than 2, b) root 3 is greater than root 2, and c) root 2 is just a bit less than one and a half; root 3 a bit more. Don’t make me tell why you need to explain this, o.k.? Finally, provide problems that help students integrate these values, long-term.

The following can work. Give students a worksheet with special triangles that have just one side length given. Tell students that, without using a calculator, they are to find the lengths of the missing sides. You don’t have to make the problems extra hard. For example, it’s helpful to show students a 45-45-90 triangle with equal legs of 10 feet. Just helping students see that the hypotenuse would measure about 14 feet is, in itself, a worthy goal. If students become capable of estimating the lengths of the sides in the special triangles, they will be that much farther along in their understanding than …  (more than the square root of 3) students.

30-60-90 triangle's side ratios

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