Algebra (Period 1)

FINDING SQUARE ROOTS

Many textbooks seem to think that since calculators can find square roots, that students don't need to learn how to find square roots using any pencil-and-paper method. But learning at least the "guess and check" method for finding the square root will actually help the student UNDERSTAND and remember the square root concept itself!

Practice at least the first method presented here. This method, "guess and check", actually works around what the square root is all about.
The square root of a number is just the number which when multiplied by itself gives the first number. So 2 is the square root of 4 because 2 * 2 = 4.

Method 1: Guess, Divide & Check
Start with the number you want to find the square root of. Let's use 12. There are three steps:

1. Guess
2. Divide
3. Average.

... and then just keep repeating steps 2 and 3.

First, start by guessing a square root value. It helps if your guess is a good one but it will work even if it is a terrible guess. We will guess that 2 is the square root of 12. ( Which does not really make sense because we all know that 3 * 3 = 9) However, this is a great example of how this method works—even if you pick a number that isn’t near.

In step two, we divide 12 by our guess of 2 and we get 6.
In step three, we average 6 and 2: (6+2)/2 = 4
Now we repeat step two with the new guess of 4. So 12/4 = 3
Now average 4 and 3: (4+3)/2 = 3.5
Repeat step two: 12/3.5 = 3.43
Average: (3.5 + 3.43)/2 = 3.465

We could keep going forever, getting a better and better approximation but let's stop here to see how we are doing. 3.465 * 3.465 = 12.006225


Method 2- Estimating your Square root

This method requires you to know your perfect squares. You should know them up to 400 by now. Start with the number you want to square root. Let’s say √183. We know that 183 is between two perfect squares 169 and 196 – or 13² and 14² So our SQ RT must also be between 13 and 14.
196
183
169

Next, find the difference between 196 and 169 ( 196-169) = 27
find the difference between 196 and 183 ( 196-183) = 13
and the difference between 183 and 169 ( 183-169) = 14

Looking at those three numbers, you notice that 183 is almost right in the middle or half way between the two perfect squares—so we can approximate
√183 ≈ 13.5

What happens if it isn’t quite in the middle, figure out the ratio and decide what a good approximation would be.

Years ago, teachers taught an algorithm for finding square roots, but these two methods are much easier and serve to approximate square roots accurately for middle school students.

If you have any questions or comments about these two ways, post a comment here.