Estimating Square Roots—Kramer Method

When I was in fifth grade I was fascinated with square roots. The square root of perfect squares are easy if you know your squares such as 1x1=1, 2x2=4, 3x3=9 and so on. Since the square of 3 is 9, the square root of 9 is 3. They are opposites: square roots are the opposite of squares; addition is the opposite of subtraction; multiplication is the opposite of division. Actually, they are inverses but it is important to identify their connections.

I was much more fascinated with square roots of numbers that are not perfect squares like the square root of 10. Of course, the square root of 10 is between the square root of 9 and 16, two perfect squares equal to 3 and 4. Therefore, we know that the square root of 10 is between 3 and 4. Because it is between 3 and 4, we know that the square root of 10 is 3 and a fraction. The algorithm I came up with in 5th grade would estimate square roots of non-perfect squares to a few hundredths and this is what I shared with our first grade and up this week. The fun part of this algorithm is that it is not on the internet as a method discovered by some famous mathematician in 1854. It is so simplistic that I am sure I am not the first to discover it but it is not marketed as such. I raise this to inspire the Mathletes to come up with algorithms of their own, knowing that it most probably has already been discovered but maybe not. The possibilities are endless. All it takes is the willingness to play with numbers and look for patterns.

The square root of 10 is 3 and a fraction, the numerator (top of the fraction) of which is the difference between 10 and 9 (the perfect square less than 10; so 10 — 9 = 1) and the denominator (bottom of the fraction) of which is the difference between the two perfect squares between which is 10 (16 — 9 = 7). So, the square root of 10 is 3 1/7. To check the accuracy of the estimate I taught the third graders and up to use graphing calculators but this can be done on a scientific calculator. Every electronic device today has one (iphones have simple calculators until you turn them 90 degrees revealing a scientific calculator). First key into your calculator the square root of 10 to get a number 3.1622 ……. This is an irrational number whose decimals will go on indefinitely without a repeat or termination. My estimate of 3 1/7 is 3.1428.. with a repeat but four decimals is enough. Comparing the two numbers 3.1622 — 3.1428 = .0194. So my estimate is within less than 2 one hundredths of the answer. As the square becomes larger, the estimate is closer to the answer. For example, the square root of 101 is actually 10.0498; my estimate is 10 1/21 or 10.0476, a difference of .0022 or 2/1000 or 22/10,000. 

It is a lot of fun to try these mentally. It is an exercise in knowing your squares and subtraction. It is also a wonderful introduction to fractions as most children that are told that a solution is between 3 and 4 will say 3 1/2. But clearly the square root of 10 is way closer to the square root of 9 than the square root of 16. 

The attached pdfs will allow the children to practice this algorithm and also to check their answers for accuracy on a calculator.

The kindergarteners practiced with square roots by creating a clock of square roots. So instead of 1-12 for the hour hand, they used the square root of 1 through the square root of 144. To go a little further, I challenged them to create the second hand by fives with square roots as well. See the pdfs attached.