A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which eighth-grade students have to locate the square root of 10 on a number line.
The square root of 10 is an irrational number, meaning it cannot be represented as a fraction: its decimals do not repeat.
On a calculator, it comes out to 3.1622776601683793319988935444327…..
That is just a little greater than 3 but less than 4, and since only one of the choices on the number line is higher than 3 (to the right of 3) and lower than 4 (to the left of 4), that one has to be the answer.
The other dots are at about 0.9, which is less than 1 and therefore can’t be the answer; 2.6, which is less than 3, not greater than 3; and at 5, which is the square root of 25, not of 10.
The steps: First, memorize all the perfect squares from 1 to 15, and then memorize those for the 5’s after that. Here they are:
| Number | Perfect Square |
| 1 | 1 |
| 2 | 4 |
| 3 | 9 |
| 4 | 16 |
| 5 | 25 |
| 6 | 36 |
| 7 | 49 |
| 8 | 64 |
| 9 | 81 |
| 10 | 100 |
| 11 | 121 |
| 12 | 144 |
| 13 | 169 |
| 14 | 196 |
| 15 | 225 |
| 20 | 400 |
| 25 | 625 |
| 30 | 900 |
| 35 | 1,225 |
| 40 | 1,600 |
Once you have those memorized, then you will be able to estimate the square root of any number that is put on a standardized test.
For example, if you are asked to estimate the square root of 200, you know the square root of 225 is 15, and the square root of 196 is 14. Therefore, the square root of 200 will be in between 14 and 15, and probably closer to 14 than to 15, since 200 is closer to 196 than it is to 225.
Locating a decimal number on a number line is a lower-level skill than estimating the square root of a number, and we will discuss this skill in sixth-grade questions.