One Way To Get Familiar With More Square Numbers

Using the algebraic formula for the square of a sum

At the high school level, most students know all square numbers between one and one hundred: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.
That is, the squares of all numbers from 1 to 10. Many students also know 121 is the square of 11 and 144 is the square of 12.
That is about how far most people go memorizing square numbers. Relatively few people know 196 off the top of their head when asked: "How much is 14 times 14?"

I find useful the following exercise for students who are learning topics such as the quadratic formula and how to factor trinomials:
I remind them of the formula for the square of a sum:
( a + b )^2 = a^2 + 2ab + b^2

Then I suggest to them applying it in the following way to figure out the squares of numbers between 10 and 20:

(14)^2 = (10 + 4)^2 = 10^2 + 2(10)(4) + 4^2 = 100 + 80 + 16 = 196

(17)^2 = (10 + 7)^2 = 10^2 + 2(10)(7) + 7^2 = 100 + 140 + 49 = 289

Working out a few of these concrete, numerical examples helps students to see a general pattern and become more familiar with both the formula for the square of a sum and, at the same time, with squares of numbers between 10 and 20. This feels like a natural expansion from their previous knowledge of squares of numbers between 1 and 10.