It’s a Square, Square World
Square numbers are very important in mathematics, and virtually at all levels from arithmetic, to algebra, geometry, calculus and statistics. So I feel young students should be exposed to them as early as possible, and that some of them should be memorized right along with the “times tables”. In fact, at the grade I teach (7th grade, Pre-algebra) I encourage — often require — that my students memorize the squares of all the numbers from 1 to 31.
Now if that seems like a tall order, I would argue that it is not, that is, after you analyze the situation carefully. First, the numbers from 1 to 10 are already in the standard times tables; and 202 (20-squared) and 302 (30-squared) should cause no great stress. So over a third can be classified as basic or very easy. The rest?
Well, I have dreamed up an organizational schema that I think makes the memorization less painful. I divide the numbers from 10 to 31 into families. Here are the families that I feel do the job nicely:
The reasoning behind some family names should be obvious, like Palindromes, 25-enders, Too-easy-to-worry-about, etc. Others may need a little clarification. Same digits means the same digits are present in the pair shown, just arranged differently. Reversals are squares in which the products and the original numbers are reversed. Consecutive Digits are squares that contain digits that could be arranged in normal counting order, but are not in that order in the number [i.e. 324 has the digits 2, 3, & 4]. Since a year has 365 days, the square of 19 gives a value that is Almost-a-year. Finally, since I could not fit 17-squared into any category in a nice manner, I just thought of it as something Left-over. Anyway, this manner of grouping the squares has helped me; I hope it might help others. It is in that spirit that it is presented here.
Mention was made above about how important squares are in advanced mathematics. Every student of the Pythagorean Theorem knows this all too well; but that topic will be discussed in another article. And when you study standard deviation in statistics, you realize how fundamental the concept of squares is. But primary students, the intended audience for this article, need an activity that uses squares in a way that they can relate to. And the Pythagorean Theorem, let alone statistics, is a little too high for them at this stage of their mathematical development. Enter at this time an idea from classical number theory: the four squares problem.
4 Squares Can Do It
Some time ago mathematicians discovered this astonishing fact: All whole numbers can be expressed as the sum of no more than four squares. Here’s what that statement means in shirt-sleeve English.
Pick any whole number. You can find 2, 3, or 4 squares (but more than 4 are not necessary) that will "add up" to the number you chose. Here are a few examples: 25 = 9 + 16 50 = 1 + 49 72 = 4 + 4 + 64 99 = 81 + 16 + 1 + 1 120 = 100 + 16 + 4 175 = 169 + 4 + 1 + 1
[Note: The statement above does not say that numbers can't be formed by using more than 4 squares; just that more than four squares are not required. There is an important difference involved.]
Some numbers have more than one way to be formed. The example of 25 above is a case in point. A second way is
25 = 16 + 4 + 4 + 1.
For 72, we can give 36 + 36. For 120, there is 64 + 36 + 16 + 4. And for 50, notice this way: 9 + 9 + 16 + 16!
So you see, there is “more than one answer” to our problem, something that young kids are not exposed to sufficiently in their math lessons.
Now we have established a basis of what we can ask the students to do: (1) to find at least one expression for all the numbers over a given range, say 1 to 100, or 100 to 120, etc.; and (2) to find as many expressions as possible — preferably ALL possible expressions — for a given number, say 500 or 1000.
I remember once the great deal of enthusiasm that was generated by a group of 4th graders who were trying to find all possible ways to do 1000. It was very gratifying that something so basically simple provoked so much work and discussion.
Try it with your students and see for yourself.
One way to broaden the work on this concept is to consider the types of squares that are used in the expression. Some expressions are composed of repetitions of a given square, like 99 above where the “1″ was repeated. This is bound to happen sometimes, like 7 = 4 + 1 + 1 + 1, and there is no other way to make 7. So the numbers can be classified as (1) those who require a repetition, or (2) those who may have a repetition but also have an expression with distinct squares. 25 is an example of that. A third class could be those numbers which have more than one expression possible, yet all expressions are composed of distinct squares (no repeats anywhere, within the expression, that is).
These are certainly ideas than can be investigated over time with the elementary student. They encourage systematic thinking, recording results carefully, and cooperating with others on a large project. I would also recommend the use of a calculator here, especially to speed up the work and ensure a greater degree of accuracy in the computations.