Square roots: A very soft introduction to irrational numbers

This is a short lesson from my daughter’s homeschooling experience. The challenge with irrational numbers is that they cannot be displayed (precisely) on a calculator. That is also true for some rational numbers such as 1/3, but regarding those she seems to agree to stick to the fraction representation — such numbers are fairly easy to represent on a number line. So the question raised by her — do irrational numbers exist if we cannot represent them on a number line, neither on a calculator display?

First, I tried with pi — a very commonly met ratio in every circle. But then I saw that square roots offered a much softer introduction to irrationals. My daughter had already had about half a year of experience with Pythagorean theorem, so she felt more or less comfortable with right triangles and finding different proportions within them.

So the main hack was to convince her that numbers are tools to measure proportions between different lengths (not so much about measuring with а ruler, which I tried to keep out of her math arsenal from the very beginning of the math training).

I am guessing it will work differently with different kids, but my daughter got fairly quickly the geometric link between squares and square roots. Once we decided that a square side has a length of 1 unit, then she could tell that the diagonal had a lenght of exactly the square root of 2 units. For the sake of proportions, of course we do not need any units of measurement, but kids tend to believe that numbers do exist if they can be measured.

Thus a square root of 2 exists. It is quite easy to find everywhere around us, just look for a square table or a square shaped piece of paper.

Somehow paradoxical, the easiest way to think about the irrationals is imagining ratios!

For the square root of 3, we had two favorite shapes. The one I like a bit more is the 3-D figure of cube. Again, its side being 1, the internal diagonal, the space diagonal, has a length of square root of 3. I like it because square root of 2 is diagonal in a 2-D square, and it kind of makes sense that square root of 3 is a diagonal of a 3-D square (cube).

The other nice shape to reveal the square root of 3 are two circles of same radius, whose centers lie on each other. The kids must be able to draw such a figure at the age of 9 or 10 and they would enjoy finding different proportions inside — only with a compass, no rulers!

Finally, at least for our soft introduction, a beautiful number is hidden behind the square root of 5. Beside being extremely easy to represent, it is a very important element of the so called golden ratio, phi, which is exactly half the sum of 1 and the square root of 5.