The magic of squares

For many kids, maths is about abstract numbers. Even though they may be able to execute formulas and work out the solutions to problems, they often don't get the concept behind these numbers. That's the reason I love maths models - they enable you to have a very concrete visualisation of what the numbers represent, which really helps understanding.

Lilian's boys have an extraordinary ability to grasp mathematical concepts, fueled by her very imaginative way of teaching (and obvious love of maths herself!) In one of her older posts, she wrote about her method of explaining squares, which I thought was absolutely brilliant.

This method helps you find the answer to large squares without doing long multiplication, but really, that's not the point. (If you want to find an answer fast, use a calculator). It helps you see very clearly how squares work and how the answer is derived.

She starts off with the basic premise of a 10 x 10 square, which in pictorial form, is basically 10 columns x 10 rows (right pic). Most kids will know 10 x 10 = 100 (which is also the area of the square).




Now, say you want to find out what is 15 x 15. Visually, what this means is that you add another 5 columns and 5 rows to your basic 10 square (right pic).

And don't forget that little square in the right bottom corner which consists of 5 columns and 5 rows. Your final 15 x 15 square will look like this (bottom pic):

Now we just need to add up all the different areas of the square. We already know that the basic 10 square = 100. Each of the additional 5 rows/columns is 5 x 10 = 50. That corner bit is 5 x 5 = 25 (bottom pic). Add all of that up and you get 225. Therefore, 15 x 15 = 225.

You can do this all the way from 11 x 11 to 20 x 20, after which you use the 20 x 20 basic square. You can read more details on how Lilian got her 6-year-old Sean to work it out here.

Just last week, I came across another intriguing pattern on squares, thanks to Adeline's precocious son. He discovered while doing some multiplication, that when you multiply any two numbers that are two apart, the answer is always one less than the square of that middle number. (Ok, ok! I know that sounds very confusing!) Let me show you what I mean:

4 x 6 = 24
5 x 5 = 25

5 x 7 = 35
6 x 6 = 36

6 x 8 = 48
7 x 7 = 49

See the pattern? This is true no matter how large the number. For instance,

246 x 248 = 61,008
247 x 247 = 61,009

(I used the calculator lah, what did you think??) There's a very simple explanation to the pattern, which I will attempt to show visually.

Here is a basic 8 square - 8 columns x 8 rows (right pic).









To change it into a 7 x 9 rectangle, you essentially take one row and move it to a column.

You'll find that you have an extra square (bottom pic) because the number in the column will always be one more than the number of rows (which has been reduced from the original by one).

See? It's so simple I don't know why I never saw the pattern before. And it took a 6-year-old to point it out :P

Once again, seeing this pattern probably won't help your kids do their sums any quicker but I believe it facilitates understanding of how squares work.