Sunday, April 19, 2009

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.


Lilian said...

Feel honoured leh :)

I clicked on the link to my old post and realised it's been more than a year. Remember I told you Sean had probably forgotten about The Method. I just got him to do 34x34, and 27x27, and he drew the boxes and got the answers, so The Method is pretty concrete and does stick in the brain.

What Ad's kiddo saw was fantastic and you explained it well. What about Adeline's question about why this doesn't work if you start with a rectangle, not a square.

I asked Sean if he sees the pattern, he does but can't explain why. I then realised Brian has done non-routine sums that use this information. Eg 499x501, easy now! Just square 500 and subtract 1, you get, 249999. 699X701? Square 700 and subract 1, 489999. You've seen such questions right? Thanks to Ad's little genius, these sums are a cinch now.

Isn't Math fun? :P

monlim said...

Yes I know! Like if you know 100x100=10,000 you can easily work out 99x101=9,999. Actually I didn't understand Ad's question about if you start with a rectangle. What does she mean? As long as you work with 2 numbers that are 2 apart, it should work right?

It's amazing that Sean can still do it, it means he really understands the concept and he isn't just following it blindly. All credit goes to you lah, for coming up with such a brilliant method! You should copyright it and make a ton of $$, hehe.

Lilian said...

She's asking if it doesn't start with a square, but rectangle.

So if it's 23x24, it doesn't work for 22x25, well at least that's what I think she's asking. I think she and her little fella have extremely inquiring minds. Anyway, I'm sure you'll be able to come up with a good explanation hehe (pressure pressure).

monlim said...

OIC. Ok, if you follow my reasoning with the moving of a row into a column, the reason it doesn't work with a rectangle is that you'll always be left with a different number of little squares. In fact, if your number of rows are more than the number of columns to begin with, you'll be short of squares when you move the top row. Only with a square can you move the top row into a column and be always left with one extra square. Does that make sense?

Lilian said...

Yes yes, you really have a knack for putting ideas into words.

Adeline: Comprende?

Alcovelet said...


Just came back from din. Headache already, after seeing the numbers. I'll comment tomorrow after I'd had time to make sense of it. *Swaying and feeling my way out of the room*

... Wait a minute. I get it! And THAT is no mean feat! Thanks, my friends! Wow! He'll be so excited to see what I have for him tomorrow. He just loves to manipulate these patterns. Both of you are brilliant! And to explain so clearly (I vaguely recall reading that post last year, LIlian. Catch no ball!) With Mon's illustrations, even I get it.

Thanks a million (1000squared)!!

Lilian said... post not clear meh? Similar to what Monica wrote what...*sulk*

monlim said...

LOL That's what I thought too! Aiyah, I praised her mah, so she's being extra nice to me, bwahahaha!

eunice said...

LOL, all your comments are so farney. Monica, once again, you manage to simplify something that could be complicated.

I'm hopeless at Maths and luckily Sean is pretty good at it. Almost pulled out my hair when I had to help him with some homework. Luckily we are not in Sin otherwise I'll be bald or I'll send him to Monica for tutoring (said that before, will say it again :))

monlim said...

Eunice: no lah, sometimes it's just interest. I'm hopeless at science and have zero interest. Luckily I happen to like math, otherwise no way to keep up with SG work!!

Alcovelet said...

Donch like that lah, Lilian. That must have been one of the first posts I had read at you blog. I thought - this woman and her kids so brilliant one. I'm the rote math type. But maybe, one year later of training by you guys and by RK, I can read beyond two-digit numbers now!

Lilian said...

Okaylah, since you said I'm brilliant, all is forgiven :)

Monica, you should seriously write a math book, sure bestseller.

Anonymous said...

All these good concepts in practical usage is good for designing games or programming routines in case some wonder why we need to know this if we know how to calculate mentally. Thanks for sharing. Also IQ is related to mainly pattern recognition. So if kids without prior training can make out patterns, it is a sign of high IQ. just my 2 cts worth.


monlim said...

Lilian: Aiyah what best seller, it's your method not mine! Or maybe we co-write and share profits eh?

QX: yup, like in A Beautiful Mind, that guy's all about patterns and he's a genius. Ad and Lilian's kids are both gifted :)

Lilian said...

Yes yes, you write and I share profits LOL!

monlim said...

LOL! Can lah, but you and your 2 geniuses come up with the concepts first! I just write :)

Anonymous said...

Mon: Yes can tell their kids are special with the observation they produce. I was just pondering over things the other day. Apart from genes factor, it must also take gifted parents to help their kids develop their gifts, otherwise do you think they would face a limitation? I don't know Ad's approach but reading Lilian and your seems to be so. Do you want to take this topic to write something about? Would be happy to understand more insights. Tks.


monlim said...

QX: Definitely a gifted kid with a gifted parent is an advantage cos the parent is able to help the child hone these abilities through discovery activities. I'm pretty sure both Ad and Lilian are gifted themselves. Ad instinctively fuels her son's curiosity by going along with what interests him, including science experiments.

I suspect in our parents' generation, there might be many gifted kids but because these were not developed, the gifts remained dormant. I believe giftedness is innate, ie you have it or you don't. But without cultivation, the kid may not be able to show what he's capable of. Likewise, all the cultivation in the world will not enable a non-gifted child to understand concepts beyond his ability.

Just my 2cts, I'm no expert so not sure I'll be able to offer much insight on this topic but this is my gut feel.

Lilian said...

I'm not gifted.

As for Eddie, definitely not LOL! Though he likes to say the boys got their smarts from him. Scary notion that hehe!

monlim said...

Don't listen to her, she's just being modest. Of course she's gifted.

Anonymous said...

Monica: Ok, I won't listen to her..LOL. Not gifted just very gifted... bwaaaaaah.


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