Monday, October 20, 2008

Using models to solve maths problems

I know there are some mums here whose kids are little mathematical whizzes. For the rest of us, maths can sometimes be a struggle, especially since many of the concepts are now taught differently from when we were in school.

In this post, I want to expound on the model method. The model method is used extensively in Singapore primary schools - it has largely taken over algebra which is no longer taught at the primary level. In my opinion, whoever thought up the model method is a genius because it allows seemingly convoluted word problems to be broken down visually, making it easier to solve (especially since most kids are visual learners). No memorising of complex formulas required.

The very basic assumptions of drawing a model are: 1) every unknown part should be equal, 2) the unit of measurement for all aspects of the model should be the same, 3) almost always, the key to solving the problem is finding out what is the value of a single part of the model.

This is a reasonably straightforward question on Fractions that you can easily solve using a model:

Margaret is 5 years older than Glenn. 1/2 of her age is equal to 2/3 of Glenn's age. How old is Glenn?

This is what Lesley-Anne did - first she drew Margeret's strip longer than Glenn's. The part that is longer is 5 units (since she's 5 years older than Glenn). She then divided Glenn's strip into 3 parts - 2 of these parts exactly cut across half of Margaret's. Since 2/3 of Glenn = 1/2 of Margaret, she knew that 1/3 of Glenn = 1/4 of Margaret.

Looking at the model, she could see that the portion marked "5" is the remaining 1/4 of Margaret. Each unit is 5 and Glenn is 3 units so 5x3 = 15.

Answer: Glenn is 15 years old. Easy peasy.

Here's a slightly more complicated problem on the topic of Money:

A shirt costs $16 more than a belt and a belt costs $65 less than a pair of shoes. Alex bought 2 belts, 2 shirts and 1 pair of shoes. He gave the cashier $300 and received $88 in change. What was the cost of the pair of shoes?

This (on the right) was the model Lesley-Anne drew. From here, you can easily see what Alex bought, that the belt is the cheapest item, each of the shirt and shoes cost $16 and $65 more than the belt respectively.

First, she found out what Alex paid altogether:

$300-$88= $212. This value represents all the parts in the model.

Remember what I said in my point 3 above, that we need to find out what is a single part of the model (ie one of those blank rectangles). So she took away the unequal parts, ie $32 ($16x2) and $65:

$212-$32-$65=$115. This value represents the 5 remaining blank parts.

Each part is equal and from the model, Lesley-Anne could see that the cost of the shoes is one part plus $65. So she did this:

$115÷5=$23 (value of one part)

Answer: Cost of shoes is $88.

The model method is especially helpful for sums that call for transferance of values. These are sums that usually make my head spin just reading them. But once you employ the model method, you'll be surprised at how easily they can be solved.

This is a problem sum in an assessment book under the topic of Volume:

Bottles A, B and C contain 4.34 litres of grape juice altogether. 1/5 of the grape juice in Bottle A is transferred to Bottle B. After that, 1/5 of the grape juice in Bottle B is transferred to Bottle C. After that, Bottle A has twice the amount of grape juice in Bottle B and Bottle B has twice the amount of grape juice in Bottle C.

a) How much grape juice was transferred from Bottle A to Bottle B?
b) How much grape juice was in Bottle C at first?

Your head spinning yet? I know if I'd seen this question before I'd heard about the model method, I would have thrown in the towel right away and muttered something about MOE being mad. This is a p4 topic, by the way.

But since I'm now a model mum (hehe), no sweat! This is the model Lesley-Anne drew:

This represents the values AFTER all the transferring of grape juice, ie Bottle C has 1 part, Bottle B has 2 parts (double of Bottle C) and Bottle A has 4 parts (double of Bottle B). All the parts together are 4.34 litres.

Because 1/5 of Bottle A's grape juice was poured out (meaning it has 4/5 left), Lesley-Anne could immediately conclude that each of Bottle A's parts contains 1/5 of its original amount of grape juice (as she has indicated on the model). So she did this:

4340ml÷7 (parts)=620ml.

Answer: 620ml was transferred from Bottle A to Bottle B.

From the model, she could see that Bottle C has 620ml now. But she needed to take away what was transferred from Bottle B. Bottle B now has 4/5 of its original grape juice, meaning that each of its 2 parts stands for 2/5 (indicated in model). So this is what Lesley-Anne did:

620ml÷2=310ml (value for 1/5 of Bottle B's original grape juice)
620ml (Bottle C's current volume of grape juice)-310ml (transferred amount)=310ml

Answer: Bottle C had 310ml of grape juice at first.

Lesley-Anne solved this problem in under 5 minutes, all thanks to the model method.

The important thing is to be able to draw the model correctly, which takes time and practice. Sometimes, you need to convert units or get rid of one unknown before you can draw the model, but if you keep all the assumptions I've listed above in mind, you'll quickly get the hang of it. Then you just need to practise deciphering and understanding the model. Once you master this, I'm sure you'll never want to go back to algebra.


Lilian said...

Thanks! Please share more. All workings too :) I'm gonna post some questions we've attempted and maybe Lesley-Anne can help to show us the "right" way to do models, cos I'm pretty sure the way we did it is not as simple as it should be. TIA :)

Alcovelet said...

Hi Monica, I know what you mean when you talk about how great the model method is. I was deadset against RK doing it initially cos I thought it would channel his thinking into a "method", which means, poof! No more thinking! But I've since realize it's a way of visualizing math, and he's good at that. That's why I really like Singapore math. Dunno if I'll be saying that when it's his turn for the PSLE!!!

monlim said...

Lilian: We're not experts in models but we'd be happy to try! (notice I said "we", sometimes the sum is so complicated we have to work it out together)

Ad: like you, at first I thought "it's yet another new-fangled method" until I saw what Lesley-Anne could do with it. It's really ingenious!

Lilian said...

Oh, definitely "we"; it's the same here too. But I'm not being much help at upper primary work; maybe being a hindrance. And the answers provided by some of these assessment books aren't exactly great either. Some solutions are so convulated; and some are basically algebra disguised as models. Anyway, thanks a bunch for sharing, it's really useful.

Anonymous said...

Lil: ""algebra disguised as models""

Oh how I've so often been tempted to do this!!


Anonymous said...

yay I did it!

Restores my confidence abit after being outsmarted by Brian :-(


monlim said...

No worries, I don't think Brian is your typical 11 yr old :D Lilian, that's a compliment!!

Lilian said...

YY, which question did you solve and did you curi tengok the answer first? hahaha, I'm worse lah, curi tengok answer and Brian tried to explain to me, still catch no ball. I only saw light for the ratio C:W:M question last night.

Anonymous said...


No lah, this one here easier than the one Brian solved mah. That one on your blog huh, draw model until siao also cannot, at last looked at answer a little bit & then do the rest.

Btw that 'politically incorrect' website, also has a discourse on why boys are better at math than girls. (oops, quickly run away)

cheh... I'm feminine, that's why!!


Related Posts Plugin for WordPress, Blogger...