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Here’s a nice problem to introduce simultaneous equations and Algebra.  I saw this on the singaporemathsplus.net blog here

There are 100 chickens and rabbits altogether. The chickens have 80 more legs than the rabbits. How many chickens and how many rabbits are there?

In GridMaths there are a couple ways to approach simultaneous equations visually – sometimes you might use a different color bean to represent each variable, lets say an orange jellybean and a brown coffee bean, and you don’t know how much each weigh, but you might be able to figure it out.  This is nice, as your not tempted to add apples and oranges together [ or jellybeans and coffee beans ].

Another way is to have rectangles or lengths and color or shade them differently – this is easy in GridMaths, you just draw color rectangles, or Cuisenaire Rod lengths.  Remember to make the point this is not to scale, you cant yet measure off the diagram, the length is standing in for the real length which youll arrive at.

Here’s a sheet with the C chickens and R rabbits problem [ Chickens in green, Rabbits in orange, known lengths in blue ] :

Apparently this is called the ‘Bar method’ in Singapore maths – I think its good to have a name so teachers, parents and kids can talk about this approach.  Someone asked me about an earlier post, whether it was the Singapore method.. and actually no, I’m not familiar with that method or series of books.. but from what I hear it seems quite good in terms of using visual keys to get the concepts across.

Here’s another problem solved using pink! jellybeans and brown coffee beans to represent X and Y :

A good order to broach these ways of looking at simultaneous equations  is :

• a word problem
• jellybean approach
• algebra / variable names
• linear graph intersection

Likewise, I think that using rectangles for multiplication is a huge thing that isn’t done enough.  This approach can help reach some learners who see algebra as just wacky rules with no meaning.  The great thing about this Visual representation is its easy to work with.. and leads directly to algebra.   Once you’ve done a couple, you can just use letters instead of colored lengths and everything works the same way.  So its seen as an efficiency, to save making the diagram each time.

I was first introduced to simultaneous equations through one of the books of W.W.Sawyer a great teacher of math teachers.  I read his book as a youngster and so got interested in Calculus.. he had a wonderful way of making things simple and interesting.  He talked about how the average speed of a car over its entire journey was not that important, if you happened to get hit by the car just when it was travelling at its maximum speed.  In court you’d want to know the average velocity of the car in the few meters and seconds before the event.. this led directly to ds/dt type discussion.  Awesome guy!

I did a couple of worksheets on Multiplication, to test out the GridMaths tools.

My feeling is that multiplication should be taught with Rectangles.  So once students are familiar with counting in groups, they can play around with something like this interactive demo to get a basic feel for the distributive rule.

[ The above animation is a custom interactive widget created in javascript, using Raphael and HTML5 divs for the draggable UI.  You drag the red dots and the diagram resizes and recalculates in realtime.  Other screenshots in this post are all done using MathsGrid itself ]

The distributive rule says we can divide up A and divide up B any way we like, and the product is just all the pieces of A times all the pieces of B, summed up together.

For humans with 10 fingers, its easy to break A and B up by 1s,10s,100s, which means we need to multiply each digit of A by each digit of B, then sum them up.

So long multiplication kind of falls into place, as a way of keeping the pieces straight :

Then the Lattice method can be seen as just another convenient system for keeping all the pieces organised and lining up the 1s,10s,100s etc :

The first time I saw this Lattice method on Hacker News it looked like Magic.. but after you do one you realize it really is the same thing, just a cool way of keeping track of the pieces.