a and b multiplied

When we first begin to learn about multiplication it can often be a help to get our the box of wooden bricks to explore the idea that multiplying 2 by 3 is the same essentially as building a rectangle of three rows of two bricks like this:

Then we probably move on and relegate the insight to the past, which is a pity, since it has a lot to offer as we move more deeply into multiplication involving algebra. Anyone who has been fortunate enough to brush up against Montessori education will know that very young children can quickly become comfortable with the idea of calculations like ${{(a+b)}^{2}}$ using flat tiles or even ${{(a+b)}^{3}}$ using colourful blocks to build a cube.

But why bother? After a while, most students become familiar with the idea that multiplying out ${{(a+b)}^{2}}$ results in ${{a}^{2}}+2ab+{{b}^{2}}$ , so what is the point of messing about with tiles or drawings. Well for me at least, the answer is twofold: firstly I love to see the parallels between the realm in which mathematics operates and the physical world, especially because so often the mathematics casts new light on why the world is as it is. The second reason is almost the opposite, that finding ways to express something on paper often casts light back on the way the mathematics works.

That’s the case here, where the three shapes in the design cast light on three simple equations which are useful to remember as building blocks for more complicated work:

$\displaystyle {{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$
$\displaystyle (a+b)\times (a-b)={{a}^{2}}-{{b}^{2}}$
$\displaystyle {{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$

The first shape in the design illustrates equation 1), with the value of a represented by the brown bar and b by the yellow bar – the bars are only there to show the length of a and b. There is no particular significance to the lengths chosen other than to create a pleasing design.

It shouldn’t be difficult to see that what we have here is a square with each side equal to the lengths of a and b added together. And by connecting the places where a and b join we end up with two squares which represent ${{a}^{2}}$ and ${{b}^{2}}$ plus two rectangles which each have one side as a and one side as b, so their size is $a\times b$ or, in other words, $ab$ . So there in one simple shape you have the physical reality behind the equation $\displaystyle {{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}$ .

The second shape is little more messy, but there’s a good reason.

When you’re multiplying together terms which involve minus signs you often end up with a result in which the same value is present in different quantities as positive and negative items. In this case, going the long way around multiplying $\displaystyle (a+b)\times (a-b)$ , where we actually multiply all the terms one by one gives the result $\displaystyle {{a}^{2}}-ab+ab-{{b}^{2}}$ . Expressing that as $\displaystyle {{a}^{2}}-{{b}^{2}}$ is quite correct but we lose some information about the way the thing works. So the second design is an attempt to show the correct final result (which is the area surrounded by the dotted line) without losing those pluses and minuses. What the design says to us is that if we were to take the whole rectangle $\displaystyle a\times (a+b)$ and divide it up as shown (stay calm) then we end up with the following rectangles: $\displaystyle {{a}^{2}}$ and $\displaystyle ab$ . to get to $\displaystyle (a+b)\times (a-b)$ we have to subtract $\displaystyle ab$ and $\displaystyle {{b}^{2}}$ , giving us $\displaystyle {{a}^{2}}-ab+ab-{{b}^{2}}$ which is the same as $\displaystyle {{a}^{2}}-{{b}^{2}}$ . Once again, expressing the calculation as a picture gives us a better idea of what is going on than simply writing down $\displaystyle (a+b)\times (a-b)={{a}^{2}}-{{b}^{2}}$ .

(Just as an aside, even if we had drawn $\displaystyle {{a}^{2}}-{{b}^{2}}$ in the shorthand form

all the information we need is actually there, since the rectangles involved are $\displaystyle {{(a-b)}^{2}}$ and two copies of $\displaystyle b\times (a-b)$ . Multiplying all those out we get $\displaystyle {{(a-b)}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ and twice $\displaystyle b\times (a-b)$ = $2ab-2{{b}^{2}}$ . Unsurprisingly, adding it all together gives ${{a}^{2}}-2ab+2ab+{{b}^{2}}-2{{b}^{2}}={{a}^{2}}-{{b}^{2}}$ .)

If you’ve coped with everything so far, the final shape in the design should be obvious. It shows that ${{(a-b)}^{2}}$ is actually made up of ${{a}^{2}}$ with two copies of $b\times (a-b)$ and one ${{b}^{2}}$ subtracted. Multiplying all that out and adding them all together gives ${{a}^{2}}-2\times ab-2\times (-{{b}^{2}})-{{b}^{2}}={{a}^{2}}-2ab+2{{b}^{2}}-{{b}^{2}}={{a}^{2}}-2ab+{{b}^{2}}$ which is correct.

With a little imagination we can get straight to the final form if we recognize that there are actually two ab rectangles meeting in the bottom right corner. If we take them both at full value to give us $-2ab$ then we have taken off too much and we have to add back the square in the corner, which is ${{b}^{2}}$ .

This design is reminder that expressing equations on paper or in physical shapes can often demystify them. Perhaps you’d like to try ${{(a+b+c)}^{3}}$ on paper or get out the woodworking tools (or the 3D printer) and produce one of those cubes beloved of children in Montessori nurseries.

The home of d/dx art

a and b multiplied

Where mathematics becomes an object of desire.