Algebra - Smarties and other unknowns

After our yarn-up on Friday we started looking at what algebra is about on Monday, using the excellent resources provided by my Maang colleagues.

The core of the idea was to start with concrete and then visual examples that make it clear why we are using a pronumeral. We begin with boxes of smarties. With the boxes unopened, the number of smarties in the box is unknown. We can call it b. Then what does b+1 look like? What does 2b look like? What about 3(b+2)? Students had whole boxes of smarties and spare smarties to make all these arrangements and then draw them. Then each student opened their box of smarties and knew the value of b, which could then be used to find the total number of smarties in each of the pictures they drew for the expressions.

The next worksheet involved trains with an unknown number of people, n, on them, and some extra people (possibly people waiting on the platform). Then mouseholes with an unknown number of mice in them and extra mice outside. Through all three of these there were expressions with matching visuals, and then substitution of different values for the pronumeral.

In hindsight it seems simple but I was amazed at how well they understood the concepts every step of the way. The bit that really blew me away was how well they understood the use of brackets. I've never thought of factorising and expanding as something you could do in a concrete way, but here it was, and they were instantly successful at it, without any examples or instructions. After all, if b is an unknown number of smarties in a box, then 3(b+2) is obviously a box with 2 spare smarties drawn 3 times. But many of them didn't even draw them in groups, they expansion was clear to them - that's 3 boxes and 6 extras.

The trains worksheet included this one: "2(n+5)-4". The ease with which the students worked this out impressed me. "Well that will be two trains and ten people, and so with the minus 4 it's two trains and 6 people." Then they drew it. While I stand there dumbfounded. All of this expression writing also gave us the opportunity to discuss "style", 1k vs k, 2k vs k2 vs k+k, etc.

Thursday we followed up with more worksheets on the same idea, and some simplifying expressions and more substitution. They cruised through it.

Now we are moving on to equations, and starting our video project. More on that later - and fingers crossed it works out!