Continuing on from the last post, where we introduced some simple operators that operate on a number, like “add 2”, I want to show more examples of composition of these operators.
The figure above shows five pairs, each pair consisting of a part on the left and a part on the right. For each, the challenge is to put something in the part of the right (replacing the question mark) so that the machine on the right behaves the same way as the machine on the left.
I did this as an experiment with seventh graders, roughly half-way through the year, organizing them in groups of four, and asking them to work together as a group. (These kids had, by and large, what I would regard as a traditional math education in their prior grades; many of them were introduced to a Connected Mathematics curriculum in sixth grade.)
I first showed them the left part of (a), and didn’t explain the notation, though it was clear none of them had ever seen this before. I asked them what they thought would come out if a “5” was put in. None of them had any trouble with this; the whole notation, as well as the idea of the output of one box being connect to the input of another seemed to need no explanation whatever. I asked what would come out if the number coming in was 1, 2, 10, 100, respectively. No one had trouble with that either.
I then showed the right side of (a) and asked them if they could put something in the box to replace the question mark that would have the right side behave the same as the left side. All tables came up with “+ 5” for this. I asked how they knew that the two machines would behave the same. In listening to each of the tables discuss this, it was clear that their approach was to try some numbers as inputs; most used the same set of 1,2,10,100 to check the right side against the left side. At some of the tables, a student would say something like: “if you first add two, and then add three more, that’s the same as adding five at once,” but not everybody at their table would be convinced by that alone.
When I gave them problem (b) and gave them the same challenge of finding something for the right side that would have it behave the same as the left side, there were some tables that came up with “× 6”, but quite a few came up with “× 5”. Though “× 5” is wrong, in that it doesn’t behave the same as the left part of (b), it nevertheless made it clear that the students thought from looking at the boxes, and didn’t work backwards from the results of putting in 1,2,10,100. When I asked all groups to check their proposal by trying out these input values, the groups that didn’t have “× 6” quickly self-corrected and changed their proposal to “× 6”.
For problem (c), most tables settled on “-1” as the thing that should replace the question mark. At the remaining tables, they had “+ -1”. In listening to their discussions, it was clear that they all got that the number coming out would be one less than the number going in. Not all these kids knew or were comfortable with negative numbers, and yet seeing that adding two and subtracting three would result in net subtracting one – that was something they could all see. (This makes me want to look at introducing arithmetic on negative numbers through working on these kinds of operators first.)
Problem (d), predictably, give students more difficulty. One of the things I’d been curious about is whether they would decide on “+ 6” as their answer, since in all the prior challenges they could have skipped the first operation symbol altogether and still come up with the correct response. (In other words, they could have handled +2 +3 by skipping the first “+” and just looking at 2 + 3; or ×2 ×3 by skipping the first “× and just looking at “2×3”.) But none of them did this. In listening to the discussion at the tables, they were all looking at the behavior of the machine on the left by trying different inputs. At one table, they were complaining that there wasn’t any number they could add, no “+ ?” that would work no matter what they would pick for “?”. At another table, they were similarly ruling out “× ?” as a workable pattern. One table came up with “+2, ×3” as the thing that would go in the single box (which I thought was pretty inventive). Though some groups were getting frustrated, they weren’t tempted to suggest a simple (but wrong) equivalent machine.
Stopping them before they got too frustrated, I switched them to problem (e), asking them if they could find an equivalent machine that had the add and multiply in the opposite place. This took them a lot longer, and not all tables finished this problem before my 90 minute experiment with them was over. To my surprise, several groups made a graph and try to match it with the biggest tool in their tool box, their y=mx+b tool. They discovered quickly that the slope was 3, so that the multiplication box had to be “× 3”. These groups ran out of time before figuring out what needed to be in the “+ ?” box. Their approach would have worked, though it seemed they lost the forest for the trees. A few groups reasoned it out entirely from imaging what was going to happen to the number coming in, if you first made it bigger by two, and then tripled it. They could see that the two was somehow also getting tripled. For sure, they could see that the machine was still all about tripling, and so the multiplication box should have “× 3” on the right as well as on the left. They tried out “+ 6” on the right, based on the idea of tripling the extra two, and found, to their surprise, that it all worked out on the numbers they tried.