Transcript

INSTRUCTOR:

There are three versions of the think of a number problem. The second version is probably the easiest. The answer is given here. So the reasoning is to write or say a mathematical expression that represents the idea of thinking of a number, multiplying by 3, then adding 2.

So for example, we can take the idea of question mark times 3, add 2 equals 17. This could be solved simply by recalling number facts. I might know it's 5, which if you times it by 3 and add 2, it gives you 17. So the algebra is in writing the expression.

But there's also an algebraic way of solving it by focusing on undoing the sequence of operations on the number. When we do the operations, first, we times question mark by 3, then we add 2. When we undo, we start with 17, and we take them and reverse order. 17 subtract 2 is 15.

Then we undo the times 3 with a divide by 3. 15 divided by 3 is 5. So the number that Harry started with was 5. This version of the problem could be solved numerically with number facts or algebraically. And in this version of the problem, there is one specific value for the unknown. It's only 5 that gives us the answer 17.

Now, look at the first version of the problem. Here, you're just asked to think of a number and then to tell me your answer. This works quite well with the teacher asking somebody from the class and then being very quickly able to tell that person the number they first thought of. It can then be used with partners writing down and hiding their original numbers, telling each other their answers, and aiming to discover their partner's number.

Asking how many of these they can find in five minutes adds some competition to the activity. In this version, the unknown takes on a range of values, because it can be whatever the learners think of. So the algebra comes from comparing the methods used to work forwards, multiplied by 3, then adding 2, to comparing the methods used to work backwards. That's subtracting 2 and dividing by 3.

Asking the learners to work forwards themselves first gives them the experience from which they can reason about how to work backwards. Finally, in the last question here, a slightly different thing happens. You think of a number, you double it, add 6, divide in half, subtract the number you started with, and the answer is always 3, no matter what number you start with.

Here, we're getting to the idea of an unknown quantity as a variable. So how can I represent that? My number could be n. When I double it, it's 2n. I then add 6. It's 2n plus 6. I divide it in half. It's 2n plus 6 divided by 2. And I subtract the number I started with.

So I have to subtract. And I'm using n again. Subtract n, because that's the number I started with. The answer will always be 3. So I can write that as equals to 3. And the algebraic question is, why is that always true? The algebra here is used in asking learners to explain convincingly why the answer is always 3.