 In a task like this, or this, where we're trying to discern a general structure or relationship, we commonly adopt what we might call a scientific approach, whereby we generate a set of examples and then look for a rule that fits the resulting numerical data. For some tasks such as this one, this might well be the best approach. For example, it is difficult here to see how these four lines have produced 11 regions. But if we now think of the next case, that of five lines, it's relatively easy to discern the effect on the number of regions of adding this extra line. However, for some other tasks, we can also adopt what might be called a generic or quasi-variable approach, whereby we focus on a single case and look for features that are general. For example, say we want to find the number of matchsticks in this three by three matchstick square. We could structure it into four rows of three matchsticks, plus four columns of three matchsticks. In other words, there are four times three plus four times three matchsticks. Now, the number of matchsticks in each row and column is equal to the side length of the square, in this case three. And the number of rows and columns is one more than the side length, so we can write our expression like this. So if we had, say, a 20 by 20 matchstick square, the number of matchsticks would be this. And for an n by n square, the number would be this, or this. In this video, I'm going to look at how we can use a generic or quasi-variable approach to analyze a pattern of dots such as this. A nice feature of a generic approach is that we can often structure the pattern in different but equivalent ways. In turn, this provides an incentive to manipulate the resulting expressions to show that they are indeed equivalent. Though this video is addressed to colleagues in maths education, it should be possible to use the rest of this video in the classroom, moderated by the teacher, who should pause the video at strategic points to give students space to work on the task or to discuss their ideas. I trust that you, when viewing the video for the first time, will also pause the video at key moments. Here is a cross shape made of five dots. And here's a chain of these cross shapes. We want to find a rule for the total number of dots in the chain when we know the number of cross shapes. Let's consider this chain, which is made of six cross shapes. As there are six crosses, we might think that there are six times five dots, but that's not the case. Can you see why? It turns out that the chain has 20 dots. We could find this number by just counting each dot, but that could become tedious if we had a longer chain. Try to find a rule for the number of dots that is quicker than counting every dot. Check that it gives the answer 20 for our chain of six crosses. Now use your rule to find the number of dots in a chain of 10 crosses. You should get the answer 32. We're now going to look at different ways of structuring the chain of crosses. Each way of structuring should help us find a rule. The rule should all be equivalent and might well include your rule. Here we start with one cross of five dots. We then keep adding sets of three dots. We can write this expression for the number of dots in our chain of six crosses. And for a chain of ten crosses, we can write this. The number of sets of three that we add to the initial five dots depends on the number of crosses in a chain. In fact, it's one less than the number of crosses. We can show this by rewriting our expressions like this. Let's return to our chain of six crosses. Here's another way of structuring the dots. We can think of the chain as a row of six dots with a row of seven dots above it and a row of seven dots below. So we get this expression. In the same way for a row of ten crosses, we get this. Again, the lengths of the rows depend on the number of crosses in the chain. And we can show this by rewriting our expressions like this and this. We can also simplify our expression. For example, like this and this. What about our earlier expressions? Can they be written in the same simpler form? Now, look again at the chain of six crosses. Try to find other ways of structuring it and consider whether the resulting expressions are equivalent to the expression three times six plus two. See how many different ways of structuring the chain you can find. And finally, here's an alternative version of the task that you might want to try. It comes from the book Algebrable that ATM has recently announced it will be publishing.