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The geometry of arithmetic sequences

MICHAEL ANDERSON: So let’s take a look at arithmetic sequences in a little bit more detail.
PAULA KELLY: OK. So if we start off with these 3 blocks– OK. So we’re going to start a pattern that goes up by the same amount each time, as we always see in an arithmetic sequence. So this is my first pattern in my sequence. If I have my next pattern in the sequence– so rather than having our 3 blocks, this time we have 5 blocks.
MICHAEL ANDERSON: OK, so it’s got larger by 2.
PAULA KELLY: Absolutely. OK. So how would our next pattern of the sequence look?
MICHAEL ANDERSON: Well, we started with 3, and then we’ve got 5. So the next pattern, or the next term in the pattern, should probably have 7 blocks. And I think it’ll look a little bit like this one.
SPEAKER 2: OK. So we can see our pattern is increasing each time. Each time, the legs of our pattern get slightly longer. So is this the only way we could think about how these patterns look?
MICHAEL ANDERSON: Well, the typical maths questions might be, can you figure out what the next term is, what the fifth term, what the hundredth term, and kind of come up with that general rule? Based on the fact that each time, each sequence is increasing by the same amount. So we might say, well, in this case, we’ve got our first term, our second term, our third term. And we might be interested in the general term, the nth term, to be able to figure out a rule or a pattern to see how it increases.
MICHAEL ANDERSON: So for this one, we’ve got 3 blocks, then we’ve got 5 blocks, 7 blocks. And we don’t really know what the kind of general rule will be, but we know that to get from one term to the next term, we just add 2 every single time.
PAULA KELLY: OK. OK. So as well as seeing it as one complete pattern, we’re looking at how it grows.
PAULA KELLY: Another way to have a look at it, if I extended the pattern– so we take this away– what’s different about how this one looks now?
MICHAEL ANDERSON: So we’ve got some different colours. I suppose the way I saw the pattern growing was that corner one, the blue one in this case, stays the same every time. And then we have a length of 1 going up and a length of 1 going out for this one. And in the next one, that length, the pink blocks, grew by 1 each.
PAULA KELLY: So we should have– again, we keep our blue fixed.
PAULA KELLY: And these ends are extending by an extra block each time.
PAULA KELLY: So rather than having just 1 on each end, we’ve got 2 on each end.
MICHAEL ANDERSON: Yeah. So if I was to write that down in some kind of pattern, I suppose we’d have that 1 block there, and then 1 above it, and 1 below it, in the first term. In the second term, we still have that 1, but then we have 2 coming up and 2 coming out. The third term, I predict, will be 1, 3, and 3.
PAULA KELLY: OK. Well, quite conveniently, we have another one. So let’s have this one.
PAULA KELLY: And again, our pattern works. We have our 1 in our corner, we have 3 either end. So our pattern remains the same. We still have 7 blocks.
MICHAEL ANDERSON: And I think thinking about it this way helps us with that kind of general idea that no matter what term we have, we’re going to have that 1 blue block. And then if it was the first term, the pink lengths were just 1. In the second term there were 2. In the third term there were 3. So in the nth term, in the general term, they’re going to be n. So if this was the 10th pattern– we’ve gone up by 2 all the way to the 10th term– we’d have 1, and then 10, and then 10.
PAULA KELLY: That’s a way of trying to predict how many cubes we’d need, how it would look, without physically making 10, 100 in our pattern.
MICHAEL ANDERSON: Yeah. But there are probably quite lots of different ways of thinking about these sequences. Did you see it growing any differently?
PAULA KELLY: Well, conveniently, yes. So we have another one. If we had some green this time– so can you tell me why I’ve chosen these colours?
MICHAEL ANDERSON: OK, so I think we’ve got two dark green ones on the bottom and then one on the top. So our first sequence would probably be a 1 there and then a 2 as the base.
PAULA KELLY: Fantastic. So next in our sequence, if we had– how would that look?
MICHAEL ANDERSON: OK, so with these 2 legs, they’re each growing by 1 each time. So the sequence overall is growing by 2, but we can think of each leg, almost, as growing by 1 each. So this 1 has gone to a 2, and the 2 has gone to a 3.
PAULA KELLY: So our next one should have 3 at the top and a 4 on the bottom.
MICHAEL ANDERSON: Yeah. We can almost see this arithmetic sequence itself. So 1, 2, 3, going up by the same amount, and 2, 3, 4, going up by the same amount.
PAULA KELLY: OK. We’ll just see how that looks. So we should have our last one, as we’re saying, 3 going up, 4 going across.
MICHAEL ANDERSON: Yeah. And in our first term, we had a length of 1 on the top, the light green ones. 2 was 2, 3 was 3. So n, our general one, is probably going to be n. And then the bottom one, the base, the dark green, was always 1 more than that. So that would be n plus 1.
PAULA KELLY: OK. So another way we could see this, then, if we took these away– if I made a very small change, so if we made it look a bit like this, how is that different?
MICHAEL ANDERSON: Well, it’s probably really similar to the previous one. But instead of seeing the base as the starting point with the dark green, you’re almost seeing the length going up as the starting point. So in this one we’ve got a 2 going up and then a 1 next to it.
PAULA KELLY: OK. So with that pattern to continue, if I use the same blocks but a different way around– so going up, we have 3 now. Going across, we have 2.
MICHAEL ANDERSON: So I can represent that as the second term, as a 3 plus a 2.
PAULA KELLY: And then finally, we should have a 4 going up, 3 across.
MICHAEL ANDERSON: So that’s a 4, a tower of 4 going up, and then 3 next to it. So I think my general rule, again– what we seem to have is something like– and I’ll put this bit in brackets– n plus 1, and then another n coming out of the side of it. So for the 10th term, we’d have a tower of 11, and then 10 going across.
PAULA KELLY: OK. One more way we could see it.
PAULA KELLY: OK. So if we extend this very slightly– so we had our first pattern like this. If I add that just in there, how does that help us with the sequence?
MICHAEL ANDERSON: Oh, OK. So I think what we’ve got now is a square. So that’s a 2-by-2 square. The red is the bit that we’re actually interested in. So we almost take away that orange square?
MICHAEL ANDERSON: So if I was to think about drawing or writing this out, it would be probably something like 2 squared take away 1 squared.
PAULA KELLY: Because this whole lot would be 2 by 2, so 2 squared. And this small orange to subtract is 1 by 1.
MICHAEL ANDERSON: OK. Does that work for the next term?
PAULA KELLY: Let’s have a look. So if you had our next one, if it’s all red, we’d have a 3-by-3 square–
MICHAEL ANDERSON: –which is the sequence we were looking at.
PAULA KELLY: So you’ve got a 3 squared. And then our orange to subtract is a 2-by-2 square, so we subtract 2 squared.
MICHAEL ANDERSON: So if I was carrying on this pattern, we have a 3-by-3 square, and we’re taking away a 2-by-2 square.
PAULA KELLY: OK. One last one. So if we have this enormous one– so we could have our– now we have a 4-by-4 square, and we’re going to subtract a 3-by-3 square.
MICHAEL ANDERSON: OK, so that was 4 squared take away 3 squared. And I think the key to all of these arithmetic sequences is to see how they’re growing, and how much they’re going up by each time. So if I look at these number patterns, with the first number we had 2 squared, 3 squared, 4 squared. So that seems to be going up by 1 each time. And then we’re taking away 1 squared, 2 squared, and 3 squared. That’s going up by 1 each time. So our general rule seems to be, whatever the term we’re at, we add 1 and square it– so the third term, it was 4 squared– and then we take away n squared.
MICHAEL ANDERSON: Now, the nice thing about these is that we’re describing the same sequence in lots and lots of different ways, but we always seem to be getting the same answer. If we look at our general rule, we’ve got 2n plus 1, 2n plus 1, and 2n plus 1. And if we expand all that out and simplify it, we’ll end up with 2n plus 1.
When students learn about arithmetic sequences they are very often just taught the procedure for finding the position to term rule: the nth term.
It can help students understand the mathematical structure of arithmetic sequences if they explore how arithmetic sequences grow using interlocking cubes.
Students can be creative, showing different ways of explaining how the sequence grows and how the position to term rule, the nth term, is generated.
In this video, Paula and Michael model what a lesson may look like when exploring the geometry of arithmetic sequences to find the nth term of the sequence. Watch the video and then attempt the challenge below.


Describe how the following sequence grows in as many different ways as you can.
Example sequence of three terms represented by interlocking cubes in a U shape. The shapes are: 1. five cubes arranged with two cubes as uprights and one between them at the base. 2. eight cubes arranged with three cubes as uprights and two at the base. 3. eleven cubes arranged with four cubes as upright on each side and three at the base
For each way show how the rule for the nth term can be generated.

Problem worksheet

Now complete question 5 from this week’s worksheet.

Teaching resource

  • Arithmetic sequence puzzle: The first puzzle involves three intersecting lines with circles marked. The challenge is to create arithmetic sequences that will fit each line. The second challenge is harder since there are four lines and more intersections.
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