Staircase patterns Stage 3
A thinking mathematically targeted teaching opportunity focussed on exploring, representing, reasoning and generalising the patterns of staircase structures
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023
You will need:
paper or your workbook
Watch Staircase patterns Stage 3 video (6:08).
[A title over a navy-blue background: Staircase patterns 3 – part 1. Small font text in the upper left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the white waratah of the NSW Government logo.
Butcher’s paper covers a table. On it is a set of 4 blocks with a pink post-it notes label below each. The first one on the left is a red block, with label: case 1. The next one is 3 orange blocks (2 blocks next to 1), with label: case 2.This is followed by 6 blue blocks (3 blocks next to 2 blocks, which is next to 1 block) with label: case 3. Followed by 10 green blocks (4 blocks next to 3 blocks, then 2 and 1), with label: case 4.
On the right side of the blocks is a marker and a blank piece of paper.]
Hello there, mathematicians. Welcome back. Now, you might have been working on a problem we sent you recently which said if this is case one…
[The Speaker circles the first block set with her finger.]
…and this is case two…
[She then circles the rest of the block sets.]
…and case three, and case four, what might the 12th, how many squares might there be in the 12th case and how could we work it out? And some of you noticed a pattern. Yes where, a-ha, there's an increase each time. Look, if we sat here with case one…
[She points to the first block set.]
…there's one block and when we get to case two…
[She points to the label of the second block set. Then the blocks above.]
…there's the one block and two more. Mm-hm. And in case three…
[She picks up the second block set and places them over the second and third column of the third block set.]
…there's the three blocks and three more. I'll leave it there so you can see it. See this? And in case four…
[She picks up the third block set and places them over the second, third and last column of the third block set.]
…there's the four, there's the blocks that we have in case three and another four. Yes, so we have, in fact, a growing pattern, don't we? So in case one, some of you worked it out like this where you said, in case one…
[On the paper, she writes: 1.]
[Below the 1, she writes: 1]
…and in case two…
[Next to the 1 above, she writes: 2]
[Below the 2, she writes: 3]
[Next to the 2, she writes: 3. She draws a line below the numbers in the top row.]
And in case three…
[Below the 3, she writes: 6]
…there's six …
[Next to the 3, she writes: 4]
…and in case four…
[Below the 4, she writes: 10]
…there's six and four more is ten. And we wanna go…
[Next to the 4, she writes: 5-12.]
…six, six, eight, nine, ten.
Oops, 11, 12.
[She extends the line across the page.]
I'll just squeeze them into my table. And so what we looked at is that you add, yes, so the distance…
[On the numbers below the line, she draws a line connecting 1 and 3.]
…from here to here is we added two…
[Below the connecting line, she writes: +2. She draws a line connecting 3 and 6. Below the line, she writes: +3.]
…then increased by three…
[She draws a line connecting 6 and 10. Below the line, she writes: +4.]
…then increased by four. So the next one would be an increase.
[She draws a line from 10. Below the line, she writes: +5.]
Yes, by five, it's a growing pattern, isn't it. So that would be…
[Next to the connecting line, she writes: 15.]
…15 and then you would increase by six…
[She draws a line from 15. Below the line, she writes: +6.]
…to be 21…
[Next to the connecting line, she writes: 21. She draws a line from 21.]
…and then you would increase by seven…
[Below the line, she writes: +7]
…yes, to be 28.
[Next to the connecting line, she writes: 28.]
[She draws a line from 28. Below the line, she writes: +8.]
…it increased by eight. Yes, so 28 and two is 30 and six more is 36.
[Next to the connecting line, she writes: 36.]
Then you have to add nine…
[She draws a line from 36. Below the line, she writes: +9.]
…Uh huh 45…
[Next to the connecting line, she writes: 45]
…'cause if you add a ten…
[She points to 36.]
…it would be 46. And then get rid of one.
[She draws a line from 45. Below the line, she writes: +10.]
Add ten more to get 55.
[She draws a line from 55. Below the line, she writes: +11.]
Then add 11 to make 66…
[Next to the connecting line, she writes: 66.]
…and then you add 12…
[She draws a line from 66. Below the line, she writes: +12.]
…and that is 78, yes.
[Next to the connecting line, she writes: 78. She draws a line down between each column of numbers.]
So we could make a bit of a table. That is very wonky. A-ha, and I was thinking, well, that's one way that some of you used to work out how many blocks there would be…
[She circles the last column.]
…in the 12 case of the pattern. So that's one way you might have worked out what the 12th term…
[She points to the last column.]
…might look like. Now, as I was working this out…
[She replaces the paper with a blank piece of paper.]
… I started to think about or noticed some other things that I could see.
Yeah, the first thing I started to notice, actually, is that there's something special about these numbers.
[She rearranges the yellow, blue and green blocks into triangular shapes.]
Look, if I just move this across a tiny bit, what can you say about the shape that they form? A-ha, they form triangles. Yes, and in fact, if I made the fifth case…
[She places some blocks on the paper.]
...which would have five on the bottom row.
[She moves the paper over the top of the block sets. Next to the block sets, she lines up 5 blocks in a row.]
And then a row of four…
[Above the 5 blocks, she lines up a row of 4 blocks.]
…and then three.
[Above the 4 blocks, she lines up a row of 3 blocks.]
And then two…
[Above the 3 blocks, she lines up a row of 2 blocks.]
…and then one more…
[Above the 2 blocks, she places 1 block. She places a label below: case 5.]
…case five. And yes, because that was ten, it should be 15. So how could we check that? Yes, so we know this structure…
[She traces the shape of block set next to case 5.]
[She points to each row of the block sets.]
So, look, there's a row four, a row of three, a row of two and a row of one. So that has to be ten and then…
[She points to the bottom row of case 5.]
…five more. So it's 15. But what I started to notice is that they're triangular numbers. And so this made me wonder about what would be the next triangular number in our sequence.
[Next to case 5, she places a label: case 6.]
But I also, as I was trying to think of more efficient strategies to work out what Case 12 would be, I think I noticed something…
[She arranges the blocks from case 2 – 5 so that the left column of each has the most blocks, and each column following is reduced by 1 block.]
…and I'm just gonna put them back into this staircase position these triangular numbers were presented to us in before.
Like a reverse staircase. Let's move that one down a bit so you can see it. Mm-hm. Yes, I think I noticed something cool. Look, if I take the triangular number here…
[She points to case 1.]
…at one and move it across to there…
[She moves the red block from case 1 to the second column of case 2.]
…it makes a square number.
[She moves the red block back to case 1.]
Look, and if I take this triangular number of three and move it to this triangular number of six…
[She moves the blocks from case 2 to the second and last columns of case 3.]
…it also makes a square number.
[She moves the blocks back to case 2.]
And so now what I wondered is if I take any two consecutive triangular numbers, will it always make a square number? Oh, my gosh, that sounds like a fun maths investigation. So it's over to you mathematicians.
[Text over a blue background: Over to you, mathematicians!
Over a grey background, the red waratah of the NSW Government logo appears amongst red, white and blue circles. Text: Copyright State of New South Wales (Department of Education), 2021.]
[End of transcript]
If we take any two consecutive triangular numbers, will their sum always form a square number?
Record your thinking in your student workbook.