Raindrops
Discussion
You might have noticed things like:
In case 2 there are more cubes than in case 1.
In case 3 there are more cubes again.
Where do you see the extra cubes adding each time?
There are many ways to answer this question as people see the cases in lots of different ways. How do others see them?
Watch the video for 3 different ways of thinking
Transcript
Alrighty everybody, I'm joined here today by two of my favourite mathematicians, Barbara and Ayesha! Hi guys!
Hi!
Good morning!
Alright! Are you ready to get started?
Yes!
So we're looking at a problem today which is asking us about how the shapes, or the squares, are growing. And so case one, sometimes referred to the case as the term in a pattern, so case one, case two and case 3... and what I'm interested in is where do you see the growth happening?
Oh ok!
Well I think for me, it's like there's another bottom row coming in each time. So if you look at case one, if we add another one on the bottom here and there and there and there, it looks like case two.
Oh, so I'm going to use these post-it notes to try to model that sort of idea. So if I make case one...this one... Which is case one.
And so you're saying then what happens for you is that the next case comes in underneath?
Yeah, that's right. And then in case three, there will be another row added to the bottom and it will be wider. It'll start like this and it would start to shift up.
Exactly right.
What about you Ayesh? How do you see it?
I'm quite the opposite to Barbara, actually. So for case one, I see that, instead, there are squares falling down like rain drops to create case two. So, if I move this down a little bit more so case one stays the same, everybody sort of has that...
And then you're saying that for you in your head, what happens is these come in like raindrops and fall down?
That's right.
And rest on top of a square that already exists.
Oh well, that's really different than what I was thinking to what I was thinking.
Yeah.
What are you thinking?
Well, I think about it differently again. So I sort of, in my head, rearrange things. So if this is case one, what I imagine is this square here moving across. And I re-position it in my head. So now I can see a square. And when the new ones come, they come across the top row... and then down the side. So if I if I re-organise it, you'll still see the same structure as case 2.
Oh, but you're noticing that it's squares every time?!
Yeah, so I'll show you with the same squares. So, this comes to here and if I use Ayesha's method, for example... like this. But what happens in my head each time is I, I can see that that would form this triangular piece and I rotate it around in my head so that comes over here like that.
I see what you mean!
Oh! That's even more different to how I was thinking [laughter]!
I wonder who has some different ways to think about how do they see the growth occurring in the pattern? And then can you use that to help you work out what would the 4th case look like, for example, or the 10th case? For example?
Over to you, mathematicians!
Collect resources
You will need:
pencils/ markers
your mathematics workbook.
More instructions
What will the fourth case look like? Draw it in your workbook.
Mathematicians often like to draw diagrams and use tables to help them identify patterns.
Can you use these strategies to help you work out what the 10th case would look like?
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