Tangrams 2

Alright mathematicians. Welcome back. I started thinking about we could now do a bit of a challenge to play around with making special kinds of rectangles, but as I started playing with this, it got me thinking about something else.

So the first thing I do want to do is think about what are some rectangles I could make that are actually the same size as this one. Using some or all, if I wanted, of my tangram pieces so this one I have the square. And I have two triangles that are formed into the same size of the square. Look, because if I cover that up you can see they are forming the same area.

So what I wondered is what other rectangles can I make of exactly those same dimensions using just my tangram pieces?

Over to you for minute mathematicians. And then of course, record your thinking. Ok.

Hello there mathematicians. Welcome back.

We have a tangram challenge for you. We were wondering something about our tangram because as we cut it up and create it, we realised that we were partitioning our tangram and making them into parts and some of the parts are equal sized like these two triangles are the same size but some of the parts are smaller.

But it all came from the same triangle, the same square originally. So what we were wondering is, if the square of the rectangle, the square of the tangram, which all the pieces inside of it partitioned. Like that, yes.

Whoops, is a bit wonky. If this whole thing is worth one, what are all of the other pieces worth? Ah ha, this represents one whole, So what that means is, what's the value of the two large triangles?

What's the value of the medium triangle? What's the value of the parallelogram? What's the value of the two small triangles? And what's the value of the square?

Over to you mathematicians.

Welcome back mathematicians. How did you go? I agree with you, some of them made my brain sweat a little bit more than some of the others. Shall we talk about it together?

Ok, good. So what I found really useful was to think about how I built my tangram to help me think about, if this is the whole. Then what is the value of each of my tangram puzzle pieces? Yeah, and so I thought about what did I do to my whole to make the puzzle pieces to start with?

Yes, so, can you remember? Yeah, the first move was that we halved our whole. That's right, so we now have two large triangles and each of these is one half of the whole.

Ok, and I'll cut it out so that we can keep manipulating it. And as I'm cutting it out, what was the next move that we made when we made our tangrams? Yeah, we partitioned this triangle into two equal triangles. Our large triangles, ah ha, so if this is a half and when I half my half I end up with quarters. Yes, so a half of a half is a quarter.

I know and that seems weird because I go from writing something that looks like this. To something that looks like this. And usually in whole numbers four means more than two and in this case it does too. It means I have more pieces, but the pieces are getting smaller. Look, here's a half and here's a quarter and this is definitely bigger. Yeah, whereas in whole numbers two is smaller than four and so sometimes, yeah, we can get ourselves muddled mathematically if we use whole number thinking when we're working with fractions. So, this then means it's a quarter, so let's just check I've got another square of paper.

And I got a different colour so that it's easier to see. And this was our half that we halved again, see. Yeah, and so, if that was there like this, I can now see I've halved it to make, yeah, quarters and I would need four of them. Look, one, if I rotate it two, three and four. Yeah, and it's the same size over here. So, the large triangle is worth one quarter.

And symbolically, we would write that like this, yeah, so this is also worth one quarter. Yeah, so the large triangle we have worked that out. I can put this over here actually, but like. So now, I still have this other half left and actually this whole half is partitioned into one, two, three triangles, a square and a parallelogram.

Ah ha, and can you remember how we made this? Yeah, so let's see if this will help us work it out. So we did a little mark down here to show half way. And we folded this down. And we cut this out like this. Alright, now this one isn't so obvious, I don't think.

Where as this was easy 'cause when you, not easy but less brain sweaty, because when you halve a half you end up with quarters, but we're not entirely sure what we did to this piece now, except that we made a triangle into a trapezium and a triangle. Yeah, and so if I lay this on here though, can you see this?

I'll just use one of another colour. So it's the same size triangle. So, that piece fits there. And Yeah, like you, I'm imagining now, this spinning around in my brain and I'm using my imagination. And if I spin this around, I think another triangle would fit there. And then I think there'd be a third one here. Uh, huh. Should we try it? So let's spin that around like that.

Ah ha, yeah, and actually that, now we can see for sure 'cause these are the same size in area and so is this one. So, so what happened actually when I cut that tip off? Yes, I've now quartered my half which means I made eighths. Yes, because for each half, I cut it into four pieces and so one half has four pieces and the other half has four pieces, which is 8 pieces all together. So this is now one eighth.

And that's this piece over here. And you're right. I could check that by laying it over, this quarter. Let's check. Because if I have two eighths, that's the same as one quarter. Uh huh. So let's use this one, 'cause it's a different colour. So. that's one eighth over the top of a quarter. Yes, and if I doubled it, I'd have two eighths, which is a quarter, ah ha, so that gives me confidence that my claim is accurate. I've got some nice proof.

And now what I'm going to do with this other part, which at the moment is three, eighths worth, isn't it? Yes, 'cause we had the triangle fit there another triangle here and another one there, see. Yeah, so one eighth, two eighths and three eighths more. Ah ha, but now with my three eighths can you remember what we did? We we partitioned it into half.

So I've now halved my three eighths. Yes, this part does make your brain sweaty. It's good our brains are growing and we now halved our trapezium to end up with two smaller trapeziums that are now both half of three eighths. I know, but then we kept partitioning it, didn't we? And we folded this little edge over here to make a little triangle.

And that gave us a triangle and a square. So, yes, I was thinking the same thing as you. I used my eighth as a measure to help me work these out. Look, here's my eighth here. And I'll use the green one. I'll just prove to you they're the same size. To help me show that, whoopsie. If I take this piece and lay it on here. Like this. What do you notice?

Uh-huh, it's half of the eighth. And if I halve an eighth I get a sixteenth. So this is one sixteenth now. Yes, and I can use that actually to work out this, look, because if I lay this over here, that covers half the surface area of my square, and if I flip it over it or rotate it, it covers the other half. So that means that this is two sixteenths which is an eighth. Ah ha, one eighth.

Ok. So now we know actually that, half of three eighths is one eighth and one sixteenth more. Yes, you're right. So if I do this action now to make my parallelogram and my triangle, so I'm decomposing my trapezium now into a parallelogram and a triangle. And these were the same size to start with, remember. This looks like this. So they're the exact same shape.

So this triangle is the same area as that triangle, which means it's one sixteenth. One sixteenth. Which means that this parallelogram, must also be one eighth. I agree with you. Because they take up, they occupy the exact same area or same portion of our original square, even though they look really weird. So that means that over here, this is a sixteenth, this is an eighth, this is a sixteenth and this is an eighth. Wow, isn't that really cool?

Yeah, and now mathematicians, after all those sweaty brains. I really want to draw your attention to these guys. So here is my triangle. Here's the parallelogram and here is the square. Now, these guys look really different as shapes, don't they? But they, we think are all the same portion of our large square, meaning they all cover the same surface area in this instance. And now what I'm wondering is, over to you mathematicians. Can you think of a way to prove, yes, that these are in fact all exactly the same or cover the exact same surface area of our original square?

Over to you mathematicians.

Before we get to your challenge, let's talk about some of the mathematics today. So here's our tangram and it revealed some really cool things for us. That when you halve a half, you create quarters. So, one half of one half is one quarter and this helps us realise two really important fractional ideas. One that, as the number of parts we break something into gets bigger the size of the parts get smaller. Yeah, and that fractional numbers don't always work the same as whole numbers and these are two really critical ideas. Yes, I've discovered today. So thanks to the tangram for that.

We also realise that we can use other fractional parts as measures. So we worked out this section was one quarter of one half which is equivalent to an eighth. And we then used it to work out the yellow section. We laid it over the top and since a half of an eighth is a sixteenth, the yellow part must be equivalent to one sixteenth of the whole area of our original square.

So now for your challenge mathematicians. How can we prove the medium triangle. the parallelogram and the square are all equal in area? And in fact they're all an eighth of our original tangram puzzle. Over to you.

Hello there mathematicians. Welcome back. How did you go with our challenge? Yes, there are, in fact a few strategies that you could have used. One would have been to use another piece of our tangram as a measure. So remember, we made one, another one in red. Yes, and that was my, triangle, that was at the top that I can never remember how to put back together. There it is.

Yes, and then from there, yes, I notice it turns out to have a parallelogram and the portion and the triangle here, and the square and the other triangle. And we had worked out that these two smaller triangles were an, equivalent in area to one sixteenth of the entirety of our square. So I could actually use this as a measure with these shapes over here, because what I can see in my mind is that this square is half the area of this triangle, yes, so if this is one sixteenth and if I spin it and this is one sixteenth, then I have two one sixteenths. So that's one sixteenth and one sixteenth and together that's one eighth or two sixteenths.

Yeah, so that's one way I could think about it and then I could use the same unit of measure my one sixteenth. So if I take my parallelogram now and if I line it up, I can see the same thing that, yes, that would be one sixteenth and that would be the second, sixteenth. Uh huh. And so that would also be two sixteenths. Which is equivalent to one eighth. So even though they look really different, they have the same area. They're both one eighth of our original square and I could use the same unit of measure for my triangle.

So, that's one sixteenth and if I spin it around, another sixteenth. Yes, and so two sixteenths is equivalent to one eighth. So that's one strategy I could have used. Another strategy I could have used is to use my square as a measure, for example, or in fact any of my other shapes but with the square, what I could do is decompose my other shapes and recompose them as a square to prove that they are equivalent.

So I think I have a square. Yes, I do. I'll use this one as my measure, just because it's easier for you to see and I'm going to think about my parallelogram 'cause if I lay that over the top of my square, what I can actually imagine here in my mind's eye is this portion that's overhanging, here.

If that was cut off and rotated around, then that would cover the same surface area. Yeah, and I could use a similar strategy for the triangle. Where if I covered it like this, I can also imagine this portion being cut off and flicking over and that's another way I could prove. But let's do it. Let's chop this guy in half.

So now my one eighth becomes two sixteenths. And if I place one sixteenth here and then this part that I imagined. Yes, it's mandatory to make those sounds, goes over here. I can see, ha, that is equivalent to the area of the square. And I could do the same with the parallelogram, look. If I do this and then this was the portion we imagined chopping. So let's chop it. Let's decompose our parallelogram into two triangles. One goes there and this section here that I imagine sliding up to there.

Yeah, and so that's how we can prove, another way we can prove that they all have the same area. They're all one eighth of a portion of our large tangram puzzle. Alright, let's get ready for your next investigation.

So what some of the mathematics here? Yeah, so we realised that we can use other fractional parts as measures. So we use the one sixteenth to prove the square parallelogram and medium triangle are all two sixteenths, which is equivalent to one eighth, the area of the original square. So there we see two of the sixteenths, to be one eighth, another two of the sixteenths in the square, one eighth and another two sixteenths, to be the area of the small, of the medium triangle.

Nice work today mathematicians.