# Staircase patterns Stage 2

A thinking mathematically targeted teaching opportunity focussed on exploring, comparing, representing and describing patterns in different ways

These videos are adapted from AAMT Top Drawer Teachers Making a staircase and AAMT Top Drawer Growing patterns. Before you begin, you might also like to watch the Numberblocks step squad episode.

## Syllabus

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

• MAO-WM-01
• MA2-MR-01
• MA2-2DS-02

## Collect resources

You will need:

• paper or your workbook

• a pencil.

## Watch

Watch Staircase pattern Stage 2 part 1 video (6:17).

Explore increasinging and decreasing patterns using coloured blocks.

### Speaker

Hello there mathematicians. How are you today? That's really good to hear. So, I have some blocks here and I'm beginning to make a pattern.

[Screen shows 6 blocks joined horizontally. There is a single brown block, 2 black blocks and 3 red blocks all joined together.]

Yeah, and I'm wondering if you can figure out what the next term or the next thing would be in my pattern if I continued it? Ah ha, so remember, a pattern is something that has a repeating core.

So, we're looking for what is the thing that's repeating over and over and over again. Yeah, and it's a bit different 'cause we've been working a lot on patterns like A, B, A, B. You know, like clap, click, clap, click, clap, click. But this is a different kind of pattern.

[Presenter claps hands then clicks fingers.]

Ah, you think you can see something? Uh, huh. I think, I think I can see what you're seeing too.

That there's one brown cube, 2 black cubes, 3 red cubes. So, the colour is changing each time, so there's nothing there that's repeating.

[Presenter points to the first brown cube, then the circles the next 2 black cubes and the final 3 red cubes, indicating the growing pattern.]

But it goes from one to 2, 3 and it's, yeah, there's one more block each time. Let's use our chunking strategy to see if that works. If I line them up. Does it increase by one more block each time?

[The presenter lines up the blocks vertically with the single brown block at the top, the 2 black blocks underneath and the 3 red blocks at the bottom.]

Ah, so what would be the next one? It would be 3 and one more. 4. Ok, I can get 4 together. And what would be the next term in my pattern?

[The presenter creates a new chunk containing 4 yellow blocks. She places this under the red blocks.]

4 and one more. 5. The number after 4 is 5. Ah ha, look at that.

[The presenter creates a new chunk of 5 orange blocks underneath the yellow blocks.]

So that's what it would look like if we put it all back together and we could keep building it out as a big tower of pattern.

[The presenter joins all of the chunks of blocks together in an ascending pattern from the singular brown block all the way to the 5 orange blocks.]

Oh, it reminds you of something else. It also. Yes, I agree with you. It reminds me too. If I turn it like this, you might have recently seen an episode of number blocks called Step Squad. Yes, and it does look exactly like our number blocks characters, doesn't it?

[The presenter disassembles the blocks and chunks them together in ascending order. She places them from smallest to largest like a staircase. From the top of the screen the presenter grabs a new set of blocks with googly eyes and features on them to represent the Number Blocks characters. She keeps these onscreen and moves the other blocks off screen.]

Let's work with them 'cause they're cute. So, when you see our number blocks characters in a step squad formation, what are some things that you notice about the shape that they have? Ah yeah, it's like a triangle.

Let me write this down.

So how many bricks? How many steps do I need to draw? One for red, another one for orange, another one for 3, another one for 4 and one for 5, and so it's like, oh, a triangle with squares in it. So, it's like a triangle shape. What else do you notice about it?

[Presenter places a white page of A4 paper on the screen. On the page she draws a right angle and creates ‘steps’ that mimic the step formation of the blocks. Presenter writes: ‘a triangle shape.’]

Oh yeah, it's like this. It goes up by one each time. Um, each column increases by one. Look, it's one, then 2, then 3. Yes, so each column increases by one. So, each time we go up a step, it gets one bigger.

[Next to the picture, she writes: ‘each column increases by one’. She draws vertical lines for the first three steps and writes the numbers 1, 2, 3 in each individual one to demonstrate that they are increasing by 1.]

What else can you see? Oh look, you're right. one, 2, 3, 4, 5 blocks wide and one, 2, 3, 4, 5 blocks high. That's cool. So, 5 blocks wide and 5 blocks high.

[The presenter draws a new right angle underneath her previous drawing. She creates 5 divisions on the bottom line and 5 on the side to represent the structure. Next to the drawing she writes: ‘5 blocks wide and 5 blocks high.’]

Oh, yes, and I can come in and mark those for you, so I'm going to come in and go, there and half and quarter. So, one, 2, 3, 4, 5.

[As she presenter counts each block she makes a small dot to represent her counting.]

One, 2, 3, 4, 5, yeah, and you're right, because it's a mathematical drawing, I don't have to draw all of the individual blocks. I can just draw the most important information. Yeah, and, and so, what our pattern is, our pattern core. How would we describe that? Yeah, the pattern core goes up by one each time. So, our core is that each step up the staircase adds one more block.

[The presenter writes the pattern core underneath the mathematical drawings. She writes: ‘Our core is that each step up the staircase, adds one more block.’]

That's really cool mathematicians.

So now I was wondering something actually, and I was wondering what would happen if we make it so we go up by one each time, but when we get to 5, what would it look like if we went down the other side?

[Presenter follows up each step of their structure, and symbolises doing the same on the other side.]

Ah. Yes, so if we had one less than 5, how many would we have? How many blocks? 4 because one less than 5 is 4.

[Presenter adds 4 green blocks to the right side, beginning to create a descending pattern. She shows the decreasing pattern by adding one red block to symbolise 5 and then taking it away to show that there are only 4 blocks left.]

There we go, look. Yes, 'cause If we had 5 and we take one away, that's one less than 5.

And what would come next in our pattern? Ah, yes, 3, because one less than 4 is 3. Uh-huh.

[Presenter adds 3 yellow blocks to the right side.]

Ok mathematicians, over to you to finish what the rest of our staircase would look like if it goes up and down the other side. And can you draw a picture of this to record your thinking?

[Presenter traces along the staircase pattern once more.]

Over to you.

So over to you mathematicians. Draw the staircase pattern we've made, continuing it down the other side. And then we'll come back together. OK over to you.

[End of transcript]

## Instructions

In your workbook, draw the staircase pattern we’ve made in the video, continuing it down the other side.

## Watch

Watch Staircase patterns Stage 2 part 2 video (4:33).

What would a staircase pattern look like using twos?

### Transcript of Staircase pattern Stage 2 part 2

[Text over a navy-blue background: Staircase patterns 1 – part 2. Small font text in the top left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the red waratah of the NSW Government logo. ]

### Speaker

Welcome back, mathematicians. How did you go?

[9 columns of blocks sit on a piece of paper. The 5 columns on the left have faces made with goggle eyes and paper cut-outs which have been stuck on. The columns form a pyramid shape, with column 1 and 9 each containing 1 red block, columns 2 and 8 each containing 2 orange blocks, columns 3 and 7 each containing 3 yellow blocks, columns 4 and 6 each containing 4 green blocks, and the central column containing 5 blue blocks. To the right of these columns there is a piece of paper with a drawing of a 5 column graph forming the shape of a staircase, with the numbers 1, 2 and 3 in the first 3 columns. Beneath this is a drawing of an X and Y axis, with the Y axis pointing to the left,. Each axis is marked by 5 evenly spaced dashes and flanked by 5 evenly spread dots on the outer edge. To the right of the staircase shaped drawing is the following text: ‘A triangle shape’ and ‘Each column increases by one’. Beside the drawing of the X and Y axis, text reads: 5 blocks wide and 5 blocks high. Beneath this reads: Our core is that each step up the staircase, adds one more block.]
Aha, yeah. Now it looks like a triangle, but in a different orientation, doesn't it?

[The speaker gestures to the rising and descending segments of the columns of blocks. She removes the last 4 columns from view and outlines the staircase pattern which is formed by the remaining blocks. She replaces the last 4 columns and outlines the resulting triangular shape.]

Before, look, the triangle shape was here, and now we see the triangle shape here.

[The speaker moves the piece of paper containing images and text from the right side of the blocks to the left side of the blocks. She adds a 10th column with2 orange blocks, then an 11th with 3 yellow blocks, then a 12th with 4 green blocks, then a 13th with 5 blue blocks, then a 14th with4 green blocks, then a 15th with 3 yellow blocks, then a 16th with 2 orange blocks, and a 17th with 1 red block.]

Yeah. Yes, and if we continued our pattern as a growing-shrinking pattern, then it would start to grow again, before it started to, whoops, gotta make it grow. And then it has to shrink again. Look. That's really cool, isn't it? It's quite beautiful too. Oops. Aha. And then I would see, yeah, you're right. It looks a bit like mountains, doesn't it now?

[The speaker outlines the blocks, forming two triangle shapes beside each other]

Yeah, that's really cool.

[The speaker removes the additional columns, leaving the original 9 columns of blocks.]

All right, so my challenge for you mathematicians, is I was thinking about something is this is what our pattern looks like if we just use one block wide.

[The speaker outlines the shape formed by the columns of blocks.]

But what would happen if we made a staircase like this using two blocks wide?

[The speaker moves the triangular formation of blocks to the top of the piece of paper. She places two red blocks beside each other in it’s place.]

Yeah, so we had one, two, and then we might have two twos.

[The speaker places 2 rows of 2 orange blocks beside the two red blocks, creating a second column.]

Not something you would dance in. Well, I mean you could. Then you might have three twos.

[The speaker places 3 rows of 2 yellow blocks beside the 2 rows of orange blocks, creating a third column.]

Uh huh. So mathematicians, over to you now.

[Text on a blue background reads: Over to you, mathematicians! Smaller text beneath reads: Draw what the staircase will look like if I continue building it up and down the other side, using twos.]

Can you continue, you can build this structure if you like, our model, or draw it in your notebook. What will it look like if I continue building it up and down the other side? 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]

## Instructions

Draw what the staircase will look like if I continue building it up and down the other side, using twos.

## Watch

Watch Staircase pattern Stage 2 part 3 video (6:00).

Use a staircase pattern to work out how many blocks.

### Transcript of Staircase pattern Stage 2 part 3

[Text over a navy-blue background: Staircase patterns 1 – part 3. Small font text in the top left-hand corner reads: NSW Department of Education. In the lower left-hand corner is the red waratah of the NSW Government logo.

At the top of a piece of paper, 9 columns of blocks sit on a piece of paper. The 5 columns on the left have faces made with goggle eyes and paper cut-outs which have been stuck on. The columns form a pyramid shape, with column 1 and 9 each containing 1 red block, columns 2 and 8 each containing 2 orange blocks, columns 3 and 7 each containing 3 yellow blocks, columns 4 and 6 each containing 4 green blocks, and the central column containing 5 blue blocks.

Beneath this is another 9 columns of blocks in the same pyramid pattern, with the same sequence of colours, but with each row of each colour consisting of two blocks.

To the left of these columns there is a piece of paper with a drawing of a 5 column graph forming the shape of a staircase, with the numbers 1, 2 and 3 in the first 3 columns. Beneath this is a drawing of an X and Y axis, with the Y axis pointing to the left,. Each axis is marked by 5 evenly spaced dashes and flanked by 5 evenly spread dots on the outer edge. To the right of the staircase shaped drawing is the following text: ‘A triangle shape’ and ‘Each column increases by one’. Beside the drawing of the X and Y axis, text reads: 5 blocks wide and 5 blocks high. Beneath this reads: Our core is that each step up the staircase, adds one more block]

### Speaker

OK, mathematicians, does your drawing look like our model here? Our structure? Oh, awesome.

[The speaker removes the 4 right-most columns of the top pyramid.]

So, what I'm curious about is, you know how when we first looked at this step squad, and we thought about some things that we noticed about it, what are some things that we notice about this staircase?

[The speaker holds up the piece of paper with drawings and writing on it. She then replaces the columns she removed.]

Some things might be the same, and some things might be different. Yeah, it still makes a triangle, doesn't it? Like this guy still makes a triangle.

[The speaker outlines both of the arrangements of blocks with her finger.]

Yeah, but it's much wider.

[The speaker traces her finger across the bottom of lower arrangement of blocks.]

In fact, it's twice as wide. Yes, because I have, instead of one for each one brick here, I have two bricks instead.

[The speaker picks up one of the units of red blocks from the lower triangle.]

So, look, we can prove it's twice as wide because if we put this here to measure it...

[The speaker moves the top triangle of blocks above the left-half of the lower triangle of blocks.]

Yes. And we've got to remember that they're not all clicked in. That's the same width.

[The speaker draws a line with both of her index fingers aligned to the outer edges of the upper triangle, with her left finger passing the left-hand side of the lower triangle and her right finger passing through the middle of the lower triangle.]

### Speaker

And if I move it across here, if I push these in...

[The speaker clicks the blocks of the lower triangle together.]

Oops. Yeah, you'll see it's also the same width.

[The speaker aligns her hands with the edges of the upper triangle and the middle and outer-edge of the lower triangole.]

So, it's twice as wide.

[The speaker moves the upper triangle back to it’s original position.]

Oh, yeah, it's the same height, isn't it? Because if we take five and lay it here, it's still only five bricks high.

[The speaker takes the column of blue blocks from the upper triangle and aligns it with the column of blue blocks in the lower triangle.]

But if it's twice as wide, how wide is it?

[The speaker places the blue column from the top triangle back in it’s place.]

Oh, 18, yeah? Because look, this is five, six, seven, eight, nine, and double nine is 18.

[The speaker points to each of the columns on the top triangle.]

Would you like to count to check? OK, ready?

[The speaker outlines each of the pairs of blocks on the lower row of the bottom triangle.]

Two, four, six, eight, ten, 12, 14, 16, 18 bricks wide, blocks wide. What else do you notice? Yeah, OK. Well, I was noticing... I'm gonna move these guys out of the way.

[The speaker removes the upper triangle from view.]

Thinking about something else that I thought was really interesting, mathematicians.

[The speaker moves the triangle of blocks over to the left slightly.]

I'm just gonna move this across a little bit, is I was noticing something about depending on how we look at the structure.

[The speaker outlines the ascending then descending edge of the triangle.]

So, when we look at it this way, it goes up and then it goes down.

[The speaker gestures from the bottom row upwards, and then outlines each row of the triangle with her fingers.]

But if we look at it this way, I can see that this row is longer than this row, which is longer than this row, which is longer than this row, which is longer than this row. Aha. Look. And if I squish it up like this way, that makes it easier to see.

[The speaker aligns all of the rows of blocks with the left-hand side, creating a descending stair-case shape.]

Oh, yes. And now it just looks like each time I go up, it shrinks each time. Or each time I go down, it grows. So, I can see it as either a growing or a shrinking pattern.

[ The speaker returns the blocks to their original pattern.]

So, let's work out how many blocks there are in each row.

[The speaker outlines the bottom edge of the triangle with her finger. She then places a sticky note pad down next to the triangle.]

What do we know about the bottom? Yes, we know that's 18. We just worked out that it's 18 blocks wide.

[The speaker writes the number 18 on the notepad and sticks it to the right-hand side of the triangle.]

And what do we know then about this row here?

[The speaker outlines the second row from the bottom with her finger.]

Yeah. We're not gonna count just yet, because we know something, don't we? We know...let me use these two bricks

[The speaker adds two blue blocks to either side of the second row, she then elevates the 3 rows above this row.]

... that if I had these two blocks here... I'll just move these out of the way for a moment.

This row would be as long as this row, wouldn't it?

[With her finger, the speaker outlines the bottom two rows.]

### Speaker

Which would mean it's 18. But I need to get rid of two.

[The speaker removes the two blue blocks she added to the right of the row.]

16. And remove another two, that makes 14.

[The speaker removes the two bluck blocks she added to the left of the row.]

Yes. Because we're counting backwards in twos.

[The speaker writes the number 14 on a new page of the notepad.]

You could have done that strategy. So, 18 and 14.

[The speaker sticks the note with the number 14 above the note with the number 18.]

And then, what about this next row? What do we know about it?

[The speaker separates the 3rd row from the bottom from the rows above and beneath.]

Yes, we know that if we had two more twos, it would be the same length as this row, but there's one two missing and another two missing, which is four.

[The speaker adds two blue blocks to either side of the row. She then removes them and places them in a block of 4 above and to the right of the triangle.]

Yeah. And you could use place value knowledge this time because one ten and four is called 14.

[The speaker points to the note with the number 14 written on it.]

### Speaker

So, just if you get rid of the four, you just have one ten left, which is ten.

[The speaker picks up the block of 4 blue blocks. She then writes down the number 10 on a new page of the notepad.]

OK. So, I know that row has ten.

[The speaker sticks the note with the number 10 written on it above the note with the number 14 written on it.]

And what do we know about this one now?

[The speaker separates the 4th row from the rows above and beneath it. She then adds two blue blocks to either end of the row, and then removes them.]

It would have ten if it had two more twos, but it has two twos missing, which is four. So, it's six.

[On a new note, the speaker writes the number 6, she then sticks it above the note with the number 10.]

And what about this guy?

[The speaker separates the top row from the rows beneath it. She then writes the number 2 on a new note and sticks it above the note with the number 6 written on it.]

Yes, yeah, I know you can see how many there are, there's two. So, I have a challenge for you mathematicians to think about. What I'm wondering about now is how could we work out how many blocks there are all together

[The speaker outlines the triangle using her fingers.]

... Yes, by using some efficient strategies to help us reason our way through.

[The speaker points to the notes with numbers written on them beside each row of the triangle.]

OK, mathematicians, over to you to have a think and record your ideas, and then we'll come back together. OK!

[Text on a blue background reads: Over to you, mathematicians! Smaller text below reads: How many blocks are there altogether? Use diagrams to record the thinking you did to work out the solution.

On a new slide, 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]

## Share/submit

• How many blocks are there altogether?
• Use diagrams to record the thinking you did to work out the solution.

## Watch

Watch Staircase pattern Stage 2 part 4 video (8:34).

Investigate representing a staircase pattern using twos.

### Transcript of Staircase pattern Stage 2 part 4

(Duration: 8 minutes and 34 seconds)

[A title over a navy-blue background: Staircase patterns 1 – part 4. 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 blocks joined together. On the right side of the blocks are post-it notes labelled ‘2’, ‘6’, ‘10’, ‘14’ and ‘18’ lined up on top of each other.]

### Speaker

Alright, mathematicians, welcome back. Let's talk about some strategies you could have used to solve that problem. So, as mathematicians, we know we always like to think about, what do I already know and how can I use that to help me? So, when you're looking at these numbers…

[She picks up note ‘18’.]

### Speaker

…you might have noticed something, like 18 and two more…

[She picks up note ‘2’ and places it over ‘18’.]

### Speaker

…combines to make 20.

[She places the notes on the top left side of the blocks.]

### Speaker

So, what we could do is have moved this block down here…

[She moves the top blocks to the right side of the bottom blocks.]

### Speaker

…and now we have one row of 20.

[She traces the bottom blocks.]

### Speaker

Yes. And we could have done that in another place as well. Look. Yeah, look, six…

[She picks up note ‘6’.]

### Speaker

…and 14…

[She places note ‘6’ over note ‘14’.]

### Speaker

…combines to make 20. Yes, because we know six and four makes 10 and then one 10 more is 20.

[She puts down the notes next to the other notes.]

### Speaker

So, I could imagine my 14 and my six, there's the 14…

[She moves the second row of blocks to align with the bottom row.]

### Speaker

…and there's the six.

[She moves the top row of blocks to join the second row.]

### Speaker

Yes. And then we have the 10 left.

[She moves the top row of blocks close to note ‘10’.]

### Speaker

Look.

[She moves them back.]

### Speaker

So, now I know I have 20, 40 and 10 more is 50.

[She moves the top row across to align with the centre of the blocks below.]

### Speaker

So, that's one way we could have worked it out.

[She moves the blocks back to their original position.]

### Speaker

Put these back where they belong. Oh, the colour's nice to help me. So, one way was we could look for 20.

[She moves the 2 last notes sets to the left.]

### Speaker

So, let's write that down together.

[She gets a marker.]

### Speaker

So, one strategy was to say, well, I know…

[In the top right hand of the butcher’s paper, she writes: 18+2=20.]

### Speaker

…18 and two is equivalent to 20 and I know…

[Below the text, she writes: 14+6=20.]

### Speaker

…14 and six more is equivalent to 20.

And then I have 10 left. And then I know that…

[Below the text, she writes: 2 tens + 2 tens + 1 ten = 5 tens = 50.]

### Speaker

…two tens plus two tens plus one ten is five tens, and that's 50. So, that's one strategy we could have used.

[She picks up the notes.]

### Speaker

I wonder if there's another one. Ah, yes. Some of you guys did some imagining. I thought about this strategy too. I'll just put these back here.

[She places the notes on the right side of the blocks.]

### Speaker

Yes. Where you imagine things moving. Yes. Like if we took the four twos…

[She pushes up the 8 green blocks on the right side.]

…and moved them. (MAKES WHOOSH SOUND) You are required to make those sounds, over to here...

[She places them over the first 2 red blocks on the left.]

### Speaker

…I now have one two…

[She points to the red blocks.]

### Speaker

…and four twos…

[She points to the green blocks.]

### Speaker

…which is five twos, and it looks like, yes, a ten frame. What would you do now to the others? Yes. If we take the three twos…

[She pushes up the 6 blue blocks on the right side.]

### Speaker

…and join it with the two twos…

[She places them over the 4 orange blocks on the left.]

### Speaker

…that makes five twos…

[She places her hand over the blocks.]

### Speaker

…which is another ten. And then, yes, these two twos…

[She pushes up the 4 orange blocks on the right side.]

### Speaker

…that always makes me think of ballerinas, come and join these three twos…

[She places them over the 6 yellows blocks on the left.]

### Speaker

…to make another five twos. And then, yes, this one two…

[She pushes up the 2 red blocks on the right side and places them over the 8 green blocks on the left.]

### Speaker

…joins the four two and then there's five twos…

[She points to the 10 blue blocks.]

### Speaker

…left. So…

[She circles the first block set.]

### Speaker

…one ten…

[She circles the first and second block sets.]

### Speaker

two tens…

[She circles the first to third block sets.]

### Speaker

…three tens…

[She circles the first to fourth block sets.]

### Speaker

…four tens…

[She circles all block sets.]

### Speaker

…and five tens, which is 50. So, what we thought about here…

[She points to the block sets.]

### Speaker

…was making groups of tens, didn't we?

[She points to the first block set.]

### Speaker

So, we first joined two and eight, so we had…

[Above the first block set, she writes: 1 two + 4 twos = to 1 ten.]

### Speaker

…one two plus four twos to make ten or one ten. Then we had…

[Below the text, she writes: 2 twos + 3 twos = to 1 ten.]

### Speaker

…two twos plus three twos to be one ten.

[Below the text, she writes: 3 twos + 2 twos = to 1 ten.]

### Speaker

Then we have three twos plus two twos to be one ten.

[Below the text, she writes: 4 twos + 1 two = to 1 ten.]

### Speaker

And then four twos and one two to be one ten.

[Below the text, she writes: 5 twos = to 1 ten.]

### Speaker

And then five twos, which is one ten.

[She circles the first ‘1 ten’.]

### Speaker

And then we have one ten…

[She circles the first ‘1 ten’ and the one below.]

### Speaker

…two tens…

[She circles the first 3 ‘1 ten’.]

### Speaker

…three tens…

[She circles the first 4 ‘1 ten’.]

### Speaker

…four tens…

[She circles the all ‘1 ten’.]

### Speaker

…five tens is 50.

[Next to the text, she writes: 5 tens = 50.]

### Speaker

There are two really great strategies for thinking about our problem. And, you know, mathematicians, this one…

[She points to the blocks.]

### Speaker

…made me think about something else that now…

[She pushes the blocks together.]

### Speaker

…look how our step squad turned into a rectangle. And if I get out our original step squad, here they are…

[She pulls down blocks with the Numberbooks Characters.]

### Speaker

…does the same thing happen? Does it turn into a rectangle? Shall we try?

[She pushes up the 4 green blocks on the right side.]

### Speaker

We'll match the one with the four because it's five bricks high…

[She places them over the red block on the left.]

### Speaker

… and then three and two.

[She pushes up the 3 yellow blocks on the right side and places them over the 2 orange blocks on the left.]

### Speaker

Two and three.

[She pushes up the 2 orange blocks on the right side and places them over the 3 yellow blocks on the left.]

### Speaker

Oh, wow!

[She pushes up the red block on the right side and places it over the 4 green blocks on the left.]

### Speaker

Look, it made a rectangle, but it's a special kind of rectangle called a square. Yeah, look, because it's five wide…

[She traces the spaces along the bottom of the blocks.]

### Speaker

…and five tall.

[She traces the spaces along the right side of the blocks.]

### Speaker

Whereas here…

[She points to the left block set.]

### Speaker

…it's ten wide…

[She traces the spaces along the bottom of the blocks.]

…and five tall.

[She traces the spaces along the right side of the blocks.]

### Speaker

And, mathematicians, this is making me think about our next challenge.

[Text over a blue background: What’s (some of) the mathematics?]

### Speaker

So, what's some of the mathematics we just saw?

[A title on a white background reads: What’s (some of) the mathematics?

A bullet point below reads:

· You can have growing patterns. We saw a growing pattern when we looked at our staircase structure like this:

Below the point is an image of blocks in a staircase pattern that grows from left to right. Under each coloured block set is a label from 2-10. Under the label is text: When we go up 1 step, the number in each column (or tower) increases by 2.]

### Speaker

We realised over these few sessions that you can have growing patterns. We saw a growing pattern when we looked at our staircase structure like this. We can see that each time you go up one step, the number in each column or tower increases by two. And we can also see shrinking patterns.

[A title on a white background reads: What’s (some of) the mathematics?

A bullet point below reads:

· You can have shrinking patterns. We saw a shrinkig pattern when we looked at our staircase structure like this:

Below the point is an image of blocks in a staircase pattern that shrinks from left to right. Under each coloured block set is a label from 10-2. Under the label is text: When we go down 1 step, the number in each column (or tower) decreases by 2.]

### Speaker

And that's when each tower decreased by two. So, shrinking patterns occur when the value in each term decreases by something.

[A title on a white background reads: What’s (some of) the mathematics?

A bullet point below reads:

· You can sometimes describe something as a growing OR a shrinking pattern...

Below the point is an image of blocks in a staircase pattern that increases and then decreases. On the right-hand side, aligning with each row are numeral texts ‘2’, ‘6’, ‘10’, ‘14’ and ‘18’. On the right side of the numerals is a blue arrow pointing down. At the top of the arrow is text: I can read this as each row increases by 4 each time if I start at the top and read down.]

### Speaker

And we also realise that you can sometimes describe something as a growing or a shrinking pattern. With this up and down staircase, I can read this as each row increases by two, increases by four each time, if I start at the top and read down. So, there's two, six, ten, 14 and 18.

[The arrow is replaced by one pointing up. At the bottom of the arrow is text: I can read this as each row decreases by 4 each time if I start at the bottom and read up.]

### Speaker

I could also read this same structure from the bottom up, meaning I'd see a shrinking pattern, because each row decreases by four if I start from the bottom.

[A title on a white background reads: What’s (some of) the mathematics?

A bullet point below reads:

· Mathematicians use what they know to help them solve problems.

Text below reads: I know 20 can be composed of 18 and 2 more.

Below the text is an image of 2 long rows of blocks and a top row half the length of the blocks below.]

### Speaker

We also realised today that mathematicians use what they know to help them solve problems. So, since I know 20 can be composed of 18 and two more, I could use that knowledge.

[A black outline appears around the 2 long blocks.]

### Speaker

I also know that 20 can be composed of 14 and six more, so I could use that as well.

[Above the image, text appears that reads: I also know 2 tens and 2 tens and 1 ten is 5 tens. I know 5 tens is renamed 50. On the image is an image of handwritten text which reads:

18+2=20

14+6=20

2 tens + 2 tens + 1 ten = 5 ten = 50.

On the image, a black outline appears around the top row of blocks.]

### Speaker

I also know that two tens and two tens and one ten is five tens, and I know that that's renamed as 50.

[At the bottom of the screen is an image of 5 columns of 5 blocks. The first block has a black outline.]

### Speaker

I could also think about this problem differently using other knowledge I know. I know one two and four twos is five twos, which is a ten.

[Above the image, a title appears: What’s (some of) the mathematics?

A bullet point below reads:

· Mathematicians use what they know to help them solve problems.

Text below reads:

I know 1 two and 4 twos = 5 twos = 1 ten

I know 2 twos and 3 twos = 5 twos = 1 ten.

The second block also has a black outline.]

### Speaker

I know two twos and three twos is five twos, which is another ten.

[Below the text, more text appears:

I know 3 twos and 2 twos = 5 twos = 1 ten

The third block has a black outline.]

### Speaker

I also know three twos and two twos is five twos, which is a third ten.

[Below the text, more text appears:

I know 4 twos and 1 twos = 5 twos = 1 ten

The fourth block has a black outline.]

### Speaker

I know four twos and one two is five twos, which is a fourth ten.

[Below the text, more text appears:

I know 5 twos = 1 ten

The fifth block has a black outline.]

### Speaker

And I know five twos is the same as saying one ten.

[Below the text, more text appears:

I know 5 tens can be renamed as fifty (50).

An image of handwritten text appears. On the left, it reads:

I know 1 twos and 4 twos = 5 twos = 1 ten

I know 2 twos and 3 twos = 5 twos = 1 ten

I know 3 twos and 2 twos = 5 twos = 1 ten

I know 4 twos and 1 two = 5 twos = 1 ten

5 twos = 1 ten

On the right side of the text, is text: 5 tens = 50.]

### Speaker

Yes. And then all I had to do was rename it as 50 because I know five tens is called 50.

[A title over a blue background: Over to you, mathematicians!

Below the title is text: Use objects to create a new staircase structure. Draw it and describe the things you notice in your workbook. Below the text is more text: Explain the growing and shrinking patterns you can see inside your new staircase.]

### Speaker

So, mathematicians, your first challenge is to use objects to create a new staircase structure. Draw it and describe the things you notice in your workbook. Then think about how you can describe your pattern or the staircase structure that you've made, both as a shrinking pattern and as a repeat and as a growing pattern. 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]

## Instructions

• How many blocks are there altogether?

• Use diagrams to record the thinking you did to work out the solution.

• Make your own staircase structure that is different to the two we made. Draw and describe it.