# Staircase patterns – Early Stage 1

A thinking mathematically targeted teaching opportunity focussed on exploring and representing increasing and decreasing patterns

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

## Outcomes

- MAO-WM-01
- MAE-RWN-01
- MAE-RWN-02
- MAE-CSQ-01
- MAE-CSQ-02

## Collect resources

You will need:

- pencils or markers
- paper or your workbook
- objects such as blocks or counters.

## Watch

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

### 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 pattern Early Stage 1 part 2 video (4:33).

[A title over a navy-blue background: Staircase patterns 1 – 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.

The blocks from Part 1 now has 2 orange blocks and 1 red one lined up.]

### Speaker

Welcome back, mathematicians. How did you go? Did your pattern look a little bit like mine? Yeah, so what are some things that you notice about when we make it…

[She traces the top of the blocks.]

### Speaker

…go up by ones and then down by ones? Oh, you're right…

[She traces the shape of the blocks.]

### Speaker

…it still does have a triangular shape. But this time, it's a different triangle, isn't it?

[She removes the blocks from the Numberbooks Characters blocks. She traces the shape of the Numberbooks Characters blocks]

### Speaker

Last time the triangle looks like this and this time the triangle peak is in the middle at five.

[She places the other blocks back next to the Numberbooks Characters. She traces the top of the steps.]

### Speaker

Yeah, and this time it goes up by ones…

[She points to the area above each Numberbooks Characters block.]

### Speaker

…one, two, three, four, five times, and then down by ones…

[She points to the area above each block following.]

### Speaker

…one, two, three, four, five times. Yeah, and this made me start thinking about something mathematicians of what if we thought about this in a different way. Yeah, 'cause what we're seeing here is it grows…

[She points to the area above each block.]

### Speaker

…and then it shrinks. And we could continue our pattern…

[She moves the sheet of the paper with writing to the left of the blocks.]

### Speaker

…actually, by saying now I'm have to make it grow again…

[On the right of the blocks, she lines up 2 orange blocks.]

### Speaker

…like this.

[She adds 3 yellow blocks, 4 greens and 5 blues.]

### Speaker

And then when I get to five, I need to make it shrink again, yeah.

[She adds 4 green blocks, 3 yellows.]

### Speaker

And so we have this really nice structure to our pattern, but when we're looking at portion, I was wondering about this.

[She removes the blocks after the last red block.]

### Speaker

When we look at it this way…

[She traces the top of the steps.]

### Speaker

…it goes up and it goes down, it grows and it shrinks. But what if we thought about it in this way?

[She turns the blocks to the right, so that its bottom side is now on the left.]

### Speaker

How many blocks do we have in this column now?

[She points to the left side of the blocks.]

### Speaker

We can check. Remember, we knew that if we looked here that it was five blocks wide. So, if I turn it back around…

[She turns the blocks right-side up.]

### Speaker

…we know that this is five.

[She points to the first 5 blocks on the bottom. She turns the blocks on its side again.]

### Speaker

So, we can use that information.

[She points to the blue middle blocks, then the following blocks down.]

### Speaker

So, five, six, seven, eight, nine. Do you wanna count them all to check?

[She points to the first red block, then the following blocks down.]

### Speaker

OK, one, two, three, four, five, six, seven, eight, nine. Now, there's nine here.

[She takes a post-it note and writes: 9. She places the note below the blocks.]

### Speaker

And what about in our next column now?

[She points to the second column of the blocks.]

### Speaker

Yes, it would be…

[She points to the area above the top block and below the bottom block.]

### Speaker

…nine if there were blocks here, but there's not blocks there. Look, if we put these guys here…

[She places a black block to the area above the top block and below the bottom block.]

…that would still be nine, wouldn't it? But I need to take one away.

[She takes the block from the bottom away.]

### Speaker

So, the number before nine is eight, and then the number before eight…

[She takes the block from the top away.]

### Speaker

…is seven.

[On the post-it note she writes: 7. She places the note below the blocks.]

So, that should have seven blocks. Would you like to count to check?

[In the second column of blocks, she points to the orange block, then the following blocks up.]

### Speaker

OK, one, two, three, four, five, six, seven. And what do we know about this column…

[She points to the third column of the blocks.]

### Speaker

…here now? A-ha, if these was here…

[She places a black block to the area above the top block and below the bottom block.]

### Speaker

… it would also have seven, but we need to take one away.

The number before seven…

[She takes the block from the bottom away.]

### Speaker

…is six, and the number before six…

[She takes the block from the top away.]

### Speaker

…is five.

[On the post-it note she writes: 5. She places the note below the blocks.]

### Speaker

So, that column has five. Oh, and are you starting to see a pattern?

[She points to each of the post-it notes.]

### Speaker

Look, nine, seven, five. Seven is two less than nine. Five is two less seven. And three…

[In the fourth column of blocks, she places a black block to the area above the top block and below the bottom block. She takes them away.]

### Speaker

… oops, is two less than five. Three, wow…

[On the post-it note she writes: 3. She places the note below the blocks.]

### Speaker

… I got so many cool patterns in it. And the last one has one.

[On the post-it note with 3, she writes: 1]

### Speaker

Wow, so mathematicians, my challenge for you is can you work out how many blocks we have altogether now? Over to you to think about that.

[A title over a blue background: Over to you, mathematicians! Below the title is text: How many blocks are there altogether? Draw a picture to record the thinking you did to work out the solution.]

### Speaker

OK, over to you, mathematicians, how many blocks are there altogether? Draw a picture to record the thinking you did to work out the solution. See you soon.

[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]

## Discuss

- How many blocks are there altogether?
- Draw a picture to record the thinking you did to work out the solution.

## Watch

Watch Staircase pattern Early Stage 1 part 3 video (3:11).

[Text over a navy-blue background: Staircase patterns 1 – K follow-up. 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, little mathematicians.

[9 left-aligned rows of blocks in the pattern of a side-ways pyramid sit on a piece of paper. Row 1 contains 1 red block with a goggle eye and a smiley face. Row 2 contains 2 orange blocks, the outermost block has two goggle-eyes in glasses. Row 3 contains 3 yellow blocks with goggle-eyes and cut-out hair stuck on the outermost block. Row 4 contains 4 green blocks with goggle eyes on the outermost block and two paper cut-out hair bands stuck on. Row 5 contains 5 blue blocks with goggle eyes and a star shaped monocle on the outermost block. Row 6 contains 4 plan green blocks. Row 7 contains 3 plain yellow blocks. Row 8 contains 2 plain orange blocks, and row 9 contains 1 plain red block. Beneath these blocks are 5 over-lapping sticky-notes with the following numbers written on them: 9, 7, 5, 3 , 1.]

What was your thinking? Ah yeah, so I thought about something similar to some of you, and I looked at these numbers and I thought to myself, 'What do you already know?' And when I see nine and one, I know something about those numbers.

[The speaker circles the numbers 9, 7, 5, 3, 1 on the sticky notes with her finger. She then points to the numbers 9 and 1.]

Yes, I know. Oh I should have drawn my... one on a different post-it note.

[The speaker picks up a new stack of post-it notes and draws the number ‘1’ on it. She then crosses off the number ‘1’ on the post-it note beneath the rows of cubes, and places the sticky-note containing the number ‘1’ which she has just drawn over the top of it]

I know something about nine of something, and one of something. That when I have nine of something and one or something, it always combines to make 10 of something.

[The speaker picks up the post-it notes containing the numbers ‘9’ and ‘1’. She sticks them above the rows of blocks and to the right.

So, that would be one 10. Uh-huh. And I could actually break off five's head and say, 'you go here' and now we have 10 in our first column.

[The speaker removes the outermost blue block from the 5^{th} row and places it above the red block in the first row, creating an initial row.]

### Speaker

And then, yes, seven. Aha. And three is also going to be 10 of something 'cause seven of something and three of something always combines to make 10 of something.

[The speakers grabs the sticky notes with the numbers ‘7’ and ‘3’ from beneath the blocks and places them to the right of the sticky notes containing the numbers ‘9’ and ‘1’.]

Yes. So, we could take these blocks here.

[The speaker removes the outermost green block from row 7 and places it beside the red block in row 2, she removes the outermost green block from row 5 and moves it beside the red block in row 10, and she removes the outermost blue block from row 6 and places it beside the other blue block in row 1.]

Thank you very much for volunteering your parts of your bodies, number blocks. And put them here, and now look. I have one 10, two 10s, look one 10, nine and 1, two 10s, seven and three and five more.

[The speaker points to the 10 blocks in the first column, then to the 10 blocks in the second column, then to the 5 blocks in the third and final column. She moves the sticky-note containing the number 5 from beneath the blocks to the right of the blocks.

And we call two tens and five more 25.

[The speaker points to the 2 columns containing 10 blocks and the to the column containing 5 blocks.]

So, there's 25 blocks all together.

[The speaker draws the number 22 on a blank sticky-note and sticks it below the blocks. Text on a blue background reads: What’s (some of) the mathematics?]

Woah, now... Some of the mathematics here.

[Text on a white background reads: Mathematicians use what they know to help them solve problems. Below this reads: I know 10 can be made of 9 and 1. To the right of the text is an image of the 3 columns of blocks, with the first column, containing 10 blocks, outlined in black.]

### Speaker

Mathematicians use what they know to solve problems.

[New text appears beneath the existing text, it reads: I know 10 can be made of 9 and 1. I know 10 can be made of 7 and 3. I know 2 tens is called twenty (20). In addition to the first column, the second column, containing 10 blocks, is outlined in black.]

I know 10 can be made of nine and one. I know 10 can be made of seven and three. I know two 10s is called 20.

[New text appears beneath the existing text, it reads: I know 2 tens and 5 more is called twenty-five. I know we write that as 25. In addition to the first two columns, the third column is outlined in black. The third columns contents become opaque.]

I know two 10s and five is called 25. And I know I can write that as 25.

[Text on a new slide reads: I know how to count so we could also have counted all of the blocks to work out how many there are. To the right of this text is an image of the rows of blocks before they were re-arranged.]

I also know how to count. So, we could also have counted all of the blocks to work out how many there are.

[Text on a new slide reads: You can have growing and shrinking patterns. We saw a shrinking pattern when we looked at our staircase structures like this:

Beneath this text is an image of the rows of blocks before they were re-arranged, with the numbers 9,7,5,3 and 1 beneath it.]

### Speaker

We also realised today that you can have growing and shrinking patterns. We saw a shrinking pattern when we looked at our staircase structures like this.

[Text appearing to the right of the image of blocks reads: It’s a shrinking pattern because the number in each column decreases by 2 each time. Taking-away 2 each time is the repeating core!]

It's a shrinking pattern because the number in each column decreases by two each time. Taking away two each time is our pattern core. Look in our first column there's nine, two less than nine is seven, two less than seven is five, two less than five is three, and two less than three is one.

[Text on a blue background reads: Over to you, mathematicians! Smaller text beneath reads: Use objects to create 2 new growing patterns and 2 new shrinking patterns. Draw and describe them in your mathematics workbook.

So it's over to you now, mathematicians! Use objects to create two new growing patterns and two new shrinking patterns.

Draw and describe them in your mathematics workbook. Have a lovely day!

[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

Use objects such as blocks to create 2 new growing patterns and 2 new shrinking patterns.

Draw and describe them in your workbook.