Let's explore patterns – Stage 1

A thinking mathematically context for practise resource focussed on creating repeating patterns using objects

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
• MA1-DATA-01
• MA1-DATA-02
• MA1-2DS-01
• MA1-2DS-02

Collect resources

You will need:

• a collection of objects
• pencils or markers
• something to write on

Watch

Watch Sorting patterns 1 (7:42), a background video about sorting.

Investigate ways to sort and categorise sets of objects.

Michelle

Welcome back mathematicians, and I'm here with one of my favourites today, Hi Sam.

Hi Michelle.

How are you?

Good.

Michelle

Very good. So I've got this box here of equipment today.

[Screen shows a box which Michelle shake. It has different items in it, 6 wheels, a toy car, 4 small animal toys and approximately 19 coloured and varying small size Lego blocks.]

Ok.

Michelle

And what I wondered is if you could sort them for me.

Sam

Alright, maybe you could write down the different types so it would it be easier. So as I sort them out you could write them then we could put their little sections.

[Sam put his hand in the box and picks up a Lego block and puts it back down.]

Michelle

So do you think there's going to be more than one way to sort them?

Yeah,

Michelle

OK. Let's have a look. Do you want to tip them out of the box or do you want to keep them in there?

[Michelle pushes the box into the centre and Sam picks up the box and tips the items out.]

Sam

We could tip them out.

Michelle

Alright, what are you thinking first Sam?

Sam

I'm thinking things that might roll.

OK.

Sam

So that definitely rolls, so we could put that over there. That would probably roll, yep, and all these are wheels too, so they would roll and this little car. So that's like the wheel section, I guess. There may be like almost animal stuff.

[Michelle places a white sheet of paper down. Sam touches items from the box before picking up each of the wheels and small toy car and placing them in a pile. Sam then moves away the rest of the items. Sam picks up an animal toy and moves it around.]

Michelle

So would you say that these are things that roll and things that don't roll?

[Michelle circle the things than roll and the things that don’t roll with her hand in the air as she checks with Sam.]

Yeah.

Michelle

Would they be our categories? So things that roll and things that don't roll.

OK, and then what else are you thinking about? If you put them all back together.

[Michelle writes on the white paper with a marker ‘things that roll’ and underneath writes ‘things that don’t roll’. Then she puts a square closed bracket around the 2 categories. Michelle pushes all the toys back together into one pile in the centre.]

Sam

Maybe things... Hmm... We could do also like almost mostly living things because I guess you could call these living things and then non-living things.

[Sam picks up the animal toys and places them in a pile.]

Michelle

Ok. Living things and non-living things. Ok, yep, what else could you do? What's another sort you could do?

[Michelle writes on the white paper with a marker ‘living things’ and underneath writes ‘non-living things’. Then she puts a square closed bracket around the 2 categories.]

Sam

You could also do things that stick together like these and stuff and then these won't stick together or anything.

[Sam pushes all the toys back together into one pile in the centre. Sam picks up some of the building block toys showing how they stick together and then some animals showing how they don’t stick together. Then he moves them into 2 piles.]

Michelle

So I'll move those over here then to help.

[Michelle moves the animals to the side away from the rest of the toys.]

Sam

These don't stick together almost, so I guess you use those, but I guess you could say that sticks together because it's made of stuff.

[Sam places the wheels in the pile with the animals and pick up a car to look at.]

Michelle

But would it stick to another car? That's a really good question.

Sam

Probably not. So, but this would stick together because like you can take it off then put it back on. Oh, not probably not.

So, yes, all these, oh no probably not, so all these things stick together onto stuff.

[Sam places the car in the pile with the rest of the things that don’t stick together. He then starts sticking some of the Lego blocks together. Sam then finds another set of wheels and puts them in the other pile.]

Michelle

Oh, yeah, I see.

[Michelle writes on the white paper with a marker ‘things that stick together’ and underneath writes ‘things that don’t stick together’. Then she puts a square closed bracket around the 2 categories.]

Sam

Then you could, I guess you could make smooth things and rough things. So that's smooth, that's smooth.

[Sam starts sorting through the Lego blocks feeling for smooth and rough surfaces.]

[End of transcript]

Instructions

• What are some different ways to sort your collection?
• Record your ways of thinking.

Watch

Watch Patterns – Part 1 (4:15) background video.

Explore a pattern using coloured blocks.

Hello Sarah.

Hi Michelle.

Michelle

How are you today?

Sarah

I’m very well how are you?

Michelle

Very good thank you.

Good.

Michelle

We thought that we would come together today to explain some of the complexities around mathematical patterning. It seems like a really simple thing to notice and wonder about patterns, but sometimes little kids find this really tricky, and it can be quite difficult to think about how to help them through identifying and noticing and extending and exploring patterns.

So, we thought we'd come with some tips and tricks for teachers in classrooms and mums and dads and family at home.

Sounds great.

Michelle

Thanks for helping me.

So, Sarah, I've prepared a pattern underneath my sheet of paper, and I wonder at the moment if you can work out what my pattern is. If I said to you what block comes next, can you answer that question yet?

[Screen shows one yellow block and next to it on the right is a white A4 piece of paper.]

Sarah

Not confidently, no.

Michelle

What if I show you my next block, can you tell me what comes next now?

[Michelle pulls piece of paper to the right revealing a green square block.]

Sarah

Not yet. I can start speculating, but I still don't really know confidently.

Michelle

Ok, what if I reveal my next block?

[Michelle pulls piece of paper to the right to reveal a yellow square block.]

Sarah

OK, yeah, I'm starting to hypothesise a few things.

Michelle

So, what are you thinking it could be?

Sarah

I'm thinking that it could be green if it's just a simple, alternating pattern.

Yes, that's what I am thinking but I'm very keen to see if potentially there is different colour being introduced, or maybe another yellow.

Michelle

So, you're thinking it could be green like this?

[Michelle places a green square block on the white piece of paper.]

Sarah

It could be, yes.

Michelle

Could it also be something like red?

[Michelle places a red square block on the piece of paper.]

Definitely.

Michelle

Could it also be something like another yellow?

[Michelle places a yellow square block on the piece of paper.]

Yes.

So, at the moment we don't have enough information about the core of the pattern to see?

[Michelle removes yellow square block.]

Yes.

Michelle

I agree with you. What if I now do this?

[Michelle pulls A4 piece of paper to the right to reveal a green square block.]

Sarah

OK yeah, I'm starting to get a bit more confident, but I still don't fully trust it yet.

Michelle

And that's actually really wise decision because what we want to see to say something has a mathematical regularity is we want to say that it happens over and over and over.

So, we've sort of seen yellow, green, yellow, green over and over, but we're waiting to see if we see it maybe for the third time to be more confident that it's definitely there. So, you're starting to think it's probably yellow?

[Michelle points to the first yellow block, then the next green block, then the next yellow block and then repeats.]

Sarah

Yes, my hunch is yes ah ok.

Michelle

OK and if I reveal again now, we have green and now you're like yeah this is the pattern and so we could describe this as a yellow, green, yellow, green, yellow, green and what would come next then?

[Michelle pulls piece of paper to the right which now reveals another green block, and she rolls a yellow block onto the carpet and picks up, she then pushes 2 blocks together one yellow and one green, and she pushes together another yellow and green block waiving her finger over them and she pushes the last yellow and green block together.]

Yellow.

Michelle

Yellow and we have now seen that 3 times, so he's sort of one chunk of the core, here is another representation of the pattern core and here is a third representation of the pattern core.

So that's why you feel a bit more confident, actually, to say I think this is now going to continue, my pattern will continue.

[Michelle removes piece of paper to reveal another yellow and green square block.]

Yes.

Michelle

And so sometimes when we talk about patterns like this with kids, it's important to acknowledge that it's yellow, green, yellow, green and the attribute that's changing here is the colour.

[Michelle moves the blocks, so they are evenly spaced out with approximately one centimeter between them.]

Aha.

Michelle

It's also important to talk about sometimes people refer to this as a one two pattern.

Sometimes that can be really tricky for students because we'd say one, two, and then we call this one again.

[Michelle points to the first block, then the second block and then the third block.]

Umm.

Michelle

Which is really confusing when they're young and learning about attributes of counting principles, and so mathematicians would refer to this as an AB pattern.

Right.

Michelle

So, I say AB, AB, AB, AB.

[Michelle points to first one and two blocks, then points to the next 2 blocks and then the next 2 indicating an AB pattern. She now adds another row of 8 square blocks underneath with red, yellow, red, yellow, red, yellow, red, yellow.]

Aha.

Michelle

And that way it becomes generalisable so I could then say and build another pattern underneath where I say red, yellow, red, yellow, red, yellow, red, yellow. And even though the colour looks really different, they're actually the same pattern structure because if I squish them together, I can see that in fact the core has two things in it before it repeats, but if I call it yellow, green it then makes it really hard for me to generalise out into this situation.

[Michelle squishes the first 2 squares blocks together and the next 2, and the next 2 and the next 2, displaying 4 lots of 2 blocks on the top and bottom row. She then points to the top left yellow and green blocks and then the bottom left red and yellow blocks underneath gesturing as she explains that it is difficult to generalise.]

Sarah

Yeah, it doesn’t apply, hey.

Michelle

Whereas if I say AB, I can still say.

[Michelle points to the 2 blocks on the top left and bottom left again as she explains that referring to them as AB applies to both rows. She then separates the blocks, evenly spacing them out with approximately one centimeter between them.]

Sarah

Yeah, it applies.

Michelle

This is why we typically refer to them as AB patterns.

[End of transcript]

Watch

Watch the background video Patterns – Part 2 (2:37). This will suggest some follow up questions you might like to ask.

Create patterns using items of different size, colour and orientation.

Michelle

We can also think about this idea of moving from an AB structure like this into another AB structure as the idea is translating a pattern into something different.

[Screen shows 2 rows of 8 square blocks. The first row shows an alternating pattern of yellow then green blocks. The second row shows an alternating pattern of red, then green, blocks. Michelle points to the top row then the bottom row.]

Aha.

Michelle

So, this one is not too tricky because we're still working with blocks, but we could also start to do something with equipment at home, like a train and a train track, a train and a train track, a train, and a train track.

[Michelle points to both rows with her finger. Michelle then adds a bottom row underneath with a toy train and a wooden train track, and then another toy train and wooden train track and then another toy train and wooden train track.]

Umm.

Michelle

And we can talk about that this is still an AB pattern.

Yeah.

Michelle

And again, if I make these spaces and do the chunking, that helps me see that in actual fact, it is the same pattern core, it's now just represented in an entirely different way.

[Michelle chunks the trains and tracks by making a gap between each pair of trains and tracks and then does the same for the coloured blocks in the row above, chunking them into pairs to indicate the same core pattern.]

Umm.

Michelle

You can also do it with body sounds.

Ok.

Michelle

So, you could do like a clap and a stomp on the ground, a clap and a stomp, a clap, and a stomp. So, a simple thing that you can do is this, ask is how else could we represent the same pattern call using different equipment or sounds or movements in their bodies.

[Michelle claps and stomps using her hands on the floor repeatedly, making a pattern.]

Sarah

I think kids would love that! They could go and find some of these other materials that they could use to replicate the same pattern.

Michelle

Why don't you try to make another AB pattern using this equipment?

Sarah

OK, I will.

[Sarah picks up a bag of clothes pegs and lays out 6 pegs in a line, alternating between red and white pegs (1 red, 1 white, 1 red, 1 white, 1 red and 1 white).]

Michelle

Oh, I see, because it's got AB, AB, and AB like our other AB patterns. You could even do something with movement too. So, if you change these ones for red. You could do it like this up, down.

[Michelle points to the first 2 red pegs then the second 2 pegs and then the last 2 pegs.]

Umm.

Michelle

Up, down, up, down.

[Michelle replaces the 3 white pegs with 3 red pegs and places the pegs in the opposite direction.]

Yes.

Michelle

So that the kids aren't always just having to think about colour, but they're looking at other attributes, and in this case their position.

Yeah.

Yeah.

Michelle

Yeah. There's another way to represent it, vertical, horizontally, vertically, horizontally, vertically, horizontally.

[Michelle then moves every second peg into a horizontal position, so the alternating pattern shows one vertical peg, one horizontal peg, one vertical peg, one horizontal peg, one vertical peg and one horizontal peg.]

Umm.

Michelle

Still AB, AB, and AB, and so, some of the questions that we can ask of students are things like.

Now what would be next in my pattern and ask them to continue it. So, what would come next in your pattern that you are making?

[Michelle moves away the trains, tracks and square blocks and then points to the space after last peg in the row.]

Sarah

It would be a vertical and a horizontal.

[Sarah adds another vertical peg and horizontal peg to the end of the pattern.]

[End of transcript]

Watch

Watch the background video Patterns – Part 3 (0:58). This will suggest some follow up questions you might like to ask.

Explore patterns using pegs.

Speaker 1

[Screen shows, 8 red pegs lined up in a vertical row. They are in an alternating pattern of one vertical peg followed by one horizontal peg.]

And then we can do things like saying - uh oh, something came out of my pattern, what might that now be?

[Presenter removes the second horizontal peg from the pattern and takes it out of screen.]

Speaker 2

OK, so you would then sort of cover it or get them to close their eyes.

Yes.

Speaker 2

Is that what you're thinking?

Speaker 1

[Presenter returns the peg to its place in the pattern.]

Yes. And so, this is much trickier for kids to see because the pattern core is disrupted.

[Presenter moves the second horizontal peg down and then returns the peg to its place in the pattern.]

Umm.

Speaker 1

And so that's where it might be worth saying - oh well, let's see if we aren't sure what comes here, let's separate the chunks that we can see like this.

[Presenter moves second horizontal peg down, leaving the vertical peg in the pattern. She then separates the pattern into 4 groups of one vertical and one horizontal peg.]

Yes, ok.

Speaker 1

And in actual fact, then I could move them and say they're exactly the same, and they're exactly the same.

[Presenter then moves the groups of pegs one at a time underneath one another in a vertical display and points to show they are the same.]

Umm.

Speaker 1

And if I move this here, now I can see what's missing.

[Presenter then moves the single vertical peg to the top of the display and points to where the horizontal peg is missing. She places the missing horizontal peg back in its place in the pattern.]

Yeah, yes.

Speaker 1

I can put the piece in and then I can re-organise and re-establish my pattern.

[Presenter moves the groups of pegs back to how they were displayed at the beginning of the video, with 8 red pegs lined up in a vertical row. They are in an alternating pattern of one vertical peg followed by one horizontal peg.]

Brilliant.

Speaker 1

Have fun patterning mathematicians.

[End of transcript]

Instructions

• Describe the first pattern.
• How do you know it is a pattern?
• How would you describe the part that repeats?
• Make an AB pattern.

• How are the patterns in the second image the same?
• How are they different?
• Make, copy or extend an ABB pattern.