# Circles and stars

A thinking mathematically context for practise resource focussed on developing flexible multiplicative strategies and renaming using place value knowledge

## 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-RWN-01
• MA1-RWN-02
• MA1-CSQ-01
• MA1-FG-01

## Collect resources

You will need:

• playing cards (we used 2, 5 and 10 only)

• a dice

• paper

• markers or pencils.

## Watch

Watch Circles and stars video (12:09).

Investigate the highest total using groups.

### Michelle

Hello Barbara.

[Screen shows 2 pieces of A4 paper, playing cards, and a marker.]

### Barbara

Hello Michelle. How are you?

### Michelle

I am great. How are you?

I'm very well.

### Michelle

We are going to play a game today from Marilyn Burns called How many stars?

Okay.

Yep.

### Michelle

So, with our game board, we need to make eighths.

Okay.

### Michelle

So, one way that I make eighths is, I halve my paper. You can halve it a different way if you want.

You don't have to do the same. And then from my half, if I halve it again and then halve it again, I'm now quartering each half and that will make me have eighths.

[Michelle folds the paper on the right length way in half, and Barbara folds hers on the left width way in half. They both now fold their paper in half and then in half again.]

### Barbara

Oh, maybe I could go this way?

Oh well, a bit too skinny.

### Michelle

Too skinny. I think I'll go this way then.

### Barbara

You know what's interesting about that, is that we folded them in a different order.

[They both unfold their papers and place then on the table, the folds have created 8 square boxes.]

### Michelle

Yes, but we still got the same.

And you know what else it looks like? An array.

It does.

### Michelle

Look, it looks like 4 twos or 2 fours. Yeah. Our game's called circles and stars so we're gonna write circles and stars.

[The 2 A4 pieces of paper are turned to a portrait orientation. In the top left hand corner they write ‘circles and stars’.]

Okay. Okay.

### Michelle

Okay, so we are using playing cards 2, 5 and 10 today and that's gonna tell us how many is in each of our groups.

[Michelle picks up the playing cards which only has 2, 5 and 10 cards in the pack.]

Okay.

### Michelle

And we roll the dice to say, how many groups do we need? So here, you go first.

Okay.

Roll the dice.

### Barbara

So, 2 groups.

[Barbara rolls the dice and rolls 2. Michelle shuffles the cards.]

Two groups.

Two circles.

Two groups.

### Barbara

In this square any square.

### Michelle

In your, or one of those.

Yep ok.

### Michelle

We don’t use that one.

### Barbara

We don’t use this one, oh ok. Ok.

Oh, one of those?

### Michelle

Yeah, we don't use that one

Nah.

Okay.

### Michelle

Okay and then pick a card and that tells you how many.

### Barbara

Awesome.

[Barbara draws 2 circles on her piece of paper. She draws a number 10 from the deck.]

### Michelle

Now if you know it, you don't have to draw it. You can just explain how many 2 tens would be.

Okay so 2 tens would be 20.

### Michelle

Mm, cause of place value

### Barbara

So just rename it?

### Michelle

Yes, so just put 20 here cuz you know this.

### Barbara

Okay.

[Barbara writes 20 under the 2 circles she drew.]

### Michelle

And then if we don't know it, we can use it to help us.

Okay.

### Michelle

So

[Michelle rolls the dice, and she rolls one.]

### Barbara

One group.

[Michelle draws a card from the deck and draws a 10.]

One 10.

Okay.

### Michelle

So, so I actually know that one 10 is 10, so, and that's 10. So, I don't need to draw them.

[Michelle draws a circle on her paper and writes 10 in the middle and 10 under the circle.]

### Barbara

Should I label mine as 10 and 10?

[Barbara writes 10 in each of her circles.]

### Michelle

Yes, that's a good idea.

Okay.

### Barbara

One.

[Barbara rolls the dice and rolls a one. She draws a larger circle on her paper in the top right-hand corner.]

Oh rats!

### Barbara

So, I'm guessing by that, that's not a good thing, right? We want to have lots.

### Michelle

Well, it could be if you get a 10 because 10 is a good move. But imagine now you get one 2.

### Barbara

Okay, so I want as many?

### Michelle

You want as many stars as possible at the end.

### Barbara

Oh, so these are stars? So, I should put in 20 stars.

### Michelle

Oh, that's a good idea! I should put stars too.

### Barbara

10 stars, 10 stars is 20 stars.

Okay, so now I've got one 2. Okay so that's just 2 stars.

[Barbara draws 1 star each under the tens in the circle and one star next to the 20, she draws a card from the deck and draws 2. She now writes 2 in her circle and underneath she writes 2 again. She draws a star next to both digits.]

### Michelle

Yes, it's a known fact so you don't have to draw it.

Okay, my go. Oh 3. Fives. Okay, so if I didn't know 3 fives, I could draw my 3 and then I could draw my 5 stars.

[Michelle rolls the dice and rolls a 3. She draws a 5 from the card deck. She draws 3 circles on her paper and draws 5 stars in the first circle.]

Two, 3, 4, 5 and what I might do is now imagine in my mind's eye that there's 5 here, 5 here, and 5 here.

Okay.

### Michelle

And what I do know is it that 5 and 5 combines to make 10.

Yep.

### Michelle

And one 10 and 5 more is 15.

Okay, yep.

### Michelle

So, I don't have to draw everything.

[Michelle points with her pen to the first circle, then the second and third circles. She then writes 15 under the circles.]

Just enough.

### Michelle

But I can if I need to. I just draw enough to help me, you know, work out how many stars it would be in total. So that would be 15.

### Barbara

Okay. 15 stars.

Where do we put that? At the bottom after you've used it.

[Barbara places Michelle’s card at the bottom of the deck and then rolls the dice. She rolls a 2 and draws a 10.]

Yes.

### Barbara

Okay, so 2. Tens.

### Michelle

Oh, 2 tens is nice. Also, nice because you can just use renaming with place value.

### Barbara

Exactly, if you know place value, then you instantly 2 tens can be renamed as 20.

[Barbara draws 2 circles and writes 10 in each circle and draws a star underneath them. Underneath the circles she writes 20 and draws a star.]

### Michelle

Okay, my go! Oh 5 is good.

[Michelle rolls the dice and rolls a 5.]

### Barbara

Imagine if you get a 10.

### Michelle

That would be good, imagine if I got a 2.

Ok let's see.

### Michelle

Oh yes! Yes! So, I can draw one, 2, 3, 4, 5, like a dice.

[Michelle draws a card from the deck and draws a 10.]

Yeah.

### Michelle

And each one is worth 10, but I actually just know that that's renamed as 50, because of place value knowledge.

[Michelle draws 5 circles and draws a star in each circle and underneath she writes 50 and draws a star next to it.]

### Barbara

Okay now I want 6 tens. Re-roll or…?

### Michelle

You can re-roll because I saw what it was.

### Barbara

All right, 2 fives. Well, I know that I know 2 fives is 10.

[Barbara rolls the dice. She rolls 2 and she draws a 5 from the deck of cards. She draws 2 circles.]

Mumm.

It's just...

### Michelle

How do you know it? Is this a fact you know?

### Barbara

It's a fact I know, but it's also my 2 hands.

### Michelle

Yes, because 2 fives together.

[Michelle displays her hands open, indicating 5 fingers on each hand add up to 10.]

### Barbara

Yeah, and also the 2 rows of a 10 frame.

### Michelle

Because 5 here and 5 there. 10.

### Barbara

Yeah. Okay.

Okay so, 10 stars. Oh, I should do 5.

[Barbara writes 5 in each of her circles and draws a star next to the digit. Underneath she writes equals 10.]

### Michelle

Okay, oh 6 is nice! Imagine if I got 10.

[Michelle rolls the dice and rolls a 6.]

### Barbara

Imagine if you get one.

### Michelle

No, there's no one in there, there's only 2, 5 or 10.

Come on 2.

### Michelle

Oh, rats!

[Michelle draws a 2.]

So, um, here's 6 and there's 2 in each one. And actually, I know that's 12 because if this was a 10 frame, say that moved to there and that moved to there.

You know, my 10 frame is going in this orientation. What I know is that for each dot there, there's actually 2.

[Michelle draws 6 smaller circles in a dice pattern, she draws a ten frame around 3 of her circles and draws arrows to signify the 2 circles going into the 10 frame. She draws an additional circle to show the one circle left over.]

Yeah.

### Michelle

And then I'd have one more left over and I know that it's 12.

### Barbara

Okay. I like how you explained that. It made sense. Okay, 6. Fives. Not bad.

[Barbara rolls a 6 and draws a 5 from the deck of cards.]

### Michelle

Oh, I know how you could work that out.

Yeah?

### Michelle

Cuz, you could say if you halved 6, that would be 3.

[Michelle places her finger over 3 of the dots on the dice, showing only 3 dots, and then points to the 5 of diamonds on the card.]

Yeah.

### Michelle

And doubled 5 you get 10.

Aww.

### Michelle

And then you just rename it.

Three 10s.

### Michelle

Because you can use your fives to work out tens.

### Barbara

Yeah, and I like how you actually, cuz I do that sometimes, but I don't always half and double.

Sometimes I just I get the result and then I halve it. I like the way you did that.

So how would I draw it then? Would I draw as 6 fives, or would I draw it as 3 tens?

### Michelle

Good question.

Because that's how we worked it out?

### Michelle

Maybe, can I draw on your paper?

### Michelle

Maybe you could draw like this, and that. So, I had 5, but I thought of them together.

[Michelle draws a rectangle on Barbara’s paper. She then draws 2 circles in the rectangles with 5 dots each in each circle. She then draws another 2 rectangles the same.]

### Barbara

Okay, oh I like that.

### Michelle

And then, there's, yeah, you had them like this and this, and then you know I'm actually thinking about fives as tens, and I only need 3 of them.

### Barbara

Okay, I really like that.

Okay I don't have to draw them all cause I know it now. So, this, so then we said that was 30 stars.

So, each one here it was 10 stars, 10 stars and 10 stars.

[Barbara now writes 10 at the bottom of each rectangle, and 30 underneath all of them.]

### Michelle

Okay, you've got one go, I've got 2 goes left. We lose, we use the last box to help us calculate.

Ah, 2. 2 twos. Well, that's 4 because I know this. But it would look like this, 2 twos, which is 4 stars all together. Your go.

[Michelle draws 2 circles with 2 dots in the circle. Underneath she writes 4.]

### Barbara

Okay. Oh, you're writing the word stars, I'm just drawing a star! Six! Yes! Twos.

[Barbara rolls 6 and draws 2 from the deck of cards.]

### Michelle

Oh, you go from this heightened state of yes 6, oh twos.

### Barbara

Oh, and well we've done this one before.

### Michelle

Still better than 6 ones.

### Barbara

Exactly, well twice as good. Okay, so, I know this number fact.

### Barbara

Yes.

But otherwise, I could just think of it, you know, the idea of double 5 is 10 and then 2 more. So, um, with 2 in each one is 12 stars.

[Barbara draws 6 circles and writes 2 in the first one. Underneath she writes equals 12.]

### Michelle

Okay alright, last go for me. I need a good roll.

### Barbara

You've got that 50. So, I think you're ok.

### Michelle

That's true. Three fives. So, you know how last time I drew it and I said I could work that out as 5 and then visualize?

[Michelle rolls the dice and rolls a 3. She draws a 5 from the deck of cards.]

Yes.

### Michelle

This time what I could think of is double 5, which is 10 and 1 more 5.

Yeah.

### Michelle

Because when you do your threes, you can work out double plus, plus one more. So how would I draw that?

I could do the same idea here. Oh, I know. I could go 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, and then one more group of 5.

[Michelle draws 2 rows of 5 circles she then draws a rectangle around the circles. Another row of 5 circles is drawn underneath the rectangle.]

Oh good.

### Michelle

And then there's the threes, look.

[Michelle draws a circle around each of the 3 columns of threes. She writes 15 underneath.]

### Barbara

Yeah, that's good recording.

### Michelle

Hmm, so that's 15 altogether.

So, now Barbara, what we need to do is work out how many stars we had and the person with the most, number of stars is declared the winner!

[Michelle circles the text and drawings on the paper and taps her fingers.]

Okay, great.

### Michelle

So, so now the fun part comes, right because we can use some really cool strategies. But we can work together to help each other.

So, what are you thinking when you look at your numbers? Because I might, what I might do is write down all of mine.

So, I've got 10 joined with 4, with 12, with 15, with 15 and with 50. And that helps me start to look for things.

[Michelle removes the dice and playing cards. In the last square of her game board she writes, 10 plus 4 plus 12 plus 15 plus 15 plus 50.]

### Barbara

Oh, I found something.

[Barbara writes 2 plus 20 plus 20 plus 10 plus 30 plus 12.]

### Michelle

Oh yeah, I can see some stuff too.

### Barbara

Okay. I ran out of space, but I'll put it down here. Okay, so when I wasn't, when I was writing them out, I realized that 20 and 20 and 10 actually makes 50.

[Barbara circles the 20 plus 20 plus 10 and writes the number 50 on top. Underneath she writes 2 plus 50 plus 30 plus 12.]

### Michelle

Yeah. So, all of that together, that's 50 there.

### Michelle

Yeah, so you could rewrite that now as 2, plus 50, plus 30, plus 12, because of our equivalent. If that helps?

### Barbara

No, that helps me, because then it's a much, it's a shorter number sentence as well. Okay, so.

### Michelle

Oh, now I can see something else.

### Barbara

Well, what I'm thinking, do you want to tell me what you're thinking? Or should?

### Michelle

Yeah, because I was thinking like 5 tens and 3 tens is 8 tens.

Yeah.

### Michelle

So, then it's 2 plus 8 tens which is 80, plus 12.

Yeah.

### Michelle

And that's even nicer to work with.

### Barbara

Yeah, or even, even get that 10 from here.

Oh yeah.

### Barbara

So, 5 tens and 3 tens is 8 tens and then 9 tens.

Yeah!

### Barbara

So, I've got 9 tens. Plus, 2, plus 2! Plus 2.

[Barbara writes underneath 9 tens plus 2 plus 2 and underneath that she writes equals 94.]

### Michelle

Oh, that is nice!

Yeah.

### Barbara

Okay and then that's 9 tens and 4, which we would rename as 94.

### Michelle

I think you've won!

So, but let's have a look. But it is close.

Very close.

### Michelle

So, so what I know actually, is that double 15 is 30.

Yeah.

### Michelle

I just happened to; I don't know why I know that, but I do. So, I'm gonna go.

### Barbara

I think you've won you know.

### Michelle

Well let's see. 10, plus 4, plus 12 plus 30, oh now I feel more confident, plus 50.

[Michelle writes 10 plus 4 plus 12 plus 30 plus 50 underneath her previous row.

Yeah.

### Michelle

Yeah, because now what I can see is that there's another hidden 50.

Yeah.

### Michelle

In there. So, if I take the 10 and the 10.

Oh yeah.

### Michelle

So, one 10 and one 10. Is 2 tens. Plus 3 tens is 50. So that would be 50, plus 4, plus 2, plus 50.

[Michelle writes 50 plus 4 plus 2 plus 5 underneath her previous equation.]

You won.

### Michelle

And then 5 tens and 5 tens is 10 tens, which you call 100. Plus 4, plus 2 and that's 106.

Aww but it was close.

[Michelle writes 100 plus 4 plus 2 equals 106 underneath.]

### Barbara

It was pretty close!

Only 12 away.

### Barbara

This one was really good.

### Michelle

That was a good lucky go!

Over to you mathematicians to enjoy Marilyn Burns' circles and stars!

[End of transcript]

## Instructions

• Divide your paper into eighths.

• Roll a dice to determine how many circles (groups) you need to make.

• Turn over a playing card (or roll the dice again) to determine how many stars to add into each circle.

• Determine how many stars there are in total. You can draw all or some of the stars in each circle - you only need to draw what you need to help you work out the product.

• Continue taking turns until each player has had 6 turns each.

• Work together to work out who has the most starts altogether.