Circles and stars
A thinking mathematically context for practise resource focused on developing flexible multiplicative strategies and renaming using place value knowledge.
Adapted from Marilyn Burns – About Teaching Mathematics, 2015.
Syllabus
Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus (2022) © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2024.
Outcomes
- 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).
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?
Barbara
I'm very well.
Michelle
We are going to play a game today from Marilyn Burns called How many stars?
Barbara
Okay.
Michelle
So, we need to organize our game board to start with.
Barbara
Yep.
Michelle
So, with our game board, we need to make eighths.
Barbara
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.
Barbara
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’.]
Barbara
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.]
Barbara
Okay.
Michelle
And we roll the dice to say, how many groups do we need? So here, you go first.
Okay.
Michelle
Roll the dice.
Barbara
So, 2 groups.
[Barbara rolls the dice and rolls 2. Michelle shuffles the cards.]
Michelle
Two groups.
So, draw your 2 groups.
Barbara
Two circles.
Michelle
Two groups.
Barbara
In this square any square.
Michelle
In your, or one of those.
Barbara
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.
Barbara
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.
Barbara
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.]
Michelle
One 10.
Barbara
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.
Barbara
Okay.
Michelle
Okay, your go.
Barbara
One.
[Barbara rolls the dice and rolls a one. She draws a larger circle on her paper in the top right-hand corner.]
Michelle
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.
Barbara
Okay.
Michelle
And what I do know is it that 5 and 5 combines to make 10.
Barbara
Yep.
Michelle
And one 10 and 5 more is 15.
Barbara
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.]
Barbara
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.]
Michelle
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.
Barbara
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.]
Barbara
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.]
Michelle
Mumm.
Barbara
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.
Barbara
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.]
Barbara
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.]
Barbara
Not bad, not bad.
Michelle
Oh, I know how you could work that out.
Barbara
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.]
Barbara
Yeah.
Michelle
And doubled 5 you get 10.
Barbara
Aww.
Michelle
And then you just rename it.
Barbara
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?
Barbara
Yes please.
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.]
Barbara
Yes.
Michelle
This time what I could think of is double 5, which is 10 and 1 more 5.
Barbara
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.]
Barbara
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.]
Barbara
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.
Barbara
Yeah.
Michelle
So, then it's 2 plus 8 tens which is 80, plus 12.
Barbara
Yeah.
Michelle
And that's even nicer to work with.
Barbara
Yeah, or even, even get that 10 from here.
Michelle
Oh yeah.
Barbara
So, 5 tens and 3 tens is 8 tens and then 9 tens.
Michelle
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.
Barbara
Very close.
Michelle
So, so what I know actually, is that double 15 is 30.
Barbara
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.
Barbara
Yeah.
Michelle
Yeah, because now what I can see is that there's another hidden 50.
Barbara
Yeah.
Michelle
In there. So, if I take the 10 and the 10.
Barbara
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.]
Barbara
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!
Michelle
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.