# Subtraction stacks

A thinking mathematically context for practise focussed on developing skills and understanding to work out the difference between two numbers

## 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
• MAE-RWN-01
• MAE-CSQ-01
• MAE-CSQ-02

• MAO-WM-01
• MA1-CSQ-01

## Collect resources

You will need:

• 10 counters or other items such as dried pasta per player

• 2 dice (you could also use playing cards, use the king as zero, or a number spinner zero to 6 (PDF 117 KB))

• A piece of paper to make a Subtraction stack game board.

## Watch

Watch Substraction stacks video (8:39).

Use subtraction to remove all markers from the game board.

### Transcript of Substraction stacks video

White text on a navy-blue background reads ‘Subtraction stacks – Jennifer Bay-Williams and Gina Kling’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government red ‘waratah’ logo.

Black text on a white background reads ‘You will need…’ Below, black text bullet points (as read by Michelle). On the right, in a still colour image, a blue rectangular piece of paper folded into 6 columns number 0 – 5 at the top in black pen.

### Michelle

Alright mathematicians, for this game you will need a gameboard. So, you just need a strip of paper that you can sixth or divide into 6 equal parts and then write the numerals 1, 2, 3, 4, 5, across the top like you can see here. You'll need two 6-sided dice but you could also use spinners or playing cards or just make your own numeral cards at home. You just need one marker to make your gameboard and you need 24 counters. But you don't have to use counters, you could use dried pasta or paperclips or Lego figurines or whatever it is that you have at home and they don't have to be the same for this one.

White text on a blue background reads ‘Let’s set up our gameboard’.

### Michelle

OK. So, let's set up our game board.

On a white desktop, 2 rectangular pieces of blue paper, a pink 6-sided dice and a red 6-sided dice, 2 black marker pens and a red dish that has small green plastic triangle counters inside it.

### Michelle

Alright there. Hello, mathematicians. Hello, Barbara.

### Barbara

Hi, Michelle. How are you?

### Michelle

I'm great. How are you?

I'm very well.

### Michelle

Welcome back to playing mathematics. I hope you're ready to... well, I know you'll win, but I'm ready to have fun losing. We're gonna play a game today called Subtractions Stacks from Jennifer Bay-Williams and Gina Kling. It's one of my new favourite games.

Michelle picks up the blue slip of paper on the left.

### Michelle

But to get started, we need a game board. So, this is gonna be our game board. We just need to partition it into six equal parts or we can call that sixthing. So, actually, what I want to do is what's your strategy for sixthing?

### Barbara

OK, so because six is an even number, I know that I can have three and three. So, what I'm about to start doing is I might half it first.

### Michelle

Yeah.

Barbara picks up the right-hand slip of blue paper and starts to fold it.

### Barbara

And now that I've halved it, I need to third it.

Third each half?

### Barbara

Third each half. But I might do it, just because I like to be efficient, rather than opening it up again, I might leave it in half.

Yeah.

### Barbara

So, I only have to do it once. And what I'm thinking is if I fold it so it looks like half, then I've got one-third here, one-third at the back, and then one-third at the front. So, even though it looks like 2 halves that are equal, I've actually got 3 pieces.

### Michelle

I see. Oh, and like a zigzag?

### Barbara

Yeah. So, like, this way.

### Michelle

Yeah. Oh, yeah. So, you halved the paper.

Yeah.

### Michelle

And then you thirded each half.

Yes.

### Barbara

Yeah. And I didn't actually think of it as a zigzag but you're right, it made a little zigzag pattern.

### Michelle

OK, so you need to just write numbers 0, 1, 2, 3, 4, 5 and just do it at the top because we'll put counters down there and as you're doing that…

Michelle starts to fold her slip of blue paper as Barbara writes 0, 1, 2, 3, 4, 5 at the top of her folded piece.

### Michelle

I've got a different strategy for thirding actually. It's similar to yours but I sometimes I take the paper like this and just get my eye in, like, and make a zigzag and I squish it. That's gonna make my thirds, see. And like you, I could now come along and half each third to make 6. But I know that once I've got thirds, it's more efficient because I can do it in one fold to half each third and that will make me sixths.

### Barbara

So, what we did was quite similar except that I halved first and then thirded, whereas you thirded first and then halved and we still got the same result.

### Michelle

Woah, see, mathematics!

Fast-motion footage of Michelle numbering her columns with a black marker.

### Michelle

So, here's how we play this game. We each need two counters, we're using triangles as counters today, on each of our numbers from zero to 5.

Fast-motion footage of the counters being placed on the blue ‘game board’.

White text on a blue background reads ‘Let’s play!’

### Michelle

Alright, let's play. Alright, you can start, Barbara. Roll the…both dice. And then you need to work out the difference between them.

Michelle and Barbara take turns to roll both the dice. Further steps explained as they go.

### Barbara

OK, so I'll do 4 and I'll take away 2. So, I've got 2 left over.

### Michelle

And you get rid of one of the counters in the ‘2’ place.

### Barbara

OK, I have to put it back in there?

Uh huh.

OK.

### Michelle

And my go. And I've got to work out the difference between 6 and 5 or 5 and 6. And actually, you know how you said 4 minus 2. I'm gonna say 5… one more than 5 is 6. So, the difference is one and I can get rid of one.

### Barbara

That makes sense because they're really close together.

### Michelle

Yeah, and I could use my knowledge of the counting sequence.

### Barbara

Yeah. Only one more.

Barbara rolls a 5 and a 6.

### Barbara

I don't even need to think about that one. So, one. There we go.

### Michelle

Oops, and I've got 4 and a one and I know the difference between 4 and one is 3. Because actually, if I just got rid of one dot on this dice, there would be 3 left. So, that's the difference. So, I can get rid of a 3. And the goal, Barbara, is to be the first person to clear the board.

### Barbara

OK, so I wanna use your strategy and imagine that that one in the middle isn't there and I have 4 left.

White text on a blue background reads ‘A little while later…’

### Michelle

OK, the difference between 4 and one is 3.

### Barbara

Oh yes.

Michelle removes a triangle from the ‘3’ column. She only has 2 counters left in the ‘5’ column. Barbara has one triangle counter left in both the ‘3’ and the ‘5’ column.

### Michelle

OK, what are you hoping for now, Barbara?

### Barbara

I think I'm hoping for a 6 and one again because I think that's probably the hardest thing to get rid of.

### Michelle

How could you get rid of your 3?

### Barbara

I've got lots of ways to get rid of my 3. So, I could do one and 4, 2 and 5, 3 and 6.

OK.

### Barbara

So, there's a few ways to get rid of the 3.

### Michelle

You've got a few choices where I can only roll a 6 and a one twice in a row.

### Barbara

So, it looks like we're even but I'm kind of winning.

### Michelle

Yeah, I think you've got a greater chance of winning.

Barbara rolls a one and a 4.

### Michelle

(LAUGHTER)

How do you do this?

### Barbara

That's luck, lady luck. So, the difference between 4 and one is 3 which I happen to have on my board.

### Michelle

You do happen to have available on your board. No go for me because the difference between 5 and 2 is 3.

### Barbara

OK. Come on 6 and one.

They continue to roll the dice.

### Michelle

Come on. Oh, 6 and 5. Difference of one. 2 and one is a difference of one. Difference of one again.

### Barbara

So, I think the lower numbers are easier.

### Michelle

5 and one is a difference of one. So, do you know what I wonder, next time, we could play is, in the game we found on Burns called Clear the Board, you get to choose where you put the counters. So, you have to have 10 counters, but you could choose where you put them…or 12 counters, but you can choose where you put them. Whereas in this game you had to have 2 for each of the differences.

### Barbara

So, I'd have to investigate but I think I would put them more towards the zero, one, and 2.

### Michelle

Yeah, because you're more likely to get...

### Barbara

More combinations, right?

### Michelle

Like you can get a difference of one between one and 2, 2 and 3, 3 and 4, 5 and 6. Yeah, whereas you can only get 6 and one to get 5.

### Barbara

Which is where we're a little bit stuck now.

### Michelle

Alright. Let's do 2 more rolls each.

### Barbara

OK. Oh no. Difference of one.

Difference of 2.

### Barbara

Come on. Come on. Come on.

### Michelle

A difference of none! Alright, come on. At least tie. I don't know if my magic breath will work. Uh, no. Alright, Barbara, in this case, I'm gonna declare you the winner by default. Congratulations!

### Barbara

Thank you very much. Good game.

White text on a blue background reads ‘Over to you, mathematicians!’.

### Michelle

Over to you mathematicians!

White text on a blue background reads ‘What’s (some of) the mathematics?’.

### Michelle

So, what's some of the mathematics?

Black text on a white background reads ‘What’s (some of) the mathematics?’. Below, bullet points (as read by Michelle). On the right, small colour images of the two 6-sided dice alongside each bullet point.

### Michelle

So, we can use a range of skills and understanding to work out the difference between two numbers and in this game, we were able to use known number facts. So, for example, Barbara knows that 4 take away 2 is equivalent in value to 2 and so she was able to use that number fact when she rolled a 2 and a 4.

We can also use what we know about using addition to solve subtraction. So, for example, Michelle, that's me, I know that the missing number needed to balance the equation or the number sentence 5 plus something is 6, is one.

We could also use our knowledge of counting sequences, including things like the number before and the number after, and we could also use our knowledge of spatial patterns and structures like dots on dice. So, for example, I rolled a one and a 4 and I could work out the difference by imagining this dot disappearing and that means that there's three dots left.

Alright mathematicians, back to you to see what range of skills and understanding you can use when you play Subtraction Stacks.

The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’

[End of transcript]

## Instructions

• Each player places their 10 counters of their subtraction stack game board. More than one counter can be placed on each number.

• Take turns by rolling the dice and finding the difference between the two numbers rolled.

• If a counter is on the number, the player removes it from the game board.

• If there are no counters to remove, miss a turn.

• The winner is the player who removes all counters from their game board first.

## Other ways to play

• Extend the subtraction stack gameboard to 9 or 11 and use a 12 or 10-sided dice. Each player will need 20 counters or items to play.

## Discuss

• What was one of your strategies for working out the difference between the numbers you rolled? Is the strategy the same or different to your opponent?

• Will you place your stacks of counters differently next time you play?Why?