Double or halve? – Stage 2 and 3

Double or halve is a thinking mathematically context for practise resource focussed on using doubling and halving to reach a target number.

Adapted from NRICH

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, 2021

Outcomes

  • MAO-WM-01
  • MA2-AR-01
  • MA2-MR-01
  • MAO-WM-01 
  • MA3-AR-01 
  • MA3-MR-01 
  • MA3-RQF-02

Collect resources

You will need:

Watch

Watch Double or halve? video (4:44).

Reach the target number using doubling or halving.

[White text on a navy-blue background reads ‘Double or halve? (Stages 2 and 3) From NRICH’. 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.]

Speaker:

Double or halve from NRICH.

[A blue text on white header reads ‘You will need…’ Three bullet points below (as read by speaker). Below, in a still colour image, a sheet of white paper has ‘Double or halve?’ written at the top in green. On the left, a black number ‘55’ is circled. On the right of the paper is a pink and a green marker and a 6-sided red dice.]

Speaker:

To play, you will need 2 different coloured markers or coloured pencils, something to write on, and a 6-sided dice or 1-to-6 spinner.

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

Speaker:

Let's play!

[The sheet of paper, markers and dice from the earlier still colour image.]

Speaker:

Hello there, mathematicians. I've got one of my favourite mathematicians with me. Hi, Sam.

Sam:

Hi.

Speaker:

And we're gonna be playing a game from NRICH called Double or Halve. Are you ready to play, Sam?

Sam:

Uh-huh.

Speaker:

So, for this game, you can have a 6-sided dice or a 9-sided dice.

[The teacher briefly places a red 9-sided dice next to the 6-sided dice before removing it.]

Speaker:

Sam, I think we'll start with a 6-sided dice today.

Sam:

OK.

Speaker:

And then we also need 2 markers. So, one for you and one for me. I'll let you choose whatever colour you like, Sam. So, what we have to do first is choose a target number between zero and 100. What would you like our target to be, Sam?

Sam:

55

Speaker:

55? Alright, let's write that up here so we know that this is our target.

[The teacher writes ‘55’ in black marker in the top left of the paper and circles it.]

Speaker:

So, Sam, the aim of Double and Halve is to be the first to get the target number without going over. We take turns rolling a dice, which we can choose to double or halve, but we can only form whole numbers. We'll also keep a running total. So, let's start. Sam, you do the honours, you can roll. OK.

[Sam rolls a 2 on the 6-sided dice.]

Sam:

I'm gonna double that.

Speaker:

You're gonna double it, and what's double 2, Sam?

Sam:

4.

Speaker:

OK, so, you write down 4. Maybe just put it here.

[Sam uses the green marker to write ‘4’ beneath the ‘Double or halve?’ header in the middle. They take turns to roll the dice.]

Speaker:

OK. Alright. Oh, so, I rolled a 2. I'm gonna double it too. So, double 2 is 4, and 4 and 4...

Sam:

Is 8.

[The teacher uses the pink marker to write ‘8’ below the previous ‘4’. They take turns to roll the dice and write a number in a column on the paper.]

Speaker:

Is 8. OK. Alright. Over to you.

Sam:

Double that, and 8 and 8 is 16.

Speaker:

OK. 3, I'm gonna double that, 6, 16 and 6. Well, I'm gonna partition the 6 into 4 and 2, 22. Over to you, Sam.

[Sam rolls a 2.]

Sam:

I'm gonna halve it this time.

Speaker:

Interesting. Why are you halving it?

Sam:

I don't know. Just to change up the game.

Speaker:

OK, fair enough.

[The teacher rolls a 3.]

Speaker:

OK. 3. Well, if I halved 3, I wouldn't get a whole number. So, I'm gonna go with 6. 23 and 6 is... Well, I know that 23 and 7 is 30, so 23 and 6 must be 29. Alright, Sam.

Sam:

2. I'm gonna double it. It's 4, 29 plus four is 33.

[The teacher rolls a 6.]

Speaker:

Ooh! That's a good one to double. I'm gonna double it. So, double 6 is 12. What? (LAUGHS) So, that is 45. Oh! I don't know if I should have done that.

Sam:

If I get a 5, I'm gonna just double it.

(LAUGHTER)

Speaker:

Yes, I know, don't get a 5...

[Sam rolls a 5.]

Speaker:

What!

Sam:

Double 5 is 10. 55.

Speaker:

Well done, Sam. That was a very lucky roll. So, over to you, mathematicians, to have a round or have a game of Double or Halve. Yeah. Have fun. Thanks, Sam.

Sam:

You're welcome.

Speaker:

So, what's some of the mathematics?

[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Two bullet points below (as read by speaker).]

Speaker:

We can use what we know about halving and doubling numbers to make strategic decisions to reach our target number. Knowing the difference between the running total and the target number helps us strategise and increase our chances of winning.

[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

  • Choose a target number between 0 - 100 and write this on your sheet of paper.

  • The first player rolls the dice and chooses whether to double or halve the number.

    • Record the roll.

  • Take it in turns to roll the dice, keeping a running total.

    • If a player can't go, they miss a turn.

  • The winner is the player who reaches the target number exactly.


Watch

Watch Double or halve? part 2 video (5:25) for another way to play it.

Double or halve variation using decimal values.

[White text on a navy-blue background reads ‘Double or halve? (Stages 2 and 3) – part 2 From NRICH’. 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.]

[A blue text on white header reads ‘You will need…’ Three bullet points below (as read by speaker). Below, in a still colour image, a sheet of white paper has ‘Double or halve?’ written at the top in green. On the right of the paper is a pink and a green marker and a 9-sided red dice.]

Speaker:

To play, you will need 2 different coloured markers or coloured pencils, something to write on, and a 9-sided dice or a 1-to-9 spinner.

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

Speaker:

Let's play!

[The sheet of paper, markers and dice from the earlier still colour image.]

Speaker:

Hello there, mathematicians. Sam and I here together again playing Double or Halve. And this time we've got a slightly different version, haven't we, Sam?

Sam:

Yeah. So, this time we're gonna be allowed to go into fractions.

Speaker:

Alright, Sam. So, our target number is going to be between 10 and 99. What are you thinking?

Sam:

99.

Speaker:

99? Yeah, why not? OK, so let's write that up here.

[The teacher writes ‘99’ with a black marker in the top left of the sheet of paper and circles it.]

Speaker:

So, there's our target number. OK, Sam, you can do the honours. If you'd like to do the first roll.

Sam:

Thank you. 1.

Speaker:

OK, so what are you thinking? That's an interesting number to start off with.

Sam:

I'm gonna halve it.

Speaker:

So, half of 1 is a half.

[Sam writes ‘0.5’ at the top of the page in black marker.]

Speaker:

We're gonna record that with decimal notation. OK.

[They take it in turns to roll the dice and write the corresponding number in a column on the sheet of paper.]

Speaker:

Ooh, 3. Do I try and make it a whole number or... Nah, I'm gonna get us going. So, I'm gonna double 3, which I know is 6. And that means that I have ‘6.5’. OK, Sam.

Sam:

One.

Speaker:

What are you gonna do?

Sam:

I'm just gonna keep it like that, 7.5.

Speaker:

Did you keep it like that, or did you double it?

Sam:

I keep forgetting, I don't double.

Speaker:

(LAUGHS) (CROSSTALK) That's OK, because now we know that because you didn't double it, that you just need to add on another one, right? So, 8.5. OK, yeah. So, we can choose to either double or halve. Alright, 8.5 so far. Oh, 1, low number again. You know what? I'm gonna double that one. So, 8.5, well, if I know that 8 and 2 is 10, then 8.5 and 2 must be 10.5. OK, Sam.

Sam:

5. I'm gonna double that, 10.

Speaker:

20.5. Oh, 4, OK, gonna double that. So, I've got 28.5.

Sam:

5, and double that, 38.5.

Speaker:

OK. I think we're both deciding to double because we're trying to get close to that target number. Is that right? OK. Ooh, OK. 6. Right. So, double 6 is 12. And I can use what I know about 8 and 2 to think about 38 and 12. So, that's going to be 50.5.

Sam:

OK. 7. Double that for 64.5.

Speaker:

Yeah, it's interesting how our brain likes numbers that end in zero. Makes it a lot easier to combine those numbers. OK. Double 3 is 6. Oh, yeah, and look, I'm using what I know about numbers that combine to make 10.

[The teacher starts a new column and writes ’70.5’ at the top.]

Speaker:

So, we'll keep going over here, so 70.5.

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

Speaker:

So, we are now at the point where the only way either of us can win is if we roll what?

Sam:

A one.

[They take turns to roll the 9-sided dice.]

Speaker:

A one. That we would then halve. Bad luck. Nope.

Sam:

Ooh, zero.

Speaker:

Now it just comes down to luck, doesn't it, Sam? Oops. Rolled off the table.

[Sam rolls a 1.]

Speaker:

(CHEERING)

Alright, Sam. So, what are you gonna do?

Sam:

I'm going to double, no, I'm gonna halve it. 99.

[Sam writes ‘99’ on the sheet of paper in black marker and circles it .]

Speaker:

Well done, Sam. That was a very lucky roll again.

Sam:

Good game.

Speaker:

Yeah, great game. I like this version where you can halve it. It definitely gets your brain sweaty. So, over to you, mathematicians, to have a go at playing Double or Halve when you can halve odd numbers because you can work with fractional numbers. Have fun.

Speaker:

So, what's some of the mathematics?

[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ Two bullet points below (as read by speaker).]

Speaker:

We can use what we know about halving and doubling numbers to make strategic decisions to reach our target number. Knowing the difference between the running total and the target number helps us strategise and increase our chances of winning.

[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]

Discuss and reflect

  • Does it matter if you go first or second? Does it increase your chances of winning? Why or why not?

  • If you could play the same game again are there any moves you would change? Why or why not?

  • What are you thinking about in order to try to win the game?

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