13 Away
Years 2 - 7


The activity includes a calculator game for 2 players using one calculator.

 

Preparation

You will need a partner for the calculator game.
They can do the whole activity with you if you wish.
Almost all modern mathematics has been discovered by two or more mathematicians working together.

  • One calculator (there's one on your phone)
  • Thirteen (13) objects such as buttons, pasta, tiny toy people, tiny toy cars or pebbles
    ...or face down playing cards (ignore the numbers, just show the backs of the cards)
    ...or paper plates
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

Getting Started

This activity begins with a calculator game.
These are the rules:

Start with 13 on the screen.

  • The only thing that happens in this game is that a player subtracts 1 or 2 or 3 whenever it is their turn.
  • The aim is to make the other person be the one who has show the answer zero (0).
  • Negative numbers are not allowed.
Decide who goes first.
  • Player 1 presses the [ - ] button and then 1 or 2 or 3, then the [ = ] button.
  • Player 2 now presses the [ - ] button, then their choice of 1 or 2 or 3 followed by the [ = ] button.
  • Player one goes again with their choice ... and so on until one player has do a take away that results in zero.
Play 13 Away a few times to get used to it.

 

Working Mathematically tells you that when a mathematician is interested in a problem they play with it to collect and organise data.
So far you are Working Like a Mathematician because:
  1. You are interested in the problem - which is to make the other person display zero.
  2. You are playing with the problem.
But what about the collecting and organising data part of a mathematician's work.
For that you need your journal.
You have already written the heading and the date.
  • Discuss the information you could record when playing 13 Away.
Play the game at least five (5) more times and record interesting and important things about it.

Have fun exploring 13 Away.

It's okay to talk with each other about what happens. You are starting a mathematical adventure together.

You might record things like:
  • who goes first
  • the take away numbers
  • who wins
  • any predictions you make as you play
  • ...or other stuff
Perhaps you will think of a way to make a diagram or table to show your recordings.

Challenge 1

The first challenge in this activity is to work out a way to always win.
You have to assume you are playing the world's best player who always makes the best move when they play.
It's not good enough to have a way of winning that depends on the other person making a mistake.
  • To complete the challenge you have to prepare a short report to explain to someone else how to always win.
Look at your Working Mathematically page again. One of the strategies in a mathematician's toolbox is to make a model.
Put aside your calculator.
  • Use your objects to make a model of the number 13.
  • Play 13 Away five more times, but now you take away 1 or 2 or 3 objects from the collection.
Playing the game with objects instead of calculator digits might help you notice more.
Record anything new in your journal.

Another mathematician's strategy is to work backwards.
That's when you start exploring from the end instead of the beginning.

  • Remember, the problem is: How can I make you face the last object so you have to lose?.
This is how to start Working Backwards:

If I make you face one counter...

you have to take it and
I win.

If I make you face two counters...

you take 1 and I have to take the last one and
I lose.

If I make you face three counters...

you take 2 and I have to take the last one and
I lose.

If I make you face four counters...

you take 3 and I have to take the last one and
I lose.

If I make you face five counters...

either you take 1 and I take 3
or you take 2 and I take 2
or you take 3 and I take 1
and no matter which choice you make
I win.

If I make you face six counters...

Copy the table so far and keep it going until I make you face thirteen counters.

 

When you finish the table:
  • You will know two other numbers I should make you face.
  • You will see a pattern connecting the four winning numbers.
  • You will know the first thing you have to do if you want to be certain of winning.
Record what you have learnt in your journal, then...
  • Use your journal notes to prepare a report explaining everything to someone else.
  • Your report could be written, or a slide show, or a poster, or a video or any other way you think will teach someone else.
It might help someone else to
understand if you arrange
your objects like this:

More Challenges

Now you know about 13 Away, choose one or two of these questions and investigate.
  • What happens if the game is 17 Away and the rules are the same?
  • What happens if the game is 16 Away and the rules are the same?
  • What happens if the game is 13 Away and the rules are the same but you win if you take the last object?
  • What happens if the game is reversed and you start with zero and the rules change so you have to add 1 or 2 or 3 and be the person to make exactly 13 objects?
  • What happens if the game is 21 Away and the rules let you take away 1 or 2 or 3 or 4?

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
  • Read again what it means to work like a mathematician.
  • In your journal list at least six things you did in 13 Away that prove you were working like a mathematician.

 

Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from 13 Away.

Send any comments or photos about this activity and we can start a gallery here.

 

Maths At Home is a division of Mathematics Centre