### Why do this
problem?

This
problem offers students a chance to analyse a game which
involves adding and subtracting positive and negative numbers, and
requires them to work out the probability of the different possible
outcomes. Teachers could explain that by knowing about the
mathematics behind a game we can sometimes determine a winning
strategy (or more realistically, a strategy that improves our
chances of winning).

Teachers may find it interesting to read the NCETM
Mathemapedia Entry:

### Possible approach

The game could be introduced using an interactive whiteboard by
asking for two volunteers to play, each linked to one of the
coloured counters, or the class could be divided into two teams.

After the introduction students could play in pairs at individual
computers, or

this
board could be printed for students to play away from the
computers.

This
spreadsheet can be used to simulate throwing the two
dice.

To switch attention from consolidation of number skills to the
mathematics behind the game, this needs to be followed by a whole
class discussion that focuses on emerging strategies, observations,
explanations and justifications.

Students can then go back to working in pairs to establish the
numbers of ways of achieving the different totals. At the end of
the lesson a plenary discussion can offer students a chance to
present their findings. The discussion can compare the merits of
the different approaches used (eg listing possibilities vs sample
space diagrams).

Following on from this problem, students could take a look at:

Weights
Consecutive
Negative Numbers
### Key questions

Are there some numbers that we should be aiming for?
Why?

Are certain numbers easier to 'cover' than others? Why?

"Have you got all the solutions?" "How do you know?"

### Possible extension

If appropriate, teachers could set up an open ended activity
for the rest of the week in which students investigate changes they
could make to the game.

For example:

* Different shaped boards

* Boards where some numbers appear
more than once

* Different dice (e.g.
dodecahedral) or different numbers on the dice

* Allowing multiplication and
division and changing the board accordingly

### Possible support

Students who struggle with adding and subtracting negative
numbers can play

First
Connect Three , a simpler version of the game that just
requires dice with positive numbers.

Teachers may like to take a look at the article on

Adding
and Subtracting Negative Numbers