# Guessing Colours Game

### Task 218 ... Years 2 - 7

#### Summary

Guessing Colours Game involves placing a known number of discs of five different colours into an opaque bag and also knowing how many of each colour there are to start with. Students then take turns to select a disc from the bag, but first they each have to guess which colour will come out. A points scoring system is involved and a running total of points is kept for each player until all the discs are removed. The game involves simple addition and subtraction which may include negative numbers and is an excellent experience of chance events which may be at an intuitive level or measured by a probability score.

#### Materials

• About 30 discs in 5 colours with at least three of each colour
• Opaque bag

#### Content

• integer arithmetic
• mental arithmetic
• probability calculations
• probability experiences
• probability, sample space / sample size
• recording mathematics
• statistics, analysing data
• statistics, collecting & organising data

#### Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

At the first level, Guessing Colours Game is a bit of fun involving addition and subtraction. An additional feature is the need to record in a way that keeps track of the guesses, whether or not they are correct, and the changing score. It is also helpful to keep a record of the changing number of each colour in the bag. Even if the game is nothing more than this, it can potentially throw up the need for negative numbers. The players start with 15 points each (one for each disc) and on each move gain or lose 2 points depending on their guess. So it only takes 8 incorrect guesses in a row to reach a score of -1 and there are still 7 more discs to draw.

One Year 1 teacher, who wasn't using this task at the time, was involved in a discussion with a colleague about negative numbers in the context of using a calculator and commented:

Oh, negative numbers, they're no problem to my class. They just call them underground numbers.
To add a little more to the arithmetic of the game expect the students to record not just who won, but the winning margin, or difference between the two scores.
• How much did you win by?
Then, if, for example, the winner scores 25 and the loser scores 5, the winning margin is 25 - 5 = 20. But, if the winner scores 25 and the loser scores -5, the winning margin is 25 - -5, which is obviously 30 ... 25 above ground and 5 below ground.

Another way to add a little to the arithmetic is to declare that the winner is the person with the higher total score after 3, or perhaps 5 rounds. What is the winning margin in each round and for the grand totals?

Beyond this, without requesting it, the game almost forces consideration of probability. For example, knowing that on the first move there are 7 Red discs in the bag of 15 discs, clearly the best chance of success is to guess Red ( 7 chances out of 15 = 7/15). The chances then change for all colours on every draw until the last draw which is a momentary experience of certainty, with a probability of 1 (1 chance out of 1 = 1/1). It is worthwhile encouraging the students to keep a record of these changing fortunes.

#### Extensions

The challenge on the card suggests the first extension.
• What happens if we change the combination of colours?
The particular example in the challenge is considerably different from the introductory activity because each colour has an equal chance of being picked. Assuming that in either version the game is played selecting the best choice based on probability each time, one thing that might happen is that this new version produces a significantly different range of possible scores. To check this hypothesis data would have to be collected for playing the game each way a number of times. But how much data is enough (that is how many times does each version need to be played) to be able to state with any confidence that the starting combination of colours can determine the range of winning and losing scores?

Perhaps each pair playing each version 5 times doesn't give a clear indication. Perhaps they should record their outcomes on a poster in the 'mathematics corner' which is a cumulative record contributed to by each pair that uses this task. Using Guessing Colours Game in its whole class investigation form (see below) would collect a lot of data about this hypothesis very quickly. Also, there is an opening here from designing software that would play many games with a chosen colour combination very quickly.

Further, if we can change the number of discs of each colour in a bag, then what else could be changed and what would that change do to the game? What happens if...

• ...we change the total in the bag?
• ...we change the point scoring system?
• ...we add a rule such as miss a turn if you make a wrong guess?

#### Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.

It is quite easy to set up Guessing Colours Game as a whole class investigation. You need need something to substitute for the black bag in the task, perhaps brown paper lunch bags will do the job, and coloured squares, discs or blocks. Write the combination of colours on the board first, then set up the first game in a fish bowl situation with one pair at a table in the middle of the room. Spend time - perhaps the whole first lesson - exploring the game for its suite of arithmetic connections described above. Then, using a question like:

• Is there anything in the game that helps you make the best possible guess at every turn?
shift the focus to the chance and probability aspects. When it comes to comparing outcomes for two different starting combinations of discs, 15 pairs in the class can collect the results for around 50 trials of each version relatively quickly. A key point to highlight in bringing the investigation to a close is how the class has worked like a mathematician.

At this stage, Guessing Colours Game does not have a matching lesson on Maths300. However, this task is related to Task 47, Red & Black Card Game, and Task 131, Walk The Plank, both of which have a Maths300 matching lesson. The Walk The Plank lesson also includes software.

#### Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.

The Guessing Colours Game task is an integral part of:

• MWA Number & Computation Years 3 & 4
• MWA Number & Computation Years 7 & 8