Take Five

In these activities, the students use sampling as the basis for predictions about the nature of an unseen whole.

Activity One

This activity is about probability, sampling, and variation. No one selection of 5 tiles can reveal what the bag contains. Each selection is likely to be different from the last. But the more samples we take, the closer we get to the truth about the contents. Question 1 is there to get the students engaged in extracting and synthesising the useful information that progressively becomes available in the sequence modelled by Malia and Natalie. This is a very good exercise in logical deduction.
The fifth sample provides the clinching piece of information. Up until now, we had been able to account for 9 of the 10 tiles. It turns out that the missing tile was blue.
The answer for question 5 implicitly warns against falsely generalising that 5 samples will always be enough. The students need to understand that Malia and Natalie could have gone on sampling the tiles for a very long time without getting a sample with 4 blue tiles in it.

Activity Two

Question 1 asks the students, working in pairs, to repeat Malia and Natalie’s experiment, using a different selection of 10 coloured tiles. If a third party makes up the contents of the bag and then gives it to the pair carrying out the experiment, the latter two can share the thinking.
Question 2 involves a similar experiment, but with double the number of tiles in the bag.  The chances of correctly predicting the contents of this bag after taking just 5 samples of 5 are extremely low. But each 5 × 5 sampling will reveal important information. After several more 5 × 5 samples, there will be even more information but probably no certainty.
It is important that the students realise that if they add all their samples together, it is likely that the most common colours in the bag will have been drawn out more frequently than the less common. This information can be used to predict how many of each colour are in the bag.
This prediction should be refined by cross-checking against the information gained from the separate 5 × 5 samples. Finally, the bag is tipped out and the contents compared with the prediction.
It is important that students understand that:

• a prediction is not the same as a guess: it is a carefully considered estimate based on the best use of the available information;
• not all predictions are equally good or valid: a prediction that ignores a piece of useful information is a careless prediction;
• using exactly the same information, people may come up with different predictions, either because the information is scanty or because of different (but reasonable) assumptions;
• a well-thought-out prediction can still be wrong: if there is enough information to be
certain, it’s not a prediction, it’s a logical deduction (as in Activity One, question 4).
Either or both of these experiments could usefully be carried out a number of times.
Note that sampling a bag of 20 tiles, 5 at a time, does not give as much information as sampling a bag of 10 tiles. If your students fi nd it too diffi cult to make reasonable predictions when working with the larger number of tiles, tell them to increase the sample size to 8 or 10 and see how they get on.

Answers to Activities

Activity One
1. Answers will vary. Possible thoughts include:
After the second selection: “Yes, it looks like they’re all blue and red. Probably equal
numbers of both.”
After the third selection: “Well, there’s obviously a yellow tile, too. It looks like 1 yellow and roughly equal numbers of blue and red.”
After the fourth selection: “Wow, so there’s a green, too. I’m surprised that didn’t come out earlier.
And 2 yellows! I’ll have to revise my ideas.”
2. This is the effect of probability (or chance). You can never be sure of an outcome when probability is involved.
3. a. There are several possibilities, but answers must include 3 blue, 3 red, 2 yellow,
1 green, plus one more. A good answer would be 4 blue, 3 red, 2 yellow, 1 green.
b. For the suggestion in a, an explanation could be: We know from the four trials
that this bag contains at least 3 blue, 3 red, 2 yellow, and 1 green, and since we
know that there are 10 tiles in the bag, there is 1 tile we don’t know the colour of.
As blue appears more often than any other colour, it may well be blue (but it could
be any of the other colours, too, or even a completely different colour).
c. Yes. There is one tile we can’t account for, which could be blue, red, yellow, green, white – or any other colour, for that matter.
4. This is the fi rst selection to contain 4 blue tiles, so we now know for certain that the tiles in the bag are 4 blue, 3 red, 2 yellow, and 1 green.
5. There is no fixed number because it all depends on what comes out of the bag each time.
Activity Two
1. Practical activity
2. Practical activity
a.–c. Predictions and results will vary.
d. Combining results is useful because the more data you have, the clearer any
patterns become.

Key Competencies

Take Five can be used to develop these key competencies:
• thinking
• participating and contributing.