Choosing Beads

Task 191 ... Years 4 - 10

Summary

The task offers experience of sampling a mixed 'population' without replacement. The initial challenge is:

What is the smallest number of beads you must select to be certain of having 3 the same colour?

The card leads into an empirical beginning to the task. With eyes closed, students choose a bead, check its colour, choose again, check the colours and so on until three are the same colour. Students may need to do more experiments than the five suggested before they begin to find the keys to the problem.

Materials

Thirty-one beads
- 15 Colour A, 12 Colour B, 4 Colour C

Content

concept of proof
probability experiences
probability, sample space / sample size
ratio & proportion
reasoning
recording mathematics
statistics, analysing data
statistics, collecting & organising data
statistics, frequency
tree diagrams

Iceberg

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

Students begin by experimenting. As they do so, it may begin to occur to them that the chances of getting each colour is not what matters in this problem. Rather, what matters is the number already the same after each selection. Clearly choosing 3 beads is not enough. The three might be the same but we can't be certain that they will be.

It is reasonably natural to begin an analysis like this:

Selection 1

Problem not solved - select again.

Selection 2

Problem not solved - select again.

Selection 3
Problem may be solved, but we can't be certain that it is. The possibilities are:

3 the same - which solves the problem

2 the same

all different

Each of these last two possibilities has to be followed separately at the next step. This is where students might need a little guidance in organising their reasoning.
Selection 4a (starting with 2 the same out of 3)

3 the same and 1 different - which solves the problem

2 pairs of 2 the same

2 the same and 2 different others

Selection 4b (starting with all different)

2 the same and 2 different others

Again there are two possibilities to follow through at the next step.
Selection 5a (starting with 2 pairs of 2 the same)

3 the same and 1 pair different - which solves the problem

2 pairs of 2 the same and 1 different other

Selection 5b (starting with 2 the same and 2 different others)

3 the same and 2 different others - which solves the problem

2 pairs of 2 the same and 1 different other

Now there is only one possibility to check at the next step.
Selection 6 (starting with 2 pairs of 2 the same and 1 different other)

3 the same, 1 pair the same and 1 different other - which solves the problem

3 pairs of 2 the same

Again there is only one possibility to check at the next step. Selection 7 (starting with 3 pairs of 2 the same)

3 the same and 2 pairs of different others - which solves the problem.

So the smallest number of beads that would have to be selected to be certain of three the same is 7. Did any of the students' trials take more than 7 beads?

A follow up question that might be explored at this point is:

What is the probability that it will take to the seventh selection to get 3 the same?

This approach to the problem is applicable to the two additional challenges on the card. However there are many ways to travel from here, so lots of room for students to choose their own additional challenge. We would be happy to publish student reports that dig deeper into:

What happens if we change the ratio of colours in the jar?
What happens if the jar contains more colours?
What happens if the problem is about getting a different number of the same colour?

If the logic above is laid out as a tree diagram (albeit complex) and the different (and altering) probabilities connected to A, B and C at each stage are followed through, the question is answerable using a theoretical approach. However, the empirical approach opens up possibilities for whole class collection of data and thoughts about designing software to explore and record the investigation.

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's worth checking with the art teacher for beads, but if you don't have them in the school use wooden or plastic cubes or linking cubes of uniform size. The bag doesn't need to be opaque, so a freezer bag will do. Seat the students in groups of four and provide a tub of mixed colours. Teams first make their bag according in the ratio you write on the board. Then remove the tubs until the challenge questions require changing the composition of the bag.
You could use a story shell such as a factory that makes up bags of beads, but why not one about a factory that makes up specialty chocolates in three flavours. They don't put the same number of chocolates in each pack because it is more expensive to make certain fancy chocolates than it is to make plain ones. It's not hard to add a line that leads into the problem...
In my family we share out the chocolates with our eyes closed, otherwise all the special ones would be taken. One day my daughter asked, I wonder how many times I would have to dip into the bag before I get three of the same chocolates?. There was only one way to find out.
Now that the initial challenge is in place, encourage the teams to plan how they will experiment together. They might assign roles such as recorder, chooser, questioner and checker.
Work the initial challenge as a class until the method of analysis is understood. Then discuss the possible What happens if...? questions. Allow each group to choose its own investigation from here and expect them to plan, execute and publish a report in some form.
At this stage, Choosing Beads does not have a matching lesson on Maths300.

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 Choosing Beads task is an integral part of:

MWA Chance & Measurement Years 3 & 4

MWA Chance & Measurement Years 7 & 8

Follow this link to Task Centre Home page.