Counters in the Middle

'Counters in the Middle' printed from https://nrich.maths.org/

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Counters in the Middle

This is one of a series of problems designed to develop learners' team working skills. Other tasks in the series can be found by going to this article.

What are you aiming to do?

For the task:

The team has to recreate the row of counters (created by the designer) by asking as few questions as possible.

As a team:

Listening and finding out what others think
Giving reasons for ideas
Pulling ideas together
Finding out whether the group is ready to make a decision.

Getting started

You will be working in a team of four or five. You will need two sets of counters. The image above uses 6 counters of 3 different colours. You might wish to start with just two colours until you have some experience of the task.

Choose someone in the group to be the designer. If you try this task more than once, team members should take turns at playing this role.

Give one set of counters to the designer and the other identical set to the rest of the team.

You may also wish to choose someone to be the observer (a really good idea if there are more than four people in the team).

Tackling the problem

Without the rest of the group seeing, the designer creates a line of counters using some or all of the counters available.

Using the rules for asking questions, and checking that they all agree first, members of the team take turns to ask the designer questions that will help them recreate the line of counters.

When all of the team think they have the correct design they can check with the designer and the task ends.

At the end, the observer gives feedback about the way members of the team worked together, highlighting strengths and ways that they could improve next time. The team discuss the feedback and how they think they worked.

Task Rules:

You must agree any question before asking the designer.
The team can only ask questions of the type listed in the rules.
The team members must take turns in asking questions.
The designer can only answer "Yes", "No" or give a number.

Try to find the pattern using as few questions as possible.

Designer guidelines:

You do not have to use all the counters.
The counters should form a line.
You can only say "Yes" or "No", or give a number as an answer.
You must only answer questions of the agreed format.
You only answer a question if the team has discussed and agreed it first.

Team question rules:

The team can ask questions about:

the number of counters,
the number of counters of each colour,
the colours of touching counters,
the symmetry properties of the design,
the colours, or numbers, of counters to the left or right of a counter.

You cannot ask about the position and colour of a particular counter, for example:

"Is the first counter red?",

"Is the second counter yellow?"

You can ask questions like:

"Is the counter to the left of the red counter green?".

Observer Guidelines:

Your role is to:

identify the times when reasons for ideas are given by members of the team,
check that everyone agrees before a question is asked,
check that the questions fit those allowed in the rules,
check that members of the team take it in turns to ask questions,
count the number of questions the team asks.

Why do this problem?

This task encourages the development of team-building skills such as sharing reasoning, allowing everyone to contribute and valuing those contributions, and coming to a consensus. This is one of a series of problems designed to develop learners' team-working skills. Other tasks in the series can be found by going to this article. In addition learners are expected to use precision in describing an arrangement of objects.

Possible approach

If learners have never worked on this sort of problem before it is best to choose no more than six counters and two colours.

You will need a screen or cloth to hide the designer's arrangement.*

Arrange learners in groups of four or five (five allows one to act as an observer).

You may wish to ask teams to record their questions before asking them. Then, as part of their review of the task, the team can discuss what may have been a more efficient set of questions to ask.

Cards for each role:

The rule cards for questioning can be printed from this document.
The role card for designers can be found here.
The role card for observers can be found here.

When teams have finished working on the task it is important that they spend time discussing in groups, and then as a whole class, how well they worked as a team, what they have learned from the experience and what they would do differently next time. Your own observations, as well as those of observers might inform the discussions.

* You may prefer to prepare some images of an arrangement first to hand out to designers. Designers then answer the team questions, making a note of the number of questions asked. The team records each question before asking it. When the groups have finished you might discuss the number of questions used and ask the team with the fewest questions to share their ideas with the rest of the group.

Why not let us know how the children have got on with their group-working skills by clicking on the 'Submit a solution' link?

Key questions

What things did you hear someone else say that you found really helpful?
How well did you listen to others in your group?
How easy was it to come to an agreement about the questions to ask?
Did everyone always agree and what did you do if you did not?

Possible extension

Increase the number of counters and the colours available.

Learners may like to try one of the other 'All for one' tasks, or other skill-building tasks which can be found by going to this article.

Possible support

Reduce the number of counters and colours available. Make a list of the allowed questions, such as:

How many counters are there?
How many red counters are there?
Is the counter to the left of the green counter red?
Is the counter to the right of the yellow counter green?
Is there a counter to the left/right of the yellow counter?