This problem is in two
parts, which can be attempted independently of each other. The
first part requires students to apply their knowledge of coordinate
geometry and the equations of straight lines. The second part
explores triangles passing through four points and encourages
conjecture about the necessary conditions for this to be
done.

The problem lends itself
to collaborative working, both for students who are inexperienced
at working in a group and students who are used to working in this
way.

Many NRICH tasks have
been designed with group work in mind. Here we have
gathered together a collection of short articles that outline the
merits of collaborative work, together with examples of teachers'
classroom practice.

This is an ideal problem
for students to tackle in groups of four. Allocating these clear
roles (Word, pdf) can help the group to
work in a purposeful way - success on this task should be measured
by how effectively the members of the group work together as well
as by the solutions they reach.

Introduce the four group
roles to the class. It may be appropriate, if this is the first
time the class have worked in this way, to allocate particular
roles to particular students. If the class work in roles over a
series of lessons, it is desirable to make sure everyone
experiences each role over time.

For suggestions of
team-building maths tasks for use with classes unfamiliar with
group work, take a look at this article and the
accompanying resources.

Hand out either or both of the task sheets (First task: Word, pdf. Second task: Word, pdf.) to each group, and make it clear that everyone needs to be ready to share what they did with the rest of the class at the end of the sessions. Exploring the full potential of this task is likely to take more than one lesson, with time in each lesson for students to feed back ideas and share their thoughts and questions.

You may want to make squared or graph paper, poster paper, and coloured pens available for the Resource Manager in each group to collect.

While groups are working, label each table with a number or letter on a post-it note, and divide the board up with the groups as headings. Listen in on what groups are saying, and use the board to jot down comments and feedback to the students about the way they are working together. This is a good way of highlighting the mathematical behaviours you want to promote, particularly with a challenging task such as this.

You may choose to focus on the way the students are co-operating:

Group
A - Good to see you sharing different ways of thinking about
the problem.

Group B - I like the way you are keeping a record of people's ideas and results.

Group C - Resource manager - is there anything your team needs?

Group B - I like the way you are keeping a record of people's ideas and results.

Group C - Resource manager - is there anything your team needs?

Alternatively, your focus for feedback might be mathematical:

Group
A - I like the way you chose to represent the situation with
a diagram.

Group B - You've drawn some straight lines - would it help to express them algebraically?

Group C - Good to see that someone's checking that each point is close enough to the triangle.

Group B - You've drawn some straight lines - would it help to express them algebraically?

Group C - Good to see that someone's checking that each point is close enough to the triangle.

Make sure that while groups are working they are reminded of the need to be ready to present their findings at the end, and that all are aware of how long they have left.

We assume that each group will record their diagrams, reasoning and generalisations on a large flipchart sheet in preparation for reporting back. There are many ways that groups can report back. Here are just a few suggestions:

- Every group is given a couple of minutes to report back to the
whole class. Students can seek clarification and ask questions.
After each presentation, students are invited to offer positive
feedback. Finally, students can suggest how the group could have
improved their work on the task.

- Everyone's posters are put on display at the front of the room,
but only a couple of groups are selected to report back to the
whole class. Feedback and suggestions can be given in the same way
as above. Additionally, students from the groups which don't
present can be invited to share at the end anything they did
differently.

- Two people from each group move to join an adjacent group. The two "hosts" explain their findings to the two "visitors". The "visitors" act as critical friends, requiring clear mathematical explanations and justifications. The "visitors" then comment on anything they did differently in their own group.

If your focus is effective group
work, this list of skills may be helpful (Word, PDF). Ask learners to
identify which skills they demonstrated, and which skills they need
to develop further.

If your focus is mathematical,
these prompts might be useful:

Does it help to draw a
diagram?

What happens to the
equation of a line when you move it slightly?

How does moving the lines
affect the distances from each point to the lines?

For Part 2: Can you draw
a configuration of four points which can NEVER lie on the edges of
a triangle?

Is everyone in your group
convinced that it is NEVER possible?

What is special about the
position of the points?

Providing
reasoned arguments for when it is and isn't possible to draw a
triangle through four, five or even six points is a very
challenging mathematical activity!

The stage 5 problem Erratic Quadratic
uses the idea of finding an object which passes close to a set of
points, but uses curves instead of straight
lines.

By working in groups with
clearly assigned roles we are encouraging students to take
responsibility for ensuring that everyone understands before the
group moves on.

The game Diamond Collector
may offer useful practice at working out the equations of straight
lines through points in advance of tackling the first part of this
problem.