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Can you find the maximum value of the curve defined by this expression?

Erratic Quadratic

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This problem is in two parts. The first part requires students to apply their knowledge of coordinate geometry and quadratic equations. The second part draws on problem solving, calculus and numerical methods.
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.

Possible approach

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.

Explain the tasks to the groups, 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.

You may want to make calculators, spreadsheets, graphing software, 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.

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?

Alternatively, your focus for feedback might be mathematical:

Group A - I like the way you chose to represent the situation with a graph. Could you use algebra to prove the result?
Group B - You've got an equation. What might be a good starting point for finding a numerical solution?
Group C - Good to see that someone's checking that each point is close enough to the quadratic.

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 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.

Key questions

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 is the general form of a quadratic?
Could you fit a quadratic exactly though any of the points and then adjust it?
Geometrically, how would we find the smallest distance from a point to a quadratic?
What values of $y$ do $x=\pm 1, \pm 2, \pm 3$ give on your trial quadratic?

Can you draw a configuration of four points which can NEVER lie within one unit of distance of a parabola?
Is everyone in your group convinced that it is NEVER possible?
What is special about the position of the points?

Possible extension

Providing well reasoned arguments for when it is and isn't possible to draw a quadratic near to four points is challenging. You might also ask when can you exactly fit a quadratic through different numbers of points.

Possible support

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 problem Close to Triangular may offer useful practice at the teamwork ideas but focussing on straight lines only.