### Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

### Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

### Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

### Why do this problem?

This month's NRICH site has been inspired by the way teachers at Kingsfield School in Bristol work with their students. Following an introduction to a potentially rich starting point, a considerable proportion of the lesson time at Kingsfield is dedicated to working on questions, ideas and conjectures generated by students.

This problem is based on ideas that have been shared on the NRICH Projects discussion site. It provides a way of engaging with parabolas which will help learners develop insights and understanding of the relationship between a quadratic function and its graphical representation.

### Possible approach

Show learners the interactivity and drop in some inputs. Ask them if they can work out what the graph is showing (the input numbers as the x coordinates and the outputs as the y coordinates). What do they notice about the graph?

Either using the interactivity or by working on paper, allow learners time to explore different mappings of the form $x \rightarrow (x - a)(x - b)$. This is a great opportunity for learners to work collaboratively and base their generalisations on their shared examples.

Encourage learners to explore the relationships between the mappings and their graphs:

Can I tell from a mapping where its graph will cross the x axis?
Can I tell from a mapping where its graph will cross the y axis?
Can I tell from a mapping where the parabola will have its vertex?
Does every parabola have a line of symmetry?

Read the article Kingsfield School - Building on Rich Starting Points, which has links to a description of a Kingsfield teacher's first lesson on functions and graphs, and a video showing how these ideas are put into practice in the classroom.

For teachers who want to create their own 'Number Plumber' mappings for use in the classroom, click here to watch an introductory video explaining how to build, load and save your own examples.

To read more about the Number Plumber, visit Grumplet's blog where you can comment on how you have used the Number Plumber and share links to files you have created. We are continuing to develop this resource so your feedback and ideas will be very useful.

### Key questions

Why do these mappings always give curves/parabolas?
Can I tell from a mapping where its graph will cross the x axis?
Can I tell from a mapping where its graph will cross the y axis?
Can I tell from a mapping where the parabola will have its vertex?
Does every parabola have a line of symmetry?

### Possible extension

Parabolic Patterns, Parabolas Again, More Parabolic Patterns and Cubics provide suitable follow-up activities.

### Possible support

Suggest that learners fix one number and see what happens when they change the other.