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.

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.

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?