### Walk and Ride

How far have these students walked by the time the teacher's car reaches them after their bus broke down?

### Fence It

If you have only 40 metres of fencing available, what is the maximum area of land you can fence off?

### How Far Does it Move?

Experiment with the interactivity of "rolling" regular polygons, and explore how the different positions of the red dot affects the distance it travels at each stage.

# Exploring Simple Mappings

### 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 straight line graphs which will help learners develop insights and understanding of the relationship between a linear 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 $x \rightarrow ax + b$. This is a great opportunity for learners to work collaboratively and base their generalisations on their shared examples.

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

Why do these mappings always give straight lines?
What affects the direction and steepness of a graph?
Can I tell from a mapping where its graph will cross the axes?
When are lines parallel?
When are they perpendicular?

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 straight lines?
What affects the direction and steepness of a graph?
Can I tell from a mapping where its graph will cross the axes?
When are lines parallel?
When are they perpendicular?

### Possible extension

Parallel Lines, How Steep is the Slope? and At Right Angles provide follow-up resources on Straight Line Graphs.

### Possible support

Suggest that learners fix either the multiplication variable or the addition variable and investigate what happens when they change the other.