### The Lily Pond

Freddie Frog visits as many of the leaves as he can on the way to see Sammy Snail but only visits each lily leaf once. Which is the best way for him to go?

### Coordinate Tan

What are the coordinates of the coloured dots that mark out the tangram? Try changing the position of the origin. What happens to the coordinates now?

### A Cartesian Puzzle

Find the missing coordinates which will form these eight quadrilaterals. These coordinates themselves will then form a shape with rotational and line symmetry.

# Dotty Relationship

## Dotty Relationship

On the grid below, join the two blue dots, A and B, together with a straight line.

Now join the green dots, C and D, with a straight line.

At what angle do the two lines cross?

Investigate the number of squares "along" and "down" from A to B compared with the number of squares "along" and "up" from C to D. What do you notice?

Using what you have found out, can you draw lines that are perpendicular (at $90^{\circ}$) to the lines drawn below?

Is there only one solution each time?

Do the two lines have to be the same length? Why or why not?

### Why do this problem?

This problem could be linked to coordinates. It could also form an introduction to vectors at a higher level. It is a powerful geometrical investigation. It has the potential to quickly lead to generalisations that pupils can apply in the second part of the question.

### Possible approach

The notion of "how long is a line?" would form an excellent discussion point in the plenary.