# Step 3

Begin to refine ways of recording and use appropriate mathematical symbols correctly

### Examples of what pupils should know and be able to do

There are six different ways to shade two squares in this shape. Can you find them all?

Try using different rectangles made up of more squares.

#### Examples drawn from 'Shading squares'

Find 10 or more of the arrangements for two squares out of six without any repeats.

### Probing questions

• Which methods of organising information have you found most helpful?
• How is this different from the way you have recorded information before?

### What if pupils find this a barrier?

#### Use 'Line crossings investigation'

• Draw other diagrams using only four (or five) lines; how many different numbers of intersections (crossings) can you get?
• How many more diagrams with a different number of intersections can you draw using four (or five) lines?
• How might you record the maximum number of crossings for three, four and five lines?
• Are there other ways?

#### Line crossings investigation

• Draw three straight lines (line segments) so that some cross over each other.
• How many crossings are there?
• Try different arrangements of the lines. What is the maximum number of possible crossings?
• Try using more lines.
• Is there a rule for the maximum for any number of lines? If so, write it down.