# Step 2

Understand a general statement by finding particular examples that match it

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

#### Hollow squares

Here is a hollow square.

• How many pegs form the square on the outside?
• How many pegs are there in the hollow?
• Draw some more hollow squares.
• Investigate.

Mathematical working 2x + 2(x − 2) = no. of pegs around the outside To work out the size of the hollow: (x − 1)2 = the size of the hollow, x = the number of pegs along the top side.

#### Examples drawn from 'Hollow squares'

Pupils can draw and make some hollow squares and count the dots in the middle (or on the sides).

• Example 1 – 16 pegs on outside
• Example 2 – 36 pegs on outside
• Example 3 – 18 pegs on outside
• Example 4 – 8 pegs on outside

### Probing questions

• Can you give me some other examples that match this statement? Can you give me some examples that don't match it?

### What if pupils find this a barrier?

#### Use 'Line crossings investigation'

• Can you draw a different arrangement?
• How many crossings are there?
• Now try another arrangement and explain how many crossings there are.
• How would you write this down?

'How many crossings? There are one, two, three, four, five crossings'

'Four lines, four crossings'

#### 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.