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.

A pegboard with the outline of a square mapped out.

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

A pegboard with the outline of a square mapped out and mathematical working showing the formula for the number of pegs and the hollow in the middle.

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.

Three straight lines crossing over each other to create three points of intersection.