Step 5 Objective

Solve problems and investigate in a range of contexts, explaining and justifying methods and conclusions; begin to generalise and to understand the significance of a counter-example.

Examples of what pupils should know and be able to do

Hollow Squares

Here is a hollow square.

diagram showing the hollow square created by circling dots on squared paper.
  • How many pegs form the square on the outside?
  • How many pegs are there in the hollow?
  • Draw some more hollow squares.
  • Investigate.
diagram showing how the formula 2x+2(x-2) equals number of pegs around the outside of a square and to work out the hollow would be (x-1)squared equals the size of the hollow, where x is equal to the number of pegs along the top side.

Examples drawn from Hollow Squares.

Pupil notices that the number of pegs on the outside increases by four as the number along each side of the square increases by one. Partially explains why the number is a multiple of 4 by referring to the number of sides of a square.

First of all the dots on the outside go up in 4, for example 12, 18, and 20


Probing questions

What have you noticed?

Will this always happen? How do you know?

Can you generalise from this?

What if pupils find this a barrier?

diagram showing three lines crossing

Line Crossings

  • 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.

Use the problem Line Crossings.

Pupils draw different numbers of lines and crossings.

What can you say about the number of crossings when there are a few lines?

Explain what happens to the number of crossings when you get more lines?


<< Previous Step | Next Step >>

Back