Discuss work using mathematical language; represent work using symbols and simple diagrams

Examples of what pupils should know and be able to do

Shading squares

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

A two-by-two square.

What about this shape? How many ways are there?

A three-by-two rectangle.

Try using different rectangles made up of more squares.

Try shading three squares.

Example drawn from 'Shading squares'

  • Try shading these squares.
  • Draw at least three different ways of shading two squares out of four.
  • Then draw all six ways of shading two squares out of four.

Probing questions

  • What have you noticed?
  • Tell me what the diagram(s)/results are showing.

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.

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