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?
What about this shape?
How many ways are there?
Try using different rectangles made up of more squares.
Try shading three squares.
Examples drawn from Shading Squares
Develops and uses a procedure guaranteeing all of the ways of shading three squares in a given number of squares, making observations based on correct results. They can systematically produce results for shading squares without drawing the diagrams.
Probing questions
What forms of presentation have you used to help analyse information and communicate findings?
What sort of improvements to your initial attempts have you made?
How did they help you to solve the problem/develop your enquiry further?
What have you found out that will be useful in future work?
What if pupils find this a barrier?
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 have rule (n/2)(n − 1) or the rule in words.)
- That is a good rule; explain to me what it means.
- Why is this rule more useful than drawing diagrams and counting intersections?
(Pupils have explained that for every extra line, the number of extra crossings is one less than the new total number of lines.)
- Can you explain why this (algebraic) rule is more helpful in finding the number of crossings than just adding the number of extra crossings each time? Can you write that down?