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.
Examples drawn from Hollow Squares.
From results generated describes in words how to work out the number of pegs inside or outside.
METHOD
The number on the outside goes up by 4 each time. You can work out the middle by if it was 6 by 6 it would take 2 off so it would be 4 by 4. Then you would times 4 by 4 and that would give you the answer.
Probing questions
How does your generalisation link to the original problem? Explain the, for example, multiply by 6, add 2, squaring (vary according to context).
What if you started with ... instead of ...? How might your generalisation/solution be different?
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.
- Look at the pattern as you add another line; how many extra crossings does it make?
- Why can’t there be any more crossings? Can you explain why?
- If you had 12 lines and you added one more line, what would be the greatest number of possible crossings? Why?
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