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.
I worked this out by using two formulas or I could have just carried on with the paterns mentioned above.
If a square is 2cm by 2cm there are 3 pegs on each side, to work out how many pegs there are around the outside
Probing questions
What conclusions can you draw from your evidence? Is it possible to generalise this further?
Is your conclusion based on mathematical explanation or experimental evidence? How do you know?
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.
The maximum number of crossings is (n/2)(n − 1), or explained fully without using algebra.
- When you add another line how many extra lines are there? Can you use algebra to explain this?
- Can you explain how you work out how many crossings there are altogether? (If pupils explain 1+2+3+4+5+6, etc., ask if they can explain how they can add this quickly; there are three pairs of 7 [1+6, 2+5, 3+4]).