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.
Makes a table of results systematically with most of results correct, may notice that the number of pegs in the hollow increases by two more each time.
There are 16 pegs on the outside. The hollow is 9 pegs
- 3 x 3
- 4 x 4
- 5 x 5
- 6 x 6
- 7 x 7
- 8 x 8
- 9 x 9
- 10 x 10
- 8
- 12
- 16
- 20
- 24
- 28
- 32
- 36
- 1
- 4
- 9
- 16
- 25
- 36
- 49
- 64
Outside goes up in fours
Hollow goes up in twos
Probing questions
What have you found out? Why do you think this is? How might you take this further? What do you think might happen?
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.
- How could you organise your work so that you go from smallest (number of intersections) to largest?
- What patterns can you see? How would you explain them?
- Why do you think this is true?
With 5 lines I can draw a lot of patterns with different number of crossings.
When lines are parallel there are no crossings.
You have to be careful because if you make some lines longer they will cross other lines.
The more jumbled the picture the more crossings there are.
<< Previous Step | Next Step >>