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.
Pupil notices that the number of pegs on the outside is in the four times table.
I noticed that all outside numbers belong to the 4 times table. In this investigation I found out that every square you draw gets bigger and there are more dots inside the squares. Also I have found out that the number of pegs in turn go odd, even and so on. I have tried many different shapes for this investigation - squares, triangles, hexagons and pentagons.
Probing questions
What have you found out?
Can you express this as a general statement that someone else could test?
Why do you think your general statement is true?
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 different diagrams with six (five) lines.
- When there are only a few crossings what can you say about the lines?
- What do the lines look like when there are a lot of crossings?
- Can you explain how you can get more (or fewer) crossings?
With five 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.
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