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.
Writes a connected chain of reasoning to explain why the number of pegs on the outside is a multiple of 4.
Or
Notices that the number of pegs in the hollow produces the sequence of square numbers.
Every time the number of dots around the outside increased by 4
After making the table I discovered that the number of squares on the inside of a square was got by squaring the sides of the previous square.
EG square 1 is 2x2 square
Probing questions
Why did you use that graph/diagram? What does the graph/diagram reveal about the problem?
How did you decide on the scales on your graph?
Would a different scale change your conclusions?
Can you explain the pattern that is shown here? Can you explain the rule? What would you expect in the next case?
Can you explain why some of the results do not follow the same pattern?
Is your generalisation always true? How do you know? What if...?
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 table of results and explain the patterns you see.
- Can you predict what the next number in the table (pattern) would be?
- Are you sure? (Check by drawing this.)
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