Beelines

Problem | Teachers' Notes | Hint | Solution | Printable page |
Stage: 4 Challenge Level: Challenge Level:1

Why do this problem?

This problem draws together knowledge of coordinates, factors and multiples and potentially Pythagoras' Theorem and gradients, with lots of scope for making conjectures and justifying them in a relatively secure mathematical context. Experimental evidence can be supported by argument and there is opportunity to share different ideas and assess other people's reasoning.

Possible approach

There are two key points to establish at the start of the investigation:
  • the focus on crossing squares. There is significant guidance in the question but you may wish to start with the diagram and ask learners to describe what they see. If necessary, focus can be drawn to the ways in which the lines cross the grid by asking what is the same and what is different about pairs or groups of lines.
  • the need to impose constraints so that variation can be controlled by the learners. Discuss, and list, what can be varied and how these variables might be controlled.
Encourage groups to write their ideas on a conjecture board before they agree where they might start and how they might demonstrate, and then prove, a relationship.

Allow time towards the end of the work for learners to share findings and feedback on the strength of each others' arguments.

Key questions

  • What happens if one coordinate is twice the other, or three times, or four... times?
  • What happens if the coordinates have a factor of two or three or four... in common?
  • Are there coordinates that build on the same rectangle? Can you predict families of such coordinates?
  • Can you see a connection between the gradients of lines and the rectangles they cross?

Possible extension

What about considering the gradients of the lines?
Can the argument be extended into three dimensions?

Possible support

Model a particular case - for example where the coordinates are equal.
Encourage learners to draw lines allowing only one degree of freedom and explore controlled situations in this way.



Published October 2000,January 2009.