Spot the Difference

Age 16 to 18 Short Challenge Level:

Why do this problem?

The problem reinforces everything already learnt about straight lines and also highlights the interplay of algebra and geometry.

It is good experience for learners to realise that they have to be cleverer than the computer and they cannot blindly accept what it reveals at 'face value'. When people make conjectures about situations and test their ideas on a computer they still have to consider whether the computer evidence is reliable.

The examples with factors $x^2+y^2=0$ and $y^2 +(x+1)^2 = 0$ merely require finding the real solutions of the equations $x^2+y^2=0$ and $y^2 +(x+1)^2$ and interpreting these as points on the graphs.

Experience with these two parts should suggest that in the final relation it is necessary to factorise the equation of the relation.

Possible approach

The Hint should be sufficient to enable learners to tackle the first 5 equations independently, which can be done for homework or as a lesson starter.

The class will then be thinking along the right lines when they tackle the final equation. Class discussion can heighten awareness that the graphs of relations can have several branches and that we need to use algebra to find all the solutions as the computer does not necessarily show all possibilities. Learners should also be aware that there may be branches that are not shown on the scale used so they might find other branches by changing the scale on the axes

Key questions

  • What is a linear equation?
  • Should we expect the graph of a relation to be a straight line if the equation is not 'linear'?
  • Why would the computer fail to show all the points of the graph?
  • How do we use the factors of an algebraic expression to find out when it takes the value zero?

Possible extension

Plot these graphs on a graphical calculator or computer graphics package. Can you think of a single relation which would produce the graph shown in the question? Can you think of a relation which would only show differences to those on the screen when the scales of the graphs are magnified greatly? How might a computer cleverly be programmed to try to spot more branches of a relation?

Possible support

Have pupils try to plot the linear graphs themselves. Then ask how their plotting process would need to vary in the more complicated examples. This would point more clearly to the fact the we need to look at solutions of the second factors as well as the first factors.