### 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.