Skip over navigation
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.
Spot the Difference
Age 16 to 18
ShortChallenge 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.You may also like
Power Up
Show without recourse to any calculating aid that 7^{1/2} + 7^{1/3} + 7^{1/4} < 7 and 4^{1/2} + 4^{1/3} + 4^{1/4} > 4 . Sketch the graph of f(x) = x^{1/2} + x^{1/3} + x^{1/4} -x
Snookered
In a snooker game the brown ball was on the lip of the pocket but it could not be hit directly as the black ball was in the way. How could it be potted by playing the white ball off a cushion?