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Guide and features
Guide and features
Science, Technology, Engineering and Mathematics
Featured Early Years Foundation Stage; US Kindergarten
Featured UK Key Stage 1&2; US Grades 1-5
Featured UK Key Stage 3-5; US Grades 6-12
Featured UK Key Stage 1, US Grade 1 & 2
Featured UK Key Stage 2; US Grade 3-5
Featured UK Key Stages 3 & 4; US Grade 6-10
Featured UK Key Stage 4 & 5; US Grade 11 & 12
Spot the Difference
Stage: 5 Short
Why do this 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.
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
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?
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?
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.
Making and proving conjectures
Manipulating algebraic expressions/formulae
Meet the team
The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities can be found here.
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NRICH is part of the family of activities in the
Millennium Mathematics Project