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Converse

Clearly if a, b and c are the lengths of the sides of a triangle and the triangle is equilateral then a^2 + b^2 + c^2 = ab + bc + ca. Is the converse true, and if so can you prove it? That is if a^2 + b^2 + c^2 = ab + bc + ca is the triangle with side lengths a, b and c necessarily equilateral?

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Parabolic Patterns

The illustration shows the graphs of fifteen functions. Two of them have equations y=x^2 and y=-(x-4)^2. Find the equations of all the other graphs.

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More Parabolic Patterns

The illustration shows the graphs of twelve functions. Three of them have equations y=x^2, x=y^2 and x=-y^2+2. Find the equations of all the other graphs.

Consecutive Squares

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2
A discussion may be needed on how to represent the eight consecutive numbers.

Students may square and add terms in an almost random way to start with and it is worth giving them time to play before disucssing what might be a more efiicient method.

Rather than squaring terms on a need-to-know basis - how about suggesting working them all out to save time re-calculating expansions as needed.