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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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Consecutive Squares

The squares of any 8 consecutive numbers can be arranged into two sets of four numbers with the same sum. True of false?

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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.

Minus One Two Three

Stage: 4 Challenge Level: Challenge Level:2 Challenge Level:2

It would be good if students knew that:
  • the product of a pair of linear brackets represents a quadratic form,
  • quadratic forms are always symmetric and
  • the axis of symmetry of a quadratic form is often worth locating.
Moving fluently between equivalent algebraic forms, appreciating what each 'view' tells us about the expression under examination, is a key problem-solving strategy. This problem benefits from encouraging learners to spend time in this way.

As with many Stage 4 algebra-based problems, there is much to gain from students making up similar problems for each other to solve, and especially if group discussion allows appreciation of the different 'ways of seeing' to be shared.

It is always good to distinguish for students identities that are 'equal' because the algebra is equivalent, from situations where the equals sign invites answers to the question 'for what values of the variable will these two expressions be equal in value?' - solving the equation?