This problem introduces students to interpolation and the concept of 'building' algebraic solutions to problems. The result is very interesting mathematically. It is based upon the idea of a 'generic example': a particular example which encapsulates in a clear way all of the properties of a more general case. The ideas in this problem pave the way for patterns of thinking which are to be found
in university mathematics courses, and the concept of interpolation is valuable in both mathematics and science. This problem also raises the idea from proof that constructing an example with the correct properties proves that an example exists, whereas inability to construct an example with the correct properties does not necessarily prove that such an example doesn't exist.
What does Steve's expression evaluate to at each of the three points?
Why has Steve not expanded the brackets or simplified the numbers?
Do you think that Steve's construction could be generalised?
Under which circumstances would Steve's construction break down?
How many degrees of freedom does a quadratic polynomial have?
There are some extension possibilities in the question. Other extension possibilities are to use a spreadsheet or computer to produce the fitting polynomials for $4$ or $5$ variable points. This is mathematically very interesting and will lead to the understanding that the fitting polynomials, whilst exactly hitting all of the points, are often very unstable and with wildly varying shape -
this is a good open investigation which might arise from this problem.
The main idea in this question concerns constructing a fitting polynomial by arranging brackets and coefficients in a sensible, organised manner. This key idea can be practised by looking at fitting quadratics through 3 points: give a few triples of points and have students construct the quadratics which go through these.