Why do this problem?
gives the opportunity for practice in manipulating
algebraic fractions and provokes discussion about two different
representations of an expression using algebra and a diagrammatic
approach. It is called 'Look Before You Leap' because a good look
at the structure of the equations, and how they might relate to the
required expression, leads to a quick answer, whereas blindly
eliminating variables will not help in this case.
Some students' natural response to the first part of the
question will be to attempt to solve the three given equations to
work out values of a, b and c in order to find the values of the
expressions. It is perhaps worth allowing them to try this, and
after a short time suggest that there is a quicker way using their
knowledge of addition of fractions to rewrite the expressions in
terms of the information they already know.
One method of calculating $a^2 + b^2 + c^2$ comes from
utilising the expansion of $(a + b + c)^2$ which forms the second
part of the problem. A useful activity is to get students to
discuss why the expansion takes the form that it does by annotating
the areas in the diagram. Then they can use this to convince
themselves of the form for the expansion of $(a + b + c)^3$ using a
diagram of the faces of a cube.
Is it necessary to work out a, b and c in order to calculate
the value of the expressions?
What is represented by each part of the square diagram?
How can we draw a similar diagram using a cube?
Work out the expansions of $(a + b + c + d)^2$ and $(a + b + c
+ d)^3$ using diagrams, and convince someone else that the
expansions are correct.
Can students actually solve for $a, b$ and $c$ in the first
part of the question?
Don't forget to offer the hint from the Hints Tab at the top
of the problem.
Focus on the middle part of the question, involving relating
the diagram of the squares to the algebra.
Can struggling students 'mark' and give feedback on the
answers of others to the first part of the question?