### Absurdity Again

What is the value of the integers a and b where sqrt(8-4sqrt3) = sqrt a - sqrt b?

### Ball Bearings

If a is the radius of the axle, b the radius of each ball-bearing, and c the radius of the hub, why does the number of ball bearings n determine the ratio c/a? Find a formula for c/a in terms of n.

### Incircles

The incircles of 3, 4, 5 and of 5, 12, 13 right angled triangles have radii 1 and 2 units respectively. What about triangles with an inradius of 3, 4 or 5 or ...?

# Look Before You Leap

### Why do this problem?

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.

### Possible approach

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.

### Key questions

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?

### Possible extension

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?

### Possible support

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?