### Modular Fractions

We only need 7 numbers for modulus (or clock) arithmetic mod 7 including working with fractions. Explore how to divide numbers and write fractions in modulus arithemtic.

Find all real solutions of the equation (x^2-7x+11)^(x^2-11x+30) = 1.

### Areas and Ratios

Do you have enough information to work out the area of the shaded quadrilateral?

# Direct Logic

##### Age 16 to 18 Challenge Level:

To prove a theorem directly we start with something known to be true and then proceed, making small logical steps which are clearly correct, until we arrive at the desired result. So, because the starting point was true and each small step clearly correct, we know the result to be true.

Breaking down a mathematical argument into small steps requires patience and clear thinking.

In the following interactivities we have written out three proofs, broken them into small steps and then shuffled up the steps. Can you rearrange them into the correct logical order?

Proof of the formula for the roots of a quadratic equation

Proof of the formula for the sum of an arithmetic progression

Proof of the formula for the sum of a geometric progression