Rationals Between...

What fractions can you find between the square roots of 65 and 67?

Root to Poly

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is $1+\sqrt 2+\sqrt 3$.

Surds

Find the exact values of x, y and a satisfying the following system of equations: 1/(a+1) = a - 1 x + y = 2a x = ay

Consecutive Squares

Age 14 to 16 Challenge Level:
A discussion may be needed on how to represent the eight consecutive numbers.

Students may square and add terms in an almost random way to start with and it is worth giving them time to play before disucssing what might be a more efiicient method.

Rather than squaring terms on a need-to-know basis - how about suggesting working them all out to save time re-calculating expansions as needed.