Good Approximations

Solve quadratic equations and use continued fractions to find rational approximations to irrational numbers.

There's a Limit

Explore the continued fraction: 2+3/(2+3/(2+3/2+...)) What do you notice when successive terms are taken? What happens to the terms if the fraction goes on indefinitely?

Not Continued Fractions

Which rational numbers cannot be written in the form x + 1/(y + 1/z) where x, y and z are integers?

Comparing Continued Fractions

Stage: 5 Challenge Level:

Why do this problem?
For experience of working with inequalities and with fractions.

Possible approach
If they can't get started students can try numerical values for $a$ and $b$.

Key question
If you increase the denominator of a fraction do you reduce or increase the value of the fraction?

Possible support
Try the problem Not Continued Fractions

Possible extension
See the article Continued Fractions 1