### 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