Successful mathematical problem solving involves insight, strategy and decision making, but also is improved through experience within many different tasks.

Here are a few problems that we have been considering in Hertfordshire; in lessons or in problem-solving events:

## Year 9 or 10

Suppose we would like to put some black and white beads in a bag so that the probability of picking 2 beads of the same colour without replacement is equal to

. One solution is 1 white and 3 black beads. Can you show that this works and find other possibilities?

**Hint:** The answers are all drawn from a standard set of numbers.

## Year 10 or 11

How much difference does it make to the answer to a compound interest problem if I wish to find my total final amount when I increase by 100% once, or by 50% twice, or by 33% three times, or by 25% four times, … , or by 10% ten times, … , or by 5% twenty times, … , or by 1% one hundred times etc? Is there a pattern or trend? How does the diagram relate to the problem?

**Hint:** Jacob Bernoulli discovered that the answers tend to a limiting value.

## Year 12 or 13

Consider a polynomial . What could graphs of the form **maths equation** look like?

Eg If **maths** then consider the graph **maths equation** which gives two straight lines **maths** and **maths. **The diagram below uses **maths equation **

Hint: Polynomials with turning points will give more interesting answers.

## Problem solving events

Discover problem solving events for your students in the East of England: