1. Mathematics: a Powerful Tool for Solving Problems
Doug French
The Mathematical Association

2. MATHEMATICS is USEFUL and INTERESTING.
MATHEMATICS is a POWERFUL TOOL
for solving PROBLEMS.
PROBLEMS provide the MOTIVATION
for learning MATHEMATICS.

3. Two cups of coffee at £1.25 with a 10% discount.
A Problem with Percentages
Two cups of coffee at £1.25 with a 10% discount.
How much to pay?
10% of £2.50 = 25p
Cost of two cups of coffee: £2.50 – 25p = £2.25
10% of £1.25 = 12.5p = 13p to the nearest penny
Cost of one cup of coffee: £1.25 – 13p = £1.12
Cost of two cups of coffee: £2.50 – 26p = £2.24

4. n² – 1 = (n – 1)(n + 1) A Curious Property 7 × 9 = 63 and 8 × 8 = 64
1 × 3 = 3
2 × 4 = 8
3 × 5 = 15
4 × 6 = 24
5 × 7 = 35
6 × 8 = 48
7 × 9 = 63
8 × 10 = 80
9 × 11 = 99
2 × 2 = 4
3 × 3 = 9
4 × 4 = 16
5 × 5 = 25
6 × 6 = 36
7 × 7 = 49
8 × 8 = 64
9 × 9 = × 10 = 100
20 × 22 = 440 and 21 × 21 = 441
n² – 1 = (n – 1)(n + 1)

5. The Difference of Two Squares
A pictorial proof

6. Difference of squares of ANY two odd numbers is a multiple of 8.
The Difference of Squares of Two Consecutive Odd Numbers n n²
(2n + 1)² – (2n – 1)² = 4n  2 = 8n
Difference of squares of two consecutive odd numbers is a multiple of 8.
Difference of squares of ANY two odd numbers is a multiple of 8.

7. REMEMBERING is NOT ENOUGH.
Need to know WHY as well as HOW.
It is EASIER to REMEMBER when you UNDERSTAND
and there is much LESS to REMEMBER.