Billiards and Snooker are some of the most mathematical of games. Why is that? This lesson explores some of the interesting patterns that are created when you track the path of a billiard ball across the table. Learning Content: Number strand: factors and ratios Space strand: angles, symmetry (reflections) Pattern and Algebra: generalisation Working Mathematically

Size of tableNumber of bounces Final pocket 6×10 4×5 6×8 9×15 8×10 99 ×100 M×N see any patterns that might help you form theories or conjectures? Draw two more tables of any size that’s going to help you prove your theory.

Size of tableNumber of bounces Final pocket 6×106Top right 4×5 6×8 9×15 8×10 99×100 M×N

As soon as you know the size of the table, can you predict: - How many bounces will it make until it goes into a pocket? - Which pocket will it finally go into?

Size of tableNumber of bounces Final pocket 6×106Top right 4×57Top left 6×85Bottom right 9×156Top right 8×107Top right 99×100197Top M×N M+N-2Depends on whether M+N-2 is an odd or even number

Summarizing Rules 1.Number of Bounces Patterns such as 6 by 10, 9 by 15, or even 15 by 25, are all identical to 3 by 5. Hence ratio and common factors are important components of the rule. which is: If M and N have no common factors then there are M + N - 2 bounces. If M and N share a common factor of f, then the number of bounces is (M + N)/f - 2. M N

2.Final Pocket This depends on which direction the ball is facing after M + N - 2 bounces (assuming M and N have no common factors. Each bounce changes the direction by 90 degrees. a) If M + N is even. An even number of 90 degree turns means the ball has turned either 180 or 360 degrees, so it must be facing the direction it came from (impossible), or towards the top right hand pocket. Hence tables such as 3 by 5, 15 by 11, 6 by 10 will all finish in the top right hand corner. b) If M + N - 2 is odd then after an odd number of 90 degree turns the ball must finish in the top left or the bottom right corner.