Newman College of Higher Education
	Bartley Green, Birmingham B32 3NT

Bounce – a mathematical investigation

Objectives

Bounce is a program that challenges able year 5 and year 6 pupils. It is an investigation into the relationships between numbers. Pupils need fluency with their tables as the ability to look for highest common factors and lowest common multiples is a key. Pupils will also need three other skills:
a) the ability to suggest rules – 'the number of bounces is equal to the two sides added together'
b) the ability to record the results of the investigations
c) a systematic approach to the problem

The Software

A snooker table is illustrated with corner pockets but no side pockets. Note the order in which the pockets are numbered. The table can be of any dimension up to a maximum of 20 units in either direction. The ball is struck at 45 degrees from the top left hand corner (pocket A). It hits the edge (at the point marked 2) and still travels at 45 degrees towards the right hand edge at the point marked 3. It continues on its way until it drops down one of the pockets. When counting the number of bounces, count 1 for the pocket the ball leaves from and 1 for the pocket it drops down. Hence on a table measuring 3 x 6 the number of bounces is 3. For the 8 x 6 table in the illustration the number of bounces is 7.

When using the software, children can also predict into which pocket the ball will fall, the length of the path that the ball has travelled (1 unit is the length of the diagonal of one square), and the number of times that the ball crosses over its previous path (number of intersections). Hence in the illustrated example, the ball will drop down pocket D, the path length is 24 units and there are 3 intersections. The ball will not begin travelling until the predicted outcomes (displayed in the left hand column under 'outputs') have been entered. Without your estimated attempts (which get recorded in the table of data) you will not be able to progress.

 
 
 
Fig. 1
 

The Teacher's Role
The teacher's role is to help pupils to structure their investigations so that they may lead to a successful outcome. For example, Fig.2 shows a sequence of possible investigations which will help pupils to discover the rules. In trying this with pupils in the past their biggest weaknesses have been disorganised approaches, as a consequence of which number patterns are camouflaged and frustration soon sets in.

Pupils should begin by trying the program several times and completing the following chart. Remember that in measuring the length of the path, the diagonal of a small square counts as 1 unit. Pupils should concentrate on one investigation at a time.
In summary, the aim is to identify the rules that give you, for any size table:

• The number of bounces

• The length of the path

• The number of intersections.

• The pocket the ball drops down

(these problems are listed in order of difficulty, the easiest first)

Fig. 2

Note that the last four tables in the list can't be tested on the computer (one or both sides are too long). Pupils can only predict these based upon the rules that they determine.

Assume that we start by trying to predict the number of bounces. Initially pupils may discover different rules for different size tables -- the real skill is finding one rule for the number of bounces that applies to all sized tables. Pupils should be encouraged to make a note of whatever rules they find, by completing a table such as the one shown in Fig. 3.
For example, progress might be as shown:

Fig. 3

Sometimes pupils find a rule that works for all cases that they have tried. I know 2 pupils who tried 12 snooker tables and were convinced that the rule for the number of bounces was the lengths of the two sides added together, but the dimensions of every table they chose was of the form n and n+1. When the teacher asked them to try 4 and 6, they couldn't believe the result! Very often the teacher's task is to find a counter example, but perhaps to drop a hint that they might also keep a note of the highest common factor of the two dimensions.

This is only one of a number of problem solving programs. The idea is to encourage pupils to hypothesise, to look for relationships and to encourage them to articulate their thinking. This short program may start pupils off an a voyage of mathematical discovery.