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This program is essentially about problem solving, relationships between numbers, factors and ratios (through the metaphor of a snooker table).
Skills involved
What will the children learn?
To work systematically at solving a problem
To be able to record their results as they go along
To look at number relations and to try to discover a pattern.
What prior experience do the children need?
Pupils will need to be familiar with their tables and have an understanding of factors and multiples.
Equipment
One computer for each pair of children
A recording sheet.
The lesson
Pupils can work in pairs. This will enable them to share ideas and to make the recording of different attempts more reliable.
Explain what the program does. A snooker ball is struck at 45 degrees from the top left hand corner and continues on its way until it drops down one of the pockets (note: there are no side pockets). The pupils count the number of bounces, but add 1 for the pocket the ball leaves from and one for the pocket it drops down. Hence on a table measuring 3 x 6, the number of bounces is 3 (even on a square table the number of bounces will be 2!)
The object of the activity is to predict the number of bounces that the ball makes before it drops down a pocket. There is one rule that applies to all circumstances. Children tend to develop a rule that is true in some circumstances and not in others. For example, children might suggest that if both numbers are odd then the number of bounces is the sum of the two numbers. This is true for 5 and 7, and for 7 and 11. The role of the teacher is to suggest circumstances where they might like to reconsider their rule. What if they try 5 and 15?
The maximum size table allowed is 20 x 20.
Children have difficulty with this program in working systematically. If they can be encouraged to do so, then they are more likely to discover the rule for the number of bounces. Hence the teachers role will be in helping pupils to structure their investigation so that it may lead to a successful outcome. For example, the table below shows a sequence of possible investigations that will guide pupils towards discovering the rule. (Initially they will only complete the column headed bounces; the other columns are for subsequent investigations).
When using the program, children should be encouraged to make an estimate of their answers first.
WidthLengthBouncesPocketPathIntersections446636485103941251537596118129151216122421281131100150
Pupils should be encouraged to make a note of whatever rules they find, by completing a table such as the one shown below (partly completed for illustration).
For example, one rule might be: the answer is always 2 when the table is square.
Rules Table  How many bounces? Dimensions Answer Suggested rule 3 x 4 7 add them together 3 x 58 still works 3 x 6 3 snag  don't add them together if one divides into the other  instead add them together and divide by the smallest number 4 x 12 4 hey, it still works 4 x 6 5 help! neither rule appears to work here  I now need another rule!! I could add them and halve the result, but why?
Hence the real test of their understanding is whether or not they can provide the correct answer to the number of bounces on these 4 tables:
12 x 24
21 x 28
11 x 31
100 x 150
References
QCA Schemes of Work ICT level 4: They use ICTbased models and simulations to explore patters and relationships, and make predictions about the consequences of their decisions
HYPERLINK "http://www.standards.dfes.gov.uk/schemes2/it/?view=get" http://www.standards.dfes.gov.uk/schemes2/it/?view=get
National Numeracy Strategy Year 6: Solve simple problems involving ratio and proportion. Recognise prime numbers and identify factors. Solve mathematical problems or puzzles, recognise and explain patterns and relationships, generalise and predict. Suggest extensions by asking What if...?
HYPERLINK "http://www.standards.dfes.gov.uk/numeracy/teaching_resouces/" http://www.standards.dfes.gov.uk/numeracy/teaching_resouces/
514 National Guidelines for the Curriculum in Scotland: Mathematics 514, Level D/E, Number, Money and measurement Functions and equations; Problem Solving Look for a pattern, produce an organised list or table
HYPERLINK "http://www.ltscotland.org.uk/5to14/guidelines/index.asp" http://www.ltscotland.org.uk/5to14/guidelines/index.asp
Where do we go next?
There are 3 other investigations that can be carried out using the same program. In order of difficulty these are:
Find the length of the path that each ball travels (assume the diagonal of each square is one unit)
Find the number of intersections
Predict the pocket that the ball goes down.
Differentiating the activity
The extra investigations listed above provide ample challenge for able upper primary children.
The less able pupils may struggle with the ideas in this program. In this case they should be encouraged to reverse the aim of the program. For example:
Find all the table sizes that give 3 bounces
If one side is 5, what length might the other side be if the number of bounces is the sum of the two sides?
These Teacher Notes are taken from the activity How Many Bounces supplied to teachers through the Becta Direct2U service. Our thanks to Becta for permission to use them here. To subscribe to Direct2U go to HYPERLINK "http://www.ictadvice.org.uk/subscribe" http://www.ictadvice.org.uk/subscribe
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