don steward
mathematics teaching 10 ~ 16

Tuesday, 11 January 2011

squares sum

Within a lesson on probability it is helpful to compare (i) an individual's results (ii) whole class results (students copy them down as they are announced) and (iii) theoretical values, worked out - partly at least - by students. This task has demanding theoretical values (well it was for me anyway) but is a neat enough experiment.

Students throw a dice 25 times to fill all the cells in a 5 by 5 grid.
They then work out the totals for each of the 16, 2 by 2 squares (the one shown has a total of 17).

They could do this twice and record the data for their totals.

Whole class data can be collected for the totals (between 4 and 24).

It's a neat symmetrical distribution and if anyone claims to have a total of 4 or 24 they have probably cheated! Get them to fill in the grid randomly before telling them the object of the task...

the theoretical values need careful recording

for example, a total of 10 will have:
6211 x 12 (two the same)
5311 x 12
5221 x 12
4411 x 6 (two pairs)
4321 x 24 (all different)
4222 x 4 (three the same)
3331 x 4
3322 x 6
a total of 80 ways

Considerations about how many different ways you can arrange four numbers, with various options of repeated digits, can be a reason for undertaking this task with a class.

The frequencies sum to 1296 (which is 6^4) so theoretical probabililities can be calculated and compared with long run (whole class) and short run (individual's) data.

Students should find that the mean of their totals is close to 3.5 x 4 = 14

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