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Why do this problem?
Learners can discover the base two number system for themselves and
gain experience of working with powers of 2 and 3. The problem can
be used to reinforce understanding of place value.
The generalisation to the problem which allows weights in both pans
leads naturally to representing numbers in base 3.
Possible approach
As a class, work together through weighing 1, 2, 3, 4, 5 and 6
units with the fewest weights possible and then ask the learners to
continue the process.
Key questions
Weights in one
pan only
What weights are needed to weigh 1, 2, 3,....15 units?
What pattern do you notice in the weights used? Can you explain
it?
When you include another weight in the set of weights used how many
objects can now be weighed?
How many weights are needed to weigh objects of 1, 2, 3 ... 100
units?
How many weights are needed to weigh 1 to 1000 units or 1 to n
units?
Weights in
both pans
What weights are needed to weigh 1, 2, 3,....13 units?
What pattern do you notice in the weights used? Can you explain it?
When you include another weight in the set of weights used how many
objects can now be weighed?
How many weights are needed to weigh objects of 1, 2, 3 ... 100
units?
How many weights are needed to weigh 1 to 1000 units or 1 to n
units?
Possible extension
Discuss with the class the idea that, using both pans, the weights
are used either as +1 times the weight (in the opposite pan to the
object), 0 times the weight (not used at all), or -1 times the
weight (in the same pan as the object). We are using a numbering
system
$$a_0 + a_1x + a_2x^2 + a_3x^3 + ...$$ where the coefficients $a_0,
a_1$ etc. are all 1, 0 or -1 and $x=3$. By a (reversible) 'shift'
this is changed to the base 3 number system.
All integers can be written
in base 3 with digits 0, 1 and 2, that is as $$b_0 + b_1x + b_2x^2
+ b_3x^3 + ...$$ where the coefficients $b_0, b_1$ etc. are all 0,
1 or 2 and $x=3$. This explains why all objects weighing an integer
number of units can be weighed with a set of weights that are all
powers of 3.
You may find a
spreadsheet
tool useful.