In these tasks we adopt the common practice of using a 10 by 10 by 10 cube to represent a unit or one.
TASK 04A: This is a fairly standard kind of task, at least as far as identifying the fractional parts is concerned: the row of red cubes forms one 100th of the large cube, the single (yellow) cube forms one 1000th of the large cube. However, interesting questions might arise when writing the fractions as decimals, especially for part c).
The decimal fractions for parts a) and b) are 0.01 and 0.001 respectively. Some pupils will happily combine them for part c) by writing 0.011. However, can they justify why we can simply 'collect' the 1s and put them in their respective 'columns' - and do pupils accept that this also represents 11 thousandths? We know that there are 11 red or yellow cubes altogether, each of which is one thousandth of the large cube, so we do indeed have 11 thousandths. Good! But when we write this as a decimal why don't we write 11 in the 1000ths column, so getting 0.0011 rather than 0.011?
In part c) we can find the fraction of white cubes by calculating 1 – 0.011, or we can go back to the beginning and determine the number of white cubes: 1000–11 = 989; these form 989 thousandths of the big cube, and so it becomes a matter of writing 989/1000 as a decimal.
TASK 04B: This task touches on the idea that if we split parts into smaller parts, and if we make the parts 10 times smaller at each splitting, we can represent the resulting fractions by 'creating' places further and further to the right of the decimal point.
As a decimal, we know that the yellow cube is 0.001 of the large cube. So we want one 1000th of that, or a 10th of a 10th of a 10th. So 0.001 becomes 0.0001, then 0.00001, and finally 0.000001.
We could of course continue this process, by imagining splitting the red cube into 1000 smaller pieces.
TASK 04C: This is a variant on the first task. It is perhaps fairly obvious which of the two stacks is proportionately 'more red', namely Stack A. The point of interest is in seeing how well the fractions, in both forms, express this fact (or how well pupils can read that the fractions express this fact).
In Stack A, 1/100 or 0.01 of the small cubes are red. For Stack B the fractions are 9/1000 and 0.009. Do students understand that 1/100 > 9/1000, even though 1 and 100 are smaller than 9 and 1000. Similarly, do pupils understand that 0.01 > 0.009, even though 1 is less than 9?
TASK 04D: This task is trying to tease out some of the elegance of the decimal system. I am not sure whether the task will work with youngish students, though it might be of interest to us as teachers of school mathematics.
The second cube is divided into 4s, then 4s, then 4s.
The third cube is divided into 10s, then 10s, then 10s.
If, for each cube, we now represent the three sets of blue pieces as fractions, it will be easier to add the fractions in the case of the second and third cube, because of the systematic relation between the denominators, and particularly easy in the case of the third cube because of our base 10 number system.
For the first cube, we can express the desired fraction as 2/5 + 1/20 + 3/200. Here we have to think quite hard about a common denominator and about the multipliers needed to make the fractions equivalent. We can re-write the fractions sum as this: (2×40 + 1×10 + 3)/200 = 93/200.
For the second cube, we can express the desired fraction as 2/4 + 1/16 + 3/48. This time it is easier to find a common denominator and the multipliers needed to make the fractions equivalent. We
can re-write the fractions sum as this: (2×16 + 1×4 + 3)/48 = 39/64.
For the third cube, we can express the desired fraction as 2/10 + 1/100 + 3/1000 which we can immediately write as 213/1000.
