Week 09b: Sum decimals and fractions

Here we embark on a gentle excursion into adding fractions and compare the effect of expressing numbers as common fractions or decimals.

Task 09A: This is a fairly straightforward task, but it is interesting to see what strategy children use to choose values for the numbers A, B and C, especially at the point where they are asked for another ten examples (a request that might seem rather excessive!).

A simple way of approaching this task is to choose fairly random values for A and B (with A and B positive and A+B<10), and then to derive C by subtracting their sum from 10. However, there are many subtle features that children could employ. For example, they might limit their choice to numbers with one place of decimals; they might allow 0 in that decimal place for some or all of the three numbers; they might try to limit the amount of 'carrying' by choosing decimal parts that sum to 1 rather than to 2; they might form new sets of numbers by re-ordering an existing set, or by modifying the set in a simple and systematic way. Where children have used restrictions that massively simplify the task, you might want to challenge children to work without them.

Task 09B: This is also fairly straightforward, and is of interest mainly for how it compares to later tasks where the decimals are replaced by common fractions.

It is likely that children will initially choose values for A and B that have only one decimal place, as in 0.1 + 0.4. However, they will soon run out of such numbers when asked to 'find ten more'. How readily do they see that they could use numbers with two decimal places, and whose sum is 0.50, as in 0.01 + 0.49?

Task 09C: Some children might find this task a good deal more challenging than the previous task. A neat way to solve it is to find a fraction equivalent to ½ (for example, ⁴⁄₈) and then to split this into two (for example, ⅛ and ⅜). An alternative approach is to choose a value for A and then derive the value for B. Children might discover that this is simpler when the denominator of A is even (as with 7/20) than when it is odd (as with 4/9). Note that the notion of equivalent fractions (and, implicitly, common denominators) is not needed in Task 09B, beyond the idea that 0.5 is equivalent to 0.50, etc.
 

Some children might solve the last part of the task, where A = 4/9, by thinking of ½ as being equivalent to 4½/9, so that B = ½/9 which is 1/18.

Task 09D: This task reinforces the idea that it is easier to find B when the denominator of A is even. It is also worth pointing out that this task would not work well if the fractions were written as decimals, as in Task 09B: most unit fractions would turn out to be recurring decimals (and when A is recurring, B is too).

As children work their way down the table, they might become more fluent in finding the values of B. If so, they might enjoy continuing the table. Children might notice that for a unit fraction A with an odd denominator, the denominator is 2 more than the numerator of B.
 

Task 09E: We open things out slightly by choosing a less familiar value for A+B, namely ⅓. Where previously it was relatively easy to find B when the denominator of A was even, now this only applies when the denominator is a multiple of 3.

Again, it can be fun to continue the table. This time children might notice that for a unit fraction A with a denominator that is not a multiple of 3, the denominator is 3 more than the numerator of B.