Time for some sustenance! Here we touch on the notion that if we view fractions as ratios, then we can compare them even if they refer to different size wholes. On the other hand, we can't simply add them in such circumstances.
Task 10A: Here Dean starts with a smaller stick of bread than Cath, and ends up with a smaller piece than Cath. However, he has 5/8 of his stick left, whereas Cath only has 7/12 of hers. Can pupils see that 5/8 is greater than 7/12, and that this applies even thought the 5/8 piece is smaller than the 7/12 piece?
1/8 is (slightly) larger than 1/12.
Task 10B: Here we reinforce the ideas in Task 10A by stating the fractional size of the left over pieces, and that the fractions relate to different size wholes.
Task 10C: Here we want to add the left-over parts, but we can't do this by simply adding the fractions 7/12 and 1/8 - why?!
There is scope here for a range of valid answers, depending on what pupils choose as their 'unit'. Some pupils might choose a small square from the grid as the unit, in which case the amount of left-over bread is 12 units. Other sensible units would be the total original length of bread, the length of Cath's original stick, or the length of Dean's original stick. This would lead to answers equivalent to ⅗, 1 and 1½ respectively.
Perhaps the simplest way of getting any of these numbers would be to use the small square as an intermediate unit: this would initially give the fractions 12/20, 12/12 and 12/8 respectively, which can of course all be simplified.
There are other methods. A more formal approach, for example with Cath's stick as the unit, would be to add the given fraction 7/12 to 5/12, where the latter is the amount of Dean's left over bread in relation to Cath's original stick! This could be derived by counting small squares (5 for Dean's left over piece, 12 for Cath's original stick), or (if one wanted to be very formal) by taking 2/3 of Dean's given fraction 5/8. (Why would this work?!)
Task 10D: A neat way of answering this is to say Erin gets ⅓ of the original amount of bread. If we want to measure this in slices, we can say ⅓ of 1⁴⁄₉ slices = ⅓ of ¹³⁄₉ = ¹³⁄₂₇ slices. This is just under half a slice.
A more grounded approach would be to measure the bread in small squares. The original amount of bread consists of 6×9 + 6×4 small squares = 78 small squares. Erin's bread consists of 2×9 + 2×4 = 26 small squares.
26/78 is of course ⅓, the fraction of all the original bread.
26/54 = 13/27, the fraction of one slice.
Visually, we could imagine taking Dean's contribution and placing the two rows of 4 small squares as a single row of 8 beneath Cath's two rows of 9. The result can be seen to be one small square less than half a slice.
Again, there are other, more formal, ways of getting to such an answer. For example, pupils might express the pieces given to Erin as fractions of a slice, and then add them: the piece from Cath is ⅓ of a slice; the piece from Dean is ⅓ of ⁴⁄₉ = ⁴⁄₂₇ of a slice; ⅓ + ⁴⁄₂₇ = ¹³⁄₂₇.
Task 10E: To round things off, this task addresses the same issues as the earlier tasks, in a fairly simple way. Pupils might well be able to see that Erin is wrong: her portion looks to be about a quarter of a pizza, whereas 3/5 is more than a half. In fact, Erin's fraction of a pizza is exactly a quarter - can pupils explain why?
