Here we continue to focus on the idea of fraction as quotient or division, in a context where we either split an object into equal parts, or where we scale or 'shrink' an object. Scaling provides a powerful way of moving from operating with natural numbers (where multiplication, for example, might be seen as repeated addition) to operating with rationals. However, this is not to suggest that as pupils become more familiar with scaling they will fully abandon earlier ideas or that such ideas will no longer be useful.
Task 14A: The first diagram below can be thought of as representing the expression 3 ÷ 8, with the second representing (1 ÷ 8) × 3. Pupils who are familiar with the notion of fraction as division, for example through tasks such as those in Week 13, might feel that both expressions are obviously equivalent to 3/8, with nothing more to be said!
Other pupils might argue that if we collect together the three blue 1/8 regions, as in the diagram below, left, then we can see that they align with the yellow region, so this too represents 3/8 of a 1m rod.
A more powerful, purely logical argument, goes something like this:
split the yellow region into 3 equal parts, as in the diagram below, right;
do the same to the other seven adjacent regions (or imagine doing so!);
this means that 3×8 = 24 of these small equal parts will fit into the 3m rod;
and so 24÷3 = 8 of these parts will fit into a 1m rod, and so each part is 1/8 of a 1m rod;
and so the yellow region is 3/8 of a 1m rod, as are the blue regions altogether.
Task 14B: This is similar to the previous task. One can again think of the diagrams as representing 8 ÷ 3 and (1 ÷ 8) × 3, or perhaps 1/8 of 3 and 3/8 of 1. Some pupils will be quite comfortable with these expressions and the idea that they are all equivalent to 3/8. However, it is still worth supporting this with the a geometric argument of the sort given earlier, where the orange region is split into 3 equal parts and each part is shown to be 1/8 of a 1m rod.
It might be useful to present the task with diagrams where the orange and green regions are not aligned, as in the layout below. Here we can't so readily see that the two regions are the same and the idea that they might be, could well give some pupils pause for thought.
Task 14C: Here we introduce a shift in viewpoint. Instead of having a 3m rod (or in this case, a strip of elastic) that is cut into equal pieces (7 in this case), we use scaling (or shrinking) to form the smaller piece. The scale factor that maps the stretched strip onto the original strip is × ⅟₇.
Scaling provides a powerful way of interpreting multiplication and division when this involves rational numbers (fractions), especially when these numbers are smaller than one, when ideas such as repeated addition (for multiplication) and sharing (for division) no longer fit very well, and where ideas such as 'multiplication makes bigger' no longer apply.
However, this does not mean that pupils will every fully abandon these earlier (and more primitive?) ideas as they become more familiar with scaling. There are many situations where these earlier ideas work perfectly well and others where they can coexist with scaling. Regarding the present task, pupils don't have to buy fully into this scaling viewpoint to be able to solve it. They could form a piece the same size as Yella's original strip by cutting a (rigid) version of the 3m strip into 7 equal pieces. This would give a piece of the desired size (3/7 of 1m), albeit with a design that looked slightly different:
Task 14D: Here we use scaling again, but this time the stretched 3m strip is partitioned into two parts, of length 2m and 1m. The scale factor that maps the stretched strip onto the original strip is × ⅟₅.
The original strip is one fifth of the stretched strip's length, so its total length will be 3/5 m and the green and yellow parts will be 2/5 m and 1/5 m respectively.
Task 14E: The scale factor that maps the stretched strip onto the original strip is × ⅟₃. As the stretched strip is more than 3m long, this means the original strip will be greater than 1m, or an improper fraction.
The original strip is one third of the stretched strip's length, so its total length will be 5/3 m (or 1⅔ m) and the red and green parts will be ⅔ m and 3/3 m (or 1m) respectively.
