Note: GERMAN versions of the Week 17 tasks are available HERE.
In this week's tasks we compare fractions with different denominators. Over the years, students will have met grounded methods of doing so, for example methods that involve partitioning a given shape into a carefully chosen number of equal parts. And by now they will also have met formal methods, based on transforming given fractions into equivalent fractions such that the resulting fractions have a common denominator. But even with such formal methods, it remains important that students have secure models of fractions that they can fall back on to support their thinking. So in this week's tasks we revisit the notion of fractions as parts of a whole, modelled by partitioning a shape into equal parts.
Monday: Here we use a paper strip of a certain known length as our whole. This allows us to 'quantify' the whole and to express fractions, and their difference, as quantities (specific lengths) before expressing these as fractions again. The approach is similar to Thursday's task in Week 15, where our whole was £1 and where we expressed various fractions of £1 as amounts in shillings and pence.
For an 18cm strip, it is easier to express ⅟₉ of the strip as a length than ⅛ of the strip. For the 24cm strip in part a) it is the other way round. Finding ⅟₉ of 24cm provides quite a strong test of students' basic understanding, as does the task of expressing the resulting difference, ⅓cm, as a fraction of 24cm.
Tuesday: Here we compare ‘adjacent’ unit fractions, starting with ⅟₈ and ⅟₉. Visually, it is difficult to ascertain their difference so we make use of a more salient comparison involving ⅟₂. In the case of ⅟₈ and ⅟₉, we first compare ⁴⁄₈ and ⁴⁄₉, where it is easy to see that ⁴⁄₈, or ⅟₂, lies exactly midway between ⁴⁄₉ and ⁵⁄₉.
In part b), the first step for Frodo's method is to compare ⁵⁄₁₀, or ⅟₂, with ⁵⁄₁₁. The difference is half of ¹⁄₁₁, or ¹⁄₂₂, so the desired difference is ¹⁄₁₁₀. Students can of course confirm this is correct by using a more formal, equivalent fractions method: ¹⁄₁₀ – ¹⁄₁₁ = ¹¹⁄₁₁₀ – ¹⁰⁄₁₁₀.
Note: This task is highly structured and it could be beneficial, and more satisfying, to loosen it. Thus, rather than simply presenting students with Frodo’s method, for them to make sense of, one could try to help them to construct the method themselves. So one could presnt students with just the diagram and then ask them to use it to find the difference between ⅟₂ and ⁴⁄₉, and then to look for a way of using the result to find the difference between ⅟₈ and ⅟₉.
Wednesday: Here is another way of comparing ⅛ and ⅟₉, but still using a partitioned paper strip to model the fractions. The task is quite highly structured. So initially, you might want to hide the part beneath the diagram, to give students the opportunity to think about the fraction represented by each white region without the prompts contained in that part of the task.
The key to Flo's method is to recognise (be it spontaneously, or through the prompts given in the task) that altogether the white regions in the given diagram cover ⅟₉ of the whole strip, because altogether the 8 tinted regions cover ⁸⁄₉ of the whole strip. The corresponding diagram for ¹⁄₁₀ – ¹⁄₁₁ would show 10 equal tinted regions, altogether covering ¹⁰⁄₁₁ of the whole strip, and 10 equal white regions altogether covering the remaining ¹⁄₁₁ of the whole strip. So one white strip, which represents ¹⁄₁₀ – ¹⁄₁₁, would cover ¹⁄₁₁₀ of the whole strip.
Thursday: Here we use a circular disc as our whole, rather than a rectangular strip. Apart from that, the method used to find the desired fraction is the same as Flo’s method in Wednesday’s task.
Students are told that each yellow region shows the difference between ⅕ and ⅙ of the disc. Can they see why? And can they see that altogether the yellow regions cover ⅙ of the disc?
Friday: This shows a variant on Thursday's task that is slightly more general and perhaps slightly more demanding. We are told that each white region covers ⅟₇ of the circular disc. But there are only 5 such regions rather than 6, so altogether the 5 yellow regions cover two- rather than one-seventh of the disc, and so each yellow region covers ²⁄₃₅ of the disc.
It can be useful and illuminating for students to find ways to check their answer.
One approach would be formally to calculate ⅟₅ – ⅟₇ using the common denominator 35.
A more grounded version of this would be to express the regions in the diagram as 35ths: we have found that a yellow region represents ²⁄₃₅ of the disc; a ⅟₇ region can be written as ⁵⁄₃₅; if we add these regions, do we get ⅟₅ of the disc?
