Note: GERMAN versions of the Week 14 tasks are available HERE.
This week we start by revisiting some of the tasks from Week 3 but this time we examine how models (such as the double number line and Cartesian graph) can help reveal whether a relationship is multiplicative or additive.
Monday: The double number line is a very powerful representation but it can take a while for students to become familiar with it and to appreciate the significance of the fact that for a multiplicative relationship the two scales are linear and the 0s are aligned.
The given double number line is for Paul's colour. Crucially, in this mixing-paint context, an increment of 1 unit on the white-paint scale does not match an increment of 1 unit on the black-paint scale. Specifically, while 12 is aligned with 7, we can see that 13 is not aligned with 8: for 13 tins of white paint, Vince would need slightly less than 8 tins of black paint to match Paul's colour. So Vince's colour is darker than Paul's.
Do students realise that the given double number line is for Paul's colour? Any pair of vertically aligned numbers, not just 12 and 7, gives us quantities of white and black paint that produce Paul's shade of grey.
Tuesday: Here we use the Cartesian plane to represent the paint mixtures. This is another powerful model, but we shouldn't assume that this power is automatically grasped by students. Using a point to represent a pair of numbers is quite a strange (and remarkable) thing to do. And while the graph produces a very tangible image, one that we can point to, talk about and embellish, we should bear in mind that it is quite abstract - it in no way resembles mixtures of tins of paint!
The line though the origin and the point (12, 7) is the set of points representing quantities of white and black paint that produce Paul's shade of grey. Points above that line, including Vince's point (13, 8) represent darker mixtures.
It can be useful to consider the line through the two given points (12, 7) and (13, 8). Some students might well think that the points on this line represent the same shade of grey. However, this becomes less and less plausible as we consider points nearer and nearer to the x-axis, such as (8, 3), (7, 2), (6, 1) and, perhaps, (5, 0). It seems likely that students will realise that, in some sense, white is becoming more and more 'dominant'!
Further, if we take the mixture represented by (6, 1), say, and combine it with the identical mixture to produce a mixture of the same colour represented by (12, 2), it is plain that this must be lighter than Paul's mixture, represented by (12, 7).
Wednesday: The graph shows that Salvo and José are climbing the steps at the same speed: the line joining the two given points has a slope of 1; as Salvo climbs another 20 steps, so does José. However, we should not assume that all students can read the graph in this fluent way.
We should again bear in mind that the graph is quite abstract and that some students might try to interpret it in a more concrete way: for example, they might see the lower point as representing Salvo’s position and the upper point as representing José’s at a particular time, with perhaps even an imagined zig-zag line of steps joining them.
Thursday: This task involves a geometric context, consisting of two different-sized T-shapes. Are the shapes similar? Is one an enlargement of the other? We use a 'ratio table' to represent the information, although this is perhaps a misnomer since the table per se can not tell us what kind of relationship there might be between the various dimensions. However, the table can be very useful for organising the given information so that one can more easily think about the possible relationships.
It turns out that the same additive relationship holds within each shape's horizontal and vertical limb, whereas there is no consistent multiplicative relationship. So the T-shapes are not similar. Students might notice that the green T-shape is more 'square' than the red T-shape. In a family of T-shapes with this additive relation, the taller the shape, the more square it becomes.
Friday: This task involves the same shapes and the same 2 by 2 table as Thursday's task. This time we look at the relation between corresponding parts of the red and green shapes. Again, we find a consistent additive rather than multiplicative relationship. So again, we can conclude that the shapes are not similar.
