[Note: GERMAN versions of the Week 3 tasks are available HERE. I welcome suggestions on how to improve the translations!]
This week's tasks focus on the Addition Strategy, a common misconception where students impose an additive relation instead of a multiplicative one on quantities that are in proportion.
Student's propensity to adopt the strategy seems to depend both on the quantities involved and on the context. Students seem to be particularly susceptible with tasks involving similar figures and where the difference in the given quantities is small, as in the 'extra' task below. [This was originally intended for Wednesday's task, before we decided on having tasks that were more closely connected.]
We revisit some of the tasks in Week 14 where we examine the affordances of the double number line, Cartesian plane, and ratio table.
Monday: This task involves a mixture, in this case of white and black paint. A useful feature of mixtures is that they can be approached qualitatively as well as quantitatively. So here we can consider whether the paint mixture becomes lighter or darker.
In this task, Vince's paint mixture contains 1 more white tin and 1 more black tin than Paul's mixture. Some students will interpret this as indicating that the colour hasn't changed. Other students might come to the same conclusion on the basis that for each mixture the difference between white and black paint is the same (5 tins).
Tuesday: Here we have a subtle variant of Monday's task. Theo changes Paul's mixture into Vince's mixture, but in the process he mixes equal amounts of white paint and black paint, which gives students the opportunity to think about the colour of this 1:1 mixture. Can it be the same as Paul's 12:7 mix?
Do students attend to Theo's initial 1:1 mixture? If so, do they realise that it will be darker than Paul's mixture and that it will therefore make Paul's mixture darker?
Wednesday: Here we have a different context, but the quantities are again increased by just 1 unit. Additive relations between the quantities are again much simpler than multiplicative relations, so some students might be attracted to them, leading to the conclusion that $14 is the right price.
It is interesting to note that the addition strategy will lead to the result $14, whether students use the within price-labels relation +3 or the between labels relation +1. Of course, the within and between multiplicative relations (×1.3 and ×1.1 respectively) also lead to the same quantity, in this case $14.30.
Note: One could argue that the shop only prices its goods in whole numbers of £ and $, in which case $14 would be a good option. Indeed, it would be the best whole-number option, even though we don't know where in the range 12.5 to 13.5 the 'true' dollar price would be for £10.
Thursday: This task gives students who used the addition strategy on Wednesday's task, the opportunity to probe it more deeply.
Friday: As students begin to become more adept at thinking multiplicatively, they might sometimes interpret situations as multiplicative when they are not. This can happen here, where the relation between the step-numbers is additive rather than multiplicative (Salvo is always 10 steps behind José but climbs steps at the same rate).
