[Note: GERMAN versions of the Week 11 tasks are available HERE. I welcome suggestions on how to improve the translations!]
This week's tasks might seem rather strange and contrived. From a calculating point of view, there is not much to be gained from writing a number as a sum if the aim is to factorise it. On the other hand, the tasks do give students useful experience of the distributive law, which can play an important role when multiplying numbers. The tasks should also help students develop their number sense through thinking about numbers and their factors.
Note: The number 78 was chosen in honour of Harrison Ford, who at the time of writing had just reached the age of 78.
Monday: This introduces the week's basic task - writing a number as a sum that can be factorised. At some point, it should emerge that this only works productively when the original number and the two numbers forming the sum have a common factor (apart from 1, of course).
Students might discover that there are families of sums that 'work', such as 72+6, 66+12, 60+18, etc (where the [largest] common factor is 6 each time) or 65+13, 52+26, 26+52, 13+65, where the common factor is 13 each time.
Tuesday: This is similar to Monday's task except that we write our number as the difference of two numbers rather than the sum.
Wednesday: Here we focus attention on the property of numbers that do or don't 'help'. If they haven't done so already, students should begin to realise that the 'nice' numbers have a common factor with the given number, 78.
Thursday: Here we consolidate the point made in Wednesday's task, by choosing a starting number that is probably not very familiar to students and that has few factors (91 = 7×13).
Two simple sums that could help Len are 91 = 7 + 84 and 91 = 13 + 78. We can find other sums that work by adding/subtracting multiples of 7 to the terms of 7 + 84, and by adding/subtracting multiples of 13 to the terms of 13 + 78.
Friday: More examples of unfamiliar numbers - some with easy to spot factors, some without.
