Led by James B. Thomson Simulacrum
About this course
You have met fractions twice already without being told. In Module 2, a division left a remainder and the question of what to do with it was deferred to this module by name. And in the same module: when a thing is divided into equal parts, the parts take their name from the number of parts into which the thing is divided — which is the definition of a fraction, delivered in the middle of a division sitting. In Module 3 you built two instruments, the greatest common divisor and the least common multiple, and were told at the end what they were for. This is what they were for.
The difficulty of fractions is not the procedures. Each of the four operations is a short rule you could memorise in a minute. The difficulty is that a fraction is not a number in the way you have been using the word — it is a relation between a quantity and a unit, and its value depends on two things at once: the size of the unit divided, and the number of parts it was divided into.
So the module opens with no figures at all. Thomson's instruction is that beginners should divide an actual object — a sheet of paper, an apple — into halves and thirds and fourths, and then put the parts back together. This course sets that as work, and it is the one place in the whole school where you are asked to do something off the screen. A score of pictures and an hour of talk will not do what one folded sheet does.
Everything procedural then follows from five statements Thomson gives in a single paragraph. Two of them are counter-intuitive and are where students go wrong; one of them is the whole of reduction, of common denominators, and of why cancellation worked in Module 3. A student who holds the five can rebuild any rule in the module from scratch. A student who has only the rules has nothing to rebuild from — which is what this module exists to prevent.
Led by James B. Thomson Simulacrum
The question
Eleven sittings, Thomson's pages 88 to 128. You begin by folding
Outcome
The student can reduce any fraction to its lowest terms or to a
Sub-units
Practice scenarios
A pupil arrives from another school able to add, subtract, multiply and divide fractions correctly and quickly. She inverts and multiplies without hesitation. She has no idea why any of it works, and when asked, says that this is how it is done, which is what she was told. Her answers are right. She sees no problem, and neither does anyone else.
Your goals