Just for a change, this post will be about a good aspect of new Curriculum. Just kidding. Sort of.
The following is an elaboration and associated content descriptor from Year 8 Measurement:
solve problems involving the circumference and area of a circle using formulas and appropriate units (AC9M8M03)
deducing that the area of a circle is between 2 radius squares and 4 radius squares, and using 3 × radius2 as a rough estimate for the area of a circle
There are two ways one might react to this elaboration. First, one might justifiably have no idea what is the meaning or intent of the elaboration, and then conclude that the curriculum was written by idiots. Or, one could recognise that the elaboration is at least attempting something good but that the attempt was an abject failure, and then conclude that the curriculum was written by idiots. All roads lead to Rome.
For those who have not bothered to decipher the elaboration, the intent is to teach some understanding of the area of circles, and thus also of π, by inscribing and circumscribing squares, as pictured:
If the radius of the circle is R then the outer, purple square has side length 2R and so has area 4R2. Next, by Pythagoras, the inner, red square has side length √2R. So, the red square has area 2R2. These areas are the “two radius squares” and “four radius squares” in the elaboration, and the area of the circle is clearly trapped between them. This is a nice thing to do.
There are significant problems with the elaboration, however, quite apart from its incredibly clumsy phrasing. Firstly and of least importance, the elaboration has essentially nothing to do with the parent content: we are not in any sense solving a problem; we are not using the formula for the area of a circle; and the instruction to use “appropriate” units is distracting and irrelevant. The second problem is that the elaboration fails to state the proper purpose of such calculations, using the trapping of the area of a circle to also trap the value of π; in this instance, π has been trapped between 2 and 4. (For this reason, the exercise is probably better done, at least to begin with, on a unit circle.) The elaboration’s “rough estimate” of 3 is also not attached to the actual trapping. Thirdly, although the application of Pythagoras’ theorem is very nice, it is also not properly possible: Pythagoras only gets introduced later, with content descriptor AC9M8M06.
The general problem is that entire framing of the activity as an exercise in “measurement” rather than abstract geometry buries the much deeper ideas that might be touched upon. The elaboration, properly presented, is a good Archimedes-like exercise in trapping π. But it could also go further, considering other polygons and considering also circumference and perimeters. The elaboration thus extended could lead to a discussion of a more formal conceptualisation of π, and at least some good hints for why “π” is the same for every circle. (We gave it a shot earlier this year, with a strong Year 8 Extension class: the notes are here.) But this is ACARA we’re discussing: it suffices to criticise the elaboration on its own terms.
Finally, it is worth noting that Archimedes is not mentioned in this elaboration, or anywhere in the curriculum. The standard absence of history and humans. There are, however, other elaborations attached to the same content descriptor, and three doors down we have,
exploring traditional weaving designs by First Nations Australians and investigating the significance and use of circles
So, ACARA’s presentation of circle geometry is not entirely devoid of tradition or culture.
Here is a diagram illustrating that Pythagoras can be avoided, as noted in the comment by Franz.