The Volume of a Cone

A few days ago, we pulled on a historical thread and wound up browsing the early volumes of The Mathematical Gazette. Doing so, we stumbled across a “mathematical note” from 1896 by Alfred Lodge, the first president of the Mathematical Association. Lodge’s note provides a simple derivation for the volume of a cone. Such arguments don’t vary all that much but, however we missed it, we’d never seen the derivation in the very elegant form presented by Lodge. Here is Lodge’s argument, slightly reworded.

Continue reading “The Volume of a Cone”

The CAS Betrayal

This post will take the form of Betrayal, with a sequence of five stories going backwards in time.


Last year, I was asked by an acquaintance, let’s call him Rob, to take a look at the draft of a mathematics article he was writing. Rob’s article was in rough form but it was interesting, a nice application of trigonometry and calculus, suitable and good reading for a strong senior school student. One line, however, grabbed my attention. Having wound up with a vicious trig integral, Rob confidently proclaimed,

“This is definitely a case for CAS”.

It wasn’t. Continue reading “The CAS Betrayal”

WitCH 70: Troubled Relations

This is not an old one. It is from the 2019 Specialist Exam 2 and comes courtesy of student PURJ, who previously contributed to the discussion here. PURJ noted one, glaring, issue with the question below and its grading, but we think there are other issues as well. As with our previous WitCH, we’ll semi-update with excerpts from the examination report once people have had a chance to ponder and to comment.

Continue reading “WitCH 70: Troubled Relations”

WitCH 64: Decreasing Intelligence

Just in case anybody got the wrong impression and hoped or feared we’d turned over a new leaf, we’ll be posting a number of WitCHes in the next few days. We’ve finally had a chance to look at the 2021 NHT exams (although the exam reports have still not appeared). As usual, the exams are clunky and eccentric, and we’ll be posting a brief question-by-question overview of the exams. But, first, some highlights. Continue reading “WitCH 64: Decreasing Intelligence”

WitCH 58: Differently Abled

Like the previous post, this one comes from Maths Quest Mathematical Methods 11, and is most definitely a WitCH. It can also been seen as a “contrast and compare” with WitCH 15.

Subsection 13.2.5, below, is on “differentiability”. The earlier part of chapter 13 gives a potted, and not error-free, introduction to limits and continuity, and Chapter 12 covers the “first principles” (limit) computation of polynomial derivatives. We’ve included the relevant “worked example”, and the relevant exercises and answers.

A Choice of Difficults

You have a choice. The two questions below come from different exams:

Each question was (arguably) last year’s most difficult exam question on the most difficult mathematics subject in that state. Each question was effectively allocated just under 20 minutes to complete (11/100 x 180 and 13/80 x 120).

Now, you must choose: which question is better, in any sense of the word “better”?


NSW  (Formula Marking guide and sample solution are here.)


VIC  (Briefly discussed here, marking guide and sample solution are in your dreams.)

WitCH 47: A Bad Inflection

The question below is from the first 2020 Specialist exam (not online). It has been discussed in the comments here, and the main issues have been noted, but we’ve decided the question is sufficiently flawed to warrant its own post.

UPDATE (10/09/21) For those who’d placed a wager, the examination report (Word-doc-VCAA-stupid) indicates that a second derivative argument was expected. Hence, thousands of VCE students no longer have any sense of what VCAA means by “hence”.

WitCH 44: Estimated Worth

This WitCH is from Cambridge’s 2020 textbook, Mathematical Methods, Unit 1 & 2. It is the closing summary of Chapter 21A, Estimating the area under a graph. (It is followed by 21B, Finding the exact area: the definite integral.)

We’re somewhat reluctant about this one, since it’s not as bad as some other WitCHes. Indeed, it is a conscious attempt to do good; it just doesn’t succeed. It came up in a tutorial, and it was sufficiently irritating there that we felt we had no choice.