Tag: indices

MitPY 13: Trigonometry and Wolfram Alpha

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March 25, 2021

9:05 am

11 Comments on MitPY 13: Trigonometry and Wolfram Alpha

MitPY

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This MitPY comes from frequent commenter, John Friend:

Dear Colleagues,

I gave a CAS-FREE question to my Specialist students whose first part was to solve (exactly) the equation \boldsymbol{\cot 2x = \sec x}. I solved it two different ways and got two different answers that are equivalent. I’ve attached my calculations.

I checked my answers using Mathematica, which lead to my question: Mathematica gives a third different but equivalent answer (scroll down to real solutions). How has Mathematica got this answer?

It may be that Mathematica ‘used’ my Method 2, got my tan answer and then for some arcane reason ‘manipulated’ this answer into the one it finally gives. If so, I can ascribe the answer to a Mathematica quirk. But it may be that Mathematica is using a method unclear to me that leads to its answer. If so, I’m curious.

Any thoughts are appreciated.

Click to access Calculations.pdf

MitPY 11: Asymptotes and Wolfram Alpha

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January 31, 2021

6:23 pm

5 Comments on MitPY 11: Asymptotes and Wolfram Alpha

MitPY

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This MitPY comes from frequent commenter, John Friend:

Dear colleagues,

I figured this was as good place as any to ask for help. I’m writing a small test on rational functions. One of my questions asks students to consider the function \displaystyle f(x) = \frac{x^3 + x}{x^2 + ax - 2a} where a \in R and to find the values of a for which the function intersects its oblique asymptote.

The oblique asymptote is y = x - a so they must first solve

\displaystyle \frac{x^3 + x}{x^2 + ax - 2a} = x - a … (1)

for x. The solution is \displaystyle x = \frac{2a^2}{(a+1)^2} and there are no restrictions along the way to getting this solution that I can see. So obviously a \neq -1.

It can also be seen that if a = 0 then equation (1) becomes \displaystyle \frac{x^3 + x}{x^2} = x which has no solution. So obviously a \neq 0.

When I solve equation (1) using Wolfram Alpha the result is also \displaystyle x = \frac{2a^2}{(a+1)^2}. But here’s where I’m puzzled:

Wolfram Alpha gives the obvious restriction a + 1 \neq 0 but also the restriction 5a^3 + 4a^2 + a \neq 0.

a \neq 0 emerges naturally (and uniquely) from this second restriction and I really like that this happens as a natural part of the solution process. BUT ….

I cannot see where this second restriction comes from in the process of solving equation (1)! Can anyone see what I cannot?

Thanks.

MitPY 10: Square Roots

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December 2, 2020

8:19 pm

18 Comments on MitPY 10: Square Roots

MitPY

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This MitPY comes from a student, Jay:

I have a question relating to polynomial equations. For context I have just finished Y11 during which I completed Further 3&4, Methods 1&2 and Specialist 1&2.

This year during my maths methods class we covered the square root graph, however I was confused as to why it only showed the positive solutions. When I asked about it I was told it was because the radical symbol meant only the positive solution.

However since then I have learnt that the graph of \boldsymbol{y=x^{0.5}} also only shows the positive solution of the square root, while \boldsymbol{y^2=x} shows both. I am quite confused by why they aren’t the same. The only reason that I could think of is that it would mean \boldsymbol{y=x^2} would be the same as \boldsymbol{y^2=x^4}, and while the points (-2,-4) and (2,-4) fit the latter they clearly don’t fit former.

Could you please explain why these aren’t the same?

Signs of the Times

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May 5, 2019

4:02 pm

25 Comments on Signs of the Times

Numbers

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Our second sabbatical post concerns, well, the reader can decide what it concerns.

Last year, diagnostic quizzes were given to a large class of first year mathematics students at a Victorian tertiary institution. The majority of these students had completed Specialist Mathematics or an equivalent. On average, these would not have been the top Specialist students, nor would they have been the weakest. The results of these quizzes were, let’s say, interesting.

It was notable, for example, that around 2/5 of these students failed to simplify the likes of 81-3/4. And, around 2/3 of the students failed to solve an inequality such as 2 + 4x ≥ x2 + 5. And, around 3/5 of the students failed to correctly evaluate \boldsymbol {\int_0^{\pi} \sin 5x \,{\rm d}x}\, or similar. There were many such notable outcomes.

Most striking for us, however, were questions concerning lists of numbers, such as those displayed above. Students were asked to write the listed numbers in ascending order. And, though a majority of the students answered correctly, about 1/4 of the students did not.

What, then, does it tell us if a quarter of post-Specialist students cannot order a list of common numbers? Is this acceptable? If not, what or whom are we to blame? Will the outcome of the current VCAA review improve things, or will it make matters worse?

Tricky, tricky questions.

What is this Crap Here?

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May 29, 2018

6:03 pm

19 Comments on What is this Crap Here?

WitCH

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OK, Dear Reader, you’ve got work to do.

So far on this blog we haven’t attacked textbooks much at all. That’s because Australian maths texts are, in the main, well-written and mathematically sound.

Yep, just kidding. Of course the texts are pretty much universally and uniformly awful. Choosing a random page from almost any text, one is pretty much guaranteed to find something ranging from annoying to excruciating. But, the very extent of the awfulness makes it difficult and time-consuming and tiring to grasp and to critique any one specific piece of the awful puzzle.

The Evil Mathologer, however, has come up with a very good idea: just post a screenshot of a particularly awful piece of text, and leave others to think and to write about it. So, here we go.

Our first WitCH sample, below, comes courtesy of the Evil Mathologer and is from Cambridge Essentials, Year 9 (2018). You, Dear Reader, are free to simply admire the awfulness. You may, however, go further, and what you might do depends upon who you are:

  • If you believe you can pinpoint the awfulness in the excerpt then feel free to spell it out in the comments, in small or great detail. You could also offer suggestions on how the ideas could have been presented correctly and coherently. You are also free to ponder how this nonsense came to be, what a teacher or student should do if they have to deal with this nonsense, whether we can stop such nonsense,* and so on.
  • If you don’t know or, worse, don’t believe the excerpt below is awful then you should quickly find someone to explain to you why it is.

Here it is. Enjoy. (Updated below.)

* We can’t.

Update

Following on from the comments, here is a summary of the issues with the page above. We also hope to post generally on index laws in the near future.

  • The major crime is that the initial proof is ass-backwards. 91/2 = √9 by definition, and that’s it. It is then a consequence of such definitions that the index laws continue to hold for fractional indices.
  • Beginning with 91/2 is pedagogically weird, since it simplifies to 3, clouding the issue.
  • The phrasing “∛5 is irrational and [sic] cannot be expressed as a fraction” is off-key.
  • The expression “with no repeated pattern” is vague and confusing.
  • The term “surd” is common but is close to meaningless.
  • Exploring irrationality with a calculator is non-sensical and derails meaningful exploration.
  • Overall, the page is long, cluttered and clumsy (and wrong). It is a pretty safe bet that few teachers and fewer students ever attempt to read it.