Free Help: Maths in the Plague Year

I don’t really know if or how this’ll work, but I figure it’s worth a try. While you’re all locked at home in your individual countries/cities/houses/rooms, you may request help here on any maths problem, of any level: just ask your question in a comment on this post.

God knows what will happen, but I will do my best to give you some guidance in a reasonably prompt manner (within a day-ish).* Others are of course free to offer help, and if they do so then I will try to ensure any subsequent discussion progresses naturally and helpfully.

A couple quick points:

  • Do your best to ask the question briefly but clearly, and indicate why you’re asking it.
  • Hopefully LaTeX works in the comments (try $ latex [Your LaTeX code] $ ).
  • If the question is small and easily resolved then the discussion can stay on this page; for more involved questions, I’ll create a separate post for the discussion.
  • Please ask new (unrelated) questions in new comments, rather than replies to existing comments.
  • My approach to this kind of teaching is to be pretty Socratic, to try lead a student to the answer, rather than just providing the answer. So, don’t be surprised if you’re asked to go away and ponder some specific aspect of the question.
  • I don’t particularly care if the question comes from an assignment or whatever, though I prefer honesty on this point. (And, the more I suspect the question is somehow officially assigned work, the more Socratic I’m likely to be.)
  • No CAS garbage, in either the questions or the replies. This will be ruthlessly enforced.

Ask away.

*) The Riemann hypothesis may take a little longer.

UPDATE (25/03/20) Here is MitPY 2 (change of base for logarithms).

UPDATE (28/03/20) MitPY 2 is done and dusted. Any offerings for MitPY 3?

UPDATE (03/04/20) MitPY 3: What to teach at the beginning on Year 7.

UPDATE (05/04/20) MitPY 4: Motivating vector products.

UPDATE (06/04/20) MitPY 5: Implied domain and range.

UPDATE (18/04/20) MitPY 6: Integration by substitution.

UPDATE (15/06/20) MitPY 7: Teaching Diophantine equations.

UPDATE (11/09/20) MitPY 8: VCE code words for proofs.

UPDATE (20/09/20) MitPY 9: Coping with colleagues.

94 Replies to “Free Help: Maths in the Plague Year”

    1. I’m happy to lend a hand too. Am currently exploring the great unknown of remote teaching …. (This will be the silver lining that many educational bureaucrats see in all this mess – the potential saving of huge amounts of money. Because it will appear that 2 teachers can do the job of 8 ….)

      Latex check: x^2 - 2x + 1 = 0.

  1. Generous offer, Marty.

    I hope we can all see the questions asked and your solutions.

    You never know, we may learn something. 🙂

    1. Thanks, Geoff. The idea is that it’s all public. As it happens I help all manner of people privately on occasion, plenty of whom I’ve never met. I generally never refuse any good faith request for help. But I can’t make a public offering of private help for all and sundry.

    1. Hi, SRK. I’m not sure whether you’re being facetious, but the basic answer is “yes”. If the question is along the lines of “What is going on with this topic/example?”, that is definitely on-topic. Many of the posts on this blog were inspired by some teacher (or student) asking such a question. If you’re talking about how one should teach a topic, for me that’s the same question. I think pretty much you can teach anything well if and only if you (really) understand what’s going on with it.

      1. No facetiousness intended. I guess I was not merely wondering about questions about teaching a specific topic, but broader questions about teaching certain courses (given that one must comply with certain requirements) or – and this seems especially pertinent at the moment – how one might effectively deliver a course remotely.

        1. Whoops. I thought I replied to this. SRK, I’m more than happy for you, or anyone, to post any such questions. I won’t be of any help whatsoever, but I’m happy for this blog to be a forum for these issues.

      2. For what can happen if one does not understand what one is talking about, go take a squiz at the Absolute Zero post…

        1. I don’t know whether it’s a problem just with my computer, but the font on these posts is very small and very faint, which makes it very hard to read particularly in dulllish light.

          Any chance of increasing the font size and brightness please?

          1. Hi Geoff, the font looks ok to me, but if others agree with you I’m happy to look into it, or into anything people think will improve the appearance or function of the blog. (The LaTex equations are fuzzy, but I have no idea what to do about that.) I took a quick look and it seems one can alter the blog “Theme”, but I’m pretty sure I’m not smart enough to go tramping around there.

  2. In last millennium C that would require some typing just to print fish or hello world…

    #include <studio.h>
    int main() {
    printf (“fish”) ;
    return 0;
    }

    Then you could experiment with the soundex algorithm

    Steve R

  3. OK. I’ll try to get a specific ball rolling and perhaps one that is slightly relevant at the moment and not really that well understood by my teacher colleagues (even some that teach mathematics classes) – how would you (Marty or anyone) go about teaching the change of base rule for logarithms to (say) a mainstream year 10 or extension year 9 class?

    1. Thanks, RF, for choosing a truly awful topic. It’s one I always dread when tutoring. The main issue with the topic is, who gives a shit? As presented in school it’s just a thing, with no purpose other than to test whether students know the thing. True idiocy. That purposelessness also makes it really hard to teach, because there is no easy value in going into depth. Nothing in VCE hinges on understanding why the change of base works.

      I think this will be thorny enough to give it a separate post. So, I’ll do that now, and then come back with the link.

      1. Thanks Marty – I will elaborate on the other post.

        SRK – I’m thinking of a hypothetical class (for now) but my thinking is that they would have been taught indices and logarithms and the change of base rule just sort of rounds off the topic in a nice way, provides an opportunity for some sort of learning by discovery and is well within the grasp of a student of reasonable ability to comprehend.

        Except that all other logarithm laws have an index law “partner” so this one just sticks out a bit.

            1. RF: Perhaps I’m misunderstanding, but I think Glen’s point is that they are more or less the same, since the change of base rule follows very quickly from that log law. (Indeed, that’s more or less how I teach the change of base rule, just as an application of that log law).

      2. Sorry – I posted before I read your reply.

        Teaching students the other day about log scales, I wondered out loud how I could do an experiment in measuring sound levels in the class. One student told me that we could do it on her watch. I shouted “Hey” and she recorded 70dB. Then I tried “HEY” and got to 75dB. Dick Tracy would be impressed.

    2. y=log_{a}x MEANS x=a^y (1)

      If you understand x=a^y then you understand y=log_{a}x .

      (“Terry is Alice’s brother” MEANS “Alice is Terry’s sister”.)

      Many high-school problems about logs, including the change of base formula, can be readily solved with (1).

      One aspect of showing the change of base rule to students is that they see (1) in action.

      1. Thanks Terry, I quite like this fundamental, reinforce the fundamental idea approach. If nothing else, as you say it gets students to keep seeing (1) before they hit VCE.

          1. Hi, JF and RF. I’ll try to find time to update MitPY2 to cover this point but, in brief, I think there are three places to start. The worst place to start is to just accept logarithms as a thing, and probably some previous log rules as a thing, and fumble your way through. One sensible alternative, and really the only sensible alternative for the vast majority of school kids, is to take Terry’s line, to think of any logarithm statement as really an exponential statement. That approach leaves theoretical gaps and, in the end, a third and more sophisticated approach works out better. I think it’s this third approach (which RF has seen me hammer) that RF is preferring, but here I think RF is expressing preference for the Terry approach over the fumble-through approach.

    1. Thanks, SR. I think you’ve pointed out that chapter before, and it’s great, from a masterpiece. Every maths teacher should read it and study it. But, it’s probably not direct enough for the specific question RF is asking.

    1. Thanks Steve, not looking for examples, just a way to get the big idea across in a meaningful way (which none of the school textbooks seem to be able to do without saying “here is the rule, now just use it”)

  4. Marti,

    sorry I am a statistician … so we may get closer to the right level by doing some of Paul’s practice questions using the change of base recipe whilst the teacher reviews sections 1 and 2 of Ch22 of the Feynmann lecture series.

    BTW I can remember using log e 10^n conversions tables in a pre calculator era

    Steve R

    1. Well, SR, stats has never been my thing … More seriously, you are of course correct, that specific examples of how things work is helpful, if not essential to the understanding. But it doesn’t get to why the rule works. So, a more pertinent use of example is as suggested by Storyteller on MitPY2.

    1. Geez, that’s weird. Thanks for reminding me how much I hate CAS. Craig, maybe indicate what machine/platform you’re using, and others may be able to say why the second, pretend-complex answer is appearing. (I’ll loosen the “No CAS” rule to permit such questions here.)

    2. I’d say your calculation mode is on some sort of complex number mode. If it’s set to real numbers only, then CAS just returns a warning of a non-real result when you enter the second expression.

        1. I promise to comment no more about CAS if this breaks Marty’s rule…

          …but in my experience the CAS does not treat -1 as a number in itself but as the number 1 with the negation operation performed on it. It then gives the exponent higher precedence in the order of operations (which I know Marty also hates… USBB) than the negation.

          Excel does the opposite in my (limited) experience.

          1. Regardless of how CAS treats this, this is an area where students often have difficulty, and I’m curious how others approach teaching this. Suppose we have f(x) = x^2 - 1 and we want to find f(-1). Often students will write -1^2 -1 = -2. So even if we charitably interpret the student’s writing at the first stage, students then proceed incorrectly. I’ve found that it’s best to insist that students write (-1)^2 -1

            1. I agree with the brackets approach; another approach I’ve used with this exact problem is to instead have them write

              f(x) = x\times x - 1

              first, and then when they compute f(-1), usually a little light turns on behind the eyes. I like that.

              1. Thanks Glen, that seems like a nice way to make the point. I did have a pang of pessimism: some of my students tend to interpret arguments like that far too procedurally and I worry about what they might do if given something like f(x) = x^{32}-1

        2. Marty, yes (modulo the issue RF picked up on below), but I was just commenting on how this works on CAS. If I set my TI (which is the brand Craig is using) to the “Rectangular” complex number calculation mode, then I replicate what Craig found. If I use the “Real” calculation mode, the result is 5^{-n}\times(-1)^n with a warning of a non-real result. (And, FWIW, if I use the “Polar” calculation mode, CAS gives \latex e^{i n \pi}\times 5^{-n})

    3. Hi, Craig and thanks to SRK and RF and Damo. I guess there are two issues (not counting the third issue, which is that I really don’t give a shit …). The first issue is why Craig’s two expressions returned different results, and perhaps the answer to that is what RF is suggesting, that the numerator is being read as -(1^n). The second issue is why the second expression has that (possibly) complexy stuff in it, and I think that’s what SRK and Damo are pointing at: If n is not an integer then the result will be either complexy or warning-sign, depending upon the machine mode.

      1. Re: The second issue. For what it’s worth, I quote from a paper I wrote some years ago:

        “The reason for these problems lies in the way in which …. cube roots …. are calculated. For example, when the TI-89 calculator is operating in real mode it uses the real branch (when it exists) for fractional powers that have a reduced exponent with odd denominator. When operating in complex mode, or when the real branch does not exist, the TI-89 uses the principle branch. Thus, for example, when operating in real and complex modes the TI-89 returns:
        ….
        \bullet the real and principle values -1 and \displaystyle{\frac{1}{2} + i \frac{\sqrt{3}}{2}} respectively for
        \displaystyle{(-1)^{\frac{1}{3}}} (a real value exists and is different to the principle value).”

        I suspect a similar explanation applies for the second issue ….

  5. Marty (and others) an idea if I may for MITPY3: If you were teaching a (very) mixed ability Year 7 class in their first term of secondary school and had COMPLETE control over the curriculum, what would you start with as the first topic/lesson sequence.

  6. Hi

    Interested to know how other teachers/tutors/academics …give their students a feel for what the scalar and vector products represent in the physical world of R2 and R3 respectively.

    One attempt explaining the difference between them is given here

    Click to access dotcross.pdf

    The Australian curriculum gives a couple of geometric examples of the use of scalar product in a plane Around quadrilaterals,parallelograms and their diagonals .

    https://www.australiancurriculum.edu.au/senior-secondary-curriculum/mathematics/specialist-mathematics/?unit=Unit+1

    Regards
    Steve R

    1. Hi Steve!

      Good question. I don’t like the first link so much because that seems to have already given up on imparting an understanding of the idea and resigned itself to teaching efficient memorisation.

      I do the dot product first and I work toward the definition using the angle and unit vectors. Then, I give the extension to non-unit vectors. I prove the more usual definition then as a computational device.

      For the cross product, I define this by first asking the general question of: given two unit vectors X and Y in \mathbb{R}^3, how can we canonically choose a third vector Z such that Z is orthogonal to X and Y? Once this canonical choice is made, we do some examples, the extension to non-unit vectors, and of course take this to be the definition. Again, the “determinant” mnemonic is treated as a computational device.

    2. Thanks, Steve. I’ve put up a MitPY 4 post for your question here. Glen, maybe repeat your comment here as a comment on that post?

    1. Thanks, Michaela. I’ll set up a post. Before I do, could you please indicate if you’re a student or a teacher, and at what year level? (I don’t care which, but it will help people to frame their answers.)

        1. OK, done here! Maybe, you could comment early on the post, indicating an example or two that confuse you. Also, I’m a bit snowed and so won’t comment immediately. But, given you’re a student, I’ll watch carefully to see if the regular commenters are replying, and helpfully. If they’re suggesting things that don’t help, please don’t hesitate to tell them!

  7. An issue with on-line teaching in mathematics is that mathematics depends on symbols. Writing mathematics is not easy on-line. Although LaTeX has made a wonderful contribution to writing mathematics, it has not caught on among teachers. I have often seen teachers struggling with writing mathematics for assignments and tests. Perhaps this is why using tests and assignments provided by publishers is so popular among teachers.

    1. It’s going to be a substantial problem with running mathematics assessments remotely, and I think it will be a problem for students moreso than for teachers. Putting authentication issues aside, for a lot of subjects, remote assessment can be done by asking students to type answers into a form. This is not feasible for mathematics. And even if there is some software available in which students can enter mathematical symbols, almost all of them will require training on how to use it.

      I’m speculating, of course, but I suspect a widely adopted solution will be to simply have students photograph written work and upload it to some sort of learning management system (Compass, Office, Moodle, etc.)

      1. Yes, photographs of written work is what I’m going with as a baseline. Although the students I have are taught LaTeX in their first year, so we could in theory ask them to LaTeX all of their work. (We only ask that certain things are TeX’d.)

  8. A question for commenters: how to explain / teach integration by substitution? To organise discussion, consider the simple case \int \frac{2x}{1+x^2}dx Here are some options.

    1) Let u = 1 + x^2 This gives \frac{du}{dx} = 2x, hence dx = \frac{du}{2x}. So our integral becomes \int \frac{2x}{u}\times \frac{du}{2x} = \int \frac{1}{u}du. Benefits: the abuse of notation here helps students get their integral in the correct form. Worry: I am uncomfortable with this because students generally just look at this and think “ok, so dy/dx is a fraction cancel top and bottom hey ho away we go”. I’m also unclear on whether, or the extent to which, I should penalise students for using this method in their work.

    2) Let u = 1 + x^2 This gives \frac{du}{dx} = 2x. So our integral becomes \int \frac{1}{u}\times \frac{du}{dx}dx = \int \frac{1}{u}du. Benefit. This last equality can be justified using chain rule. Worry: students find it more difficult to get their integral in the correct form.

    3) \frac{2x}{1+x^2} has the form f'(g(x))g'(x) where g(x)=1+x^2 and f'(x) = \frac{1}{x}. Hence, the antiderivative is f(g(x)) = \log (1+x^2). This is just the antidifferentiation version of chain rule. Benefit. I find this method crystal clear, and – at least conceptually – so do the students. Worry. Students often aren’t able to recognise the correct structure of the functions to make this work.

    So I’m curious how other commenters approach this, what they’ve found has been effective / successful, and what other pros / cons there are with various methods.

  9. Hi, Marty and friends.

    If one needed to convert an object given by a vector say \textbf{r}(s) = \sin(2t) \textbf{i} + \cos(2t) \textbf{j} into cylindrical coordinates, I’m confused as to what to do. Normally if there is an x for example I can plug in \rho \cos\theta but I’m unsure what to do with a parametric function with t as the variable. All I have is \textbf{r} = [basis vector stuff] but here \textbf{r} is what the function is called. Thanks in advance for any help. I hope all that LaTeX works out okay. (I’m having trouble moving the \textbf{i} and \textbf{j} away from the sin and cos. Guess I’ll have to usbb.)

    1. Hi, Craig. Just to make sure, do you really have components \sin 2t and \cos 3t? The different arguments makes the question weird.

    2. Hi Craig!

      You know r here is just the unit circle, right? So in polar coords it would be r = 1 (note that this is a different r).

      1. Thanks, Glen. Are polar coordinates the 2-D analogue of cylindrical coordinates? If so, do I just use those even when it says to convert it to cylindrical coordinates? I should also say it asks for spherical coordinates also, but I’m unsure what the 2-D analogue of that would be.

        1. Hi Craig, I’ll try not to get in the way, but cylindrical coordinates is just polar coordinates with a z thrown in. So, you’d either ignore z here or just put z = 0. But, the exercise is not well expressed, and I can think of a couple different things they might be wanting here.

          1. I’d prefer anyone to ‘get in the way’ as much as possible. I replied to John below with the exact original question, that might clarify what exactly I’m asking. If you have seen that, then if you’re unsure I don’t know how I can ever be sure what to answer.

            1. OK then I’ll get in the way a little more.

              So, let’s ignore z (or think of z = 0). Then, as Glen suggested, polar coordinates is thinking of points in the x-y plane in terms of the distance r from the origin, and an angle \theta. Usually (but definitely not always) we choose \theta to be the angle “made” with the x-axis. Then we have the conversion formulae

              x = r\cos \theta and y = r\sin \theta.

              So, you’re then trying to figure out the r and the \theta in terms of t. The r is fine: that’s pretty obviously r = 1, as Glen indicated. (There’s more one can say but I won’t say it here).

              Is that clear so far? If so, we can talk about the t and the theta, which is where what the question intends is not so clear.

              1. It’s clear that r = 1 (it’s a circle), but I guess my point of confusion is I’ve always used \rho, \theta, and z as the ‘format’ of cylindrical coordinates, and I’m not sure where to get it from sin and cos. So if x = cos t in this case, the transformation formula sheet says that x = \rho cos t, so that means that cos t = \rho cos t and then…. I’m lost.

              2. The question talked about converting a vector. As I remarked earlier, a vector expressed in cartesian coordinates is being converted into a vector expressed in cylindrical coordinates. So the final answer has to be a vector. r = 1 is a scalar and so cannot be the final answer – there is more to be said.

                It looks to me like the question is interpreting the vector as the coordinates of a point and people are talking about what the coordinates of this point are when expressed in cylindrical coordinates.

                Can we please get some clarification – are we talking about a vector or a point??

                Edit: OK, I’ve just read the clarification from Craig. We’re dealing with a point that is specified by a position vector. So we’re changing the cartesian coordinates of a point into cylindrical coordinates, spherical coordinates etc. I think …?? There still seems some ambiguity (in the question) to me.

                1. Hi John,

                  I’m about 60% sure that the asker is intending for students to work out the equations of motion for a frame in a mechanics subject — but I’ll let Craig give any further clarification.

                  I just wanted to reply here and say that although the equation r = 1 looks like an equation for a scalar, in the context of this comment, it is shorthand for the set of vectors (or points) (cos\theta, \sin\theta, 0) where \theta\in[0,2\pi). That’s why I called it a circle.

                  However, since Craig gave a more complete description of the question below, this is basically irrelevant to the task of actually helping Craig, so my comment is becoming close to worthless, and this one even less so :).

                  Cheers

                2. Sorry for all the confusion. I’m confused as well. But I think I’ve figured out that I need to use basis vectors in the conversion. A listed basis vector in the appendix is \textbf{e}_\rho = \cos(\theta)\textbf{i} + \sin(\theta)\textbf{j} which is the one I think I’m meant to use. But I’m confused if I can still do it even though t is the argument and not \theta.

    3. I’m confused by your statement of the question. Can you please can post the exact original question.

      You have a vector in the x-y plane defined in cartesian coordinates. Cylindrical coordinates are a three dimensional set of coordinates that are an alternative to xyz-coordinates and often used when there is an underlying cylindrical symmetry (in order to exploit that symmetry).

      In your case you are in two dimensions, in which case talking about cylindrical coordinates makes no sense – you would be talking about using polar coordinates (to exploit the underlying circular symmetry).

      And since you have a vector it’s not a simple matter of saying r = 1 because this is a scalar. You need an answer that is a vector, in which case the answer is 1r where r is a radial unit vector.

      But please, post the exact original question and the level of the question (which I suspect is not a secondary school level question). Knowing the level of the question is important so that we know what knowledge and skills to assume.

      1. Thanks for responding, JF. It’s a question from 2nd year university.

        Here’s the exact wording of the question. “Consider an object whose position is given by \textbf{r}(t) = R \cos(t)\textbf{i} + R\sin(t)\textbf{j}. Convert \textbf{r}(t) into cylindrical coordinates.” It then asks for \rho'(t), \theta'(t), z'(t). As you noted this is a 2-D problem, I assume that you hold the coefficient of the “k vector” to be 0.
        Might not matter, but the vectors in the question were represented by tildes under the ‘letter’.

        1. OK, I missed this before. Note that the cos t and sin t are in different spots from what you wrote earlier, which makes a massive difference. I still don’t like the wording of the question, but I think the question is clearer now.

          The coordinates of the object at any time t are

          x = R\cos t and y = R\sin t and z = 0

          So, then how do you find \latex, \rho, \theta, z, all as functions of t?

          I’m not sure of your difficulty here.

          1. So I set x=\rho\cos\theta\ = R\cos t and y=\rho\sin\theta\ = R\sin t and then I’m unsure what to do next. Am I doing that part right at least? Or do I use basis vectors? And I’m confused as to where i and j go, do you have to keep it as a vector equation?

            Just want to say also that I think the answer should written like
            \textbf{r}(t)_{cyl} = ...
            if that’s any help.

            1. R is a constant. You want rho and theta as functions of t, so that the two expressions for x are equal, ditto for y. This kind of thing can be difficultlt, but here it is easy. Think fo R and t as given, then what do rho and thetas have to be (in terms of R and t) to make the equations match?

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