WitCH 13: Here for the Ratio

The WitCH below is courtesy of a clever Year 11 student. It is a worked example from Jacaranda’s Maths Quest 11 Specialist Mathematics (2019):

Update (11/08/19)

It is ironic that a solution with an entire column of “Think” instructions exhibits so little thought. Who, for example, thinks to “redraw” a diagram by leaving out a critical line, and by making an angle x/2 appear larger than the original x? And it’s downhill from there.

The solution is painfully long, the consequence of an ill-chosen triangle, requiring the preliminary calculation of a non-obvious distance. As Damo indicates, the angle x is easily determined, as in the following diagram: we have tan(x/2) = 1/12, and we’re all but done.

(It is not completely obvious that the line through the circle centres makes an angle x/2 with the horizontal, though this follows easily enough from our diagram. The textbook solution, however, contains nothing explicit or implicit to indicate why the angle should be so.)

But there is something more seriously wrong here than the poor illustration of a poorly chosen solution. Consider, for example, Step 5 (!) where, finally, we have a suitable SOHCAHTOA triangle to calculate x/2, and thus x. This simple computation is written out in six tedious lines.

The whole painful six-step solution is written in this unreadable we-think-you’re-an-idiot style. Who does this? Who expects anybody to do this? Who thinks writing out a solution in such excruciating micro-detail helps anyone? Who ever reads it? There is probably no better way to make students hate (what they think is) mathematics than to present it as unforgiving, soulless bookkeeping.

And, finally, as Damo notes, there’s the gratuitous decimals. This poison is endemic in school mathematics, but here it has an extra special anti-charm. When teaching ratios don’t you “think”, maybe, it’s preferable to use ratios?

19 Replies to “WitCH 13: Here for the Ratio”

  1. OK, so there is no justification for why step 1 is even logically valid.

    Hence, the rest of the argument is pointless.

    (Am I even close this time?)

    1. RF, I think you’re looking too hard for errors, and not hard enough for crap. Think carefully of how you would present the solution to that problem, either to yourself or to your students. And then compare to what is done here.

  2. Adding a construction line.

    And, just to be nit-picky, the question never specified two decimal places, so I would have assumed an exact answer was called-for (as per VCAA *tries to find nice adjective but can’t* requirements).

    1. Ah, I see. Yes, for all their pedantry, they left out the obvious first step. I wouldn’t say their step 1 is invalid, but it’s mighty confusing. It’s a textbook question, so the 2DP seems reasonable enough.

    2. For what it’s worth, the similar triangle diagram is meaningless in my opinion until the points A, B, C etc. have been defined on the diagram in the question. So the construction line that RF mentions is vital. The you’d make a simple justification for the angle being x/2 etc.

  3. A triangle connecting the centres of the two circles, with a horizontal length of 120 and a vertical length of 10 would give tan(x/2)=10/120. But this seems to obvious. Am I missing something?

    1. Apart from the second “o” in “too”, I don’t see that you’re missing anything. The only other ingredient that I can see is justifying that the angle in your triangle would be x/2, which the text skips as well. I’m not sure if that step would be expected in this context, though it’s easy.

    2. Damo, the only thing you’re missing is all of the superfluous crap that this example has.

      An example intended to illustrate the use of particular mathematical concepts (such as similar triangles, ratios etc.) shouldn’t have an alternate solution that is much, much simpler. Otherwise it’s just another example of using an elephant gun to kill a fly.

      (Yet another reason for adding a construction line like RF suggests – it makes the much simpler solution bleedingly obvious).

      It’s sad that the example goes to soooo much trouble trying to spell out the key problem solving steps but fails to include the most important ones (add a construction line, add appropriate labels to the diagram). It completely shoots itself in the foot (with an elephant gun).

  4. I actually noticed that missing “o” as soon as I posted, breathed deeply for a couple of minutes and let it go. Mostly. With regards to the x/2, I figured that this had already been established and could be assumed. Although, when it comes to textbooks, I really should know better.

  5. OK, a few more points. I know that diagrams don’t have to be too scale, but in both of those diagrams the scale is way off. I don’t necessarily mind it if we were to consider the diagrams independently, although it would still make me anxious, but I am far less comfortable with the inconsistency across diagrams – particularly as it leads to an angle x/2 that looks bigger than the original angle x. Apart from anything else, it makes it more difficult for the student to understand where the diagram has come from, something not helped by the fact that the points A, B, … have not been defined on the original diagram.

    Having said that, the big problem with their first step is not the diagram, but the fact that they decide to find y, rather than drawing a horizontal line from C to AB.

    Particularly considering that this was from a Specialist textbook, why the laboured descriptions on how to set up and solve equations in Steps 4&5.

    Why use 1.25 instead of 5/4? And the follow up, why divide by 0.25 rather than multiply by 4? Why Step 6?

    Unrelated, why am I not writing reports?

    1. Damo, they’re all great points, but all I can think about is that you cleverly made up your “o” deficit.

  6. Marty, in response to your update paragraph beginning “The whole painful six-step solution…”

    I have found, to some discouragement, that there are students who easily feel overwhelmed or lost if worked examples aren’t presented in such excruciating (and distracting!) detail. Not so much in Specialist, but definitely in Methods. MathsQuest has a reputation of catering to these students, and there is also a fairly widely held view amongst students that MathsQuest has “good” worked examples (which I interpret as: requiring no active thought when reading).

    1. SRK, I know and understand the argument, and I don’t buy it. I don’t buy it for Methods, and I definitely don’t buy it for Specialist. I don’t buy it all. I know that some teachers and some texts push this anal-algorithmic approach to “help” weaker students, but they’re not helping them. You teach thinking or you don’t. But you cannot teach syntax and pretend it’s semantics.

  7. The thing that bothers me right off the bat is the diagram not being to scale. Like the 50 roller is twice the size of the 40. Would probably at least explicitly inform the students of that unless they are pretty sophisticated and the drill problems routinely pull that. As it is, I wonder if the artist/author just didn’t think of that.

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