This is our post for discussion of the 2023 NHT Mathematical Methods exams, which have now been posted, here and here. We haven’t looked, and don’t particularly intend to, unless something is flagged.
To be fair this is the first time I’ve looked at a VCAA exam in over six months now, so I might be a bit off. From memory it does look a lot more similar to standard VCAA exams than previous NHT exams.
Re 3(a): The wording is very clumsy, but doesn’t the use of “at random” suffice for your point?
Re 8(a): It is certainly something, but why do you think it’s silly? (Not necessarily disagreeing, just curious about your thoughts.)
Re 8(b): The wording is atrocious, real Fenimore Cooper stuff, but why do you regard it as ambiguous? (If I squint, I think I can see what you’re suggesting.)
MCQ Q16 seems to be similar to that of VCAA 2019 Exam 1 Question 8b, where a composite function is considered to have a ‘maximal’ or ‘largest’ domain to be defined on.
MCQ Q17 asks for values of k for which a function has more than one ‘solution’, instead of ‘root’ or something like ‘f(x) = … = 0 has more than one solution when…”.
If they had asked for the largest set or interval, they may have escaped on a technicality. But the word “domain” indicates they are regarding g(f(x)) as a new (and composite) function, and thus they lose by their own stupid rules. I’ll add it to the list.
Re MCQ17: Ugh! I’ll think about whether to list it as an error, since such sloppiness is pretty common and sort of accepted. But it is an error, and it is gauche.
MCQ 17 – It’s not gauche. It’s diabolical (and wrong).
Given that a common MCQ asks “Which of the following are the roots of …” and at least one of the wrong options always lists factors (and the vice versa question), it’s not unreasonable to expect a much better standard of examination wording when it comes to the difference between a root and a solution. (Doubly so since the DET has mandated literacy in all subjects as a teaching focus and countless hours have been invested/mandated in staff PD). As RF has already said, functions have roots, equations have solutions.
It’s an extreme example that nevertheless captures the essence of what is a generally poorly worded exam.
Yeah, you’re right, BiB. I think of the root-factor thing as akin to a grammar error, and am not overly fussed by it. BUT, VCAA are of course fussed by it when a student screws up, and will happily score a kid 0/1 or 1/2 for such an error.
I haven’t hunted, but can you give examples of: (a) a question on this exam (or this year’s NHT exams) where such a question asks for a root/factor, and they would clearly penalise a student for giving a factor/root; (b) an example of an MCQ where root and factor answers are both offered; (c) a question on last year’s (or the most recent year’s) reports for which the examiners whined about this issue?
Adding on to those, MC Q4 forgets that the positive whole numbers are already larger than zero: “n is a positive whole number larger than zero.” Could have just said “n is a positive integer” or n∈ℤ.
Not a huge fan of MC Q18, feels dodgy. They didn’t need to mention either the fee or the intial investment and the question still works.
I think I was being lenient to make sure I wasn’t missing something and that it was as bad as I thought it was.
I’d feel stinged if I invested $10k and had to pay 4% in fees before getting any interest out of it, let alone only ~2% p.a. Takes two years just to get back to your initial investment 🤦🏻♂️
I’d feel stinged if I misinterpreted a question because I took it to be what it appeared to be, a bog standard compound question, or if I forgot to consider the idiotic fifth decimal place.
What the hell do they think such nastiness is teaching kids other than to hate “mathematics”?
There can be little doubt that MCQ18 was intended to trick students. For exactly those two reasons but particularly for the fifth decimal place business. Its another (one of the many over the years) MCQ thats doing too much. But at least there’s a correct answer 🙂
Section B Q1 they need to decide if they are going to state the sets each parameter is in or not. They do it for part a and b, not c or d, then do for f. Also, “using algebra” in part d is completely unnecessary with the answer to part b.
Q5 needed a better through line. This one is all over the shop. The f(x)-1 seems silly just to remove the +1 on 2sin(3x)+1 when they could just write the integral of 2sin(3x). Some context of why we’re equating an average value to g(a)/a would be nice (might be an application I haven’t seen before, but seems arbitrary). And part e is…something.
Thanks, Alex. Q1 is pretty funny. Of course it demonstrates that declarations such as “where x is an element of the entirely obvious world” are nothing other than ritual.
Q5 is appalling throughout. Part (d)(iii) smells of some deeper meaning, but I haven’t had the stomach to look closely.
Q5 (d) (iii) is equating the average value of over two different intervals, viz and . But I can’t immediately see it’s raison d’etre . Is there an obscure connection to part (e)?
Thought about using the Reals qualifier (there is even a package called RealOnly which I’ve seen VCE students use) but decided that this was an exercise in speed as much as anything else.
Did use NSolve though. Seemed to work fine.
Whether it gave the correct/intended answer of course is another matter!
I think the problem with some of those packages is that they require internet connectivity, which students won’t have in the exams. As for the correct/intended answer, time will tell.
You install them once, then you just need to “activate” similar to the TI NSpire library/UDFs.
Finding the packages in the first place is the challenge as the software is not exactly designed for school exams.
That aside, defining the v(t) function on any CAS and then solving simultaneously for a, b will give an answer (hopefully).
I still think this may have been what VCAA was intending when the question was written. NHT papers don’t give any commentary on solutions, especially with MCQs so we may never know.
And the packages are entirely unnecessary. Only basic code is needed but most students think that they need a nuclear bomb to kill a fly. Worse:
1) They don’t understand the nuclear bomb.
2) They can’t even see the fly.
It is very disheartening. Students don’t want to understand how to answer a question using mathematics. They impatiently say “Yes, but how would I do it on [Mathematica, TI-nspire, ClassPad].”
To state the bleeding obvious, the education system is producing generations of students that are subservient to coding and button pushing without any understanding of what’s going on. Pseudo-understanding. All under the guise of ’21st century thinking’.
I wrote “A” as the answer, but with VCAA papers I am never confident that I have interpreted the question and answers in the same way the paper setter has.
So, the answer is clear. Provided it is the answer.
e) What is the point of specifying “for t>0” ? A function can be continuous, or continuous on an interval: the wording here suggests that the continuity of a function is some pointwise condition, when of course it isn’t.
g) The derivative clearly doesn’t exist everywhere, and I don’t like glossing over this fact (“find the derivative C_2′(t)”) even if there is some talk of “relevant domains”.
Just some gripes about the wording.
The question states “students suggest” a model and the rest of the question is written as if this model is definitely correct. The intention is clear, but there’s no need for such clumsy wording. Just make the model definite from the beginning.
On page 14 the model is for “…the amount of caffeine…”, on page 15 a new model is suggested for “… the absorption of caffeine…”, and parts f and h of the question refer to “…the amount of active caffeine…”. Why?
I’ve added the Exam 2 errors to the Methods Error list. I’ve also added an NHT error to the Further error list, and I’ve noted some cluelessness from the 2022 Further Exam 2 report.
Exam 1:
3.a) There is no specification that the bags chosen on each day are independent of each other. It’s somewhat implied, but it’s not definite.
8.a) This is silly.
8.d) There are two possible interpretations, though I suppose it’s clear enough. They should be more precise with language, though.
Those are the only mistakes I’ve noticed from a precursory glance of the first exam.
Thanks, Bob. I’ll look later. Apart from mistakes or their absence, do you have any impressions?
Seems similar to most VCAA methods exams in previous years. 8.c) is the only slightly interesting question, and only slightly so.
Thanks. Usually NHT exams are weird, which is why I don’t take much interest. Maybe they’re becoming more standard.
To be fair this is the first time I’ve looked at a VCAA exam in over six months now, so I might be a bit off. From memory it does look a lot more similar to standard VCAA exams than previous NHT exams.
Thanks, Bob.
Re 3(a): The wording is very clumsy, but doesn’t the use of “at random” suffice for your point?
Re 8(a): It is certainly something, but why do you think it’s silly? (Not necessarily disagreeing, just curious about your thoughts.)
Re 8(b): The wording is atrocious, real Fenimore Cooper stuff, but why do you regard it as ambiguous? (If I squint, I think I can see what you’re suggesting.)
Some quick ones on E2:
MCQ Q16 seems to be similar to that of VCAA 2019 Exam 1 Question 8b, where a composite function is considered to have a ‘maximal’ or ‘largest’ domain to be defined on.
MCQ Q17 asks for values of k for which a function has more than one ‘solution’, instead of ‘root’ or something like ‘f(x) = … = 0 has more than one solution when…”.
Thanks, anon.
Re MCQ 16: Yep, that’s a mistake.
If they had asked for the largest set or interval, they may have escaped on a technicality. But the word “domain” indicates they are regarding g(f(x)) as a new (and composite) function, and thus they lose by their own stupid rules. I’ll add it to the list.
Re MCQ17: Ugh! I’ll think about whether to list it as an error, since such sloppiness is pretty common and sort of accepted. But it is an error, and it is gauche.
MCQ 17 – It’s not gauche. It’s diabolical (and wrong).
Given that a common MCQ asks “Which of the following are the roots of …” and at least one of the wrong options always lists factors (and the vice versa question), it’s not unreasonable to expect a much better standard of examination wording when it comes to the difference between a root and a solution. (Doubly so since the DET has mandated literacy in all subjects as a teaching focus and countless hours have been invested/mandated in staff PD). As RF has already said, functions have roots, equations have solutions.
It’s an extreme example that nevertheless captures the essence of what is a generally poorly worded exam.
How dare you claim, correctly, that I’m wrong!
Yeah, you’re right, BiB. I think of the root-factor thing as akin to a grammar error, and am not overly fussed by it. BUT, VCAA are of course fussed by it when a student screws up, and will happily score a kid 0/1 or 1/2 for such an error.
I haven’t hunted, but can you give examples of: (a) a question on this exam (or this year’s NHT exams) where such a question asks for a root/factor, and they would clearly penalise a student for giving a factor/root; (b) an example of an MCQ where root and factor answers are both offered; (c) a question on last year’s (or the most recent year’s) reports for which the examiners whined about this issue?
Adding on to those, MC Q4 forgets that the positive whole numbers are already larger than zero: “n is a positive whole number larger than zero.” Could have just said “n is a positive integer” or n∈ℤ.
Not a huge fan of MC Q18, feels dodgy. They didn’t need to mention either the fee or the intial investment and the question still works.
Will get to the Section B later
Thanks again, Alex.
Re MCQ4: Yeah, that is remarkable. It’s not like it’s the worst sin one could commit, but how does wording like that survive proper vetting?
Re MCQ18: Jesus H. Christ. What a stupid, nasty little question.
I agree that Q18 is dodgy, but I’m guessing the examiners are just wanting a simultaneous-equation-2-variables type situation.
Q17 – the function doesn’t have
it has 
or 
.
Section B Q5e. GIVE? To whom? Does
now mean the same as 
?
I think you and Alex are being way too lenient. I think MCQ18 is actively designed to trick students. Twice.
Yes, MCQ17 was pointed out above.
Re B5(e): An appallingly worded question. the double “give” is the least of its issues, but is definitely a really stupid cherry on top.
I think I was being lenient to make sure I wasn’t missing something and that it was as bad as I thought it was.
I’d feel stinged if I invested $10k and had to pay 4% in fees before getting any interest out of it, let alone only ~2% p.a. Takes two years just to get back to your initial investment 🤦🏻♂️
I’d feel stinged if I misinterpreted a question because I took it to be what it appeared to be, a bog standard compound question, or if I forgot to consider the idiotic fifth decimal place.
What the hell do they think such nastiness is teaching kids other than to hate “mathematics”?
There can be little doubt that MCQ18 was intended to trick students. For exactly those two reasons but particularly for the fifth decimal place business. Its another (one of the many over the years) MCQ thats doing too much. But at least there’s a correct answer 🙂
Section B Q1 they need to decide if they are going to state the sets each parameter is in or not. They do it for part a and b, not c or d, then do for f. Also, “using algebra” in part d is completely unnecessary with the answer to part b.
Q5 needed a better through line. This one is all over the shop. The f(x)-1 seems silly just to remove the +1 on 2sin(3x)+1 when they could just write the integral of 2sin(3x). Some context of why we’re equating an average value to g(a)/a would be nice (might be an application I haven’t seen before, but seems arbitrary). And part e is…something.
Thanks, Alex. Q1 is pretty funny. Of course it demonstrates that declarations such as “where x is an element of the entirely obvious world” are nothing other than ritual.
Q5 is appalling throughout. Part (d)(iii) smells of some deeper meaning, but I haven’t had the stomach to look closely.
Q5 (d) (iii) is equating the average value of
over two different intervals, viz ![\displaystyle \left[a, \frac{\pi}{2} \right]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-9e220da3cec71cf2d0bb9c586b1c17db_l3.svg "Rendered by QuickLaTeX.com")
and ![\displaystyle [0, a]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-95a857196883ab33ef60520c3f20c626_l3.svg "Rendered by QuickLaTeX.com")
. But I can’t immediately see it’s raison d’etre . Is there an obscure connection to part (e)?
How is g(a)/a an average value?
Refer to the definition of g(a).
Oh, I see. Thanks. That’s nuts.
Re: MCQ18.
Just for some fun, I asked Mathematica to solve
and 
simultaneously for 
.
Amongst a large number of complex solutions (expected) I got
This ignores the second and fourth sentences, but if I was in a rush… choose answer A and move on.
You should include a domain:
Clear[V, a, b, f]
V[t_] := (b – f)*(1 + a)^t
NSolve[{V[10] == 12000, V[20] == 15000, b == 10000}, {a, b, f}, Reals]
and get rid of all those pesky non-real solutions:
{{b -> 10000., a -> 0.0225652, f -> 400.}}
(Since Mathematica treats everything as complex, sometimes NSolve won’t even give an answer unless the domain Reals is included).
Thought about using the Reals qualifier (there is even a package called RealOnly which I’ve seen VCE students use) but decided that this was an exercise in speed as much as anything else.
Did use NSolve though. Seemed to work fine.
Whether it gave the correct/intended answer of course is another matter!
I think the problem with some of those packages is that they require internet connectivity, which students won’t have in the exams. As for the correct/intended answer, time will tell.
You install them once, then you just need to “activate” similar to the TI NSpire library/UDFs.
Finding the packages in the first place is the challenge as the software is not exactly designed for school exams.
That aside, defining the v(t) function on any CAS and then solving simultaneously for a, b will give an answer (hopefully).
I still think this may have been what VCAA was intending when the question was written. NHT papers don’t give any commentary on solutions, especially with MCQs so we may never know.
And the packages are entirely unnecessary. Only basic code is needed but most students think that they need a nuclear bomb to kill a fly. Worse:
1) They don’t understand the nuclear bomb.
2) They can’t even see the fly.
It is very disheartening. Students don’t want to understand how to answer a question using mathematics. They impatiently say “Yes, but how would I do it on [Mathematica, TI-nspire, ClassPad].”
To state the bleeding obvious, the education system is producing generations of students that are subservient to coding and button pushing without any understanding of what’s going on. Pseudo-understanding. All under the guise of ’21st century thinking’.
“Pseudo-understanding” will be my new catchphrase when I give a quiz on the new pseudocode… nice one.
I don’t understand. Isn’t the intended answer clear?
I wrote “A” as the answer, but with VCAA papers I am never confident that I have interpreted the question and answers in the same way the paper setter has.
So, the answer is clear. Provided it is the answer.
Exam 2 page 15
e) What is the point of specifying “for t>0” ? A function can be continuous, or continuous on an interval: the wording here suggests that the continuity of a function is some pointwise condition, when of course it isn’t.
g) The derivative clearly doesn’t exist everywhere, and I don’t like glossing over this fact (“find the derivative C_2′(t)”) even if there is some talk of “relevant domains”.
Thanks, A.
I also don’t like the wording of (e), but why do you consider continuity to not be a pointwise condition?
On (g), yes I don’t like this kind of framing either, but I think it’s pretty standard in VCE.
Exam 2, Section B, Question 2.
Just some gripes about the wording.
The question states “students suggest” a model and the rest of the question is written as if this model is definitely correct. The intention is clear, but there’s no need for such clumsy wording. Just make the model definite from the beginning.
On page 14 the model is for “…the amount of caffeine…”, on page 15 a new model is suggested for “… the absorption of caffeine…”, and parts f and h of the question refer to “…the amount of active caffeine…”. Why?
Yes, VCAA has made an extra special effort to make the wording terrible.
I’ve added the Exam 2 errors to the Methods Error list. I’ve also added an NHT error to the Further error list, and I’ve noted some cluelessness from the 2022 Further Exam 2 report.