# Incorrect teaching of units - will you lose marks for using the actual correct units? (1 Viewer)

#### blyatman

##### Well-Known Member
The issue I'm raising here has to do with the dimensional inconsistency of variable units in the HSC, particularly in the topic of exponential growth/decay.

Consider the following problem involving the exponential decay of a mass $\bg_white m$ with initial mass 10 kg, which decays to a mass of 8 kg after 1 hr. The general solution is, of course, $\bg_white m=Ae^{-kt}$, and the goal is to solve for $\bg_white A$ and $\bg_white k$.

Now, $\bg_white A$ must have the same units of $\bg_white m$ (from basic dimensional analysis), which means that $\bg_white A=10~\mathrm{kg}$. However, schools don't teach students to add units onto $\bg_white A$, and they often leave it as the dimensionless result $\bg_white A=10$.

Similarly, the exponential must be dimensionless, and the argument $\bg_white -kt$ must also be dimensionless. This means that $\bg_white k$ is not dimensionless, but has units of $\bg_white \mathrm{time}^{-1}$. So, the value of $\bg_white k$ depends on the students choice of the time unit. If the question doesn't specify that $\bg_white t$ represents the time in hours (like in this example), then the student is free to use another time unit (e.g minutes). This will, in turn, lead to a different value of $\bg_white k$, with units of $\bg_white \mathrm{min}^{-1}$ (as opposed to $\bg_white \mathrm{hr}^{-1}$).

I'm not sure how many HSC teachers/markers know of this dimensional inconsistency when this topic is taught, so my main concern is that they'll mark $\bg_white k$ as incorrect if it doesn't line up with what they have on their marking sheet.

So the questions I have are:
1. Will students lose marks for adding units onto $\bg_white A$ and $\bg_white k$?
2. Will the student lose marks for a different (but still valid) $\bg_white k$ value if they used a different time unit (in this case, if they used minutes instead of hours)?

The first question is my primary concern, since I would prefer to teach my students to use the actual "correct" units (as opposed to leaving those quantities dimensionless), but I will avoid it if it means them losing marks.
The second question is mainly out of curiosity, since there's no logical reason one would use minutes over hours in this example problem.

Are there any HSC markers who can shed some light on this?

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#### Drdusk

##### π
Moderator
IIRC, I think I remember my teacher saying you just use whatever units they give you. Like the question will never require you to convert units.
Regardless I'm no HSC marker, so someone should confirm this.

#### blyatman

##### Well-Known Member
IIRC, I think I remember my teacher saying you just use whatever units they give you. Like the question will never require you to convert units.
Regardless I'm no HSC marker, so someone should confirm this.
Yeh you can't go wrong with that, since that's what all the schools teach and it's what the examiners will expect and have on their answer sheet. But what if (for arguments sake) someone where to change out the units for something else? It might be inefficient and strange, but it's not wrong lol, as long as they append the correct units onto it. Instead of $\bg_white k=60~\mathrm{hr}^{-1}$, you have $\bg_white k=1~\mathrm{min}^{-1}$. It's no different than saying "It takes 60s" vs "It takes 1min" in other problems. Most teachers will be able to recognise the equivalence between "it takes 60s" and "it takes 1min", but they probably won't be able to recognise the equivalence between the different $\bg_white k$ values unless the answer has units of $\bg_white \mathrm{min}^{-1}$ appended onto it (even then they'll probably be like wtf is this weird unit).

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#### Trebla

If the logic is all correct and the teacher can follow the unit conversions then there is no issue and you’ll get full marks. The so called ‘answer’ sheet they are given is meant to be used as marking guidelines and are not prescriptive (in maths are many ways to achieve the same answer).

If a teacher were to deduct marks simply because they couldn’t recognise an equivalent answer, I would question their competence lol

#### blyatman

##### Well-Known Member
If the logic is all correct and the teacher can follow the unit conversions then there is no issue and you’ll get full marks. The so called ‘answer’ sheet they are given is meant to be used as marking guidelines and are not prescriptive (in maths are many ways to achieve the same answer).

If a teacher were to deduct marks simply because they couldn’t recognise an equivalent answer, I would question their competence lol
Yeh I'm just wondering if they'd incorrectly assume that $\bg_white t$ had to be in hours (since there's a unit of hours in the original question) and not realize that $\bg_white k$ is not absolute, but rather depends on their choice of time. There's a lot of incompetent teachers out there, so I'm uncomfortable with assuming that they're all on the ball.

Do you think a student could attach units to $\bg_white A$ and $\bg_white k$ without losing marks? If a variable was actually meant to be dimensionless, and a student added units onto it, they may potentially lose marks (e.g. adding units^2 when evaluating an integral). So my main concern is if HSC markers only accept dimensionless values of $\bg_white A$ and $\bg_white k$.

#### Trebla

Yeh I'm just wondering if they'd incorrectly assume that $\bg_white t$ had to be in hours (since there's a unit of hours in the original question) and not realize that $\bg_white k$ is not absolute, but rather depends on their choice of time. There's a lot of incompetent teachers out there, so I'm uncomfortable with assuming that they're all on the ball.

Do you think a student could attach units to $\bg_white A$ and $\bg_white k$ without losing marks? If a variable was actually meant to be dimensionless, and a student added units onto it, they may potentially lose marks (e.g. adding units^2 when evaluating an integral). So my main concern is if HSC markers only accept dimensionless values of $\bg_white A$ and $\bg_white k$.
In answer to your first point they should give you full marks if the answers are equivalent (unless particular units are explicitly requested in the question). If your concern is whether the teacher is competent enough to notice that equivalence, even if your working is clearly set out, then that would likely be a marking error on their part and that is outside your control. It’s no different to how teachers should still award marks for students using a different but still correct pathway (compared to whatever sample answers they are given) to get to the same result.

On your second point, the assumption that A and k are dimensionless is not correct. They do have dimensions in the way you outlined in your original post. It is possible they may have no specified measurement units in the context of the question but that does not make them ‘dimensionless’.

In the context of HSC exam marking (not necessarily school marking), as long as the required value is correctly obtained in your working (in whatever units) then you’ll get full marks as markers will always give the student the benefit of the doubt. If the question was specified in minutes and you converted it to hours then that conversion should be pretty obvious in your working anyway - and if your final answer is in hours then that is perfectly fine.

FYI don’t think you lose marks for not saying “square units” in an area question - as long as the value and working is correct then you get full marks (in the HSC exam anyway - for individual school assessments some teachers are more pedantic than others).

#### blyatman

##### Well-Known Member
Fair enough, I suppose I should have more faith in HSC markers (I've seen my fair share of clueless teachers which is why I'm always hesitant on non-conventional matters).

In my experience, teachers have always emphasised to add units, especially in calculus applications. If the question asks you to calculate a rate, and you gave a dimensionsless number, then it doesn't really make any sense: is their answer in mm/s or m/s? Likewise, wrong units may cost you marks: 100mm isn't the same as 100m.

On your second point, the assumption that A and k are dimensionless is not correct. They do have dimensions in the way you outlined in your original post. It is possible they may have no specified measurement units in the context of the question but that does not make them ‘dimensionless’.

FYI don’t think you lose marks for not saying “square units” in an area question - as long as the value and working is correct then you get full marks (in the HSC exam anyway - for individual school assessments some teachers are more pedantic than others).
I think you misinterpreted the intent of my questions, which I've reworded below.

I wasn't saying A and k were dimensionless, the whole point of the post (like you said) was that they have dimensions. I just meant IF you had a quantity that was meant to be dimensionless, then adding dimensions to them would be incorrect. So my concern was that, in a similar manner, if teachers (incorrectly) assumed A and k were dimensionless (since they were not aware that they had dimensions), then I was worried that they might consider the added dimensions as incorrect. So, what I said about the square units in the integral was referring to the opposite situation: where you add units to a problem that DIDN'T require them (as opposed to forgetting units in a problem which required them). E.g. 1 + 1 = 2 kg or something, which doesn't really happen often, so I don't know how they'd approach it. So, since most markers probably assume A and k are dimensionless, I'm just worried they might think "why is this muppet adding units to these dimensionless constants?" lol.

I should've worded it better originally, but hope that makes more sense now.

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#### Trebla

Whilst strictly speaking (i.e. outside of HSC world) the correct expression of units gives a more complete answer, this usually isn’t part of the HSC marking criteria so they are generally quite lenient on it (i.e. they wouldn’t dedicate 1 mark out of say 3 marks for something so trivial).

If you incorrectly added units to an otherwise correct answer that shouldn’t have units then you will not lose marks - as you’ve already gained all the marks you needed through your working.

In your exponential decay example it’s the other way around in that A and k should have units depending on the context of the question. But again whether or not you express the units correctly is irrelevant to markers compared to the correctness of the actual working.

I am not sure where you’re getting this idea that most markers would assert A and k must be strictly dimensionless - it’s more likely that they don’t care for it.

#### blyatman

##### Well-Known Member
Whilst strictly speaking (i.e. outside of HSC world) the correct expression of units gives a more complete answer, this usually isn’t part of the HSC marking criteria so they are generally quite lenient on it (i.e. they wouldn’t dedicate 1 mark out of say 3 marks for something so trivial).
Yeh fair enough.

I am not sure where you’re getting this idea that most markers would assert A and k must be strictly dimensionless - it’s more likely that they don’t care for it.
It goes back to my high school days when I'd have my fair share of clueless teachers who know the bare minimum to teach the course.

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