#### blyatman

##### Active 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 with initial mass 10 kg, which decays to a mass of 8 kg after 1 hr. The general solution is, of course, , and the goal is to solve for and .

Now, must have the same units of (from basic dimensional analysis), which means that . However, schools don't teach students to add units onto , and they often leave it as the dimensionless result .

Similarly, the exponential must be dimensionless, and the argument must also be dimensionless. This means that is not dimensionless, but has units of . So, the value of depends on the students choice of the time unit. If the question doesn't specify that 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 , with units of (as opposed to ).

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 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 and ?

2. Will the student lose marks for a different (but still valid) 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?

Consider the following problem involving the exponential decay of a mass with initial mass 10 kg, which decays to a mass of 8 kg after 1 hr. The general solution is, of course, , and the goal is to solve for and .

Now, must have the same units of (from basic dimensional analysis), which means that . However, schools don't teach students to add units onto , and they often leave it as the dimensionless result .

Similarly, the exponential must be dimensionless, and the argument must also be dimensionless. This means that is not dimensionless, but has units of . So, the value of depends on the students choice of the time unit. If the question doesn't specify that 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 , with units of (as opposed to ).

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 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 and ?

2. Will the student lose marks for a different (but still valid) 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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