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I am a teaching assistant for a winter school course in basic algebra and I was explaining the meaning of the slope of a line to a student today using the following example: $$y = \frac{5}{3}x$$ where $y$ is measured in units of meters and $x$ is measured in units of time. I told them that the slope of the line is the rate of change of $y$ with respect to $x$, however, I then followed up by saying for this line in particular the rate of change is "five-thirds meters per second". This got me thinking, is this the same thing as saying "the rate of change is five meters every three seconds"? I concluded that these two statements do in fact mean the same thing since $\frac{5}{3} \cdot 3 = 5$ which precisely what you should have if you think about "your position" after $3$ seconds has elapsed.

I understand this is a little informal, but the concept to new students is important to me. Any feedback is welcome.

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    $\begingroup$ Yes, you can say that. $\endgroup$ Jan 13, 2021 at 22:51
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    $\begingroup$ A derivative is an expression of rate of change at and instant. There is no interval of $3$ seconds or $1$ second of $\frac 12$ a second or $7$ seconds. In this case $1$ second is just a unit of time. If you went to another planet where all their clocks measured time in a unit called the "three second" and it was three times as long $\frac 5{3}$ per second would equal $5$ per three second. Likewise you could go to a strange country where they have something called "minutes" that are $20$ times slower than a "three-second" then it would be $100$ per minute.. .. wooo..... $\endgroup$
    – fleablood
    Jan 14, 2021 at 0:17
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    $\begingroup$ Yes, you would say exactly that. $\endgroup$
    – fleablood
    Jan 14, 2021 at 0:41
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    $\begingroup$ ... assuming the function is one whose input are second units of time and whose output are meter units of measure. $\endgroup$
    – fleablood
    Jan 14, 2021 at 0:43
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    $\begingroup$ But "five meters per 3 seconds" is okay (but weird) . Also $\frac 53$ per second is okay, but $\frac 53$ every second is not correct. $\endgroup$
    – fleablood
    Jan 14, 2021 at 1:07

2 Answers 2

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Is saying “the rate of change is five-thirds meters per second” equivalent to saying “the rate of change is five meters every three seconds”?

Yes. To make things even simpler, the following two notions are equivalent:

  • five-thirds meters per second
  • five meters every three seconds

In general, if $p$ and $q$ are two positive integers,

  • $p/q$ meters per second
  • $p$ meters every $q$ seconds

are equivalent. Essentailly, you are looking at the function $y=\frac{p}{q}x$ in two different way by using two different "time unit":

  • $\displaystyle y\ (\textrm{meters})=\frac{p}{q}\ (\textrm{meter}/\textrm{second})\cdot x(\textrm{seconds})$
  • $\displaystyle y\ (\textrm{meters})=p\ (\textrm{meter}/\color{red}{q\ \textrm{seconds}})\cdot x(\textrm{seconds})$

If you want to use the second interpretation, the "slope" needs to be rational. Imagine how you want to interpret $y=\pi x$.

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  • $\begingroup$ Thank you for your feedback. Quick follow up: I’ve noticed with the derivative of a curve at a point we simply say “$m/n$ meters per second” rather than “$m$ meters every $n$ seconds”; why is that? $\endgroup$ Jan 13, 2021 at 23:05
  • $\begingroup$ @TaylorRendon: I edited my answer. Saying (a) “𝑚/𝑛 meters per second” or (b)“𝑚 meters every 𝑛 seconds” depends on what "time unit" one wishes to use: in (a) the time unit is "one second" while in (b) the time unit is "n seconds". In physics, they are simply two different ways to measure things while in mathematics, they are equivalent. $\endgroup$
    – user9464
    Jan 13, 2021 at 23:10
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    $\begingroup$ @TaylorRendon: well, they are equivalent as you observe. (b) works only when you have a rational number if you require the denominator to be an integer. (a) sticks to the original time unit, which is more desirable I suppose, since everybody can agree on one uniformly used. This is particularly useful when one adds two velocities. $\endgroup$
    – user9464
    Jan 13, 2021 at 23:18
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    $\begingroup$ @TaylorRendon: JimJim was probably talking about $y=\frac{5}{3}x^2$. But your function is linear. $\endgroup$
    – user9464
    Jan 13, 2021 at 23:27
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    $\begingroup$ @TaylorRendon: in the nonlinear case, you are then talking about "instant rate change" and the slope of the tangent line. The two are still equivalent. $\endgroup$
    – user9464
    Jan 13, 2021 at 23:34
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Yes, the two statements are equivalent. Since you can think of the unit for rate of change in this case as fraction with meters divided by seconds. So the measurement $\frac{5}{3}$ mps is the same as $\frac{5 \text{meters}}{3 \text{seconds}}$, or 5 meters per 3 seconds as you said.

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