I was having this debate with a colleague. The test we have to use has a problem where it states the slope without units just a numerical value. It also gives the horizontal distance with units. The problem asks to find the vertical distance but does not mention the units.

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My colleague argues that since slope is a ratio units are not needed and that implies the units must match at the end.

I argue that you do need the units since the vertical units are never specified. Not to mention we want to be clear in our message. You wouldn't go up to someone and say the slope is .25. You would say something like you raise .25 ft vertically for every 1 ft. horizontally. I realize this means the units reduce out but the context matters.

Thoughts?

The units for slope are the units for the vertical axis divided by the units for the horizontal axis. For example, if the horizontal axis represents time and the vertical axis represents distance traveled, then the slope has units of distance per time, i.e. velocity. This follows from the fact that the slope is the change in the y-value divided by the change in the x-value.

In the case where the horizontal and vertical axes have the same units, i.e. if both represent distance, then the slope is a dimensionless quantity.

For those of us that have been learning and doing math for such a long time I feel like that is easier for them to infer. For students though it is this kind of subtle stuff that confuses the crap out of them. When we could just be straight forward and express it.

When would you ever say "feet per foot" in conversation? A unitless presentation matches natural language

*and*is the best mathematical interpretation.

How do we know that the vertical axis is not in inches? The slope could be 3 in./ 2 in.. The mathematical interpretation would be 1.5. But our horizontal is given in feet.

At which point you could argue that the context is mixed up and units don't match. Which is really what my argument is about. How do we know for sure the units of the slope are feet.

I think that the context is very important. I think it helps to solidify the idea that a graph isn't separate from an function, but a visual representation that can help understand the behavior of a function. Whether it's feet per second or inch per inch. The simplest I've ever thought of it was still rise over run for contextless bookwork and that's still a unit of distance per distance.

When I taught slope I never taught the simple mathematical ratio first. I would always introduce the concept by allowing the students to just explain the rise and run in English; this *necessitated* using units and clarifying horizontal or vertical.

At some point in this process we would discuss how saying "feet vertical" / "feet horizontal" can be simplified to just saying rise/run (dimensionless) as long as we're using the same units for the two dimensions (as we should in a unit Cartesian plane). We just state how this is easier and move on. Eventually, the idea of "slope" is now a ratio with units *implied* but not *stated* just out of convention of being easier.

As such, I think the answer to your question is less about "are units needed" but more of "can the student answer the question asked?" I would say this means that in the context of a "physical" problem, we should make sure students know that the idea of slope and other mathematical concepts are simply abstractions, not the answer.

See more: Convert 28 Inches Equals How Many Feet, 28 In To Ft

To that end, I'd say units are required since (a) the problem specifies units, and (b) slope is a rate of change of one dimension vs another, and has units implied for both dimensions... and we should make sure the student knows that.