Teaching Slope as Rise/Run

Our lesson today in the basic concepts of algebra course had to do with the slope concept. We had already spent some time on linear functions and investigated the notion that the rate of change of a linear function is constant – and that is what the slope of the line is. Side note: a few students were confused about the difference between a constant function and a constant rate of change – that can be a topic for future discussion here – I can’t afford to digress because as a lazy mathematician, I may never get back to the current topic – slope of a line.

So, we were going about it quite happily and figuring out our rises and our runs – focusing for today only on lines with positive slope. All of our rises were positive, an our runs were also positive. We worked with pictures like this one:

Line showing slope as rise/run

Positive slope -positive rise - positive run


Then our author did something that threw at least one student for a loop. He put this picture up as an illustration of the rise/run process for figuring out slope:
Positive slope - negative rise - negative run

Positive slope - negative rise - negative run

My very astute and inquisitive student wondered out loud: “Shouldn’t the rise and the run both be negative?”

Well, the lazy mathematician would have to agree – the rise and the run as pictured should indeed both be negative. To have our picture match what we lazy mathematicians do, we need to have both the rise and the run as negative numbers – else we get confused.

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2 Responses to “Teaching Slope as Rise/Run”

  1. Christopher Says:

    Interesting. As a not-very-lazy-at-all mathematician, I see a different thing to wonder about in this diagram.
    I notice that there is no direction on the rise and run at all. Figure 1 has arrows:

    The arrows are what indicate positive rise and run to me. Figure 2 has no arrows:

    So now I don’t know whether to indicate positive or negative rise and run.
    Your student was attending to whether the rise and run were indicated above or below the line segment; is this right? What an interesting observation.
    I’m glad your class is structured in such a way that your student could ask about something you and I didn’t really notice.

  2. Mary Daunis Says:

    Yes, that is what I think my student was getting at. She expected to see a diagram like the first one above, with the RISE/RUN segments above the slanted segment – I think her math textbook will be better than the one we are using in this class.

    This particular student tends to be the class spokesperson – she is the brave one who is willing to admit what she dose not know and so always asks the best questions – and in the end shows how thoughtful a student she really is. Give me a class full of students like her any day.

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