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I am going over rate of change with my 8th graders tomorrow and I foresee a difficulty. I would like to see different ways to explain this for my students who have trouble with it.

In terms of magnitude, which function has the greater rate of change? $$y=−2x+1 \qquad{\text{or}}\qquad y=2x+1$$

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  • $\begingroup$ Sorry, in terms of magnitude. $\endgroup$
    – Ash
    Oct 9, 2016 at 16:13
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    $\begingroup$ Asking which function has the greater rate of change could be misleading, as students may miss the possibility of equality or feel that it is ruled out for some reason. Currently, the question could be interpreted as a trick question. $\endgroup$
    – J W
    Oct 9, 2016 at 16:26
  • $\begingroup$ In a way it is a trick question. On the actual sheet one function is a table and the other is a graph. The goal is to be able to identify the slopes and compare them. This is our first example where the rate of change is the same but with different signs. I want them to think about it first before I phrase it as "are they the same? why or why not?" but now I may do just that to help guide their thinking. $\endgroup$
    – Ash
    Oct 9, 2016 at 16:45
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    $\begingroup$ Really, this isn't a question about rate of change. As currently stated ("In terms of magnitude, Which function has the greater rate of change?"), this is a question on whether they know the meaning of the word "magnitude" or not. $\endgroup$
    – Daniel R. Collins
    Oct 9, 2016 at 18:07
  • $\begingroup$ Have you given the students precise definitions of "rate of change" and "magnitude"? If so, what are those definitions? (It's hard to foresee what difficulties the students will have without knowing what the words in the question will mean to them.) $\endgroup$
    – Daniel Hast
    Nov 2, 2016 at 2:23

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Without first discussing what the word magnitude means in mathematical context, it's really pointless to ask them this question. How can they think and learn about the magnitude of a rate-of-change if they don't know what magnitude means?

Maybe for 8th graders, you could just say that the magnitude of something is its size/force/power with any information about it's direction thrown away. Talk about speed being the magnitude of velocity. Where velocity can be negative or positive depending on whether you are going forwards or backwards (like a roller-coaster on its tracks), speed doesn't care about direction; it's just how fast the thing is going.

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Magnitude implies absolute value. While a vector heading west has the opposite sign of one heading east, the magnitude may still be the same. In your example, 2 > -2 but they have the same absolute values.

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