## An easy way to remember the difference between a line with zero slope and a line with no slope

Monday, October 8th, 2012A lot of students get the concepts of “zero slope” and “no slope” confused when they’re first introduced.

Most students think something along the lines of, “They’re the same thing, right? Because zero equals nothing…..?????????? Wait… no, they’re totally different — BUT HOW DO I REMEMBER WHICH IS WHICH?”

Here is a super easy way to remember the difference:

Zero slope means that the line is horizontal. Just like the line that makes the top of a “Z” is horizontal.

No slope means that the line is vertical. Just like the line that makes the beginning of a “N” is vertical.

(If you’re interested in a mathematical explanation to go with the visual reminder, check out Elizabeth Stapel of PurpleMath’s lesson on slope. The part about zero slope and no slope is towards the bottom of the page.)

Many of my students have used this tip with great success — so spread the word! No one needs to be confused about this anymore!

*Do you wish someone would just explain math in a way that really makes sense to **you**? Do you yearn for the confidence that comes from really GETTING it? Give me a call at 617-888-0160 or send me an email at rebeccazook@gmail.com, and I’d be happy to set up a time for us to have a complimentary conversation to explore whether or not it would be a good fit for us to work together!*

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I think the phrase ‘no slope’ is a serious mistake. (I know, many textbooks use it…) I think it make more sense to say the slope is undefined on vertical lines.

Most of your ideas focus on meaning, and I love what you write. But this one is a memory device for something that could have meaning. I want my students to really understand slope. If there’s no hill (a flat line we could walk on) then the slope is 0. If you couldn’t possibly walk up or down the line, its slope is undefined.

Dear Sue, It’s so great to see you here!

I agree that the phrase “no slope” is confusing and problematic. But since a lot of my students have to deal with textbooks that use that term all over the place, that’s why I created this way to remember the difference, and because it has helped a lot of students, I wanted to share it.

We also talk a lot about how vertical lines have a slope that is undefined and why, so this is all taking place in a larger context. For some students, having this memory device that immediately reminds them of what the graph looks like is a way to connect to the concept. Others like to think about the formula for calculating slope, the definition of undefined, etc…

You could also adopt this memory device and do it with the vertical line going through an uppercase U of the word “Undefined”.

Thank you always for what you share!

Sincerely,

REBECCA

A very helpful post. I remember mixing these up all of the time. And it was great to hear reasons for why the term “no slope” isn’t actually that handy anyhow, a whole problem that I wasn’t even aware of.

Nichole, I’m so happy to see you here! Thank you so much for your thoughts. And congratulations on your translation being published! I am SO excited for you!!!

This is such a great tip! Sometimes students can get stuck on a concept that they don’t understand and then simply give up. Approaching the problem in a different way, like you have illustrated, can make all the difference.

Great post to read rebecca

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The real difficulty arises from the association of the number zero with the concept of nothingness. In colloquial usage this is a non-issue. In the context of mathematics, this is a problem. Really the NUMBER zero is something. It is a NUMBER. While nothingness is a CONCEPT. In mathematics the empty set is the representative for nothingness. Zero is associated to the “quantity” connected with nothingness (or in fancy terms the cardinality of the empty set is zero) but as far as sets and mathematics are concerned the empty set is not the same as zero.

As an example, if an equation has the NUMBER zero as its only solution, no one would ever say that said equation has no solutions. For example no one should ever say the solution to the equation x+5=5 is nothing just because its only solution is the NUMBER zero. It solution is SOMETHING.

As such no one should be confused about the situation with slopes. The slope of a line is a NUMBER. If when we try to compute the slope of a line we get a NUMBER, then we would say that the line has a slope. If somehow, when we try to compute the slope of a line we do not get a NUMBER, then we say the line has no slope. The slope of a horizontal line is SOMETHING! When we try to compute the slope of a horizontal line we get the NUMBER zero. While for vertical lines, the slope is “nothing” because there is no number associated to the slope of such lines.

In the same vein, imaginary numbers do exist and improper fractions are not “bad.”

peace

jeff

Dear Jeff, wow, thanks so much for sharing your thoughts so extensively! I’m glad to “meet” you here.