Calculating slope of a line is pretty standard fare for most students around grade nine. Slope is a simple word and is one of those times where correct math terminology aligns with their general understanding of that word. Even the idea of positive, negative, and zero slopes are lined up well with student thinking and intuition. Of course, all of this is done while calculating slope between two points.
Is it possible to calculate the slope of just…one point? Yes/No?
Isn’t that the perfect question to ask these young students? Pick an answer. Defend your choice. Let students tread mathematical water here for a while. Give a look that shows you are listening. Let a visible smirk out…
That’s what I did over 10 years ago, the first time I taught a grade 9 math class. I had taught many courses before that — including grade 9 science and senior physics — but this was my first time teaching an introductory high school math course. And of course, like many of my ideas, there was no research or beta-testing. I just winged it. If it didn’t work, then no worries, back to the drawing board. I found so many teachers consumed by trying to make sure the perfect lesson went perfectly.
The perfect lesson always occur when you are just honest with your students and how/why you teach the subject you do.
Anyways, I told my grade 9 students that it is quite possible to find the slope at one point — even though their protests were shouted through a vague, but right understanding that a single point wont’ have any rise or run. You can’t divide by zero was the rallying cry from the mathematical mob.
I wish Siri was around to jump into the fray.
Their curiosity was peaked. The classroom was brimming with resolution.
I told the class that what I am going to show you is something you are not supposed to see for another three years. The mystery just intensified.
Here is the how I went about introducing calculus to them.
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I asked students to fill out a table of values for the simple function y = x². I also gave them graph paper and told them to sketch it. This took all of 5 minutes.
I then asked them to locate the point (2,4) on the graph. I told them to draw, to the best of their eye-balling ability, a line tangent to this point. I also drew the graph on the board and participated in drawing a tangent. We then made a right angle from this. We took our rulers out and each measured our legs of the triangle and calculated the rise/run.
The answers varied, if I remember correctly, from around 3.3 to 4.5, with most of the answers around 4.
I said “there, we did it!” We found the slope at one point — it’s like 4ish!
They weren’t impressed with this sloppy approach. Plus, they rightfully told me that I kind of cheated since I used two points. I asked them what do you think the actual answer is? The consensus was 4.
Would you like to for me to prove that it is 4? The classroom was teeming with an impatience now. Imagine if I would have said “nah, you’re going to have to wait…”.
I located a random point above and below our point of (2,4). I said that we are going to move some unknown distance away from 2 on the x-axis, to the left and to the right. I called it h. There was strangely little fuss as to that. They didn’t know I wasn’t being random in my selection of that variable.
I used the idea Q and R being marbles — as they would later slide up and down towards P.
Drawing three lines and seeing that our tangent would have to be a slope between our two secants was a strong visual representation of what was about to happen. The squeeze play. And here, was the key point.
Letting h go to zero, flirting with danger, and damned to be determinate!
Before I even give the students to catch their breaths and stop becoming bug-eyed. I told them there is 5 second way to do this and it will work every single time. I “dropped” the 2 from the exponent, put in front of x and plugged into our x coordinate of 2. The answer was also 4.
If kids had some understanding of the binomial theorem and a little more algebra under their belts, I think I might have persuaded myself to prove the power rule of differentiation.
However, I did draw this curve and talked about slopes and change, giving a little teaser as to what happens to the second derivative in the red circle zone. Our change of the change was negative, and then one other side it went positive. Hmmmm. I rubbed my chin just to irritate their curiosity even more.
The problem with K to 12 math education is that so many topics could be taught earlier. NOT for acceleration purposes, but for lighting the mathematical fire of awe, wonder, and curiosity.
In another universe with another grade 9 class I showed Riemann Sums instead…:)
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This story will appear in Chasing Rabbits: A Curious Guide to a Lifetime of Mathematical Wellness in the The Looking Glass chapter.