# Let THEM Figure Out the Power Rule

I have been reading and enjoying (though not fully buying everything in) Daniel T Willingham’s book Why Students Don’t Like School: A Cognitive Scientist Answers Questions About How the Mind Works and What It Means for the Classroom. One of the ideas that I think is really useful in planning instruction is that humans are wired to enjoy learning – some scientists believe that the brain releases a little bit of dopamine every time we solve a problem. We actually physically get pleasure from solving problems.

As an example, check out these two word picture puzzles (figure out the common expression indicated by the words and their placement):

Which of the two puzzles did you enjoy more? If you’re anything like me, or most human beings, you didn’t really enjoy the one that had the answer right above it. Even if you didn’t figure out the other one, you probably at least thought about it more than the other one (though Willingham points out that the physical response only occurs when a person solves a problem). How often do we give the answers to the riddles first in math instruction?

Here is a rule and here are examples of every type of problem you will have to do with it, now do problems like those even though you kind of already know the answer.

### An example of posing math as a riddle instead:

It took a few days for students to learn the power rule this year, as opposed to me just writing $f(x)=x^{n}$ so $f'(x)=nx^{n-1}$, which takes about 10 seconds (if you talk while you are doing it and write very, very slowly, and have to erase something in the middle because you forgot what you were doing). Despite the time needed, I felt that the cognitive payoffs with the progression I used were great, and students really internalized the idea because THEY FIGURED IT OUT THEMSELVES. Figuring out the Power Rule is something that is totally in their reach, and I would have been robbing them of some learning pleasure had I just given them the power rule at the beginning.

PHASE 1: What is a derivative? We started just by drawing tangent lines to $f(x)=x^2$ at a bunch of points, estimating the slope and then making a table of values. I chose this function specifically because with the derivative of $f'(x)=2x$, it’s easy to see the pattern for the slopes in the numbers without graphing them (saving one level of abstraction). Yay, the slope at any point is just twice the x-value!

Then we did this two more times, once on a small sheet of paper for $\sin x$, and then once, in groups, on a huge sheet of butcher paper for $\frac{1}{3}x^{3}$. This was laborious and took a ton of time in class, but by the end I felt like students really understood well the idea of a derivative. More importantly, were ITCHING for an easier way to find it. They had all these great ideas that they were proposing, so it was easy to funnel their energy into the next phase.

PHASE 2: Finding the Rules. Then, I introduced the derivative tracer, a GeoGebra applet that does in seconds what they did in 15 minutes. I gave them a sheet of functions (below) for them to find the derivative of using the derivative tracer (kind of like collecting data in a typical canned high school science lab) and asked them to make conclusions about the derivatives that they found.

Though it was interesting to talk through with them the idea of a constant function’s derivative being 0, and a linear function’s derivative being a constant, the highlight of the lesson was seeing students figure out the power rule. When students got to that section, they seemed really proud that they could see the pattern. I had numerous students raise their hand to call me over to ask me if their idea worked, and then were so excited that it did that they immediately gave me a high-five. Students raised their hand so that I would come give them a high-five… in math class. I know the power rule is kind of easy, but I felt like they were so much more invested in the quest of learning mathematics because they figured something out themselves. Further, instead of trying to get ideas from my math notation, they had the ideas first and then I formalized it with math notation (though many students could do this no problem for themselves).

#### Long story short: The excitement in the room while the students were discovering something mathematical was palpable, even though that thing had been discovered many many many times before, including by their classmates sitting a few seats down. There was no “real world” motivation in this progression, no gimmicks – just the pure pleasure of mathematical discovery. So, to add to my ever lengthening list general goals for the year: I hope to avoid at all costs robbing students of the pleasure of figuring something out for themselves.

Here is my derivatives “lab” using the GeoGebra derivative tracer. Note that I’m not quite as adventurous as some and still want some structure in the classroom while “discovery” is happening. This is part of my controlling personality – tell me if you think this is too guided given my goals.

By the way, the answer to the other word picture puzzle is “mathematical induction.”

Posted on September 29, 2012, in Calculus, Derivatives, Teaching and tagged , , , . Bookmark the permalink. 15 Comments.

1. I love what you did here. I had my kids conjecture the power rule through their observations of doing a variety of different functions to look for a pattern. What I really like about your approach was that you had a great graphical representation to along with all of this. I think that is something I neglected. Can I steal this for next year?

2. I like the no-limits let’s see what we have here introduction, and the inductive pattern finding approach to the power rule. I’ll do this next time I teach derivatives, I think. However, what I’ll do is add the derivative by first principles in parallel/in between the activities you described here. After drawing a bunch of tangents, let’s see if students can figure out the gradient of a tangent if they don’t have a graph but only the function equation. Proceed to limit definition of derivative. Then the power rule just as you did here, but complement it with a heavily scaffolded formal proof that takes the opportunity to remin kids about binomial expansion. It’ll take longer of course, but will include deduction, which I feel is essential to both mathematics and math teaching and learning.

• Totally. I actually did that one year when I had already taught the limit definition. I just said “what’s the pattern? Now try to prove with the limit definition”. This year I’m experimenting with doing limits second, so I took that off. I think you’re right that doing the proving stuff in parallel would be really effective.

3. Blaise

Great post. I wish there were more quality lessons like this one available to us teachers who aren’t as creative and forward-thinking.

4. Jim

To me this sort of class activity always sounds (a) wonderful and (b) like it’s going on in a parallel universe. How do you make time for this sort of thing and still complete what the course outline requires within one term?

• Heather

I see we have two choices; teach it the quick way and the students can remember it for a day and dont understand it, OR teach it the discovery way and the students remember and understand it for a long time, even years. It is an activity you can refer to to jog their memory. I find doing the graphs and by computer and tangents by hand is best as if they do they it all on computer it is too abstract. If each pair of students do a different function each and they share the results, the pattern and rule are made, 15 examples done with graphs, in ONE LESSON.

• yeah, agree with what Heather wrote. this is also the third year i’m teaching calc, and the third year i have done this lesson and i have really streamlined each time i’ve done it. i think that exploratory lessons can get off topic really fast and waste time, but if you can hone and hone and hone an activity until you get to the point where the students are still thinking but directed in a proper way, it’s not as much of a time suck as you think. so long story short… just try things! and they will get better and more efficient each year. also you can frame questions like this in much quicker ways. like w chain rule, as a warmup, i gave my students a sheet with functions and their derivatives and just asked them to figure out with the rule is. it takes 5-10 minutes and is totally worth them constructing their own meaning.

5. The power law is a great place to start. I’m writing interactive textbooks that teach in a similar way. Another fun one you might want to try: After the power law, challenge the students to find (or draw) a function that’s the same as its derivative (other than y=0). And boom — another three days later, they’ll be telling you all about exponential functions!

• huh, interesting. i have always done that they other way around. cool idea.

I’d like to point out that the “proof” in almost all of the introductory books is completely bogus anyway. They claim “induction” then proceed to use the rule for non-integer powers! The “classic” textbook method is not mathematically sound, so there’s really no argument against using a discovery process (over the binomial expansion, which only works for integers).

• i kind of waved my hand at non-integer powers too… makes me think that i could at least use wolfram alpha or something like that in conjunction with this activity just to let them figure out how to get the correct derivatives that W|A gives.

7. Dylan Boyd

Like Ian, I had them “discover” the initial rules for derivatives with several carefully chosen functions but not the structured graphing. They were repeatedly approximating the derivative through slopes of secants over small changes in x. I thought this too tedious, and it proved an obstruction to the pattern-seeking I was hoping they would engage in.

This year – lifting from your graphing exercise here – I created a few interesting graphs (an exaggerated cubic, a quartic, and a composition of functions with weird stuff) in a grapher (Desmos), printed copies, and had them use a ruler to sketch tangents and approximate their slopes, plotting just like you did. The goal then was to get a fine-tuned sense of sketching the derivative from the function (eg. exactly how high/low do the bumps go?). We spent another day sketching derivatives, especially special cases like sharp corners and asymptotes, and discussed plots of time vs. distance from the start for an exciting race they wrote a story for. Then we went back to manually plotting slopes of tangents like you did here – this time with the explicitly explained goal of finding rules for these sketches so we could stop approximating from poorly drawn manual graphs or using the slope formula for nearby secants. Tomorrow, it’s to the lab for the Geogebra Checker 🙂

As the comments from Jim and Heather point out, these can take a lot of time. I have the luxury of not teaching this calculus course toward the AP exam, so however far I get is however far I get and no one has any issues. However, there were some *really* good, forward-thinking questions posed during the manual portions – (“Is every function a derivative of some other function, but to a higher degree?” – antiderivatives! with no prompting! woo hoo!)

And for the physics students how came in already bragging about the Power Rule, I remained non-committal and insisted that they confirm or refute, beyond doubt, that their theory fit the practice. They remained game and I hope will have more connections between these rules and the meaning of change as a result. Time will tell…

8. Reading this I have mixed feelings:

1. This is awesome! We should all teach more like this.

2. I started the year teaching 5th graders the order of operations like this. I’m currently out of a job.

Seems “using the right vocabulary” is more important than getting children to learn. I didn’t teach OoO using PEMDAS (we were working through discovery and learning about the power of parenthesis) and I used “job” to describe the operation between two numbers: “3+5 is 3 and 5 doing the job of adding together and getting 8.”

I’m still heartbroken.

But reading this and seeing others still trying gives me hope.

Keep it up!