The Concept of Signed Areas
In calculus, after first introducing the concept of signed areas, I came up with the “backwards problem” which really tested what kids understood. (This was before we did any integration using calculus… I always teach integration of definite integrals first with things they draw and calculate using geometry, and then things they do using the antiderivatives.)
I made this last year, so apologies if I posted it last year too.
Some nice discussions/ideas came up. Two in particular:
(1) One student said that for the first problem, any line that goes through (-1.5,-1) would have worked. I kicking myself for not following that claim up with a good investigation.
(2) For all problems, only a couple kids did the easy way out… most didn’t even think of it… Take the total signed area and divide it over the region being integrated… That gives you the height of a horizontal line that would work. (For example, for the third problem, the line would have worked.) If I taught the average value of a function in my class, I wouldn’t need to do much work. Because they would have already discovered how to find the average value of a function. And what’s nice is that it was the “shortcut”/”lazy” way to answer these questions. So being lazy but clever has tons of perks!
Motivating that an antiderivative actually gives you a signed area
I have shown this to my class for the past couple years. It makes sense to some of them, but I lose some of them along the way. I am thinking if I have them copy the “proof” down, and then explain in their own words (a) what the area function does and (b) what is going on in each step of the “proof,” it might work better. But at least I have an elegant way to explain why the antiderivative has anything to do with the area under a curve.
Note: After showing them the area function, I shade in the region between and and ask them what the area of that bit is. If they understand the area function, they answer . If they don’t, they answer “uhhhhhh (drool).” What’s good about this is that I say, in a handwaving way, that is why when we evaluate a definite integral, we evaluate the antiderivative at the top limit of integration, and then subtract off the antiderivative at the bottom limit of integration. Because you’re taking the bigger piece and subtracting off the smaller piece. It’s handwaving, but good enough.
In Precalculus, I’m trying to (but being less consistent) have kids investigate key questions on a topic before we formal delve into it. To let them discover some of the basic ideas on their own, being sort of guided there. This is a packet that I used before we started talking formally about polynomials. It, honestly, isn’t amazing. But it does do a few nice things.
Here are the benefits:
- The first question gets kids to remember/discover end behavior changes fundamentally based on even or odd powers. It also shows them that there is a difference between and … the higher the degree, the more the polynomial likes to hang around the x-axis…
- The second question just has them list everything, whether it is significant seeming or not. What’s nice is that by the time we’re done with the unit, they will have a really deep understanding of this polynomial. But having them list what they know to start out with is fun, because we can go back and say “aww, shucks, at the beggining you were such neophytes!”
- It teaches kids the idea of a sign analysis without explaining it to them. They sort of figure it out on their own. (Though we do come together as a class to talk through that idea, because that technique is so fundamental to so much.)
- They discover the mean value theorem on their own. (Note: You can’t talk through the mean value theorem problem without talking about continuity and the fact that polynomials are continuous everywhere.)
The Backwards Polynomial Puzzle
As you probably know, I really like backwards questions. I did this one after we did So I was proud that without too much help, many of my kids were really digging into finding the equations, knowing what they know about polynomials. A few years ago, I would have done this by teaching a procedure, albeit one motivated by kids. Now I’m letting them do all the heavy lifting, and I’m just nudging here and there. I know this is nothing special, but this course is new to me, so I’m just a baby at figuring out how to teach this stuff.
I really like the questions having them draw the graph that would produce a particular definite integral – very clever. You could also impose another constraint as a follow up question: a section of the graph is very steep; or the slope is “mostly” 0; or the graph has many x-intercepts (the more the better).
One question that I like for the polynomial handout is “what would be a nice x-value to evaluate”. Maybe followed by a question about whether the corresponding point is unique in some way (often a y-intercept, x-intercept, or vertex).
I like the idea of more constraints!!! My sheet was a 10-15 minute activity in class, but the concept of signed area is so crucial that I think the more they play around with it, the better it is for them. So I hope to add more on next year in your vein.