Cartoon Calculus: Ch 3 (Area)

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

RS: I read Chapter 3 and I like it lot. The only thing is that the word “integration” doesn’t actually seem to appear. In fact “integral calculus” seems to get introduced by itself on page 40, but it seems like it could be jarring. Instead how about saying something like “calculating the total amount of something by combining (aka ‘integrating’) all of the parts just makes sense. And that’s why we call it integral calculus.” YB: Well, we do mention “integrals” in the first chapter, pages 5-7. But you’re right that this is a bit jarring. It might be tough to include an explanation (it’s a lot of text, and IMHO it might just add to the confusion because “integrating” isn’t really that common of a word and can be mistaken because of phrases like “integral to” and “radical integration”) but perhaps it would help to change the top line of p31 to “the big idea in integral calculus is that…”? Update: I’ve suggested to Grady that we add something about “integration” to the top of p32.

MT: For Chapter 3 (area): We like that the chapter starts with material that was covered in the previous chapter and builds on to what students already know or have just learned. We also really like how on page 31 the central idea of the chapter is explicitly stated and presented; this gives students (including ourselves) a clear understanding of what is about to be covered in this chapter… Overall, we really enjoyed reading chapter 3, and think that all of the explanations are both engaging and interactive. Like with chapter 2, we think a few application problems at the end of chapter 3 would be invaluable to students who struggle with math. YB: Good! But I don’t think we’ll have space for problems.

Page 29: Nothing could be simpler…

Pages 30-31: In the last chapter… // The big idea here…

RC: Nice explanation of the purpose of the book in the beginning, but I noticed some things are quite well explained, especially in the beginning, others are lightly touched on, and some of things are in the glossary and some not (for instance epsilon-delta). Sometimes the glossary expounds or clarifies a term, sometimes it’s actually better explained by your text. How does the novice math learner know whether to stop reading and learn more? Ignore it? Look in the glossary? Does a “big idea” signal the fundamentals?

Pages 32-33: Using this chopping technique… // The basic idea was nice expressed…

MB: I like Cavalieri! I had never heard of him, And the approximations to pi; very nice. YB: Thanks!

MT: We appreciate the historical context given on pages 33 and 34 and how the characters who are introduced in those 2 pages are included for the rest of the chapter. YB: Good!

Pages 34-35: In fact, the same basic idea… // Their basic approach was to take a circle…

MT: One thing that stood out to us is on page 35, when it says “the area of any circle = pi * r^2…” and then “…so the area of this circle = pi.” It makes sense to us why the area of that circle is pi, because 1 squared just equals 1, but we feel like it could stump some students, especially those who do not have a solid foundation in geometry. To prevent any possible confusion, we think it could be helpful to write out the actual math for why the area = 1, similar to how the math is written out on page 39 for the Pythagorean theorem. I’ve attached to this email a screenshot of what it could possibly look like (excuse my terrible drawing skills; I’m awful at drawing with a mouse on a computer!). YB: Good point. I don’t think the exact solution you suggest is going to work (because we’d need to put all that text inside the circle, and there’s no space), but maybe we can change the text to: [outside circle] The area of any circle is pi*r2, and for this circle r=1… [inside circle, as-is] …so the area of this circle = pi. It’s not quite as hand-holding as what you suggest, but I think it might be the best we can do.

MT: Still on page 35, in the comic second from the top, we were slightly confused as to why it is not explained how the area of each triangle = 1/2. Although the formula for the area of a triangle is given, students who struggle with math do not know the base and height of the triangle when only two arms are provided. Not sure how this can be resolved, but it can definitively confuse some students. (We do acknowledge that the math is explained on page 39, but think it should be included on page 35). YB: I will discuss with Grady. We could edit to something like “area = 1/2 * 1 * 1 = 1/2″. The question is whether it’s worth the extra text, especially if we also explain (per PH’s suggestion above) that the area of the square is 4. But I’m inclined to adopt both of these suggestions. (Grady, maybe it’s possible to do some of this in the light font you use on p39?)

PH: Explain that area of square with sides of length 2 is 4 because 4 = 2 x 2. YB: The side here is not 2, but I see your point and will suggest a fix to Grady.

FC: P.34-40 – Q: Is the “ancient Chinese gent” wearing Japanese getas? – Traditional footwear cultural sensitivities. YB: Good catch! I asked a Chinese friend, and he said: “I agree that the two high heels on Liu’s right shoe’s make him look like a Japanese. You can cut the heels for an easy fix.” We will work to fix.

DLJ: I’m 44 pages in, and one objection, one suggestion. (Overall it’s excellent — and I think your idea of getting the ideas first before the uses is absolutely first rate.)… Here’s the objection. Historically the way they estimated PI was from both sides, i.e. calculating bigger than the inscribed polygons and smaller than the enclosing polygons of increasing numbers of tangents. You’ve elided this, and you shouldn’t… One of the Great Lies Teachers Tell is that lying is simplifying and simplifying is making things easier to understand. The appropriate simplification is sometimes, but not always, pedagogically useful, but a lie is never a simplification. An untruth necessarily makes the whole story more complicated. (Remember Harry Truman, “I don’t have to take notes ’cause I don’t tell lies.”)… You’re going to get one kid in every classroom who knows you’ve cut corners and he’s going to put up his hand and cause a fuss with the teacher, and all the usual stuff from there on out…I’m not suggesting a major revision. It shouldn’t need more than one or two panels of the comic, saying “And a bunch of other guys were doing it from the outside in…” Quite possibly you can use the outside and inside to make the point that although the two never meet, still they are approaching the same limit… Here endeth my complaint. Now, my suggestion: Pythagoras appears in one panel as they’re calculating those smaller and smaller triangles, but you never say what he’s all about. It’s worth a note. There are thousands of proofs of a^2 + b^2 = c^2. (I don’t know whether you can fit one into the margin. Maybe squeeze one in by lubricating it with some Eul.) Einstein came up with one when he was a teenager and Ulysses. S. Grant did a different one, though I don’t know whether it was as General, President, or student… [Later: I was wrong about Ulysses S Grant coming up with his own proof of Pythagoras. It was some other President, maybe Grover Cleveland. When I come across it again I'll get it right and to you. On Pythagoras, my idea was that he might be explained as a marginal note when you first us him to say a^2 + b^2 = c^2. I saw him as a head sticking itself out from the side of the page, bending the cartoon borders back to stick his head out -- so that you don't have to renumber all your pages among other things. Once you've had him stick his headout once, then you could make a running gag out of him, perhaps talking about math in the ancient world, or different proofs of the Theorem through the years, or...] Can you fit something about it in? One way might be to have a totally different character turning back the visual edge of the “paper of the page” and sticking its head out. It could maybe say “Hey, ya wanna hear about Pythagoras? I’ve hidden him on page…”) YB: (1) Note that on page 35 we hint at the idea that you can also bound pi from above: “We can also show that pi < 4!" (2) Note also that we're already making other simplifications, e.g., Archimedes (and I think Liu Hui as well) started with a hexagon, not with a square. (3) The idea about Pythagoras is interesting, but I'm not quite sure what you're suggesting. What would you like us to say about him?

RC: On the subject of limits introduced on page 19, 27, 73, then detailed throughout the book: I like the way it’s introduced slowly and than repeated throughout. Additionally, one the easiest ways I’ve found to understand limits is to ask what happens if one suggests dividing by zero, which is undefined. I like it because it’s easy to explain, very intuitive, and easy to graph. YB: Hmm… I think I’m going to avoid the dividing by zero idea because that suggests (at least to me) a continuity argument, i.e., that dividing by zero should be the limit of dividing by smaller and smaller fractions. In my mind that conflates limits and continuity.

Pages 36-37: Their real genius… // …to a hexadecagon…

BG: yay for pizza! YB: Indeed :)

Pages 38-39: There are two obvious questions… // First, calculate the height.

WM: I like the details on Archimedes’ estimate of pi and how it relates to finding areas of nonlinear figures. ( I also like that you cued the reader that he/she could skip the details on how the areas of the triangles were calculated.) YB: Thanks!

MT: On page 38, because the triangle inside of the square is called “the inside one,” we were a little confused as to why the second triangle in the octagon is called “the other triangle” and not “the outside triangle.” Therefore, we suggest that “other” should be replaced with “outside”–to make: “To get the area of the outside triangle.” YB: Agreed! Great suggestion.

Pages 40-41: The second obvious question is… // The way we use integral calculus…

BG: nice using the word “limited” here. YB: Thanks!

RS: I think the joke about “limited sort of way” on page 41 is deflating. It seems to be saying to the naïve reader “we are closer, but we’ll never get there.” What about a different joke, such as “We’re almost to the limit of what we want to cover in this chapter!” YB: Hmmm… well, so far that’s one vote Yes and one vote No. Let’s see if there are any more votes, but for now I’m voting to leave it as-is.

MT: On page 40, “integral calculus” needs to be defined in the glossary, especially because this chapter dumps a lot of new terms on students, and therefore as much clarity as possible is a must. YB: Yes, we will do this. (Comment moved to glossary.)

Page 42: Next we’ll tie everything together…

RS: On the last page of the chapter you reference the “standing on the shoulders of giants” quote. I think this is kind of an in-joke for people who know about this quote from Newton, and tracing it back to Bernard of Chartres. It’s a tremendously important concept for human progress and I think it deserves more than a reference. Like, maybe an asterisk and a small box referencing that? YB: We can reference this in the online page notes or maybe in the glossary entry for Newton, but we try not to include asterisks or boxed explanations. But thanks for the idea, and especially for the reference to Bernard of Chartres!

Quick links to Front matter, Back matter, and:
Part One: Ch 1: Introduction, Ch 2: Speed, Ch 3: Area, Ch 4: Fundamental theorem, Ch 5: Limits
Part Two: Ch 6: Derivatives, Ch 7: Toolkit, Ch 8: Extreme, Ch 9: Optimization, Ch 10: Economics
Part Three: Ch 11: Hard way, Ch 12: Easy way, Ch 13: Revisited, Ch 14: Physics, Ch 15: Conclusion

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