Find the smallest natural number radius so a sphere's volume with this radius equals the volume of a cylinder of equal radius & height 4.
To solve this problem, we clearly need the volume formulas for a sphere and a cylinder, which we find in Lessons 10-8 and 10-5 of the U of Chicago text, respectively:
Sphere volume: V = (4/3)pi r^3
Cylinder volume: V = pi r^2 h
(4/3)pi r^3 = pi r^2 h
We are given that the height of the cylinder is h = 4, so let's substitute this into the equation:
(4/3)pi r^3 = pi r^2 (4)
(4/3)pi r^3 - 4pi r^2 = 0
Let's clear out the fractions and then factor:
4pi r^3 - 12pi r^2 = 0
4pi r^2 (r - 3) = 0
4pi r^2 = 0 or r - 3 = 0
The first equation gives r = 0, but of course our sphere doesn't have a zero radius. So we solve the other equation to get r = 3. Therefore the correct radius is 3 -- and of course, today's date is the third.
Chapter 13 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Analogies." Here is how it begins:
"We have seen that abstraction is how we get to the world where logic works. The abstract world is a world of ideas and concepts, removed from our concrete, messy world of objects."
And as the title implies, one major method of abstraction from the real world to the logical world is through the use of analogies. Cheng reminds us that there can be many different ways of finding an abstract version of the same situation:
"This doesn't mean that some are right and some are wrong, it means that different abstractions show us different things, and we should be aware of what we have lost and gained by doing it."
According to the author, we learn to abstract as soon as we learn to count. The number "two" is an abstract concept. Concrete instances of "two" include "two cookies" and "two bananas."
Cheng draws many diagrams in this chapter. Her first diagram looks something like this:
2 things
/ \
/ \
v v
2 cookies 2 bananas
She tells us that this diagram of abstraction looks a bit like a pivot:
"The number two enables us to pivot from a situation involving two cookies (perhaps one and then another one) to a situation involving two of something else."
Cheng now provides us a framework for analogies:
"The general situation is that we are making an analogy between concepts A and B, via an abstract principle X that is often implicit rather than explicit. The diagram looks like this:"
X
/ \
/ \
v v
A B
These diagrams don't look good in ASCII, and so I'm no longer drawing them on the blog. Instead, I'll describe each diagram to you.
For example, her next two diagrams have a + b and a * b on top, respectively. She writes:
"People often tell me that they lost [understanding of math] when 'numbers became letters.'"
But then she combines these two diagrams into a tree. At the top of this tree is a ( . ) b, where the symbol is actually a dot within a circle (that's even harder to draw in ASCII) that represents a binary operation that could be +, *, or something else. The author proceeds:
"One of the important lessons my PhD supervisor Martin Hyland taught me was the importance of finding the right level of abstraction for the situation."
For example, Cheng draws a tree with "2 apples" and "2 bananas" at the bottom. These are linked together by "2 fruits." Then this in turn, along with "2 chairs," are linked together by "2 things":
"If we only go up to the level of '2 fruits' we will get an analogy between 2 apples and 2 bananas, but will omit 2 chairs. In order to include '2 chairs' we need to go up further, to the level of '2 things.'"
Cheng's next chart is based on the factor "cubes" (trees) of Chapter 6. These trees demonstrate an analogy between the factors of 30 and the factors of 42. So she draws a new analogy tree with "cube of factors of 30" and "cube of factors of 42" on the bottom, linked by "cube of factors of a * b * c." In her next chart, this in turn, along with "cube of privilege" (as in rich, white, male), are linked together by "cube of subsets of {a, b, c}.
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next week and skip all posts that have the "Eugenia Cheng" label.
The author continues:
"Much of my argument about math comes from my view that math is a little removed from normal life, so if we pick low pivots we will not get very far out of math, perhaps only a far as physics."
Cheng's next example is very real in life. In this tree, "black people" is linked by "visible minorities," which in turn, along with "gay people," are linked by "minorities." This in turn, along with "women," are linked by "oppressed people." This in turn, along with "straight white men," are linked at the top by "people."
Now she asks how long to draw each of the arrows -- that is, should all the classes of people be written on the same level at the bottom of the chart? This leads to divisive arguments because people pick the level of abstraction that best suits their own argument.
The author reminds us about axioms:
"In Chapter 11 we talked about finding axioms for our personal belief systems, that is, the fundamental beliefs from which all our beliefs stem."
Cheng's example here is also divisive:
"When arguing about affirmative action on grounds of race, some people are opposed to it on the grounds that there are people of color who come from well-off backgrounds who need help much less than some disadvantaged white people. I still believe that we should try to help all people of color, and all people from less privileged schools, even if some of them don't 'need' it."
And her next example also relates to the current events of Washington DC:
"Being falsely accused of sexual misconduct is indeed a trauma that nobody should have to go through, but I believe we need to be concerned with the quantity of sexual misconduct going unstopped."
The author draws diagrams for these two examples. In the first tree, "social services" and "affirmative action" are joined together by "help people even though they might not need it." In the second tree, "cancer screening" and "sexual harassment" are joined by "take evidence seriously although it might cause unwarranted action." Then she writes:
"Opposite is the diagram of the different levels of abstraction producing different analogies:"
Then she combines the two previous trees, joining "help people even though..." and "take evidence seriously although..." with "avoiding false negatives is more important than avoiding false positives," which is one of her basic axioms.
In fact, she realizes that she favors compulsory voting, as in Australia, for the same reason -- she fears that without compulsory voting, there could be voter suppression:
"It is yet another situation where I care about preventing false negatives the most; I just hadn't realized that was the issue until someone pointed it out to me."
Two weeks ago, I mentioned how those who didn't support Obama were accused of racism. Now Cheng repeats this example along with the counterargument -- they didn't support him because "he was inexperienced." She also discusses the 2016 election, in which those who didn't support Clinton were accused of sexism. A counterargument is that "she was a liar."
"We can clarify this using diagrams. Someone might think they're applying a principle about inexperienced people in general, regardless of the fact that person A is a woman:
In her first chart, "an inexperienced woman A" is linked by "inexperienced people." In her second chart, "an inexperienced man B" is also linked by "inexperienced people." But in her third chart, sexism is at play, with "inexperienced women" placed as an intermediate level between "an inexperienced woman A" and "inexperienced people."
Cheng's next chart is the abstract version of this. "A person from group A doing X" is linked first by "people from group A doing X." This in turn, along with "a person from group B doing X," are now linked directly by "people doing X." She writes:
"If the person from group A is treated differently from the person in group B, it's a sign that the intermediate principle is at work, not the general one."
And she repeats her example from Chapter 3 about black people shot by police in the US. Now she moves on to the debate between science and religion.
In her first chart, "shouldn't believe religion" is linked by "shouldn't just believe books and teachers," and in her second chart, "shouldn't believe religion" is also linked by "shouldn't just believe books and teachers." But in her third chart, a "more nuanced principle" is placed as an intermediate level between "shouldn't believe religion" and "shouldn't just believe books and teachers." She adds:
"That more nuanced principle might be that we shouldn't just believe books and teachers unless they are backed up by reproducible evidence, but that still leaves the question of how we can tell if the evidence is reproducible."
For the next set of examples, Cheng introduces another symbol to denote the power relationship between a powerful group and an oppressed group. This is actually a triangle or delta symbol, but in ASCII let's represent it by the letter V.
So in her next diagram, "men V women" and "Oxbridge V non-Oxbridge" are linked by "privileged group V oppressed group." Notice that here "Oxbridge" refers to Oxford and Cambridge, the two most prestigious universities in England -- and indeed, Cheng admits that she has privilege as a Cambridge alumna. An American would likely replace this with "Ivy League V non-Ivy League." (I point out that Ivy League privilege has also come up during the current hearings and investigations in DC, because the accused is a graduate of Yale.)
Also, Cheng reminds us that she is an Asian person. Because of white privilege, she draws "white people V non-white people," but she also concedes that Asians are arguably more privileged than others among non-white people. Thus she draws "Asian people V black people."
So she uses this analogy to perform a pivot. "White people V Asian people" and "Asian people V black people" are linked by "privileged group V oppressed group."
And Cheng abstracts this even further. Everyone is less privileged than someone and more privileged than someone else. So group A: more privileged than you V you V group Z: less privileged than you.
So she uses this analogy to perform another pivot. "Group A V you" and "you V group Z" are linked by "privileged group V oppressed group."
In her next example, the author writes about the idea that everyone should take responsibility for themselves, therefore we should oppose universal healthcare. Again, she uses charts here.
In her first chart, "oppose universal healthcare" is linked by "everyone should take responsibility for themselves," and in her second chart, "oppose roads" is also linked by "everyone should take responsibility for themselves." But in her third chart, "everyone should take responsibility for their own optional extras" is placed as an intermediate level between "oppose universal healthcare" and "everyone should take responsibility for themselves." She adds:
"This actually explains the sense in which the healthcare denier thinks healthcare and roads are different."
Now Cheng tells us that we must pick the right level of analogy. She begins:
"My wise friend Gregory Peebles says analogies are like bridges that can take us anywhere -- so we'd better be careful what bridge we choose."
Here's how we usually use analogies in discussions:
A is analogous to B.
B is true.
Therefore A is true.
This is much less watertight than using an actual logical equivalence:
A is logically equivalent to B.
B is true.
Therefore A is true.
In other words, there is some implicit principle X:
A is true because of principle X.
B is also true because of principle X.
B is true.
Therefore A is true.
But as Cheng points out:
"There is now a logical flaw in the argument, which is that just because B is true it doesn't mean that principle X is true. In a sense we are trying to move backwards up the right-hand arrow."
X
/ \
/ \
v v
A B
For example, she draws a tree in which "straight marriage" and "gay marriage" are linked by "2 unrelated adults." But opponents of gay marriage disagree with this tree. For example, their erroneous tree might have "straight marriage," "gay marriage," and "incest" all linked by "2 adults." Or they might have "straight marriage" linked to "an unrelated man an woman."
Perhaps a correct way to draw this tree is to begin with "straight marriage" linked to "an unrelated man and woman." Then this in turn, along with "gay marriage," is linked to "2 unrelated adults." This in turn, along with "incest," is linked to "2 adults."
It's even possible to draw a tree incorporating various slippery-slope arguments. We continue by linking "2 adults" and "pedophilia" to "2 humans." Then this in turn, along with "bestiality," is linked to "2 creatures." Cheng points out that there was once a level below "an unrelated man and woman," namely "an unrelated man and woman of the same race." In all cases, the claim that going up one level necessarily involves going up more than one is an erroneous argument.
The author wraps up the chapter by discussing these implicit levels. She writes:
"Disagreements over analogies basically take two forms, as in the argument about gay marriage. It starts by someone invoking an analogy of this form:"
X
/ \
/ \
v v
A B
However, typically X is not explicitly stated. Now someone objects, either because they see a more specific principle W at work (between A and X), or they see a more general principle Y at work (that links X along with some objectionable thing C).
For example, is racism by whites against blacks the same as racism by blacks against white. She answers with another diagram. "Racism of white people against black people" is linked to "prejudice of privileged people against oppressed people." This in turn, along with "racism of black people against white people," is linked to "prejudice of people against people." The argument is really about which level of principle we should go up to.
Cheng concludes the chapter, as usual, with a summary and a preview:
"In the end the whole aim is to reach greater understanding of in what way situations are equivalent and in what way they are not. This is the subject of the next chapter."
Lesson 3-5 of the U of Chicago text is called "Perpendicular Lines." This corresponds to Lesson 3-8 in the modern Third Edition of the text. Meanwhile, Lesson 3-7 of the new edition is actually an introduction to "size transformations" (or dilations). This introduction is basically the same as Lesson 12-1 in my old version, and dilations are studied in more detail in Chapter 12 of both editions.
[2018 update: Last year I changed the way I covered Lesson 3-5, and so I had a long discussion about my old and new ways of teaching the lesson. I preserve some of this discussion here, if only because I'm still posting those old worksheets.] But first, let me post a YouTube video from Square One TV on perpendicular lines.
Officially, I'm doing Section 3-5 now, but then again, not really. Let's consider the contents of this particular section:
-- The definition of perpendicular has already been covered. I'm moved it to Section 3-2 when I defined right angles, because I wanted to get it in before jumping to Chapter 4 on reflections, since reflections are defined in terms of the perpendicular bisector.
Now as I imply in this post, most of what I write on this blog is derived from mathematicians like Dr. M and Dr. Wu, who have written extensively about Common Core Geometry. But my plan to include Perpendicular to Parallels as a postulate appears to be original to me. I've searched and I have yet to see any text or website who will derive all the results of parallel lines from a Perpendicular to Parallels Postulate. Then again, what I'm doing here is, in some ways, as old as Euclid.
Let's look at Playfair's Parallel Postulate again:
Through a point not on a line, there is at most one line parallel to the given line.
This is a straightforward, easy to understand rendering of a Parallel Postulate, and Dr. M uses this postulate to derive his Parallel Consequences. But let's look at Euclid's Fifth Postulate, as written on David Joyce's website:
http://aleph0.clarku.edu/~djoyce/java/elements/bookI/post5.html
That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles.
No modern geometry text would word its Parallel Postulate in this manner. For one thing, even though the use of degrees to measure angles dates back to the ancient Babylonians, Euclid never uses degrees in his Elements. So the phrase "less than two right angles" is really just Euclid's way of writing "less than 180 degrees." Indeed, in Section 13-6, the U of Chicago text phrases Euclid's Fifth Postulate as:
If two lines are cut by a transversal, and the interior angles on the same side of the transversal have a total measure of less than 180, then the lines will intersect on that side of the transversal.
But let's go back to the Perpendicular to Parallels Theorem as stated in Section 3-5:
In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other.
Now count the number of right angles mentioned in this theorem. The transversal being perpendicular to the first line gives us our first right angle, and the conclusion that the transversal is perpendicular to the second line gives us our second right angle. So we have two right angles -- just like Euclid! So in some ways, making the Perpendicular to Parallels Theorem into our Parallel Postulate is making our postulate more like Euclid's Fifth Postulate, not less.
Of course, if we wanted to make our postulate even more like Euclid's, we could write:
If a plane, if a transversal is perpendicular to one line and form an acute angle (that is, less than right) with another, then those two lines intersect on the same side of the transversal as the acute angle.
But this would set us up for many indirect proofs, which I want to avoid. So our Perpendicular to Parallels Postulate is the closest we can get to Euclid without confusing students with indirect proofs.
So this is exactly what I plan on doing. Since we're in Section 3-5, the section that has Perpendicular to Parallels given, I could include it here -- we don't need to worry about how to prove it since I want to make it a postulate. But I already said that I want to wait until Chapter 5 before including any sort of Parallel Postulate. And so the new postulate will be given in that chapter.
Returning to Lesson 3-5:
-- The Perpendicular Lines and Slopes Theorem must also wait. Common Core Geometry gives an interesting way to prove this theorem, but the proof depends on similar triangles, which I don't plan on covering until second semester.
So that leaves us with only one result to be covered in 3-5: the Two Perpendiculars Theorem:
If two coplanar lines l and m are each perpendicular to the same line, then they are perpendicular to each other.
This theorem doesn't require any Parallel Postulate to prove. Indeed, even though I just wrote that I don't want to use indirect proof, in some ways this theorem is just begging for an indirect proof:
Indirect Proof:
Assume that lines l and m are both perpendicular to line n, yet aren't parallel. Then the lines must intersect (as they can't be skew, since we said "coplanar") at some point P. So l and m are two lines passing through point P perpendicular to n. But the Uniqueness of Perpendiculars Theorem (stated on this about two weeks ago) states that there is only one line passing through point P perpendicular to n, a blatant contradiction. Therefore l and m must be parallel. QED
But this would be a very light lesson indeed if all I included is this one theorem. Because of this, I decided to include a theorem that I mentioned back in July -- the Line Parallel to Mirror Theorem (companion to the Line Perpendicular to Mirror Theorem mentioned a few weeks ago):
Line Parallel to Mirror Theorem:
If a line l is reflected over a parallel line m, then l is parallel to its image l'.
Lemma:
Suppose T is a transformation with the following properties:
- The image of a line is a line.
- Through every point P in the plane, there exists a line L passing through P such that L is invariant with respect to T -- that is, T maps L to itself.
Now all Common Core transformations satisfy the first property -- that the image of a line is itself some line. (Forget about that Geogebra "circle reflection" where the image of a line can be a circle, since that's not a Common Core transformation.) As it turns out, there are five types of Common Core transformations that satisfy the second property:
- Any reflection
- Any translation
- Any glide reflection
- Any dilation
- A rotation of 180 degrees
Here is a proof of the lemma. (By the way, "lemma" means a short theorem that is mainly used to prove another theorem.) Let l be the original line and l' its image, and let P be any point on l. Since l doesn't contain any fixed points, the image of P can't be P itself -- so instead, it must be some point distinct from P, which we'll call P'. So of course P' lies on l'. Now the point P lies on some invariant line L -- and by invariant, we mean that P' lies on it. Now through the two points P and P', there is exactly one line, and that line is L, not l. Since P lies on l, it means that P' can't lie on l. But this is true for every point P on l. For every point P on l, P' is not on l. So l can't intersect its image l', since every point on l fails to have an image on l. In other words, l and its image l' are parallel. QED
OK, that's enough from four years ago -- we don't need to go any further because we haven't actually learned reflections this year. So here are all the old worksheets from that year -- except that I replaced Line Parallel to Mirror with a worksheet that is more nearly aligned with Lesson 3-5. There are now questions referring to Perpendicular to Parallels, and Perpendicular Lines and Slopes.
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