What is the sum of the midpoint's coordinates of
By the way, here is the given info from the diagram: A(-2, 0), B(3, 7).
The only reason I consider this to be a Geometry question is that the Midpoint Formula appears in Lesson 11-4 of the U of Chicago text. The formula given there is as follows -- the midpoint of the segment with endpoints (a, b) and (c, d) is ((a + c)/2, (b + d)/2).
Using that formula, we obtain ((-2 + 3)/2, (0 + 7)/2) or (1/2, 7/2) as the midpoint. The sum of these coordinates is 1/2 + 7/2 = 4. Therefore the desired sum is 4 -- and of course, today's date is the fourth.
Chapter 14 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Equivalence." Here's how it begins:
"One long-standing myth about mathematics is that it is all about 'getting the right answer.' That everything is simply right or wrong. Another pervasive myth is that it's all about equations."
And of course, one group of believers in those myths are the traditionalists. But Cheng mentions equations here to lead us to the subject of this chapter, equivalence. Equations tell us that two mathematical objects are equal, and now we wish to find out when two thoughts are equal:
"This idea continues into research-level math, where the senses of 'sameness' become more and more subtle, and increasing amounts of technical effort have to be put into finding and describing appropriate notions of sameness."
Cheng's first example involves handwriting:
"If I write the letter 'a' several times they'll all look slightly different, but a handwriting expert should be able to tell that they're all written by the same person."
There's no point in my trying to reproduce the author's a's here. So instead, let's proceed to her next example, which is all about Geometry:
"You might remember that two triangles are called congruent if they are exactly the same shape and size, that is, they have the same angles as each other and the same length of sides. If they have the same angles as each other but possibly different lengths of sides then one is a scaled version of the other and they are called similar."
Cheng includes a picture of two similar triangles -- you don't need to see that one. She also draws one more triangle:
"It's just the first triangle but flipped over sideways. Does flipping a triangle make it a different shape?"
Our upcoming Chapter 4 of the U of Chicago text is all about reflections. We learn that two objects are congruent if there is an isometry mapping one to the other. A reflection is an isometry, and so two mirror images of each other are indeed congruent. The idea that reflection images aren't "the same" is encapsulated by the idea of "orientation." The author's example includes a backwards S -- it doesn't have the same orientation as a forwards S.
Now Cheng begins to describe the famous recently solved Poincare conjecture. Last year, we had a side-along reading book all about Poincare, so I don't need to repeat it here. Recall that the Poincare conjecture is all about topology, and interestingly enough, Cheng starts writing about that field without ever using the word "topology." She even gives the example of a doughnut being equivalent to a coffee cup without any form of the word "topology":
"It doesn't mean that those spaces are the same; it just means that viewed in this particular light they can be seen as the same."
At this point, Cheng moves on to false equivalence as used outside of mathematics. She writes about how some people argue that boys and girls can play with the same toys if they want, while others declare that we should just "let boys be boys and girls be girls":
"It seems to me that they are equating the argument 'boys and girls can play with the same toys' with a desire to turn boys into girls and girls into boys. This is a false equivalence."
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.
Anyway, regarding this debate about gendered clothes and toys, she continues:
"When people get very upset about someone else's point of view it is worth trying to work out what the underlying issue really is, which might not be logical at all. It might be personal."
And so the author proceeds to write about how personal taste clouds our thinking and leads us to make false equivalences. Here is her first example:
I don't like toast.
I don't like people who like toast.
This is a false equivalence because Cheng hates only toast, not toast-lovers. On the other hand:
I don't like stealing.
I don't like people who like stealing.
In this case, the author hates both theft and thieves.
I don't want to be fat.
I think people who are fat are bad.
This is a false equivalence. Cheng abstracts from this:
A: I don't want to be X.
B: I think people who are X are bad.
Most of the time, B => A is true, but then people try to conclude the converse A => B. This is a logical error.
I like doing X every day.
I think everyone should do X every day.
If X is "playing the piano" then the first statement is true while the second statement is false (for Cheng, that is). But if X is "brushing one's teeth," then both statements are true (for the author):
"So much for the logic of the situation. It is an example of how I can always be right, by restricting my statement to my own personal taste or aspirations about myself."
Cheng now moves on to damning accusations, many of which are based on false equivalence:
"A sign that someone is about to make a false equivalence is when they launch in with 'You're basically saying...' which is a sign they're about to twist your words into something they're not."
Her examples include equating not wanting to be fat with fat-shaming/misogyny, and finding inheritance of large amounts unfair with desiring confiscation of all wealth at death. In general, an antagonistic argument driving by a false equivalence goes like this:
You are saying A.
A is equivalent to B.
B is bad.
Therefore you are a terrible person.
The logical error could appear in the second line, or third line, or even both. Cheng finds this to be the case for comprehensive sex education. (This issue is all over the news here in California due to a bill that comes into effect this year.) She writes:
"I don't think it is equivalent to condoning sex outside marriage, but I also don't think sex outside marriage is evil."
Cheng's next example of a logical fallacy is false dichotomy. For example:
A: Label some toys "for boys" and others "for girls."
B: Force boys to be girls and girls to be boys.
According to the author, a false dichotomy is in fact a false equivalence between one statement and the negation of the other. Another example is:
A: I am skinny.
B: I am fat.
Cheng draws several diagrams to illustrate this. In her first example, she shows a true dichotomy, with the universe shown as a pie divided completely between A and B halves. Then she draws two types of false dichotomies -- one in which A and B are two circles in the universe with a large "neither" region outside them both, and the other where A and B are two overlapping circles with a "both" region in between. She tells us that it's possible to be a false equivalence in both ways.
Cheng’s example of a false dichotomy involves dieting:
A: Some people should watch what they eat (because it helps them stay healthy)
B: Some people should not watch what they eat (because it hinders them)
She explains that this is a false dichotomy because both A and B are true – some people should indeed diet while others shouldn’t.
Now the author draws a new type of diagram. The following four phrases are in the four corners of this diagram:
Upper-Left: some people should watch what they eat
Upper-Right: nobody should watch what they eat
Lower-Left: everyone should watch what they eat
Lower-Right: some people should not watch what they eat
Along the top and bottom are double-arrows labeled “true dichotomy.” That’s because the UL and UR really are negations of each other, as are the LL and LR. Along the left and right are double-arrows labeled “false equivalence.” That’s because UL and LL aren’t equivalent, and neither are UR and LR (but detractors on both sides make these equivalences). Along the diagonal from UL to LR is a double-arrow labeled “no disagreement.” That’s because these two statements don’t contradict each other, yet Cheng writes:
“The really funny thing about this argument is that it usually turns into a meta-argument about whether or not we’re disagreeing. I try to point out that we’re both making the same point, and the arguer usually insists that we’re not.”
In other words, we go from the reasonable points:
A: Some people should watch what they eat (because it helps them stay healthy)
B: Some people should not watch what they eat (because it hinders them)
To these two absurd and antagonistic ones:
A: Everyone should watch what they eat.
B: Nobody should watch what they eat.
Cheng now redraws the diagram but highlights the other diagonal, from LL to UR. Another double-arrow is placed there, labeled “needless antagonism.” She writes:
“I suppose in many cases people are criticizing different choices, but it doesn’t have to be that way, if we don’t let false dichotomies push us to extremes.”
She now proceeds to discuss straw man arguments. False equivalences are a source of these, as people replace their opponents’ arguments with “straw men” that are easy to knock down. One example is dear to our hearts – the STEM debate. This stems from a more fundamental false dichotomy between:
A: Being creative.
B: Being logical.
That is, some people oppose our emphasis on STEM because they fear that they don’t allow students to be creative. The author draws another chart:
UL: creativity
UR: logic
LL: art
LR: science
Once again, there are false equivalences on the left and right. But this time, the top and bottom are false dichotomies. In reality, we aren’t forced to choose between creativity and logic, and neither must we choose between art and science.
Now here’s the big one that Cheng discusses: “Black lives matter” and “All lives matter.” She explains this in more detail:
“What the slogan ‘Black lives matter’ really means is ‘Black lives matter just as much as other lives, but are currently being treating as if they do not matter as much, and we need to do something to correct this injustice.”
But unfortunately, opponents of “Black lives matter” set up and attack a straw man instead – they interpret the name to mean “Black lives matter and other lives don’t.” So once again, Cheng draws a chart:
UL: black lives matter
UR: black lives don’t matter
LL: some lives don’t matter
LR: all lives matter
Once again, there are false equivalences on the left and right. Just as with the dieting chart, the top and bottom are true dichotomies. The diagonal from UL to LR is “no disagreement.” But again, she redraws the chart with the diagonal from LL to UR as “needless antagonism.”
Cheng tells us that opponents of “Black lives matter” should really debate the following:
A: Black lives matter just as much as other lives.
B: Black lives are currently being treated as if they do not matter as much.
C: We need to do something to correct this injustice.
The argument being made is:
A and B and C.
Cheng has little respect for those who refute A – she regards them as explicit racists. As for the other two statements, we can discuss why they disagree with B and C:
“Is it because they think that black people are bringing it on themselves? Is it because they do not think it is anyone’s responsibility to help other people, in general?”
She also tells us that we may need to have a discussion about:
1. Whether or not the Black Lives Matter movement is really synonymous with anger and aggression, and
2. When anger and aggression are reasonable.
But unfortunately, such discussions are too complex for interactions on social media:
“It is also too complex for people who are riled up by anger or fear. Complex arguments require a certain level of calm.”
The author now tells us that analogies often lead to false equivalence:
A is (falsely) equivalent to B.
B is true.
Therefore, A is true.
Or more likely, we’ll see:
A is (falsely) logically equivalent to B.
B is good/terrible.
Therefore, A is good/terrible.
Cheng returns to an example from earlier:
Saying we should remove gender labels from children’s clothes (A)
Is logically equivalent to
Saying we don’t want to just let girls be girls and boys be boys (B).
B is terrible.
Therefore, A is terrible.
She returns to the diagrams from the last chapter. We might have A and B be linked directly by X, or we might have A linked to Y first, and then have Y and B be linked to X.
Her next example is all about “mansplaining.” She writes:
“But sometimes the evidence that you don’t need the explanation is that you already said that very thing yourself.”
Some people try to make the counterargument that women might try to explain things to men just as much as men explain things to women. And Cheng draws a diagram. “Mansplaining” is first linked by “a man telling a woman something she already knows as part of a general pattern of men failing to give women credit for intelligence.” Then this, along with “a woman being patronizing to a man,” are linked by “a person being patronizing to a person.” Hence here’s how she addresses the counterargument:
“If we go all the way to the top level of this diagram then [mansplaining] is indeed analogous to women being patronizing to men, but this is a level of abstraction too high.”
The author warns us to be aware of false accusations of false equivalence. For example, she writes about a discussion over whether government should pay for higher education, as compared to government paying for healthcare. The difference would seem to be that higher education is “optional,” while going to the doctor isn’t “optional,” which is why government should pay for the latter but not the former. But is it as simple as it seems?
“I don’t think it’s as clear cut as ‘hypochondriacs vs. normal people,” and ‘sick people vs well people.’”
She now draws two charts. The first one (black and white logic) has rows for “hypochondriac” and “normal” and columns for “well” and “sick,” with “no doctor” in the normal-well cell and “doctor” in the other three. The second chart (fuzzy logic) contains “worrier” between “hypochondriac” and “normal.” “ropey” between “well” and “sick,” and “maybe doctor” between “no doctor” and “doctor.”
Returning to education vs. doctor, Cheng goes back to the arrow charts. On one hand, we might link “education” to “you decide whether to go” and “doctor” to “you go when it’s necessary,” but on the other hand, someone else might link both “education” and “doctor” to “you go when it’s necessary.” She agrees with the latter, but:
“I will concede that there might be some exceptions – rich people who go to university just for the sheer fun of it, as they will never need to depend on their education in their lives.”
Cheng’s final example involves manipulation. Recall that she’s British, and so one issue important to her is Brexit. Proponents regard their opponents as “unpatriotic,” but they counter that they think that Remain is best for the UK:
“It then comes back to an argument about what is best for the UK, which is what the argument should be about, rather than simple name-calling.”
For an American example, Cheng also discusses kneeling during the national anthem and whether or not this is “unpatriotic.” In the end, this is really an emotional argument.
Therefore, as usual, Cheng concludes the chapter with review and preview:
“Unfortunately, because of the power of emotions over logic, there are many people who are either unable to think logically enough, or prevented from doing so once their emotions are stirred up. In the next chapter, we will explore what a better interaction between emotions and logic could be.”
Lesson 3-6 of the U of Chicago text is "Constructing Perpendiculars." (It appears as Lesson 3-9 in the modern edition of the text.)
This is what I wrote last year on today's lesson:
Section 3-6 of the U of Chicago text deals with constructions. So, we're finally here. The students will need a straightedge and compass to complete this lesson.
Here's a good point to ask ourselves, which constructions do we want to include here? The text itself focuses on the constructions involving perpendicular lines. Well, let's check the Common Core Standards, our ultimate source for what to include:
CCSS.MATH.CONTENT.HSG.CO.D.12
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.
CCSS.MATH.CONTENT.HSG.CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.
But let's also go back to what David Joyce writes about constructions:
The book [Prentice-Hall 1998 -- dw] does not properly treat constructions. Constructions can be either postulates or theorems, depending on whether they're assumed or proved. For instance, postulate 1-1 above is actually a construction. On pages 40 through 42 four constructions are given: 1) to cut a line segment equal to a given line segment, 2) to construct an angle equal to a given angle, 3) to construct a perpendicular bisector of a line segment, and 4) to bisect an angle. Later in the book, these constructions are used to prove theorems, yet they are not proved here, nor are they proved later in the book. There is no indication whether they are to be taken as postulates (they should not, since they can be proved), or as theorems. At the very least, it should be stated that they are theorems which will be proved later.
David Joyce, after all, emphasizes that at the very least, constructions should be proved. He writes here that they can be proved later -- but of course, he prefers that theorems not be stated until they can be proved.
So which of the theorems in the Common Core list can be proved so far? Let's look back at that list one by one:
Copying a segment: This should be trivial to construct and prove. The student simply uses the straightedge to draw a line, marks a point O on it, and opens up the compass to the length of the given line segment AB to mark the second point P. The proof that these segments AB and OP have the same length simply follows from the definition of straightedge and compass. It's a bit surprising that the U of Chicago text doesn't begin with this as the first construction, as this one should be easy for the students.
Copying an angle: This is the one that we can't prove yet. The usual construction requires SSS to prove. This is one reason why Joyce would prefer that Chapter 8 of his text occur before these constructions in his Chapter 1.
Bisecting a segment; constructing perpendicular lines, including the perpendicular bisector of a line segment: This is the focus of Section 3-6 of the U of Chicago text. Notice that as soon as we've constructed the perpendicular bisector, we've already done the other two constructions (bisecting the segment and drawing its perpendicular). And so, as soon as we prove the perpendicular bisector construction, we are done.
Given: Circle A contains B, Circle B contains A, Circles A and B intersect at C and D.
Prove: Line CD is the perpendicular bisector of AB
Proof (in paragraph form -- can be converted to two columns later):
Just as in the proof of Euclid's first theorem (Section 4-4), since both B and C lie on circle A, AB = AC by the definition of circle, and since both A and C lie on circle B, AB = BC. Then by the Transitive Property of Equality, AC = BC -- that is, C is equidistant from A and B. So, by the Converse of the Perpendicular Bisector Theorem, C lies on the perpendicular bisector of AB. In the same way, we can prove that D also lies on the perpendicular bisector of AB. And through the two points C and D there is exactly one line -- and that line is the perpendicular bisector of AB. QED
In many texts, it's pointed out that the compass opening for the two circles need not be exactly the same as AB. All that's necessary is for the opening to be greater than half of AB -- that guarantees that the two circles intersect in two points.
The text states that the midpoint of AB has been constructed as a "bonus" -- so we've bisected the segment, as requested. All we need now is to construct perpendicular lines -- and that's exactly what the text does in the next example, construct a perpendicular to line AP through point P on the line, using our perpendicular bisector algorithm as a subroutine.
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