Point P lies on a lateral face of this prism. Find the volume of the pyramid with vertex P and with base lateral face ABCD.
[Here is some given information from the diagram: This is a rectangular prism, or both, with dimensions 8, 2, and 4 1/2.]
Notice that I didn't specify which lateral face of the box is also the base of the pyramid, even though I can see which face it is on my own calendar. That's because it's irrelevant to the problem -- the volume of the pyramid is one-third that of the box no matter what. (There are three different possible pyramids depending on which face is the base -- the three pyramids each have a different base and height yet the same volume.)
Thus the volume is:
(4 1/2)(2)(1/3)(8) = 24.
(This is the order in which I'd multiply these four numbers in my head.) Therefore the volume is 24 cubic units -- and of course, today's date is the 24th. The volume of a pyramid appears in Lesson 10-7 of the U of Chicago text.
Chapter 6 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Relationships." Here's how it begins:
"In the previous chapter we saw how crucial it is to consider whole systems of interactions rather than people or events in isolation. The idea of considering thins in relation to each other is one of the important basic principles of modern mathematics."
So this chapter is all about relationships. The first interaction between events that Cheng considers is the vicious cycle. She draws a diagram with arrows, but I can sum it up in words right here:
When I feel bad, then I overeat.
When I overeat, then I feel bad.
Cheng explains:
"Some people don't suffer from either arrow: emotions don't cause them to eat, but also, when they do overeat they don't feel bad about it. Suffering from one arrow is unfortunate but at least it doesn't cause the escalation that the two cause in conjunction."
She tells us that one arrow ("I overeat" -> "I feel bad") represents "feelings," while the other ("I feel bad" -> "I overeat") represents "action." Here's her next example:
When Alex feels disrespected, then Alex can't show love. (action)
When Alex can't show love, then Sam feels unloved. (feelings)
When Sam feels unloved, then Sam can't show respect. (action)
When Sam can't show respect, then Alex feels disrespected. (feelings)
Cheng's next example is very relevant to recent events in the news:
When police feel threatened by black people, police aggressively defend themselves against blacks. A
When police aggressively defend themselves against black people, blacks feel threatened by police. F
When black people feel threatened by police, blacks aggressively defend themselves against police. A
When black people aggressively defend themselves against police, police feel threatened by blacks. F
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 two weeks and skip all posts that have the "Eugenia Cheng" label.
How do we end such a vicious cycle? Cheng writes:
"Some people argue that black people should simply 'do what they're told' by the police. But tragically, there are well-documented examples of black people being shot by police even when they were doing what they were told."
She tells us that it's important to understand that this is a cycle, and claiming there is one root cause is an oversimplification, unless we acknowledge that the cycle itself is the root cause.
In the next section, the author writes about category theory. As we recall from her first book How to Bake Pi, Cheng is a category theorist. So it's understandable that she continues to write about category theory in her next two books. She explains her field as follows:
"Category theory is a field of modern mathematics that brings relationships to the forefront. In this approach, the framework for thinking starts with deciding what objects and relationships we're going to focus on."
In her first example, she writes down the factors of 30:
1, 2, 3, 5, 6, 10, 15, 30
Then she draws a diagram in which arrows join numbers to their factors. Again, I don't draw the diagram, so let's follow the author as she describes it:
"We now see that this has the structure of a cube -- a more interesting structure than just some numbers listed in a straight line. We can then think about the hierarchy of these numbers in the picture."
(We just saw a similar hierarchy in last week's Lesson 2-7 -- the Triangle Hierarchy. The big example of a hierarchy, of course, is the Quadrilateral Hierarchy of Lesson 5-2.)
At the bottom of the factor hierarchy we have 1. Then Cheng writes:
"At the second level we have the factors 2, 3, and 5 because nothing goes into them except 1. That is, they are prime numbers."
The next level up has:
6 = 2 * 3
10 = 2 * 5
15 = 3 * 5
Finally at the top we have:
30 = 2 * 3 * 5
Cheng tells us that the cube structure appears because 30 has three different prime factors. Other such numbers will produce the same pattern, such as 42. (At the Dozens Online forum, we'd say that 30 and 42 have the same abstract prime factorization or APF, namely {1, 1, 1}. Another accepted name for APF is "prime signature.")
42 = 2 * 3 * 7
has the following factors, in order of size:
1, 2, 3, 6, 7, 14, 21, 42
When she draws the same cube diagram for 42 as she does for 30, she points out that 7 is on the second level and 6 on the third, even though 6 < 7. She explains:
"If we also represented the size hierarchy, we would have to skew the diagram to look like this, a cuboid rather than a cube."
Here Cheng uses "cuboid" to represent what Pappas calls a (rectangular) "prism" and what the U of Chicago text calls a "box." Officially, a cuboid need not even be a prism (much less a box) -- it's only necessary that its six faces be quadrilaterals.
Returning to 30, Cheng redraws the picture so that each number is represented by the set of its prime factors -- so that {2, 3, 5} is at the top, and then {2, 3}, {2, 5}, {3, 5} right below it. At the bottom of the diagram she draws the empty set, a symbol often rendered as
"Here
The author draws a similar diagram for 42 ({2, 3, 7}), and then generalizes to {a, b, c} where these can be prime factors or any objects.
And of course, Cheng's next example is all about privilege. So the three objects that she considers for a, b, and c are "rich," "white," and "male." She draws this diagram twice -- first with "rich white male" at the top, and then subsets of this through the diagram with
Cheng points out that perhaps a cuboid diagram (as for 42) may be appropriate for privilege:
"Many people would argue that rich white women have higher status then rich black men, for example, and that rich black men in turn are better off in society than poor white men (not just in terms of wealth). It turns out that money goes a long way towards mitigating other problems."
And so she redraws the privilege diagram yet again as a cuboid. "Rich white male" remains at the top, followed closely by "rich white non-male." Then "rich non-white male" is next, followed by "rich non-white non-male." Below this is "poor white male" and so on. As Cheng writes:
"In particular, this provides a logic-based account of why some poor white men are so angry in the current socio-political climate -- because they are considered to be privileged from the point of view of number of types of privilege (white and male) but they are in reality less advantaged than many people who count as having fewer types of privilege than them."
Cheng finally draws one more cube diagram -- this one is more relevant in the context of feminism (such as last year's Women's March). All the members of this diagram are women, but now the three dimensions of privilege are "rich," "white," and "cis." She explains:
"This helps us understand why there is a lot of anger towards rich white cis women, among women activists who feel excluded by mainstream feminism."
She summarizes these charts as follows:
"Animosity tends to occur when someone is prone to thinking of themselves in a context that makes them underprivileged (a "victim") while others tend to view them in a context that makes them overprivileged."
And Cheng concludes the chapter with some advice for all of us:
"If we all become more adept at seeing things from both a privileged and a non-privileged point of view, we will achieve greater understanding of disadvantaged people's struggles but also of the actions, whether malicious or ignorant, that cause bigotry and oppression."
Today marks the start of the review for the Chapter 2 Test. Notice that Chapter 1 of the U of Chicago text had nine sections, and so there was no true review day for the Chapter 1 Test. But Chapters 2 through 6 each have only seven sections, and so the opposite happens -- there are two review days for each section.
In the past, I kept juggling around how I wanted to assess the first three chapters. The worksheet from last year prepares the students for a test with 11 questions -- and some of these questions are from Chapter 1, not just Chapter 2. But there's no harm in retesting Chapter 1 material again.
With two days of review, some teachers may use the extra day differently. Some, for example, might choose to cover Lesson 2-7 from the Third Edition of the text today. This section, "Conjectures," doesn't appear in my old Second Edition of the text. There's even an activity -- the Conjectures Game (or "Who Am I?") that I refer to several times on the blog -- that fits here. (How ironic is that? I have a Conjectures Game even though my book doesn't have a conjectures section.) Then the review worksheet can be given tomorrow instead.
This is what I wrote last year about today's worksheet:
Here is the rationale for which questions I decided to include on this review worksheet -- just as I did for the Chapter 1 Quiz, these problems come directly from the "Questions on SPUR Objectives" appearing at the end of each chapter.
For Chapter 1, I begin with Question 21, the three undefined terms (point, line, and plane), and then move on to Questions 26 and 32, two of the properties from arithmetic/algebra (Multiplication Property of Inequality and Substitution Property of Equality). Next are Questions 36-37, order on the number line -- except that I made the distances whole numbers, not decimals, and also I omitted point V from the second question, which serves no purpose other than to confuse and frustrate the students. Question 39 directs students to find the two points R on the number line that are the right distance from Q, and Question 41 is another distance question. Finally, I jumped to Question 61, another absolute value question similar to one that appeared on the Chapter 1 Quiz.
For Chapter 2, I begin with Question 16, which asks why the following definition is not a good definition of triangle: "A triangle is a closed path with three sides." The problem is, what exactly is a "closed path"? We're not allowed to give definitions containing words that also themselves need definitions. Question 20 asks the students to rewrite a statement in if-then form, then Question 30 reminds students that just because a conditional p=>q is true, it doesn't mean that its converse q=>p must be true.
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