If an edge of this octahedron is 4 units, find its volume to the nearest whole unit.
Yes, we study volume in Chapter 10 of the U of Chicago text. But there's hardly any mention of an octahedron anywhere in the text, much less a formula for its volume.
Indeed, in Lesson 9-7 (in the Exploration Exercises), the five regular polyhedra are listed. Also called Platonic solids, these are the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Even though no volume formulas are given for these polyhedra, we do notice that the cube counts as a prism, while the tetrahedron counts as a pyramid. Thus the prism and pyramid formulas can be used to find the volumes of the cube and tetrahedron.
The octahedron is the third simplest of the five Platonic solids to find a volume. It has two properties that will help us in calculating its volume
- It is the disjoint union of two solid pyramids. Thus we can use the pyramid volume formula.
- It is the dual polyhedron of another Platonic solid -- the cube. This means that the center of each face of a cube is a vertex of a octahedron (and vice versa).
Of course, by "octahedron" Pappas means regular octahedron -- we wouldn't be able to find a volume unless we assume that it's regular. After all, the centers of each face of a general parallelepiped are the vertices of an octahedron, but that won't be a regular octahedron (unless the original solid parallelepiped is a cube). We use the symmetry of the regular polyhedra in order to derive a volume.
We notice that four of the vertices of the octahedron are coplanar -- and by a symmetry argument, they are also the vertices of a square. And the side of this square equals the edge of the octahedron -- namely 4 units. So its area is 16 square units. This square also equals the base of the two pyramids which join to form the octahedron. Thus all we need to find the volume is the height.
But now we view the octahedron from other perspective. Two opposite vertices of the square base of the pyramids, along with the two vertex points (top/bottom) of the pyramids, are also coplanar, and are also the vertices of this square. Half the diagonal of this square equals the height of the pyramids, and this is easy to calculate. The side of the square is 4, the diagonal is 4sqrt(2), and thus 2sqrt(2) is the height of the pyramids.
So the volume of the pyramid is:
V = (1/3)Bh
V = (1/3)(16)2sqrt(2)
V = 32sqrt(2)/3
And the octahedron contains two of these pyramids, so the total volume is 64sqrt(2)/3. A calculator tells us that this works out to be about 30.17 cubic units. Pappas asks us to round this off to the nearest whole unit -- of course, she means the nearest whole cubic unit.
Therefore, the desired volume is about 30 cubic units -- and of course, today's date is the thirtieth.
Yesterday was the little-known Christian holiday known as "Michaelmas." I mention holidays on the blog only in connection to academic calendars, and indeed, there are two types of schools which observe Michaelmas:
Notice that this is not that much different from the definition given in the U of Chicago text. Of course, a proof must be a finite sequence of statements -- proofs can't go on forever! The U of Chicago states that valid justifications in proofs include postulates -- note that "axiom" is basically another word for "postulate," as we just found out in Cheng -- and theorems already proved (the "previous statements" mentioned above). But what about the third justification mentioned in the text -- definitions? Strictly speaking, a definition is also a special type of axiom, called a "definitional axiom." And of course, the last statement in the sequence is "Phi" -- that is, the statement that we're trying to prove!
Now Section 3-2 of the text mentions two theorems, the Linear Pair and Vertical Angle Theorems. But I left these out, since they didn't fit on my Frayer model page from last week. But those theorems certainly fit here in 3-3, for after all, the first example of a proof in the text is that of the Vertical Angle Theorem.
The proof of the Vertical Angle Theorem in the text is sort of a hybrid between a paragraph proof and a two-column proof. The conclusions and justifications aren't written in two-column form, but since each conclusion is followed by its justification, it might as well be a two-column proof. Subsequent proofs in this section are essentially paragraph proofs -- actual two-column proofs don't appear until Section 4-4.
One thing I like about this section is that it gives the reasons why anyone would want to write a proof. The first one is:
"What is obvious to one person may not be obvious to another person. Sometimes people disagree," like the Monty Hall Problem, for example.
As I mentioned before, when asked what the least favorite part of geometry class is, a very common answer is, proofs. But this is what professional mathematicians do all day -- a common joke is that a mathematician is a machine for turning coffee into theorems. (This line is usually attributed to the 20th-century Hungarian mathematician Paul Erdős.) Recall that a theorem is a statement that has been proved -- so Paul is telling us that mathematicians are machines that prove things.
Indeed, some of the most famous math problems in the world are proofs. About 400 years ago, a French mathematician named Pierre de Fermat (actually he was a lawyer -- but then again, both lawyers and mathematicians are known for proving things) made a very innocent-looking statement:
"No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two."
But Fermat was unable to write a proof of this statement -- at least, not a proof that he could fit in the margin of the book he was reading. It was not until 20 years ago when a British mathematician named Andrew Wiles finally proved of Fermat's Last Theorem. His proof is extremely complicated -- no wonder it took over 350 years for anyone to prove it!
Even today there are statements that appear to be true, but no one has proved them yet. The Clay Mathematics Institute has offered a prize of one million dollars to the first person who can prove each of the seven Millennium Problems (so called because the prize was first offered at the start of this millennium). So far, only one of the problems has been proved, so six million dollars remain unclaimed.
And we can go from problems that take years -- or even centuries -- to prove, to some which take a few hours to solve. Every year on the first Saturday in December, college students from around the country participate in the Putnam competition. There are twelve questions -- most or all of which are proofs -- and six hours in which to solve them. And if you can get even one of the twelve questions correct, then you will have one of the top scores in the country!
Now let's compare this to the attitude of many high school geometry students -- mathematicians may spend hours, years, even centuries to write a proof, yet the students can't spend a few minutes proving the Vertical Angle Theorem?
There's a wide range of beliefs on how much proof there should be in a geometry course -- from David Joyce, who believes that anything that can be proved should be proved as soon as possible, all the way to Michael Serra, who doesn't prove anything in his text until Chapter 14. On this blog, I'll take Joyce's approach, but only for proofs emphasized by the Common Core.
Yesterday was the little-known Christian holiday known as "Michaelmas." I mention holidays on the blog only in connection to academic calendars, and indeed, there are two types of schools which observe Michaelmas:
- British universities such as Cambridge and Oxford have a "Michaelmas term." It corresponds to "autumn term," "autumn quarter," or "fall quarter" at other schools or colleges. (I mentioned this back when we were reading the Stephen Hawking comic.)
- Waldorf schools have a "Michaelmas festival."
I once described Waldorf schools on the blog (since some traditionalists were discussing Waldorf as an alternative to public schools), but never "Michaelmas" in connection to Waldorf.
In fact, since yesterday was Michaelmas, I decided to read a few articles about Waldorf schools, and stumbled upon the fact that this month was the 100th anniversary of Steiner's original movement. The actual anniversary wasn't on Michaelmas, but ten days earlier. I didn't post yesterday, but I did post on the actual anniversary. I might have mentioned Waldorf that day if only I'd been aware that it was the centennial. Instead, I blogged mostly about palindromes that day. (Notice that Steiner's original Waldorf date was 9-19-1919, a palindrome in the m-dd-yyyy format.)
Waldorf schools are known for their particular jargon. In addition to "Michaelmas," if you're at a school and you hear terms such as "eurythmy," "main lesson book," or "pentatonic recorder," it's probably a Waldorf school. For some reason, Waldorf math classes use the phrase "four processes" to mean the four operations of arithmetic (addition, subtraction, multiplication, and division).
The schools are also unusual in that the cutoff date for school entry is June 1st. Although redshirting for kindergarten or first grade is becoming more common, this is usually for children with birthdays in July or August, not June. The strict Waldorf cutoff date means that that students born in June can't complete 12th grade until their 19th birthday.
But to me, what sticks out the most at Waldorf schools is "looping." In theory, a student is supposed to have a single teacher throughout "the grades," which means Grades 1-8. This means that a first grade teacher who enjoys little kids will find herself in seven years teaching teenagers. And the mathophobe who doesn't mind teaching the first "process," addition, to young kids will eventually be forced to teach Pre-Algebra.
I'm not sure whether I could ever agree with full eight-year looping as in Waldorf. On the other hand, perhaps some looping of subject teachers (like math, for example) might counteract the problems associated with having more than one teacher in elementary school. (This is mainly geared towards traditionalists who wish to see elementary students tracked in each subject.) The only "looping" I've ever experienced as a young student is having the same teacher for second and fifth grades. (I also had the same teacher for specialized high school classes such as Calculus and French.)
Anyway, Waldorf schools are completely unlike anything I experienced. Some Waldorf ideas may agree with traditionalists, while others clash with traditionalism. Still, it's worth revisiting what Waldorf schools are during this centennial month.
Meanwhile, today is Rosh Hashanah, the Jewish New Year. Schools in some districts, such as LAUSD, are closed today. My old charter school closed for Rosh Hashanah -- and indeed, this is the latest in the school year that the holiday has occurred since 2016, the year that I taught at that charter.
Lesson 3-3 of the U of Chicago text is called "Justifying Conclusions." This actually corresponds to Lesson 3-5 of the new Third Edition. (Meanwhile Lesson 3-4 in the new edition is called "Algebra Properties Used in Geometry." This is broken off from Lesson 1-7, "Postulates," in my old edition.)
(Last year, I was reading Eugenia Cheng's third book on logic during Lesson 3-3. I'll preserve some discussion of what I wrote about her book during this cut-and-paste.)
This is what I wrote last year on this lesson:
Section 3-3 of the U of Chicago text is an introduction to proof. Because the Common Core Standards specifically mention statements that students are supposed to prove, that makes this section one of the most important sections of the book -- and that's why I cover this section before skipping to Chapter 4.
But what exactly is a proof? The following definition of proof comes from a professional mathematician:
A proof of a statement Phi consists of a finite sequence of statements, each of which is either an axiom, or follows from previous statements by logical inference such that Phi is the last statement in the sequence.
No, the "professional mathematician" I mentioned here isn't Cheng, but it's very similar to how she defines "proof" in her Chapter 2: "A proof is basically a whole series of implications strug together like this: A => B, B => C, C => D. We can then conclude that A => D."
(Last year, I was reading Eugenia Cheng's third book on logic during Lesson 3-3. I'll preserve some discussion of what I wrote about her book during this cut-and-paste.)
This is what I wrote last year on this lesson:
Section 3-3 of the U of Chicago text is an introduction to proof. Because the Common Core Standards specifically mention statements that students are supposed to prove, that makes this section one of the most important sections of the book -- and that's why I cover this section before skipping to Chapter 4.
But what exactly is a proof? The following definition of proof comes from a professional mathematician:
A proof of a statement Phi consists of a finite sequence of statements, each of which is either an axiom, or follows from previous statements by logical inference such that Phi is the last statement in the sequence.
No, the "professional mathematician" I mentioned here isn't Cheng, but it's very similar to how she defines "proof" in her Chapter 2: "A proof is basically a whole series of implications strug together like this: A => B, B => C, C => D. We can then conclude that A => D."
Notice that this is not that much different from the definition given in the U of Chicago text. Of course, a proof must be a finite sequence of statements -- proofs can't go on forever! The U of Chicago states that valid justifications in proofs include postulates -- note that "axiom" is basically another word for "postulate," as we just found out in Cheng -- and theorems already proved (the "previous statements" mentioned above). But what about the third justification mentioned in the text -- definitions? Strictly speaking, a definition is also a special type of axiom, called a "definitional axiom." And of course, the last statement in the sequence is "Phi" -- that is, the statement that we're trying to prove!
Now Section 3-2 of the text mentions two theorems, the Linear Pair and Vertical Angle Theorems. But I left these out, since they didn't fit on my Frayer model page from last week. But those theorems certainly fit here in 3-3, for after all, the first example of a proof in the text is that of the Vertical Angle Theorem.
The proof of the Vertical Angle Theorem in the text is sort of a hybrid between a paragraph proof and a two-column proof. The conclusions and justifications aren't written in two-column form, but since each conclusion is followed by its justification, it might as well be a two-column proof. Subsequent proofs in this section are essentially paragraph proofs -- actual two-column proofs don't appear until Section 4-4.
One thing I like about this section is that it gives the reasons why anyone would want to write a proof. The first one is:
"What is obvious to one person may not be obvious to another person. Sometimes people disagree," like the Monty Hall Problem, for example.
As I mentioned before, when asked what the least favorite part of geometry class is, a very common answer is, proofs. But this is what professional mathematicians do all day -- a common joke is that a mathematician is a machine for turning coffee into theorems. (This line is usually attributed to the 20th-century Hungarian mathematician Paul Erdős.) Recall that a theorem is a statement that has been proved -- so Paul is telling us that mathematicians are machines that prove things.
Indeed, some of the most famous math problems in the world are proofs. About 400 years ago, a French mathematician named Pierre de Fermat (actually he was a lawyer -- but then again, both lawyers and mathematicians are known for proving things) made a very innocent-looking statement:
"No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two."
But Fermat was unable to write a proof of this statement -- at least, not a proof that he could fit in the margin of the book he was reading. It was not until 20 years ago when a British mathematician named Andrew Wiles finally proved of Fermat's Last Theorem. His proof is extremely complicated -- no wonder it took over 350 years for anyone to prove it!
Even today there are statements that appear to be true, but no one has proved them yet. The Clay Mathematics Institute has offered a prize of one million dollars to the first person who can prove each of the seven Millennium Problems (so called because the prize was first offered at the start of this millennium). So far, only one of the problems has been proved, so six million dollars remain unclaimed.
And we can go from problems that take years -- or even centuries -- to prove, to some which take a few hours to solve. Every year on the first Saturday in December, college students from around the country participate in the Putnam competition. There are twelve questions -- most or all of which are proofs -- and six hours in which to solve them. And if you can get even one of the twelve questions correct, then you will have one of the top scores in the country!
Now let's compare this to the attitude of many high school geometry students -- mathematicians may spend hours, years, even centuries to write a proof, yet the students can't spend a few minutes proving the Vertical Angle Theorem?
There's a wide range of beliefs on how much proof there should be in a geometry course -- from David Joyce, who believes that anything that can be proved should be proved as soon as possible, all the way to Michael Serra, who doesn't prove anything in his text until Chapter 14. On this blog, I'll take Joyce's approach, but only for proofs emphasized by the Common Core.
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