This is my second visit to this classroom. My first post was fairly recently, and indeed I wrote about my first visit in my December 19th post.
In the Eleven Calendar, today is Saturday, the second day of the week. Therefore we should look at the second New Decade's Resolution today:
Decade Resolution #2: We make sacrifices in order to be successful in math.
But it's difficult for me to do anything with this resolution today, since I'm in a class that's completely unrepresentative of what I'd like to teach someday. This resolution specifically mentions math, and today the students don't even have a math lesson. Two of the hours today have guest speakers to replace the regular lessons -- one is called "Get Safe" (where the lecturer explains the SAFE acronym Scan, Assess, Forecast, Execute) and the other is about occupations (where today's is "cleaning").
Since the students leave for P.E. and various other electives (including vocal music class -- the same one where students watched The Little Mermaid last week), for only one hour are all the students in the classroom for a lesson. That hour is filled with a packet on Martin Luther King, Jr. -- today is the civil rights leader's actual birthday, and his holiday is coming up on Monday.
I suppose I could have twisted this resolution so that it doesn't refer to math -- the students must make sacrifices to be successful at reading and completing the packet (which includes a word search). Or I could have mentioned that MLK himself made personal sacrifices -- allowing himself to be jailed for civil disobedience in an effort to get civil rights laws passed. But as usual, there are plenty of aides to work with the students, and so I don't have much opportunity to discuss sacrifices.
Lecture 8 of Michael Starbird's Change and Motion is called "Galileo, Newton, and Baseball." Here is an outline of this lecture:
I. Aristotle wrote that heavier objects fall faster than lighter ones.
II. In 1665, the plague closed Cambridge University and Newton spent a couple of years on his family farm thinking about mathematics and the universe. Perhaps years with less instruction would improve our creativity. (This is part of the reasoning behind the modern "gap year.")
III. Using Newton's laws, we can analyze the motion of a falling body.
IV. Throwing balls lets us analyze the paths that projectiles take.
V. Let's analyze the path of a fly ball, with the help of a graph. (I won't post the graph here of course, but at least here's something positive about baseball, amid all the recent news about MLB cheating.)
Today's lesson is all about the motion of objects. Starbird begins his lesson by telling us that Aristotle was wrong -- Galileo (1574-1642) proved that objects fall at the same rate. Later on, Isaac Newton (1642-1727), a co-founder of Calculus, formalized the laws of gravity.
The professor shows the following chart to show how fast objects freely fall on earth's surface:
In 1 second, an object is falling -32 ft./sec.
In 2 seconds, an object is falling -64 ft.sec.
In 3 seconds, an object is falling -96 ft./sec.
In t seconds, an object is falling -32t ft.sec.
v(t) = -32t
d/dt(-16t^2) = -32t
So now we can add total distance fallen to the above chart:
In 1 second, an object is falling -32 ft./sec. and the distance is -16 ft.
In 2 seconds, an object is falling -64 ft.sec. and the distance is -64 ft.
In 3 seconds, an object is falling -96 ft./sec. and the distance is -144 ft.
Now Starbird considers throwing a baseball upwards with initial velocity of 48 ft./sec.:
v(t) = 48 - 32t
p(t) = 48t - 16t^2
The professor moves on to a baseball that has been hit. We suppose that a baseball has an initial vertical velocity of 48 ft./sec. but its horizontal velocity is 100 ft./sec. Thus after t seconds, its position is (100t, 48t - 16t^2). The ball lands when height = 0.
48t - 16t^2 = 0
(48 - 16t)t = 0
t = 0 or 48 - 16t = 0
t = 3
Therefore the ball lands 300 feet away. At this point, Starbird shows several more graphs, so that we can imagine the perspective of an outfielder trying to catch the ball:
Slope of line of sight at time t is (48t - 16t^2)/(300 - 100t) = 16(3 - t)t/(100(3 - t)) = (4/25)t
So if the fielder is standing in the right place, the slope increases at a constant rate -- in other words, if d/dt(slope of line of sight) is constant. The professor tells a story about Willie Mays, who ran backwards to catch a fly ball in the 1954 World Series. Conclusion: Every outfielder knows Calculus.
Lesson 9-2 of the U of Chicago text is called "Prisms and Cylinders." Our text refers to both prisms and cylinders as "cylindric surfaces."
This is what I wrote last year about today's lesson:
This lesson consists mainly of definitions. Terms defined in this lesson are surface, solid, box, rectangular solid, faces, opposite faces, edges, vertices, and skew lines -- and that's just the first page! Another term defined in this lesson is parallelepiped.
Let's get back to Euclid's Elements, since I've started discussing it yesterday. We can finally look at the first two definitions in Book XI:
https://mathcs.clarku.edu/~djoyce/java/elements/bookXI/bookXI.html
Both Euclid and the U of Chicago distinguish between the boundary and the interior.
Here are the definitions of the two surfaces mentioned in the lesson title:
It's interesting to compare Euclid's definitions to the U of Chicago's. Euclid imagines a cylinder as a solid of revolution -- we take a rectangle and rotate it around one of its sides. (Don't forget that a rotation in 3D has an axis, not a center.) But in the U of Chicago text, we perform a very different isometry -- we begin with a circle and translate it out of the plane. David Joyce points out that Euclid's definition doesn't generalize -- it describes only right cylinders. Not only does the U of Chicago definition include oblique cylinders, but the idea of cylindric solids (or surfaces) extends to prisms as well.
By the way, Euclid defines a prism in terms of its faces. Since the lateral faces of Euclid's prisms are parallelograms (and not necessarily rectangles), oblique prisms are included.
Let's return to the theorems. Since we looked at Proposition 4 yesterday, let's try Proposition 5 today:
Here's a modern version of the proof. Notice that Euclid writes "For suppose they do not...," which implies that this is an indirect proof.
Given: Line AB perpendicular to BC, BD, and BE.
Prove: Lines BC, BD, and BE are coplanar.
Indirect Proof:
Assume to the contrary that they aren't. By the Point-Line-Plane Postulate part f, any three points are coplanar, and so let P be the plane containing B, D, E, and let Q be the plane containing A, B, C.
By Point-Line-Plane part g, since planes P and Q intersect (in B), they intersect in a line. Let F be another point on this line. So lines AB, BC, and BF are coplanar as they all lie in plane Q.
We are given that AB is perpendicular to both BD and BE, and so by yesterday's Proposition 4, AB is perpendicular to the entire plane containing BD and BE, namely P. By the definition of a line perpendicular to a plane, AB is perpendicular to every line in P through B, which includes BF.
But we are given that AB is also perpendicular to BC. Therefore AB is perpendicular to two lines, BC and BF, both in plane Q. This is a contradiction, since through a point on a line, there is exactly one line in the plane perpendicular to the line. Thus the assumption that BC, BD, and BE aren't coplanar is false. Therefore BC, BD, and BE are coplanar. QED
This proof isn't as difficult as yesterday's, but it is an indirect proof -- and we don't cover indirect proofs in our text until Lesson 13-4. It also requires yesterday's Proposition 4 in order to prove -- and Proposition 4 has a difficult proof.
Here is the worksheet for today's Lesson 9-2:
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