Notice that this isn't the first time I've subbed in this class. In my May 31st post, I wrote about a special ed teacher for whom I subbed that day -- except that during first period conference, I had to cover today's media teacher because he was running late. This time, I'm in the computer room the entire day.
8:15 -- This is the middle school that starts with homeroom and first period everyday. First period turns out to be a digital video class. The students already have projects to work on, so all I do is make sure that they don't start playing games or YouTube on the computers.
9:20 -- All periods except for first rotate at this school. On Thursdays, fifth period is after first, and fifth is the teacher's conference period. This leads directly into snack.
10:30 -- Sixth period arrives. This is the first of two web design classes. The students are to create a web page all about how they would celebrate their ideal birthday.
11:25 -- Sixth period leaves and second period arrives. This is the second of two digital video classes.
12:20 -- Second period leaves for lunch.
1:05 -- Third period arrives. This is a multimedia class. It's also the only eighth grade class -- all the other classes are for seventh graders. (Just as with last week's art class, the seventh grade media classes are part of an exploratory wheel. Hence they last for only one trimester.)
The eighth graders are to create their own song using Garageband software (for the Mac). If they finish, they should write two more songs. In order to inspire these students, I decide to create my own song -- except instead of Garageband, I use -- you guessed it -- Mocha!
And now you're probably thinking -- don't say I found another excuse to mention Mocha music! But once again, I play Mocha in an actual classroom, so it's not as if I'm blogging about music that I'll never play on a school campus. (As usual, I have more to say about what I play later in this post.)
2:00 -- Third period leaves and fourth period arrives. This is the second of two web design classes.
2:55 -- Fourth period leaves, thus ending my day.
Normally, I'd like to focus on the next New Year's Resolution in order:
4. Begin the lesson quickly instead of having lengthy warm-ups.
But it's difficult to justify focusing on this resolution today, since there's nothing like a Warm-Up in today's computer class. Let's save this resolution for my next math class. Instead, I wish to write about a special issue today.
In the regular teacher's notes, he warns me that second and third periods are the two classes I need to watch out for. So we'll revisit those two classes today.
And there's an additional instruction that he wants me to enforce. In addition to forbidding YouTube and games (especially Fortnite), I must require the students to sit in their assigned seats and not allow them to help students in other rows. And this is without actually leaving me a seating chart.
Fortunately, for the seventh grade classes, I quickly figure out that all the seats are for the most part in alphabetical order. But of course, the students find excuses to change seats anyway.
I know that trying to explain why they should sit in their seats leads only to argument. And so I write the following on the board:
Here are the exact words left by your teacher:
"They can only help those people sitting next to them, not their friend in the front row, without your permission."
He knows that many of you students will try to take advantage of a sub, and so his rules for today are much stricter than normal. I'm only enforcing the rules left by your teacher, so please don't make me out to be the bad guy! I'm sure that if your teacher returns tomorrow, you'll be allowed to help students in other rows as usual.
Actually, I don't write it on the board since there is no whiteboard in this classroom. Instead, I create a temporary Word document and then project it on the screen.
Notice that this response is intended to counter the common student argument, "But the regular teacher normally lets us move seats!" On previous subbing assignments, sometimes the regular teacher expects me to enforce rules that he doesn't often enforce himself, out of fear that the students might take advantage of me. But then this leads to the students' complaint that I'm wrong to enforce these extra rules not enforced by the teacher. And in the past, I'd respond by arguing -- which then leads to yelling.
(A common example of this is a restroom rule -- the regular teacher allows passes because he or she knows which students to allow and which ones ask to go too often. But then that teacher tells the sub not to let anyone go, to which the students respond that the regular teacher usually lets them go.)
It appears that this message works. Most of the seat moving ends at this point, and the only students whose names I must write have nothing to do with changing seats. (I end up naming sixth period as the best class of the day, followed closely by first period.)
One group of students starts playing loud music on YouTube as well as the forbidden students. As it turns out, the boys complete the project and thus have nothing else to do. This leaves me in a rather tricky situation. First, I like to reward students who work hard and finish quickly by writing their names on a good list. Second, the teacher said not to go on YouTube or games, but what else are the students to do after they've finished? And third, even if I arbitrarily decide to allow students who finish early to go on YouTube, surely playing music loudly enough to disturb others is enough to land the names on a bad list.
In the end, I decide to leave the names for the teacher as "questionable" and then explain exactly what has happened today. Then it's up to the regular teacher to decide whether to reward the boys for working hard on the project or punish them for going on YouTube and playing loud music.
Third period -- the other class I'm warned to watch out for -- is the only eighth grade class. Unlike the seventh grade seating charts, this chart apparently has nothing to do with alphabetical order. (This might be in part because there are about three or four special ed students in the class, and all of them must sit together with their aide.) Thus enforcing the seating chart (which once again, I don't actually have a copy of) is much more difficult here than in the Grade 7 classes.
One person tells me that he's already finished his song. After the second period experience, I tell the student to write yet another song (which he can submit as extra credit).
When I play my song on Mocha, many of the students enjoy it. The Mocha song does its magic -- the students are inspired to write their songs, and overall behavior in this class improves.
What exactly does today's Mocha song sound like? Since I'm right in the middle of the class, it's too much trouble to write the actual song randomly generated by Mocha. Also, I don't recall whether I used the 12EDL or 18EDL scale. I do remember using the version that I posted back in my Halloween post -- the one that includes eighth and sixteenth notes.
So let me post a song similar to what I play in class earlier today. I already mentioned my May 31st post (from the day I first visited this classroom) and in that post, I introduced the 18EDL scale. Thus I'll use 18EDL for today's song. (I know, I know -- when I first posted these scales, I claimed that the scale I'd use the most would be 12EDL. And yet every time I'm in an actual classroom, I keep going back to 18EDL.)
The assignment given by the regular teacher is for the first song to have 24 measures. The program that I wrote on Halloween generates only 16 bars -- and that's in 2/4 time. The songs the students create are mostly in 4/4 time, so I should count this as only eight bars, not 16. Yet I know that I only run the program once in the actual classroom, and so we'll just stick to eight bars for today's post:
10 CLS
20 N=8
30 FOR X=1 TO 44
40 READ A,T
50 SOUND 261-N*A,T
60 NEXT X
100 DATA 13,2,15,4,14,2,11,1,9,1
110 DATA 17,2,18,2,17,2,13,4,9,3
120 DATA 14,1,14,4,11,3,14,1,11,4
130 DATA 9,3,17,1,17,4,11,1,15,3
140 DATA 13,4,14,3,16,1,9,4,15,3
150 DATA 11,1,12,2,18,2,16,4,15,2
160 DATA 17,6,18,4,15,4,14,4,12,2
170 DATA 10,2,10,3,12,1,13,3,11,1
180 DATA 13,2,18,2,13,4,18,16
Since this song contains more short notes (sixteenth and eighth notes), it's much more suitable for adding lyrics to than those generated by my old program (with many dotted half and whole notes).
But notice how difficult it is to create and play music in Mocha. We run my Halloween code to create random music -- and then I write more code that places these notes into the DATA lines so that running this program over and over plays the same song. Otherwise, running the original code (the random generator) over and over creates a new song, with the first song gone forever.
That, of course, isn't how Garageband works. The user decides which notes he or she wants to add to the song. When the user presses "Save," the music is preserved. One doesn't have to write new code just to preserve the song he or she created.
So a new project for us (albeit a very difficult one) is to write a program in Mocha that works the same way as Garageband. We can let the user choose which notes to play, or add random notes in predetermined scales (such as 12EDL or 18EDL). But then the player should be able to save these notes so that they can be played over and over, as real songs are. I might do so in future posts.
Before we leave today's computer class, I can't help but think back to the coding teacher at the old charter school from two years ago. Today's class is yet another example of what his class might have looked like if only our students had been better behaved.
Today on her Mathematics Calendar 2019, Theoni Pappas writes:
The area of this circle is 48pi sq. in. The circle is tangent to the sides of [i.e., inscribed in] an equilateral triangle. How long is each side of the triangle?
First, we notice that since the area of the circle is 48pi, we have:
pi r^2 = 48pi
r^2 = 48
r = 4sqrt(3)
So 4sqrt(3) is the apothem of the equilateral triangle. The one tricky thing about equilateral triangle problems (and these are frequent on Pappas) is that we know that 30-60-90 triangles appear somewhere in this triangle, but it's tricky to prove. A full proof goes back to the fact that the angle bisectors of any triangle meet at the center of the inscribed circle. This implies that the apothems and angle bisectors divide the triangle into six congruent 30-60-90 triangles, with the apothem 4sqrt(3) appearing as the shorter leg.
Thus the longer leg of the 30-60-90 triangle is sqrt(3) times as long, or 12 inches. A side of the equilateral triangle is twice as long, or 24. Therefore the desired side length is 24 inches -- and of course, today's date is the 24th.
Lesson 9-7 of the U of Chicago text is called "Making Surfaces." In the modern Third Edition of the text, making surfaces appears in Lesson 9-8.
This lesson is all about making nets that can be folded to form polyhedra and other surfaces. Some figures have much simpler nets than others.
Today's lesson naturally leads to an activity, but unfortunately, today isn't an activity day. (Last year it was -- recall that last year's Friday activity lessons land on Thursdays this year.) And so I must create a new worksheet for this year. But I retain some of last year's activity as part of today's bonus questions.
25. A regular polyhedron is a convex polyhedron in which all faces are congruent regular polygons and the same number of edges intersect at each of its vertices. There are only five regular polyhedra; they are pictured here.
a. Determine the number of vertices of each regular polyhedron.
b. Determine the number of edges of each regular polyhedron.
Ah -- we've seen these before. The five regular polyhedra are also called the Platonic solids. I've mentioned these in previous posts -- three summers ago we explained why there are only five of them, and two years ago we discovered that there are six regular polytopes in four dimensions. The Platonic solids are the tetrahedron, hexahedron (cube), octahedron, dodecahedron, and icosahedron.
The Exploration section continues thusly:
In 26-30, use cardboard and tape to construct a model of the regular polyhedron from the net provided. The patterns below should be enlarged. Cut on solid lines, fold on dotted lines.
Many teachers have given Platonic solid lessons in their classes. Since I don't want to try to create the nets myself, I link to previously made lessons. The first page, based on Question 25 (counting the vertices and edges) comes from the following link -- an elementary school in Washington State:
http://wilderptsa.ourschoolpages.com/Doc/Math_Adventures/Platonic_Solids.pdf
Question 26-30, the nets themselves, come from the following link:
https://www.math-drills.com/geometry/net_platonic_solids.pdf
The Math Drills link provides two nets for the dodecahedron. I chose the second one, since it more closely resembles the net in the U of Chicago text. On the other hand, their icosahedron net is very different from ours in the U of Chicago text.
Several members of the MTBoS have had Platonic solid activities in their own classes. Let's link to some of them:
Our first link is to Pamela Lawson, a Maine charter high school teacher. She taught her class about the Platonic solids about three years ago today:
https://rawsonmath.com/2016/01/26/how-do-we-know-that/
https://rawsonmath.com/2016/02/07/more-3d-geometry/
Notice that these posts were part of the 2016 MTBoS Blogging Initiative. (And no, there's still no sign of a 2019 Initiative.) Lawson begins:
I’m teaching this 12 week geometry class focusing on 3-dimensional figures. It’s a brand new class, like many at Baxter Academy, so I get to make it up as I go. Since our focus is on 3-dimensional figures, I thought I would begin with some Platonic solids. So I found some nets of the solids that my students could cut and fold. Once they had them constructed, there was a lot of recognition of the different shapes and, even though I was calling them tetrahedron, octahedron, and so on, many of my students began referring to them as if they were dice: D4, D8, D12, D20. Anyway, I must have made some statement about there only being 5 Platonic solids, and they now had the complete set. One student asked, “How do we know that? How do we know that there are only 5?” Great question, right?
(She's teaching a 12-week Geometry course? That's right -- hers is one of the rare high schools that uses trimesters!) Of course, I'd already give a full explanation here on the blog, just after Independence Day in 2015. Let me repeat parts of that post here:
Legendre's Proposition 357 states that the sum of the plane angles that make up a solid angle must be less than [360 degrees]. He proves this essentially by "flattening out" the solid angle -- he takes a plane that intersects all sides of the solid angle and uses the previous Proposition 356 (which we've already proved here on the blog) to show that each plane angle of the solid angle is less than the same angle projected onto the new plane. A good way to visualize this is to imagine that the solid angle is formed at the vertex S of a pyramid -- the points A, B, C, etc., mentioned Legendre can be the vertices of the base of the pyramid, and the point O can be any point in the plane of the base -- for example, the center of the polygonal base.
I won't take the time to show the full proof of Proposition 357, but I will mention an application of this theorem. Suppose we want to figure out how many Platonic solids there are. Recall that a Platonic solid is a completely regular polyhedron -- all of its faces are congruent regular polygons. As it turns out, we can use Proposition 357 to find all of the Platonic solids.
We start with the equilateral triangle, with each angle measuring 60 degrees. Now each vertex of our Platonic solid forms a solid angle. We need at least three plane angles to form a solid angle, but there is an upper limit to how many plane angles there can be. Proposition 357 tells us that the plane angles must add up to less than 360 degrees, and since each angle is 60 degrees, there must be fewer than six of them (since 6 times 60 is 360). So there can be three, four, or five 60-degree plane angles. The Platonic solid with three 60-degree plane angles is the tetrahedron, with four is the octahedron, and with five is the icosahedron.
If we move on to squares with their 90-degree angles, we can have three 90-degree plane angles, but not four (since 4 times 90 is 360). Three 90-degree plane angles gives us the cube. Regular pentagons have 108-degree angles. Again, we can't have four of them (since 4 times 108 is more than 360), and three 108-degree angles gives us the dodecahedron. Regular hexagons have 120-degree angles, but 3 times 120 is already 360. Since each solid angle must contain at least three plane angles, we are done, since increasing the number of sides in the polygon only increases the angle. Therefore, there are only five Platonic solids -- tetrahedron, cube, octahedron, dodecahedron, and icosahedron.
Returning to 2019, let's go back to Euclid, who gives definitions of the Platonic solids:
We notice that the tetrahedron is missing. According to David Joyce, Euclid refers to the tetrahedron merely as a triangular pyramid. In Book XIII, he also proves that these are the only five Platonic solids -- and there, he refers the tetrahedron simply as "pyramid."
Since I don't wish to jump to Book XIII of Euclid, let's look at the next proposition here in Book XI:
As usual, let's modernize the proof:
Given: l, m intersect at B, n, o intersect at E, l | | n, m | | o (lines not all coplanar)
Prove: The angle between l and m is congruent to the angle between n and o.
Proof:
Statements Reasons
1. bla, bla, bla 1. Given
2. Choose A, C, D, F on l, m, n, o 2. Point-Line-Plane, part b (Ruler Postulate)
so that AB = DE, BC = EF
3. ABED, BCFE are parallelograms 3. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
4.
5. AD = BE, BE = CF 5. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
6.
7. AD = CF 7. Transitivity of Congruence
8. ADFC is a parallelogram 8. Parallelogram Tests, part d
(one pair of sides is parallel and congruent)
9. AC = DF 9. Parallelogram Consequences, part b
(opposite sides of a pgram are congruent)
10. Triangle ABC = Triangle DEF 10. SSS Congruence Theorem [steps 2,2,9]
11. Angle ABC = Angle DEF 11. CPCTC
We can't help but notice that the six points A, B, C, D, E, F are the vertices of a triangular prism. And indeed, we see that the translation that appears in the U of Chicago definition of prism is the same translation that maps Triangle ABC to Triangle DEF.
Of course, this requires us to show that if two lines are parallel, then a translation must map one line to the other. I've alluded to the proof of this in posts from previous years, but I no longer include it as part of our curriculum.
This year, I'm keeping two of last year's prisms -- the tetrahedron and the icosahedron -- along with a new lesson worksheet:
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