For once, today is a regular schedule -- no Digital Citizenship Tuesday, no Common Planning Wednesday, but just an ordinary Thursday. The rotation for Thursday is 0-1-5-6-2-3-4. For today's "Day in the Life," our focus resolution is:
3. Move on from past incidents instead of bringing them up with students.
7:15 -- Zero period is the first Algebra I class. This time, the students are working on a review worksheet with all three methods of solving systems -- graphing, substitution, and elimination.
8:15 -- Zero period leaves and first period/homeroom begins. This is the second Algebra I class.
9:20 -- First period leaves and fifth period begins. Since today is Thursday, the official rotation for the day is (0-)1-5-6-2-3-4. This is the teacher's conference period, followed by snack.
10:35 -- Sixth period arrives. This is the third Algebra I class.
11:25 -- Sixth period leaves and second period arrives. This is the final Algebra I class.
12:20 -- Second period ends and it's time for lunch. But this is an adventure. Because the regular teacher is in charge of ASB, he runs all activities -- including today's pumpkin pie eating contest held in honor of Thanksgiving.
I don't watch any of the contest. Instead, I supervise the ASB classroom. Some students enter the classroom, speak loudly, and try to hit other students. None of them are actual ASB students. I kick out a few of them, and in the end, security forces the other non-ASB kids to leave.
1:05 -- Third period ASB class proper begins. There are actually two large pumpkin pies left over, and so I serve a pie to the ASB students. They continue to work on posters.
2:00 -- Third period leaves and fourth period arrives. This is the Math Support class. I actually serve these students the second leftover pie. They continue working on ALEKS.
2:55 -- Fourth period ends, thus concluding my day.
There are two main classroom management issues today. The first is that the Algebra I classes are rather loud today. I've stated before that students tend to act up when they don't have something specific to do -- and from their perspective, giving them a worksheet is "nothing to do," at least when compared to the new lessons the last two days. Some students ask "Is this homework?" when I pass out the paper -- and technically, what they don't finish is indeed homework. But I fear that this will cause them to avoid doing any work at all so that today can be a non-academic free period. (This often occurred at the old charter school, and this led to arguments.)
In the past, I might have required the students to have certain questions finished in class, and write the names of those who don't on the bad list for the regular teacher. But I've found out that this leads to problems -- some students might skip the graphing questions and proceed directly to substitution, or skip substitution in favor of elimination. Even if I state which questions I'll check in advance, some students will inevitably solve different ones. Then it will take forever for me to check all the papers to determine whether they answered sufficiently many questions. Then the ones who don't will say that it's because they needed help and I didn't help them -- because I was too busy checking the other students' papers.
Perhaps the only thing I could have done better is repeat what worked yesterday. Even though this isn't new material, I should have treated it as if it were indeed new. During the entire period, I call on students to solve systems on the board and require them to be quiet while their classmates are working on them.
By the way, this worksheet directs the students to solve some systems by graphing, yet doesn't provide much room to graph. Some students use the ever-popular Desmos software to graph -- and this is good, because the SBAC calculator will be powered by Desmos. But then they keep on graphing the other systems on Desmos instead of using substitution or elimination.
The second management issue is in the non-Algebra I classes. Students believe that today should be a non-academic free day -- and it's all because of the pumpkin pie. It's difficult to work on posters or ALEKS when eating pie, but then they keep playing around even after finishing the pie.
Of course, today's pie is unexpected (or semi-expected). But it makes me think about days in a future classroom when I might intentionally bring pie to the class -- such as on Pi Day. Perhaps it's not a good idea to serve pie right at the start of class.
If I had my own classroom, when would I serve pie on Pi Day? That's easy -- 1:59 PM. Notice that at this school, 1:59 PM is just before the start of fourth period -- during passing period. (The upcoming Pi Day will fall on a Thursday, and so the Pi Day moment will indeed be during passing to fourth.) So today, by serving pie right at the start of fourth period, I mimic what I'd do in this class on Pi Day.
A real math teacher serving pie on Pie Day should know the schedule in advance. If 1:59 is right at the start of a period (or during passing period), then the teacher should probably expect the entire period to be a party period with no written work completed. In my own classroom, I'd probably declare the entire period to be "music break" and sing Pi Day songs the whole time.
At schools where 1:59 is a few minutes after the start of the period, then there might be time for a Warm-Up before pie is served. If 1:59 is deeper in the period, then there's time for a full task -- but of course, the task should be related to the number pi. I've posted suggestions for such activities on previous Pi Days. If there's nothing else to do, at least have the students measure the pie that they're about to eat.
But today, midway through the period I try to get the students to return to ALEKS (since the regular teacher's lesson plan demands it). A few minutes later, one group of girls claim that they just barely finished eating the pie -- in reality, only one girl has a pie plate on her desk. She keeps giving excuses why she eats her pie slowly -- but the honest truth is that from her perspective, eating fast means more math work, so she has every incentive to eat her pie as slowly as possible.
Because the morning math classes are on the loud side today, I tell third and fourth periods that "the best class of the day" race is between them. This is before I know there'd be leftover pie. In the end, I require the students to answer three ALEKS questions that I see on another student's Chromebook -- they must answer three questions by naming appropriate metric units. Since they finally complete the questions, I declare this class to be the "best class of the day." First period is the least obnoxious among the Algebra I class, so this is "second best class of the day"/"best Algebra I class."
The focus resolution is to avoid bringing up past incidents. This is a tricky one -- when telling classes that they're in the running for "best class of the day," I'm informing them what the other classes did so the current class can try to beat them.
But the emphasis of this resolution is to avoiding bringing up the past when telling a class that they're "just as bad" or "worse" than another class. It's not terrible to compare present students to the past if it's a favorable comparison (that is, "you're better than the other class" or "you can beat them").
I recall one recent subbing day when students were talking loudly and misbehaving. I was just about to tell them that I wrote students in another class on the "bad list" for doing the same thing. But fortunately, I stopped myself -- I told them that I'd write their names without mentioning anything about the earlier class, thus fulfilling the third resolution.
OK, that's enough Algebra I for today -- let's get back to Geometry. Today on her Mathematics Calendar 2018, Theoni Pappas writes:
Find x.
Not only is all the given info in the diagram, but it's given only as lengths where none of the points are labeled. So let me name all the vertices:
In Triangle ABC, AB = AC = 25. BC = 40.
BD = 24, where D is the foot of the perpendicular from B to line AC (which lies outside the triangle).
AE = x, where E is the foot of the perpendicular from A to line BC.
This is clearly intended to be one of those problems where the area of a triangle can be found in two different ways, since two distinct base-height pairs are given:
A = (1/2)hb
(1/2)(AE)(BC) = (1/2)(BD)(AC)
(1/2)(x)(40) = (1/2)(24)(25)
20x = 300
x = 15
Thus the desired value is x = 15 -- and of course, today's date is the fifteenth.
Notice that it's possible to solve this problem without using the triangle area formula. We observe that Triangle ABC is isosceles, and so altitude AE is also a median and BE is half of 40, or 20. This makes Triangle ABE a right triangle with legs x and 20 and hypotenuse 25. By the Pythagorean Theorem:
(AE)^2 + (BE)^2 = (AB)^2
x^2 + 20^2 = 25^2
x^2 + 400 = 625
x^2 = 225
x = 15
Fortunately, we obtain the same answer using either area or Pythagoras. But I've seen inconsistent Geometry problems where solving the problem in two different ways produces two distinct answers (because the question writer overlooked the second method). Typical problems where two different base-height pairs are given usually involve scalene triangles, not isosceles triangles.
Lesson 6-4 of the U of Chicago text is called "Miniature Golf and Billiards." In the modern Third Edition, we must backtrack to Lesson 4-3 to play miniature golf.
This is what I wrote last year about today's lesson:
Today we proceed with the next lesson in the text. Lesson 6-4 of the U of Chicago text is all about applying reflections to games such as miniature golf and billiards. I don't need to make any changes to the lesson, so I can just keep what I wrote last year for this lesson almost intact:
One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired exactly 28 years ago today (November 15th, 1990), was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."
Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.
The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:
"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce offAB at P and go toward the hole."
We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:
Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG
Statements Reasons
1. H refl. over line AB is H' 1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG 4. Transitive Property of Equality
Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.
Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:
"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]
"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"
In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.
In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.
For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page, so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.
Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.
Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.
Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.
One of my favorite TV programs is The Simpsons -- I've been watching it for decades. One of its earliest episodes, having aired exactly 28 years ago today (November 15th, 1990), was called "Dead Putting Society." In this episode, Bart Simpson is preparing for a miniature golf competition. His sister Lisa shows him how he can use geometry to help him make a difficult shot. After saying this, Bart proclaims, "You've actually found a practical use for geometry."
Lesson 6-4 of the U of Chicago text discusses miniature golf and billiards. Just as Bart learns in this above video, one can use geometry to determine where to aim.
The key is reflections -- one of the important transformations in Common Core Geometry. It is often said that when a ball bounces off a wall, the angle of incidence equals the angle of reflection. The text describes where to aim a golf ball G so that it bounces off of a wall and reaches the hole H:
"In this situation, a good strategy is to bounce (carom) the ball off a board, as shown [in the text]. To find where to aim the ball, reflect the hole H over line AB. If you shoot for image H', the ball will bounce off
We can write a two-column proof to show why the angle of incidence -- the angle at which the ball approaches the board, which is BPG -- equals the angle of reflection APH:
Given: H' is the reflection of H over line AB.
Prove: Angle APH = Angle BPG
Statements Reasons
1. H refl. over line AB is H' 1. Given
2. Angle APH = Angle APH' 2. Reflections preserve angle measure
3. Angle APH' = Angle BPG 3. Vertical Angle Theorem
4. Angle APH = Angle BPG 4. Transitive Property of Equality
Notice that for this proof, I've skipped a few steps. Technically, we should write that the reflection images of both A and P are the points themselves, since they lie on the mirror (Definition of Reflection), and so angle APH' is the image of APH (Figure Reflection Theorem). But I'm tired of writing that over and over again -- how much less, then, will the students want to write that.
Of course, this only works if the ball caroms only once. The U of Chicago text describes the game of billiards, where the player is required to bounce the ball off of three cushions. The text writes:
"Pictured [in text] is a table with cushions w, x, y, and z, the cue ball C, and another ball B. Suppose you want to shoot C off x, then y, then z, and finally hit B. Reflect the target B successfully over the sides in reverse order: first z, then y, then x. Shoot in the direction of B"' [...]
"Notice what happens with the shot. [...] On the way toward B"', it bounces off side x in the direction of B". On the way toward B", it bounces off y in the direction of B. Finally it hits z, and is reflected to B. [End of quote]"
In the video clip above, four caroms are required for the golf ball shot by Bart to find the hole. But unfortunately, the path of the ball as drawn on the show is impossible. To see why, let's label the direction from the starting triangle to the hole "North," and all the walls appear to meet at right angles, so they are all oriented in the north-south or east-west directions. Bart begins with the ball slightly to the right side of the starting triangle, so the initial direction of the ball is northwest. After hitting the first east-west wall, the ball is now traveling southwest. But then, after hitting the second east-west wall, the ball should be traveling northwest again. Indeed, we can use the Alternate Interior Angles Consequence and Test Theorems to prove that the path of the ball after hitting two walls should be parallel to the original direction of the ball. Yet the show depicts the ball as travelling due north after hitting two walls.
In fact, we can show that the only correct path to make the ball arrive in the hole involves hitting a north-south wall -- most likely the wall to the far west (in front of "Do not sit on statuary"). Only by hitting a north-south wall can the direction change from anything-west to anything-east, which is necessary for the ball to approach the hole. Lisa's advice to her brother is sound, but the way that it is animated is geometrically impossible.
For this worksheet, I decided to reproduce Bart's golf course, but tilt two of the walls in order for the path of the ball to be geometrically correct. I used equilateral triangle paper to create this page, so that the students will be able to figure out the paths without needing a ruler or protractor, for classrooms in which these are not available. Just as for Bart, four caroms are needed to get from G to H.
Working backwards on this worksheet, we can determine the path from A to H by reflecting H to the point H', then aiming from A to H'. But to determine a path from B to H using two caroms, we can't reflect H to first H' and then H", because H" would be well off the page. It may be better to aim from B to A', the reflection of A in the necessary wall.
Notice that if there are two walls and the paper is large enough, it may be actually possible to perform both of the necessary reflections. If the two walls meet at right angles, it is fairly easy to perform both reflections -- because the composite of those two reflections is a rotation. So to find the direction to aim at, we take the target point and rotate it twice 90, or 180, degrees around the point where the two walls meet. And if the ball is bouncing off of two parallel walls, then the composite of the two reflections is a translation, so we can just translate the target twice the distance between the two walls.
Officially, this lesson is the Guided Notes for Lesson 6-4 of the U of Chicago text. But this lesson naturally lends itself into a group activity -- the teacher can provide additional golf courses for the students to solve, or even allow the students to make up their own via the Bonus Question. It is a nice activity to give right before the week-long Thanksgiving break.
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