Tuesday, January 26, 2016

Lesson 12-2: Size Changes (Day 90)

Lecture 16 of David Kung's Mind-Bending Math is called "Surprises of the Small and Speedy." In this lecture, Dave Kung describes two of the most counterintuitive results in all of physics.

The first is Einstein's Theory of Relativity. I briefly mentioned Albert Einstein back in November, during the 100th anniversary of the publishing of this theory. Then in December, I wrote that I'd discuss Einstein in more detail on the day that I reach the corresponding Kung lecture. Well, that day has finally arrived.

Kung begins with the notion that light always travels at the speed of -- well, light. This seems obvious, but this notion has several implications. The first is called Lorentz dilation. (Hmmm, that word dilation sure sounds familiar, doesn't it?) He describes a light inside the top of a moving train and asks how long it takes for the light beam to reach the bottom of the train. To an observer on the train, the beam travels straight down, but to someone standing on the ground, the beam travels slanted at an angle. If this object had been a ball, the observer on the ground would come up with a faster speed for the ball, since the ball is traveling a longer distance (the hypotenuse of a right triangle) in the same time that an observer on the train sees the ball move vertically (a leg of that triangle).

But light, unlike a ball, can't travel at two different speeds. It can only move at the speed of light. So instead, the two observers see time move differently. The observer on the ground sees that a clock on the train ticks more slowly. Starting with the Pythagorean Theorem, Kung calculates that for a train moving at 98% of the speed of light, the Lorentz dilation factor is about 5. This leads to the famous Twin Paradox -- if two twins are separated at birth, with one of them traveling on a round-trip ride through space at 98% of the speed of light, and the traveling twin returns at the age of 10, then the earthbound twin won't be 10, but 50 years old.

Kung describes more paradoxes related to Einstein's relativity, then moves on to the Quick Conundrum, where he suspends a board from three strings tied to the ceiling. If he turns the board 360 degrees, the strings tangle and are impossible to untangle, but if he rotates it an additional 360 degrees, it's now possible to remove the tangle. Kung says that this leads to the other counterintuitive physical theory, Quantum Mechanics, which states that electrons can have either a half spin (tangled) or a full spin (untangled). (Recall that Relativity and Quantum Mechanics are the two theories that Stephen Hawking tried to unite in his proposed Theory of Everything.)

Kung moves on to show how Quantum Mechanics leads to more surprising results. The famous Double Slit experiment shows how light or electrons act as a wave passing through two slits at the same time, but when we try to measure them, then they behave as particles that can only pass through one slit at a time. The final result that Kung describes is the Heisenberg Uncertainty Principle, which states that we can't know the position and velocity of a subatomic particle at the same time. This leads to the following joke:

Cop: Do you know how fast you were going?
Driver: No, but I know where I am!

Today is the second day of my three-day subbing assignment in a sixth-grade class. And as promised, today I played the Conjectures/"Who Am I?" game that I mentioned in MTBoS Week 2. I knew that the students will have a quiz on Friday, so I wanted to make sure that the students are ready for it.

This teacher assigns warm-up questions, so I incorporated them into the game -- each of the three questions is worth a point, so I already awarded three points before asking the usual questions about my age and weight. But these questions were review from lessons much earlier in the year -- one required them to know how many feet are in a yard, and another involved percents. So students who couldn't remember these questions struggled just on the warm-up.

The main part of the game involved the assigned MathLinks worksheet which, as I mentioned yesterday, is all about expressions and equations. In particular, the front side of the worksheet had students convert from language to math, using a burger stand as an example. So, for example, the first question describes a customer who orders a hamburger and a regular fries. First the students must convert this into the expression h + r, then substitute in $5.25 for h and $1.85 for r to obtain the total cost of the order, $7.10.

According to the rules of my game, some questions are for all groups to answer, while others are more like a race. The first question from the worksheet is always for all to answer, so that all students can understand a new concept. I was going to make the fourth question the next question for all, but then first period struggled to figure out the third question, the expression 3(c + r + d) -- three combos, each with a cheeseburger, regular fries, and a regular drink. So I changed it from a race question to one for all, and made each group copy my explanation to get the point. This worked out well because then the next question for all would be the sixth question, which was a Common Core "explain your answer" question -- why are two bills calculated as 2(v + d) and 2v + 2d (where v is a veggie patty) equal? The answer, of course, is the distributive property -- one which students always have trouble with (as several groups tried to answer "commutative"). Then in subsequent classes, I adjusted so that questions 1, 3, and 6 are always the questions for all groups to answer.

Time was running out, and I had to rush a few questions from the second side of the worksheet -- one where the students are given an expression and must identify the number of terms, the constant term, the variable terms, and the coefficients. For example, the first question was 4a + 3b + 2c + 1. On one hand, I wouldn't have minded just skipping this side so that we can focus on the front side -- but I suspect that these will appear on Friday's quiz. But in rushing these questions, I can't be sure that the students fully understand the concept.

In fifth period -- the "Honors" class -- the students got through the examples faster, and so I was able to squeeze in an extra question from the back side of the worksheet. But this question provided an expression such as 2r + 3 + 14v + 1, where students must combine the like terms 3 and 1 before identifying the number of terms as 3 and the constant term as 4. At this point I had to rush it, and so even if the students combined the terms 3 and 1, they looked a little confused as to why 4 was the constant term (as opposed to there being two constants, 3 and 1). It also didn't help that in my first example, the constant was 1 -- making it appear that finding the "constant" means counting the number of terms without variables, so that the "constant" of 2r + 3 + 14v + 1 would be 2.

In some ways, playing this game takes extra time -- especially when I'm wasting time asking about my age and weight. But I think it's worth it -- the incentive of earning points encourages students to participate, whereas they might have just ignored me and talked the whole period -- maybe stopping just to copy my writing without thinking -- if I had just sat down and lectured to them. I feel better that they can answer a question from the front side on Friday's quiz, even if they have trouble with a question about constant terms on the quiz.

Let's get back to Geometry. Lesson 12-2 of the U of Chicago text is called "Size Changes Without Coordinates" -- and recall that "size changes" are what the text calls dilations.

Aha -- so that's where we've heard that word before. In a way, Lorentz dilations really are similar to Common Core dilations -- Lorentz scales all intervals of time by the same factor (depending on the speed of the object) while Common Core scales all distances by the same factor.

Much of today's lesson doesn't need to change from last year's lesson. But there's one activity about dilations that I found on the MTBoS. Yes, I mentioned that the teacher who posted above me during Week 1 (and therefore to whom I responded as part of my Week 1 obligation) had a dilation activity that I'd post on this blog as soon as we reached dilations. Well, that day is today:


Here's what the author Olivia writes about her activity:

At the beginning of the week, I assigned a dilation project in geometry.  Students were to pick a picture from the internet, draw a grid over top of it, then redraw the picture following the grid on a larger piece of paper.  My district does not have an art program, so many of the students are definitely not comfortable when it comes to art.  I heard a lot of negative comments that day from students saying they sucked at drawing, it was going to turn out horrible, and many pleas of students asking me to “please not hang them up!”  They were especially adamant that they WOULD NOT be putting their names on their pictures.  I told them it would be okay and they would turn out great.  I said that if I could do it, anyone could do it!  Well I gave them 2 full days of class time to work on their posters.  I hung up a couple of the posters after the first day, because two of my students finished theirs by working on it in study hall.  The next day, my other classes were all asking who drew what.  I said, sorry guys this class wanted to remain anonymous.  Well, my geometry class came in later in the day.  I told them all that people kept asking about who drew what but I did not rat anyone out about who drew what.  Then, a very exciting thing happened! Students started saying, “Mine looks so good, I’m definitely putting my name on mine” or “I want everyone to know who drew mine.”  They told me that it wasn’t as terrible as they thought it would be and it was actually fun.  I loved seeing students so excited and proud of their own work. It always makes my day brighter when students realize just what they can accomplish.  Everyone ended up putting their names on their finished products, and I was left as one happy math teacher.

Well, there's nothing stopping us from assigning this activity today. I obtained the pictures simply by performing a Google image search for "cartoon character" -- those just happened to be the ones that came up.

There's one thing about activity -- it works better on a coordinate plane. But note that Lesson 12-2 of the U of Chicago text uses Slope and Distance Formulas just to prove that the mapping from (x, y) to (kx, ky) is a dilation with scale factor k. Just as with translations earlier (which we proved using SAS Congruence), we can prove this dilation formula using SAS~ -- so it's good that I already covered SAS~ in yesterday's lesson.

Stay tuned for tomorrow's MTBoS, which will focus on my third day of subbing. And Kung's next paradox will be about one of my favorite topics -- Geometry!

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