Activity: The Four-Color Theorem and Reflections (Day 93)

Lecture 19 of David Kung's Mind-Bending Math is called "Crazy Kinds of Connectedness." In this lecture, Kung introduces us to the world of topology.

I mentioned topology back in October, while discussing Mandelbrot's fractals. Recall that a topologist is someone who can't tell the difference between a doughnut and a coffee cup. In other words, a doughnut and a coffee cup are topologically equivalent, because they each have one whole. There exists a transformation mapping one to the other called a homeomorphism.

In this lecture Kung describes some weird shapes -- indeed, he quips that a topologists' job is to find stranger and stranger figures. One such figure is the Mobius strip -- a figure created by taking a strip, putting a half-twist in it, and taping the two ends together. A Mobius strip is considered to have just one side and one edge. As the title of the lecture implies, a Mobius strip has a "crazy kind of connectedness," as it's connected, but not simply connected. On a plane or a sphere, a curve can be shrunk down to a point, but on a Mobius strip, the edge can't be continuous deformed to a point.

Another property of the Mobius strip is that it's non-orientable. We've seen the word "orientation" before regarding transformations -- reflections reverse orientation, translations preserve it. Here's why we say that a Mobius strip is not orientable -- take a triangle on a Mobius strip and reflect it, which would switch its orientation. It's now possible that we can slide (translate) this mirror image all the way around the Mobius strip until it coincides with the original preimage -- which would imply that the triangle has opposite orientation from itself! This contradiction implies that the Mobius strip is, in fact, non-orientable.

The Quick Conundrum involves a special type of knot -- a braid. Kung shows how to take a sheet of fabric with two slits and fold it until it creates the braid -- thereby implying that the double-slitted fabric and the braid are topologically equivalent.

Then Kung moves on to a famous theorem -- the Four-Color Theorem. Just like Lesson 12-7 "Can There Be Giants" from yesterday, the Four-Color Theorem appears in the U of Chicago text -- in fact it's Lesson 9-8. We skipped over it only because we omitted Chapter 9 entirely (as the rest of the material, 3D figures, are covered more thoroughly along with volume in Chapter 10).

The Four-Color Theorem states that any map can be colored with at most four colors. The U of Chicago tells us that the theorem holds on either a plane or a sphere. Kung points out that on other surfaces, different numbers of colors are required. On a Mobius strip, six colors are needed. Both Kung and the U of Chicago text mention that for a special doughnut shape (or is it a coffee cup shape?) called a torus, seven colors are necessary.

Today I subbed for a middle school special ed class who teaches reading and history. But during my sixth period conference, I was assigned to a sixth grade math class -- it's at the same school, but right next door to the class I covered earlier this week.

The teacher for this class left on short notice, so I wasn't sure of the correct assignment. Apparently, the previous night's homework was an introduction to exponents, but there was a stack of worksheets involving reflections on the front desk. I thought that reflections are a bit advanced for sixth grade, but the TA, an eighth grader, confirmed that they really were learning about reflections. And so I gave HW answers quickly and played Conjectures/"Who Am I?" game using the reflections worksheet -- points on a coordinate plane are given and students must reflect them over one of the axes.

The class was already divided into six groups of six students each. Of these, one of the groups finished all of the questions quickly and were way ahead of the questions I was asking during the "Who Am I?" game. Another group, meanwhile, was struggling. My game is designed to help me assist struggling groups, but it was tough because this group was lost while two of the other groups were misbehaving -- one by making too much noise, another by tearing up the worksheet into small scraps of paper and throwing them like spitballs.

Here I identify some of the problems with today's game -- the start of the class was chaotic as I was trying to figure out what to do with the homework. Also, I believe that this game works most smoothly with groups of four, not six. But I had to go with the table groups into which the class was already divided.

It's time for my traditionalist post for this week. Actually, I've been thinking about what Dr. Ze'ev Wurman and other Common Core opponents have been saying about testing the students less often than annually.

I believe that my proposed standardized test can provide useful information to students, teachers, and parents -- but if I were trying to convince the public to adopt my standardized test, the public is unlikely to believe that it will provide useful information. It is just like the fable of the Boy Who Cried Wolf. The boy lies so much about a wolf that no one believes him when the wolf comes. And likewise the public has distrusted standardized testing so much -- believing that the main direction of information flow is from students, teachers, and parents toward either the government or the large testing companies -- that no one will believe me when I saw that my test moves information in the opposite direction.

And so in order to gain everyone's trust, I can't give my test to the students every year. Instead, I would have to prove that my test provides useful information to students, teachers, and parents. If I do it correctly, the public would want to administer my test every year, if not even more often.

How do we decide which years to administer the test? One way is to spread it out so that a different test is given each year -- say English in third grade, math in fourth grade, science (as in the Next Generation Science Standards) in fifth grade, then repeat the pattern in middle school. But some may prefer there to be years within Grades 3-8 when there are no standardized tests. By this line of reasoning, all three tests should be given during one elementary and one middle school year. But then students would have to spend so much time during those years preparing for tests.

So let's compromise -- take two tests one year, one another year, and none yet another year. Now there are test-free years without there being three-test years. We can give the math and science tests in the same year, as these subjects should reinforce each other. As I mentioned before, a great year to give the middle school math test can be either seventh grade (to determine Algebra I vs. Math 8 placement for eighth grade) or sixth grade (to determine Pre-Algebra vs. Math 7 placement for seventh grade). I prefer the latter, as three years earlier we can test third graders on their times tables.

The sixth grade math test can consist of around 14-16 questions so that students can finish the test in regular class period (of course, if it's computer-adaptive then the number of questions can vary greatly from student to student). The third grade math test can contain many more questions, since some questions should be times tables questions that can be answered in seconds.

Today is an activity day. This is what I wrote last year about two certain activities first thought up by Dr. Hung-Hsi Wu. I've just decided that I will no longer post these activities as I'm breaking off from Wu's similarity lessons, but I am posting the discussion about the activity anyway because  even back then I had something to say about the traditionalists. (I cut out all references to classes that I was teaching or tutoring last year, but I retained a reference to my student teaching.)

The general idea comes from generalizations of yesterday's Side-Splitting Theorem. If a trio of parallel lines splits a transversal proportionally, then it splits any transversal proportionally. In particular, if the parallel lines are equidistant, then any transversal is split equally. This extends to any number of equidistant parallel lines.

So all we need are a bunch of equidistant parallel lines. And many students just happen to be carrying not just one, but several pages filled almost completely with equidistant parallel lines.

I'm speaking, of course, about lined notebook paper.

Wu's activity is simple. He begins by picking a point A near the middle of the top line of the notebook paper, and then two points B and C on a certain line below the top -- Wu chooses the fifth. Then he finds the points where rays AB and AC intersect another line -- in this case the seventh -- and labels these points B' and C', respectively. Then it can be proved using the Side-Splitting Theorem that B'C' is exactly 7/5 as long as BC.

This activity can be made more dramatic by presenting a segment on one of the lines and asking the student to draw another segment that is exactly 7/5 as long as the first, without measuring. That's the tricky part, since the straightedges that the students use to draw the lines are likely to be rulers that allow the students to measure them.

So far, I haven't mentioned the U of Chicago text. Notice that so far in Chapter 12, I haven't been giving the Exploration Questions at the end of each lesson as a bonus. But notice that, right at the end of yesterday's Section 12-10, Exploration Question 22 is similar to the Wu activity.

Here the student is asked to take a segment AB and divide it into fifths. We see that as given, the steps can be constructed with a straightedge and compass, including the equidistant parallel lines. But instead, I want to use the equidistant parallel lines of the notebook paper.

The best way to do the U of Chicago activity is to have the students take something small, such as an index card, and divide it into something like thirds or fifths. (Halves and quarters are trivial.) To divide the index card into fifths, the student places one corner of the card on the top line of the notebook paper and another corner on the fifth line down from the top. Then the four lines in between touch the paper at exactly the one-fifth marks of the card, so the student labels these points. Finally, the card is lifted from the paper and folded. If these are done exactly (but this is difficult), then the card has been successfully divided into fifths.

As I mentioned before, giving a word problem to get the students thinking about slope does occur in pre-Common Core texts. For example, Section 3-5 of the Glencoe Algebra I text -- the book I used for student teaching -- is on "Proportional and Nonproportional Relationships." The lesson begins with a chart showing the number of miles driven for each hour of driving, and students have to figure out that every time the number of hours increases by one, the number of miles increases by 50. And the very next lesson, Section 4-1, "Rate of Change and Slope," the students calculate the speed -- and therefore the slope -- by dividing the change in distance by the change in time.

But when actually teaching Chapter 3, my master teacher had me teach Sections 3-1 and 3-2 on relations and functions, then skip directly to Chapter 4 -- there was no Chapter 3 test. And I don't remember how much of Sections 4-1 and 4-2 we actually taught -- we might have just given the students a separate worksheet on slope before diving right in to Section 4-3, on graphing equations in slope-intercept form. I believe that this practice was common for most Algebra I classes before Common Core forced us to emphasize these sections that we used to skip.

The students discovered nothing because few of them were calculating the rates correctly, so none of them realized that the rate was always the same no matter which points they chose. Instead, according to the traditionalists, the "sage on the stage" should just tell them the slope formula and have the students practice plugging in points and finding the slope.

But then again, the students in my student teaching class still struggled with the slope lesson even when taught traditionally.  Chapter 4 was a turning point for many students in that class, as many students who performed well on previous chapters struggled from this point on. Many students simply did not remember the slope formula, no matter how many times I gave the formula -- and some simply chose not to listen to me when I told them the formula.

Of course, students will struggle no matter how slope is taught if they are generally weak on operations with integers. As we already know, many students simply don't remember how to add, subtract, multiply, and divide integers from year to year -- that is, a student can be taught to excel on an integer test one year, yet won't remember the integer rules the following year. It is one of the two main topics from grades 4-7 that students can't remember -- the other, of course, is fractions. But unfortunately, slope is heavy on both integers and fractions. To calculate slope, one must add -- actually subtract -- integers for both the rise and the run, and when one divides these, the answer may be a fraction.

Is it possible to compromise between the traditionalist and progressive philosophies here to obtain a slope lesson that will get most of the students to do well? I can't help but notice that the example that I mentioned earlier has a rate, or slope, of 50 mph. Therefore, students are more likely not to make a mistake and reach the Aha! moment that the rate is the same no matter which times they happen to choose.

And so we, as teachers, must be careful when designing the sort of lesson that the Common Core Standards encourage. The traditionalists are correct that discovery lessons won't work unless the students get the basic calculations correct, so let's design "Opening Activities" and "Anticipatory Sets" so that the students are likely to get the answers correct. This means delaying problems with negatives and fractions until after the opening activity, by which time the formula is given.

Then again, my slope lesson next week is from a geometric, not algebraic, perspective. I won't be focusing on how a constant speed of 50 mph is related to the constant slope of the line, but rather how similarity will be used to prove that a line has constant slope. So this will be tricky.

CCSS.MATH.CONTENT.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

Notice that eighth grade is the first in which transformations appear.

Here are the two lessons that I'm posting instead. One is based on the reflections worksheet that the students worked on in class, and the other is based on the Four-Color Theorem -- another lesson inspired by Kung's lecture. Next week, Kung will continue with topology.

Lesson 12-10: The Side-Splitting Theorem (Day 92)

Lecture 18 of David Kung's Mind-Bending Math is called "Filling the Gap Between Dimensions." In this lecture, Dave Kung continues his description of paradoxes related to Geometry -- obviously a topic dear to my heart considering the title of this blog.

Kung begins by discussing a three-dimensional solid called Gabriel's horn. As it turns out, Gabriel's horn has a finite volume, yet an infinite surface area. He shows this using Calculus -- this is a bit beyond AP Calculus, but it might be understandable by an AP student.

The Quick Conundrum involves balancing a yardstick on two fingers, one on each hand. If the hands approach each other quickly, they meet at the midpoint (the 18" mark), but if he moves them slowly, the ruler slides first one way, then the other.

Then Kung moves on to figures which, as the title goes, fill the gap between the dimensions. Of course, I'm talking about fractals. Kung calculates the similarity dimension of three fractals -- all of which I discussed on the blog back in October when I read the fractal book written by the Polish mathematician Benoit Mandelbrot:

Cantor Set (mentioned in the other two links below)
Sierpinski Triangle (Sierpinski was also Polish, born in the same city as Mandelbrot.)
Menger Sponge

In describing how to find the similarity dimension of a fractal, Kung mentions what the U of Chicago calls the Fundamental Theorem of Similarity in Chapter 12 (as these fractals are self-similar). Since we are currently working in Chapter 12, we ought to be discussing this right now on the blog -- except that we aren't.

Notice that we're skipping from Lesson 12-3 (yesterday's lesson) to 12-10 (today's lesson). Well, actually we'll be covering some of the skipped lessons next week. But the material mentioned in Kung's lecture comes from Lessons 12-6 and 12-7, which we aren't fully covering on the blog.

So what's going on here? The problem is that when I first wrote these similarity lessons last year, I based them on the writings of Dr. Hung-Hsi Wu. He used the name "Fundamental Theorem of Similarity" to refer to a different theorem from the U of Chicago -- the text uses the name FTS to refer to the idea that areas of similar figures vary as the square of the length and volumes as the cube of the length, while Wu uses the name FTS to refer to what I called the Dilation Distance Theorem in yesterday's post.

I had labeled Wu's FTS proof as "Lesson 12-6" even though it referred to a different theorem. But Wu's proof was highly complicated -- and, as I later decided, too difficult for high school. And so I ended up dropping Wu's FTS proof and using the PARCC proof (using SAS~) instead. But Kung points out that it's the U of Chicago's FTS that leads to the similarity dimension of fractals -- the Cantor set doubles in content as its length triples, so its dimension is log_3(2).

As for Lesson 12-7, I never included it as a lesson on the blog. But this lesson is actually fun -- and Kung refers to it in his lecture. The question is, "Can There Be Giants?" The answer, as it turns out, is no, for the weight of a giant increases as the volume (the cube of the height) while its strength varies as the area of a bone's cross-section (the square of the height). So a giant isn't strong enough to stand up. On the other hand, as Kung explains, a tiny person can't exist because the amount of blood its heart can pump drops fast, like the volume (the cube of the height) -- so not enough blood can make it all the way to its feet and head (the first power of the height). Large and small animals must have completely different shapes -- they can't be geometrically similar to humans.

Well, here is the section of the U of Chicago text for which we do have a worksheet all ready and made -- Lesson 12-10, the Side-Splitting Theorem. This is what I wrote last year about the theorem:

The U of Chicago version of the theorem is:

Side-Splitting Theorem:
If a line is parallel to a side of a triangle and intersects the other two sides in distinct points, it "splits" these sides into proportional segments.

And here's Dr. Wu's version of the theorem:

Theorem 24. Let triangle OPQ be given, and let P' be a point on the ray OP not equal toO. Suppose a line parallel to PQ and passing through P' intersects OQ at Q'. ThenOP'/OP = OQ'/OQ = P'Q'/PQ.

Notice that while the U of Chicago theorem only states that the two sides are split proportionally, Wu's version states that all three corresponding sides of both sides are proportional.

Moreover, the two proofs are very different. The U of Chicago proof appears to be a straightforward application of the Corresponding Angles Parallel Consequence and AA Similarity. But on this blog, we have yet to give the AA Similarity Theorem. So how can Wu prove his theorem?

We've seen several examples during the first semester -- a theorem may be proved in a traditionalist text using SSS, SAS, or ASA Congruence, but these three in Common Core Geometry are theorems whose proofs go back to reflections, rotations, and translations. Instead, here we skip the middle man and prove the original high-level theorem directly from the transformations. We saw this both with the Isosceles Triangle Theorem (proved from reflections in the U of Chicago) and the Parallelogram Consequences (proved from rotations in Wu).

So we shouldn't be surprised that Wu proves his version of the Side-Splitting Theorem using transformations as well. Naturally, Wu uses dilations. In fact, the names that Wu gives the points gives the game away -- O will be the center of the dilation, and P and Q are the preimage points, while P' and Q' are the images.

Here is Wu's proof: He considers the case where point P' lies on OP -- that is, the ratio OP'/OP, which he labels r, is less than one. This is mainly because this case is the easiest to draw, but the proof works even if r is greater than unity. Let's write what follows as a two-column proof:

Given: P' on OP, Q' on OQ, PQ | | P'Q', r = OP'/OP
Prove: OP'/OP = OQ'/OQ = P'Q'/PQ

Statements                                           Reasons
1. P' on OP, Q' on OQ, PQ | | P'Q'      1. Given
2. OP' = r * OP                                    2. Multiplication Property of Equality
3. Exists Q0 such that OQ0 = r * OQ  3. Point-Line/Ruler Postulate
4. For D dilation with scale factor r,    4. Definition of dilation
    D(Q) = Q0, D(P) = P'
5. P'Q0 | | PQ, P'Q0 = r * PQ               5. Fundamental Theorem of Similarity
6. Lines P'Q0 and P'Q' are identical      6. Uniqueness of Parallels Theorem (Playfair)
7. Points Q0 and Q' are identical          7. Line Intersection Theorem
8. OQ' = r * OQ, OP' = r * OP,            8. Substitution (Q' for Q0)
    P'Q' = r * PQ
9. OP'/OP = OQ'/OQ = P'Q'/PQ = r    9. Division Property of Equality

Now the U of Chicago text also provides a converse to its Side-Splitting Theorem:

Side-Splitting Converse:
If a line intersects rays OP and OQ in distinct points X and Y so that OX/XP = OY/YQ, then XY | | PQ.

The Side-Splitting Converse isn't used that often, but it can be used to prove yet another possible construction for parallel lines:

To draw a line through P parallel to line l:
1. Let X, Y be any two points on line l.
2. Draw line XP.
3. Use compass to locate O on line XP such that OX = XP.
4. Draw line OY.
5. Use compass to locate Q on line OY such that OY = YQ.
6. Draw line PQ, the line through P parallel to line l.

This works because OX = XP and OY = YQ, so OX/XP = OY/YQ = 1.

The U of Chicago uses SAS Similarity to prove the Side-Splitting Converse, but Wu doesn't prove any sort of converse to his Theorem 24 at all. Notice that many of our previous theorems for which we used transformations to skip the middle-man, yet the proofs of their converses revert to the traditionalist proof -- once again, the Parallelogram Consequences.

Another difference between U of Chicago and Wu is that the former focuses on the two segments into which the side of the larger triangle is split, while Wu looks at the entire sides of the larger and smaller triangles. This is often tricky for students solving similarity problems!

Let me review the three MTBoS links to which I referred in yesterday's post:

https://fractionfanatic.wordpress.com/2016/01/27/better-questions/

Julie Morgan is a Scottish high school math teacher. Here's what I wrote about her questions, which are related to a lesson on the area of a sector:

Yes, I heard of the idea of an “Anticipatory Set,” a introductory question that motivates the students to start thinking. I’m glad that your Anticipatory Set question worked well!

I’ve tutored students in the past, and when they reached this lesson, I motivated them by bringing them an actual pizza slice and make them calculate the area before I let them eat it!

As an American, I find it interesting to read about education in other countries. So it’s nice to read your blog about teaching in Scotland!

http://forbetterproblems.blogspot.com/

This is the second time that I referred to fellow California math teacher Marisa Aoki. Her question often shows up in the traditionalist debates. This is what I wrote:

Don't worry about being unable just to "follow the prompt" -- some of my MTBoS posts end up going off on a tangent as well.

I suppose this was the thinking behind the Common Core Standards -- idea is to increase math comprehension and make sure that the students do understand what the numbers mean. But so many people are afraid that students are spending too much time on comprehension and not enough on learning the addition and multiplication tables.

And this is Aoki's response:

I don't think fluency is bad - but when kids tell me that 9x8 is impossible because they can't remember it, that is an issue. If a kid really knows what multiplication means, even when facts are forgotten, there is a strategy to figure it out (instead of just giving up or guessing). I wish there weren't such a pervasive idea that comprehension and fluency are necessarily at odds. If anything, I feel like they should compliment each other.

http://www.8ismyluckynumber.blogspot.com/2016/01/a-better-question-for-quadratics-mtbos.html

Massachusetts teacher writes about her Advanced Algebra I class:

Yes, this must be an Accelerated class if they're already using complex numbers in Algebra I!

That being said, now that the students have learned complex numbers, they should know that ALL quadratic equations have a solution -- which means that the answer "no solution" that they tried to write is NEVER correct. (This, of course, is the Fundamental Theorem of Algebra.)

But yes, I know that many students often wonder why they must learn more than one method. I was once a long-term sub in an Algebra I class and when we were learning about factoring, this very bright girl had her father show her the Quadratic Formula, so when asked to factor x^2 + 5x + 6 (which is not an equation, just an expression to factor), she'd just write -2 and -3 and didn't want to learn how to factor.

Here are the worksheets. After watching Kung's lecture, I decided to include an extra worksheet on Lessons 12-6 and 12-7 -- the only problem is that since we switched the order of the chapters, students following the blog haven't truly learned about area or volume yet. A disclaimer is included at the top of that page.

Kung will continue with Geometry paradoxes in his next lecture.