Lesson 5-3: Conjectures (Day 53)

I suppose that as a Californian, I can't avoid mentioning the wildfires. Once again, I must describe our state's unique weather -- this is the time of the year for high temperatures and wind that often lead to fires. Even though temps have cooled down a little this week, the blazes keep on going.

One of these is the Getty Fire, named after a local museum -- the Getty Center. This museum is only a few miles from my alma mater, UCLA. Even though there is no fire at UCLA, the school ended up canceling classes yesterday. (Classes started up again today.) Meanwhile, several LAUSD schools (those in the valley, which is near multiple fires) also lost days of school due to fire and smoke.

Oh, and since I mention the U of Chicago text all the time, I might as well add that there is also a teachers' strike going on in the Second City. (I only wrote a little about the LAUSD strike back at the start of the year.)

Chapter 10 of Ian Stewart's The Story of Mathematics is called "Impossible Quantities: Can Negative Numbers Have Square Roots?" Here's how it begins:

"Mathematicians distinguish several different kinds of number, with different properties. What really matters is not the individual numbers, but the system to which they belong -- the company they keep."

As the title implies, this chapter is all about the history of imaginary numbers. I've blogged before that, contrary to popular belief, imaginary numbers like sqrt(-1) were not invented to solve quadratic equations without real solutions. Let's take it from Stewart here:

"Among the first mathematicians to wonder whether it had a sensible meaning were the Renaissance algebraists, who ran into square roots of negative numbers in a surprising indirect way: the solution of cubic equations."

I don't need to repeat what Stewart writes here, because we've seen it in a Mathologer video. The main point was that if we use the Cubic Formula to find the three real roots of a cubic equation, the formula produces imaginary parts that cancel out to produce the real solutions:

"This was all very interesting, but why did it work? To answer this question, mathematicians had to develop good ways to think about square roots of negative quantities, and do calculations with them."

One of the first mathematicians to formalize imaginary numbers was John Wallis, who used a sort of Cartesian plane with real numbers on one axes and imaginary numbers on the other. He used the symbol i to denote the square root of -1. But not all mathematicians embraced this notation:

"In 1758 Francois Daviet de Foncenex, in a paper about imaginary numbers, stated that it was pointless to think of imaginaries as forming a ling at right angles to the real line."

The author moves on to discuss some complex analysis -- functions of complex numbers:

"Square roots of complex numbers are marginally trickier, but there is a pleasant reward for making the effort: every complex number has a square root."

But Stewart wants to evaluate more complicated functions using complex numbers:

"At this stage, things started to get very interesting, but also very puzzling, especially when it came to logarithms. Like i itself, logarithms of complex numbers turned up in purely real problems."

Johann Bernoulli believed that the logarithm of a negative number is real (which he tried to prove by integrating a differential equation). But it was Leibniz who first proposed that the logarithm of a negative number might be complex:

"This particular controversy was sorted out by Euler in 1749, and Leibniz was right. Bernoulli, said Euler, had forgotten that any integration involves an arbitrary constant."

In other words, Bernoulli forgot the "+ C." It was Euler who finally defined logs of negatives. He came up with the formula that ln(-1) = i pi -- that is, that e^(i pi) = -1. (This is Euler's identity.) He was able to find log z for almost any complex number z:

"(When z = 0, the value log 0 is not defined.) Mathematicians were used to functions that could take several distinct values, the square root being the most obvious example: here, even a real number possessed two distinct square roots, one positive and one negative."

But the log function, as it turns out, produced infinitely many values, each differing by exactly 2pi i.

We now move on to Cauchy's Theorem of complex analysis (that is, Calculus):

"Today we call it Cauchy's Theorem, because it was published by Cauchy, but Gauss had the idea much earlier in his unpublished writings. This theorem concerns definite integrals of complex functions: that is, expressions..."

...um, you already know what definite integrals look like. There's no need for me to attempt to write the integral from a to b of f (z)dz in ASCII. Gauss discovered that definite integrals work differently in complex analysis, because their values depend on "how" we get from a to b in the complex plane:

"'But if this does not happen...I affirm,' Gauss wrote to Bessel, 'that the integral has only one value even if taken over different paths provided [the function] does not become infinite in the space enclosed by the two paths."

But it was Cauchy, not Gauss, who ultimately proves this. The author continues to write about the imaginary unit i:

"Its status remained uncertain until it was realized that there is a logically consistent extension of the traditional system of real numbers, in which the square root of minus one is a new kind of quantity -- but one that obeys all the standard laws of arithmetic. Geometrically, the real numbers form a line and the complex numbers form a plane; the real line is one of the two axes of this plane."

On that note, Stewart concludes the chapter as follows:

"Today, complex numbers, and the calculus of complex functions, are routinely used as an indispensable technique in virtually all branches of science, engineering and mathematics."

The sidebars in this chapter include a biography of Augustin-Louis Cauchy, what complex numbers did for them, and what complex numbers do for us.

Lesson 5-3 of the U of Chicago text is called "Conjectures." There is no real equivalent in the modern Third Edition, as Chapter 6 of that text goes directly from the equivalent of the old Lesson 5-2 to the equivalent of the old Lesson 5-4.

This is what I wrote last year about today's lesson:

Lesson 5-3 of the U of Chicago text discusses conjectures. The text defines a conjecture to be "an educated guess or opinion." It is a statement that has yet to be proved.

David Joyce has a low opinion of conjectures in a high school geometry class. In his scathing review of the Prentice-Hall text, Joyce writes:

(And this occurs in the section in which 'conjecture' is discussed. "Test your conjecture by graphing several equations of lines where the values of m are the same." What's the proper conclusion? That theorems may be justified by looking at a few examples?)

In summary, the material in chapter 2 should be postponed until after elementary geometry is developed.

That is to say, the statements should be postponed until they can be proved.

Now Michael Serra takes a diametrically opposite approach from Joyce. In Serra's Discovering Geometry text, a great many statements are given well before they are proved -- since the proofs don't occur until the final three chapters of the book, Chapters 14-16 (old version -- Chapter 13 in the new version). What would Joyce say about having so many of these unproved theorems in his text?

The theorems can be proven once a little actual geometry is presented, but that's not done until the last half of the book. A little honesty is needed here. Why not tell them that the proofs will be postponed until a later chapter? Or that we just don't have time to do the proofs for this chapter. Even better: don't label statements as theorems (like many other unproved statements in the chapter).

Well, Serra doesn't label statements as theorems -- he calls them conjectures! In all, 114 conjectures are in Chapters 3-13 of Serra's text -- Conjecture 1 is the Perpendicular Bisector Conjecture (which we've already proved on this blog) and Conjecture 114 is the Law of Cosines. Clearly, discovering and stating conjectures are the heart of Serra's learning philosophy.

As I've stated before, I want to lean towards Joyce's preference of proving all theorems as soon as they are mentioned. I've rearranged the order of the U of Chicago text in order to make sure that proofs precede applications of the theorems -- especially when I want to highlight the Common Core proofs, which may differ from traditional proofs.

But I'm also sympathetic to Serra's philosophy. I want to show some interesting results of geometry without being limited to what the students can prove. Also, one way to pique a student's interest is to show the result and ask, "Is this always true?" or "Why is this always true?" Proving many boring low-level theorems just to make sure that nothing is used before it is proved would result in students losing interest and wondering why they are forced to write endless proofs. And of course, Serra's text helps out students who may be weak at proof-writing and would easily forget how to write a proof once taught -- the main results are stated so the students can learn them, and then the proofs are given at the end of the book, right before the PARCC exams.

The point of this lesson is to get the students thinking about the properties of special quadrilaterals without worrying about how to prove them. In other words, I want to get the students engaged and thinking about the quadrilateral properties so that they can make the conjectures.

We begin by dividing the class into groups -- say of three or four students. Each group is assigned a worksheet -- or the members can write down answers on a common blank sheet. Then my usual set of ten questions are assigned -- but there are some differences between this and the usual individual worksheets that I post.

First of all, let's look at the first two questions:

1. What is the teacher's __________?

2. What is the teacher's __________?

Beforehand, the teacher fills in the blanks with words -- I'd fill them in with age and weight. I have no problem with giving this much information to the students -- but many people, especially women, are highly sensitive to revealing such personal data. This is why I left blanks in the questions -- so that the teachers fill in the blanks with words that they are comfortable revealing in class.

The teacher asks the question, "What is my age?" (or whatever is in the first blank). The groups signal when they want to answer. The teacher calls upon the group that signaled first to answer -- and since this answer will almost certainly be wrong, the teacher then calls upon another group. When a group finally gives the correct answer, the teacher awards this group a point. (In case you're as curious as the students are about my age, I am currently 38 years old.)

Notice several things about this game so far. The first team to give a correct answer -- and the answers in my version of this activity are numerical so far -- is the one to get the point. And after the first two questions, two groups have one point each -- or possibly one team already has two points -- and the rest have none.

Certainly the groups without points so far are eager to earn one. And so they are faced with the next question in the activity:

3. True or false: the diagonals of a rectangle are always equal in length.

Recall that this activity is all about conjectures. The students have already spent time making conjectures (that is, educated guesses) about the teacher's age and weight -- now it's time to make a conjecture about geometry!

This question serves several purposes. First, the students in groups that are trailing in points -- the same students who would have complained about doing math after the long exam -- now suddenly want to answer a math question because they want to catch up to the leaders. Second, this question is a true-or-false question, so students who might have tuned out if given an open-ended question will want to try this one at least since there are only two possible answers. The students are likely to guess at the answer -- and they're encouraged to do so, because a conjecture is a guess! Third, the conjecture in question involves rectangles -- and students who tend to forget what a rhombus or trapezoid is will still remember what a rectangle is. The only problem word that might be a barrier to participation is diagonal -- so the teacher reminds them that the two diagonals of a rectangle run from a corner to the opposite corner.

In my activity, every third question (that is, the third, sixth, and ninth) is a true-or-false question. I use these to give the students more opportunities to earn points. The teacher allows every group to give an answer of true or false before revealing the answer, and every group that gives the correct answer earns a point. In this way, groups can earn points without worrying about being the fastest group to get the answer.

Of course, the answer to Question 3 here is true. Hopefully, most, if not all, of the groups were able to guess that the diagonals of a rectangle are equal, so that every group is on the scoreboard. Now we move on to the next questions.

4. The diagonals of a square always divide the square into four triangles of __________ size.

5. The diagonals of a kite are always __________.

Now these questions are open-ended, just like the first two questions (but there are no more personal questions -- from now on, all are geometric). So we return to having the groups compete, and only one group will receive the point.

The students should test the fourth conjecture by drawing several squares -- by several, let's say one for each member of the group. So the first group to have drawn enough squares for the group as well as give the correct answer "equal" is the group to earn the point. Since these problems are increasing in difficulty, a teacher may choose to give two points, rather than one. (Notice that the four triangles are in fact congruent, but since congruence has not been taught yet, we instead say that they have equal "size" -- where the students can probably get an idea of what that might mean.)

The difficulty in the fifth question is that after having seen the diagonals in Questions 3 and 4 turn out to be equal, the students may jump to the conclusion that "equal" is correct yet again. The teacher should remind the students to draw the kites to make sure -- and to drive the point home, the teacher should draw a counterexample to the claim that the diagonals of a kite are equal. The students might not think to say that the diagonals are "perpendicular," which is the correct answer. So the teacher can give the hint that they should check the angle between the diagonals. By now, I'd award the point(s) to the group telling me that the angle is 90, even if the actual word perpendicular is not used.

Now we move on to our next true-or-false question:

6. True or false: consecutive angles in a parallelogram are always equal.

Of course, the teacher should define consecutive angles (or "adjacent angles"). If necessary, the teacher can draw a parallelogram on the board and point out where the consecutive angles are -- and naturally, that parallelogram should be a rectangle (or nearly so), in order to avoid giving away that the correct answer is false.

7. If ABCD is a parallelogram and angle A has measure 30, then angle B has measure _____.

8. Opposite angles in a parallelogram are always __________.

Notice that the seventh question is an extension of the sixth -- ABCD is a parallelogram, and the measure of angle A is 150 degrees. Because of question 6, the answer isn't 150 degrees. A point (or points) are awarded to the group correctly answers that the angle B measures 30 degrees. No conjecture is stated in this question, but the implied conjecture is that the consecutive angles in a parallelogram are supplementary. Since the answer to question 7 is numerical, I don't require the groups to draw a parallelogram for each student in the group.

 Of course, the answer to question 8 is that opposite angles in a parallelogram are always equal.

9. True or false: opposite sides in a parallelogram are always equal.

This one is self-explanatory -- the answer is also true.

10. A square has _____ lines of symmetry.

I've mentioned this one earlier on this blog. A square has four lines of symmetry -- if the sides of the square are parallel to the coordinate axes, then a square has one horizontal, one vertical, and two diagonal lines of symmetry. The diagonals are the ones that are often missed. Once again, this is a numerical answer, so I don't require a diagram for each member of the group. Of course, the students will want to draw at least one square in order to find its symmetry lines.

And now, as I often like to do, here's a Bonus Question. As I pointed out last week, I don't like it when students are eliminated from passing when there is plenty of time left in the semester, and in the same manner, I don't want students to be eliminated from winning this game too early. And so this question can be worth many points -- enough for the last (or maybe the next-to-last) place team to catch up. (In my game, I may deduct points for behavior -- so I might not want the last place team to be able to win if their behavior doesn't warrant it.)

Bonus Question: Take a quadrilateral and find the midpoints of its four sides. Join these four points to form a new quadrilateral, the Midpoint Quadrilateral. Midpoint Quadrilaterals are always what type of quadrilateral in the hierarchy?

Students will need time to figure out how to draw this Midpoint Quadrilateral. After each member of the group draws the Midpoint Quadrilateral, the winning team will be the one that correctly identifies the Midpoint Quadrilateral to be a "parallelogram."

For the rest of this post, I write about two memorable times that I played this game in class. One was as a sub, in fact a few days after Halloween. The other was as an actual teacher at the old charter school, on my birthday.

First, as a sub....

But yesterday, I had an opportunity to play my game. Most of the eighth grade math classes were assigned to take notes, but the Math/Computing class had a worksheet to finish. And so I began by asking the students to guess my age, and then my weight. So two of groups already had a point, while the other seven were scoreless.

Then the third question I asked was simply the first question from the worksheet -- namely to graph the equation x + y = 5 using intercepts. Just as I mentioned from my original Conjectures worksheet, every third question was a chance for each and every group to earn a point. I think that only about half of the groups earned the point. Some of the groups drew the graph incorrectly, while others had the correct graph but identified the slope as 1 instead of -1. My fourth question was the second question from the worksheet, x + 2y = 8 -- which, just as planned, allowed for only one group to earn the point.

I admit that graphing isn't necessarily the best sort of question for this game -- especially when the students had to do so much work to answer each question (finding both intercepts and the slope). The game worked out better on Monday, when the Computing class was working on the computers and the rest of the classes had equations to solve, so I played the game only in the other classes. When solving equations, nearly all the groups earned the point on the third question, which is what I want.

The worksheet consisted of about a dozen graphs, yet I only had time to play the game with six. So someone might point out that if I had simply passed out the worksheet and asked the students to work on it, they might have completed many more than six of the graphs. The game wastes so much time when the students can't work on the next problem until the class reaches it -- especially on every third question where I must check every group before proceeding.

But let's recall the situation the class was in. It was the second straight day the class had a sub. So the students, already knowing coming in that there was a sub, were already thinking about how much mischief they could get in -- things they'd never dream of doing with the regular teacher. And once they arrived, they were probably hoping that they could play around on the computer, only to find out that the teacher had locked the computers away and assigned the worksheet.

So we can see that the students weren't in much mood to work. And yet, I believe that there was a game, the allure of earning points motivates them. Many of the students might have just thrown the paper away, or worked on it at a snail's pace and still be on the first graph late into the period. Also, checking every third question keeps those who might have drawn every graph incorrectly or calculated every slope with the wrong sign.

Some people -- especially traditionalists -- dislike group work, and believe that students learn much, much more effectively when doing individual work. But I often find that as a sub, classroom management is easier when I only have to keep track of nine groups, not 35 students whose names I don't know. In order to earn a point, every member of the group must have drawn the graph on the paper, and so the students end up motivating each other at least to draw the graph on the paper (even if all they're doing is copying the other group members).

I remember that the group who debated with me about a slope of 1 vs. -1 failed to earn a point for that third question, and consequently fell behind the other groups. Even though by the end of the fourth question, I'd convinced them (using Slope Dude) that their slopes had the wrong sign, the group never caught up to the leaders. As I announced the winners, I overheard a member of that group saying to one of his teammates, "But we're the real winners because we did eleven problems and the others did only six." Yes, imagine that! The group kept on working on the graphs well ahead of my pace in an effort to catch up to the other groups' score. (This was my intent of including the Bonus Question on my original worksheet.) And these are students who would -- had I not played a game -- at best, have calculated all the slopes with the wrong sign, and at worst, have just thrown the papers away and not worked on the graphs at all!

And so I will continue to play this game in class. One difference between my original vision and the way the game has played out in the classroom is that I almost never make the questions for the entire class to answer true-or-false. After all, how often will the students be given a worksheet where every third question is true-or-false? One thing I might consider is, on every third question, have the groups work on a problem, and then present one group's answers to the class. Then all of the groups must determine whether that answer is correct (true) or incorrect (false). Indeed, even if I don't explicitly ask a true-or-false question, I might said to the group that insisted that the slope was 1, "Well, this other group says that the answer is -1. So let me take a point away from them and give it to you." So this threat would force the other group to defend their answer of -1 -- now the groups are debating the answer with each other, rather than the usual situation of me vs. the students.

Notice that I never actually give this game a name. I'm prone to name this game either Conjectures -- after the post in which it appears -- or Who Am I? -- which is the first part of the title of the post that appeared the day after I first posted the game. The only problem with either name is that they both refer only to the first two questions -- unless it's from my Conjectures worksheet -- since, for example, the fact that the slope is -1 isn't merely a "conjecture," but should be known. Of course, even on the Conjectures worksheet, "Who Am I?" only refers to the first two questions.

By the way, what exactly was the prize that I gave the winning groups? Actually, all I did was leave their names for the regular teacher with a positive note!

Second, as a teacher....

The answer to the first question "What is the teacher's age?" is 36. That's because today is -- you guessed it (or remembered from last year) -- my 36th birthday! And so I knew that if I was going to play a game which starts with my age, it might as well be on my birthday.

What lessons do I include in today's game? Well, just as in the version of the game I posted as a sub, I want to focus on geometry questions. As it turns out, the game fits the current seventh grade lesson like a glove. Yesterday, the students cut out triangles out of straw, and Illinois State even asks the students to make conjectures about the triangles they created. So it's easy to fit some of those right into the game.

Today is Wednesday -- always a scheduling adventure at our school. For once, we actually follow the same schedule as last week -- but again, it means that I don't see the seventh graders as much as the other grades. I try having them come up with Triangle Inequality as a conjecture. A few of them are able to get on the right track, especially after I give them the hint (or "lead them by the nose," much to David Joyce's dismay).

For eighth grade, I notice that the STEM project mentions the measures of angles that are vertical, adjacent, corresponding, and so on. So I play the game using these conjectures. One big problem is that some students can't use a protractor correctly, so many don't arrive at the conjecture that vertical angles have the same measure. (Actually, the seventh graders also had to conjecture Triangle Sum, but I don't even try to reach that conjecture, knowing that if the eighth graders won't use the protractor correctly, neither will the seventh graders.)

Meanwhile, for sixth grade, the animals project ultimately relates to guessing how much room animals need, so it fits into the game as well. They are learning about how to find the dimensions of a rectangle given its area -- that is, factoring.

I like this game as a sub because it gives the students something to do. But if I use it in the regular classroom, it might be better to do some preparation. Once again, I just took the STEM project and added my own "What is the teacher's age?" questions. But instead, I could have come up with some questions such as just measuring random given angles. If I award points in the game, then the students should be motivated to find them. Then after that I segue to finding specific angles such as vertical angles or those of a triangle. That should lead them to make the conjectures.

So far in 2019, I have only played the Conjectures/"Who Am I?" game a few times. I'm now wondering whether singing songs is a more effective way to motivate the students as a sub, since I've been singing more and playing "Who Am I?" less. (Sometimes it depends on the exact assignment given by the regular teacher -- last week, I found out that "write a five-sentence paragraph as a summary" doesn't convert well to this game.) But if I ever become a regular teacher again, I might find a place for "Who Am I?" once more.