Friday, August 29, 2014

Activity: The Daffynition Game (Day 17)

It's tough trying to find activities that fit this chapter. One source that I like to use for activities is Michael Serra's textbook, Discovering Geometry. Just like most other geometry texts, in Chapter 2 he discusses the concept of definition (Section 2-3, "What is a Widget?") Then the text introduces a project, "the Daffynition Game," where the students take turns making up definitions to real words.

A few comments I'd like to make about the game as introduced by Serra. Step 3 reads:

3. To begin a round, the selector finds a strange new word in the dictionary. It must be a word that nobody in the group knows. (If you know the word, you should say so. The selector should then pick a new word.)

The problem is that this depends on the honor system -- how do we know that a student who knows the word will actually admit it? Rather than depend on the students' honesty, why not make knowing the word an actual strategic move? That student will then earn a point for knowing the word -- and the student can still make up a fake definition in order to earn even more points? This means that the selector must be very careful to choose a word that isn't in the dictionary.

Another question is, what affect would this project have on the English learners? I'd say that this would be a great project for them, since they can learn both English and math in this lesson. But English learners might be at a disadvantage in this game, since if they choose a word that a native English speaker knows, the English speaker would earn a point (since I'm not counting on the honor system here). One debate that always comes up in a group activity is whether to group homogeneously or heterogeneously. For this project, it may be best to group homogeneously, but by English, rather than mathematical, ability.

Finally, this project requires students to look up words in a dictionary -- but what dictionary? I threw out the problems in earlier sections that depended on the availability of a dictionary. Perhaps the night before this activity, part of the homework assignment could be to look up a word in the dictionary and write down its definition -- but that assumes that the students will actually do the homework, and besides, there's no guarantee that the students have access to a dictionary at home (or online) either.

My solution is for the teacher to have enough index cards with words and definitions on them. Therefore the selector chooses an index card, not a word from the dictionary. Indeed, the teacher can give an index card to each student even before dividing the class into groups! But the selector should still follow the other steps as originally written in the Serra text. My worksheet with the modified rules will appear at the very end of this blog post.

Now I've been thinking about Wu's proposal from yesterday -- that there should be specialized math teachers at elementary schools. Even though I want to focus on high school math on this blog, I do want to give my opinion on Wu's proposal.

As I mentioned yesterday, back when I was in the upper elementary grades, I had a different teacher for math from the teacher I had for English/language arts. Actually, with separate teachers for math comes the opportunity to separate students into classes by mathematical ability. And this is exactly what my elementary school did, by implementing a "Path Plan." I'll describe a modified version of this "Path System" that incorporates some of Wu's ideas as well.

My elementary school ranged from kindergarten to sixth grade, but instead of dividing the classrooms into grade levels, they are divided into "paths." Students are placed into one of the various paths based on their ELA (not their mathematical) ability. The approximate grade levels corresponding to each path were:

Primary: Grades 1-2
Transition: Grades 3-4
Preparatory: Grades 5-6

but it was possible to be placed into a different path than the nominal path for the grade level -- so there could be an above-grade-level second grader in the Transition Path, and likewise a below-grade-level third grader in the Primary Path. Officially kindergartners, along with four-year-old children in Headstart, were placed in the Early Learning Path, but in practice these classes were simply called "kindergarten."

In my version of the Path Plan, students stay in their homerooms to study ELA, and this lasts until every group has had recess. Then the students change classes. There are 40-minute blocks when the students spend time with a different teacher. How many such blocks there are depends on the path:

Primary: 1 block
Transition: 2 blocks
Preparatory: 3 blocks

For Primary Path students the block is used for math, of course. For Transition Path students, I recommend that the second block be a fun class like art or possibly an exploratory wheel of classes that rotates every semester or trimester. In the Preparatory Path, the third block could be for science -- especially since fifth graders take a standardized test in science under the No Child Left Behind requirements. It may be better to reverse the Transition and Preparatory Paths here, with science in the Transition Path, since many fifth graders might be in that path (and besides, I have no idea how testing requirements will change under the Next Generation Science Standards).

This is where teacher specialization can take place. Since Primary Path students have only one block, specialization isn't possible, but starting in the Transition Path, stronger math teachers can teach math during both of the blocks while others teach science (or whatever subjects were chosen for the blocks).

After the proper number of blocks, the students return to homeroom to pick up their sack lunches, lunch money, or lunch cards for free/reduced lunch. After lunch recess, the students spend the remainder of the day in homeroom to pick up the missing subjects, including P.E. and others. Kindergartners spend the whole day in one classroom -- once again, I strongly disagree with the idea of making five-year-old children move from class to class during the day.

There is a very similar plan to this Path Plan, well-known as the "Joplin plan." But the major difference between the Joplin and Path Plans is that for Joplin, all grade levels spend ELA (and sometimes math as well) in another classroom during a block of time, whereas in the Path Plan ELA is part of homeroom (since the homerooms are already divided by reading ability). Notice that if math is included in the Joplin plan, teacher specialization may be possible if some teachers teach math during both blocks while others teach reading during both blocks. Also, under both plans, all teachers would still need a multiple-subject credential to teach, since they all still teach more than one subject during the day. It's just that only some teachers will have to teach math -- ideally, the strongest math teachers.

My plan gives a gradual increase in the number of teachers a student has in a day as that student progresses in age. In particular, kindergartners have one teacher, Primary Path students have two (homeroom and math), Transition Path students have three, and Preparatory Path students have four. In my district, this pattern continues in middle school, since seventh and eighth graders have only five teachers -- ELA and history being combined into a two-period "Core" class. Not until high school do the students have six teachers, a different teacher for every period. (As an aside, notice that after that, the number of teachers a student has at one time begins to decrease again. Juniors and seniors in my district only had to take five classes, many community college students take four classes a semester, upper division students take three classes, Masters candidates take two, and Ph.D. candidates have only one adviser for their dissertation...)

Of course, plans such as this are controversial. This plan is very similar to academic tracking, and the problem is that no one wants to be the parent of a student placed on a lower track, or a third grader on the Primary Path, or a fifth grader on the Transition Path. One thing I noticed about the Path Plan at my school was that although students were placed on lower paths for reading -- no student was placed into a lower math class than their actual grade level. So a third grader might be placed in third grade math, or possibly fourth grade math, but never in second grade math -- not even a third grader on the Primary Path. And so the classes were not completely homogeneous -- notice that a third grader in a fourth grade math class is at a fourth grade level, but the fourth grader in the class might not be. But actually, I don't believe that completely homogeneous classes are even possible -- after all, whenever a lesson is taught, some students will understand it while others won't. So a class that was homogeneous before the lesson is no longer homogeneous after the lesson.

And now I present my worksheet for the Daffynition Game. Remember that only one of these worksheets need to be given to each group -- in particular, to the scorekeeper in each group. The students write their guesses for Rounds 1-4 (or 5) on their own separate sheet of paper. I recommend that it be torn into strips so that they are harder to recognize. And the teacher provides the index cards, one for each student. Make sure that the students give back the index cards so you can reuse them for the next period. The students may keep their "guess cards," so there should be one for every student in every period.

Thursday, August 28, 2014

Section 3-1: Angles and Their Measures (Day 16)

Chapter 2 of the U of Chicago text is complete. But I've decided to lump the first half of Chapter 3 with Chapter 2, since it focuses on angles. As I've mentioned before, Chapter 3 is awfully late to introduce angles, as most texts do so in Chapter 1. This blog compromises by including angles in Chapter 2.

I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.

The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.

So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":

Now from time to time, I read articles commenting on how mathematics should be taught. Normally, I don't post these, but this one intrigued me because it was written by Dr. Hung-Hsi Wu -- one of the major inspirations for this blog.

The topic of this article was not high school geometry, though, but elementary school arithmetic. Here Wu argues that elementary school math should be taught by math teachers -- as opposed to teachers who hold multiple subject credentials. Elementary school teachers typically choose that career because they like little children, not because they like math -- and if elementary teachers weak in math are unable to teach it well, the students undergo a domino effect where they can't learn middle math or high school geometry, and end up in dead-end remedial classes in college.

My feelings about Wu's proposal are mixed. I can see students in the higher elementary grades such as fifth grade having more than one teacher during the day. Indeed, my own elementary school did exactly this when I was a fifth-grader -- and it later extended to third graders. But I find it especially awkward to implement such a program in kindergarten. I prefer to see five-year-old children experiencing school for the first time in a single classroom the entire day -- the teacher and her aide are like mother figures during the day. And besides, the specific example mentioned in the article for having math specialists involves the multiplication and division of fractions -- and that's way beyond kindergarten.

So I'd say that Wu has a point with respect to the higher elementary grades, but this may be tough with the lower elementary grades. There may be a way to implement Wu's plan more gradually across the grades, and I may discuss this in future posts.

Wednesday, August 27, 2014

Section 2-7: Terms Associated with Polygons (Day 15)

Section 2-7 of the U of Chicago text deals with polygons. Notice that this lesson consists almost entirely of definitions and examples. But this chapter was setting up for this lesson, since a polygon is defined (Section 2-5, Definitions) in terms of unions (Section 2-6, Unions and Intersections) of segments:

A polygon is the union of three or more segments in the same plane such that each segment intersects exactly two others, one at each of its endpoints.

It follows that this section will be very tough on -- but very important for -- English learners. I made sure that there is plenty of room for the students to include both examples and non-examples of polygons. The names of n-gons for various values of n -- given as a list in the text -- will be given in a chart on my worksheet.

The text moves on to define a polygonal region. Many people -- students and teachers alike -- often abuse the term polygon by using it to refer to both the polygon and the polygonal region (which contains both the polygon and its interior). Indeed, even this book does it -- when we reach the chapter on area. Technically, triangles don't have areas -- triangular regions have areas -- but nearly every textbook refers to the "area of a triangle," not the "area of a triangular region." Our text mentions polygonal regions to define the convexity of a polygon -- in particular, if the polygonal region is convex (that is, if any segment whose endpoints lie in the region lies entirely in the region), then the polygon itself is convex.

The text then proceeds to define equilateral, isosceles, and scalene triangles. A triangle hierarchy is shown -- probably to prepare students for the more complicated quadrilateral hierarchy in a later chapter.

Many math teachers who write blogs say that they sometimes show YouTube videos in class. Here is one that gives a song about the three types of triangle. It comes from a TV show from my youth -- a PBS show called "Square One TV." This show contains several songs that may be appropriate for various levels of math, but I don't believe that I've ever seen any teacher recommend them for the classroom. I suspect it's because a teacher has to be exactly the correct age to have been in the target demographic when the show first aired and therefore have fond memories of the show. So let me be the first to recommend this link:

Another song from Square One TV that's relevant to this lesson is "Shape Up." Notice that many geometric figures appear on the singer's head -- though not every shape appearing on her head is a polygon:

With so many definitions in this important section, I decided to leave out the Review Exercises, but I did include the Exploration as a Bonus Question. There are many obsolete names for polygons, such as enneagon and pentadecagon, whose definitions the students are supposed to guess. (Notice that the text gives dodecagon as an obsolete name, yet I've seen modern texts that include it. But duodecagon is definitely obsolete.) An interesting case is trigon. Believe it or not, we still use the word trigon -- the study of trigons is called trigonometry!

The text suggests that students check the guesses in a large dictionary -- that's so 1991! Of course students should try a Google search to find these terms. I was able to find all of them online -- but trigon was a bit difficult because most of the results referred to the DC Comics character Trigon. I had to scroll down towards the bottom of the first page before I found a relevant result.

Tuesday, August 26, 2014

Section 2-6: Unions and Intersections of Figures (Day 14)

Section 2-6 of the U of Chicago text focuses on unions and intersections. This is, of course, the domain of set theory.

In many ways, set theory is the basis of modern mathematics, and so many textbooks -- including higher math such as Precalculus and beyond -- mention set theory early on. Of course, the focus in this text is on unions and intersections of geometric figures. In particular, unions are used to define both polygon and angle, while intersections are used to define parallel lines.

The first three examples in the text, where the underlying sets contain natural numbers, real numbers, and points -- are OK. But I didn't like the fourth example, on airlines. I've decided to throw this one out -- if we want a non-mathematical example, why not just use sets of letters, such as {a, e, i, o, u}, the set of vowels?

One of the most important sets in mathematics is the null set, or empty set. According to the text, this set can be written as either { } or an O with a line through it (often called O-slash by students). Once again, since I can't represent that symbol on Blogger, let's use the strikethrough instead: O.

Now the text mentions that the intersection of two sets might be the empty set. But it doesn't mention what happens when one finds the union, or intersection, of the empty set and another set. As it turns out, the union of the empty set and another set is that other set -- so the empty set acts as the identity element for union, just as 0 is the identity for addition and 1 is the multiplicative identity. But the intersection of the empty set and another set is the empty set -- so the empty set acts as the absorbing element for intersection, just as 0 is the absorbing element for multiplication.

One question students often ask is, if { } is the empty set and O is the empty set, what's {O}? When I was young, I once heard a teacher point out that this is not the empty set because it's no longer empty -- it contains an element.

Sunday, August 24, 2014

Section 2-5: Good Definitions (Day 13)

Section 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.

Consider the following non-mathematical example:

Given: My friend is Canadian.
Prove: My friend comes from Canada.

The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.

And so, let's take the first definition given in this section -- that of midpoint -- and consider:

Given: M is the midpoint of AB
Prove: AM = MB

The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.

The text proceeds with the definition of a few other terms -- equidistant, circle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.

Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. I also notice that his Section 2-4 on biconditionals has an (H) symbol -- which stands for honors. It seems interesting that Dr. M would consider this an honors-only topic -- but then again, we, as teachers, don't necessarily want the students to be bogged down in formalism and proofs.

Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.

Friday, August 22, 2014

Activity: Logic Puzzles (Day 12)

Some people believe that math classes in school should focus only on direct instruction of math, and projects and other group assignments mean less time to study math, so less math would be learned. The teenagers might dislike the monotonous math classes focused only on direct instruction, but, the thought process goes, years later they'll thank you because they actual know enough math to pursue a career while their peers in the less focused math classes aren't being hired.

But I don't follow that philosophy on this blog -- simply because a believer in the direct instruction method doesn't even need to read a blog to find adequate teaching material. If a teacher is searching for the blogs of fellow math instructors online, then that teacher most likely is looking for special projects to implement in the classroom -- not just lecture notes. The most successful math teacher blogs focus on special projects, and so I wish to include such on my blog as well.

I like to gather such activities from various sources, to see which ones I like -- for if I draw only from a single source, the reader would simply go to my source and not read this blog. My main source is, of course, the U of Chicago text. But I also draw from other geometry texts as well as other math teacher blogs.

Now I remember once reading a geometry text and, interspersed throughout the second chapter on logic, there were a number of interesting logic puzzles. Since I'm right now in the logic chapter, why shouldn't I include some logic puzzles as well?

In searching for logic puzzles, I found the website of John Pratt, an astronomy professor from Utah:

What I like about this list of 20 problems is that two of them are identical to the problems I saw in that other geometry text -- Puzzles 4 and 8. And so I decided to take the first ten problems and make them into an activity worksheet.

Now how can teachers use this worksheet? I don't expect any student to discover the answers to all ten problems during a single class period. But perhaps a teacher can divide the class into ten groups and give a different puzzle to each group. Indeed, it's possible to use differentiation here -- since Dr. Pratt writes that he arranged the puzzles from easiest to hardest, a teacher can assign the first few problems to the lower performing groups. Pratt states that elementary students (and hopefully, the lower performing high school students) can solve the first few problems.

I don't include the answers here. Teachers can either solve the puzzles themselves or get the answers directly from Pratt's website.

Thursday, August 21, 2014

Section 2-4: Converses (Day 11)

Section 2-4 of the U of Chicago text deals with an important concept in mathematical logic -- converses. We know that every conditional statement has a converse, found by switching the hypothesis and conclusion of that conditional.

The conditional "if a pencil is in my right hand, then it is yellow" has a converse, namely -- "if a pencil is yellow, then it is in my right hand." The original conditional may be true -- suppose all the pencils in my right hand happen to be yellow -- but the converse is false, unless every single yellow pencil in the world happens to be in my right hand. A counterexample to the converse would be a yellow pencil that's on the teacher's desk, or in a student's backpack, or even in my left hand -- anywhere other than my right hand.

If converting statements into if-then form can be confusing for English learners, then writing their converses is even more so. Here's an example from the text:

-- Every one of my [Mrs. Wilson's] children shall receive ten percent of my estate.

Converting this into if-then form, it becomes:

-- If someone is Mrs. Wilson's child, then he or she shall receive ten percent of the estate.

Now if a student -- especially an English learner -- blindly switches the hypothesis and the conclusion, then the following sentence will result:

-- If he or she shall receive ten percent of the estate, then someone is Mrs. Wilson's child.

But this is how the book actually writes the converse:

-- If someone receives ten percent of the estate, then that person is Mrs. Wilson's child.

In particular, we must consider the grammatical use of nouns and pronouns. In English, we ordinarily give a noun first, and only then can we have a pronoun referring to that noun. Therefore the hypothesis usually contains a noun, and the conclusion usually contains a pronoun. (Notice that grammarians sometimes refer to the noun to which a pronoun refers as its antecedent -- and of course, the text refers to the hypothesis of a conditional as its antecedent. So the rule is, the antecedent must contain the antecedent.)

And so when we write the converse of a statement, the hypothesis must still contain the noun -- even though the new hypothesis may be the old conclusion that contained a pronoun. So the converse of another conditional from the book:

-- If a man has blue eyes, then he weighs over 150 lb.


-- If a man weighs over 150 lb., then he has blue eyes.

Saying the converse so that it's grammatical may be natural to a native English speaker, but may be confusing to an English learner.

Now let's look at the questions to see which ones are viable exercises for my image upload. An interesting one is Question 9:

A, B, and C are collinear points.
p: AB + BC = AC
q: B is between A and C

and the students are directed to determine whether p=>q and q=>p are true or false. Now notice that the conditional q=>p is what this text calls the Betweenness Theorem (and what other books call the Segment Addition Postulate). But p=>q is the converse of the Betweenness Theorem -- and the whole point of this chapter is that just because a conditional is true, the converse need not be true. (Notice that many texts that call this the Segment Addition Postulate simply add another postulate stating that the converse is true.)

Now the U of Chicago text does present the converse of the Betweenness Theorem as a theorem, but the problem is that it appears in Section 1-9, which we skipped because we wanted to delay the Triangle Inequality until it can be proved.

Here, I'll discuss the proof of this converse. I think that it's important to show the proof on this blog because, as it turns out, many proofs of converses will following the same pattern. As it turns out, one way to prove the converse of a theorem is to combine the forward theorem with a uniqueness statement. After all, the truth of both a statement and its converse often imply uniqueness. Consider the following true statement:

-- Barack Obama is currently the President of the United States.

We can write this as a true conditional:

-- If a person is Barack Obama, then he is currently the President of the United States.

The converse of this conditional:

-- If a person is currently the President of the United States, then he is Barack Obama.

This converse is clearly true as well. By claiming the truth of both the conditional and its converse, we're making a uniqueness statement -- namely that Obama is the only person who is currently the President of the United States.

So let's prove the converse of the Betweenness Theorem. The converse is written as:

-- If A, B, and C are distinct points and AB + BC = AC, then B is on AC.
(I explained why segment AC has a strikethrough back in Section 1-8.)

Let's let AB = and BC = y, so that AC = x + y. To do this, we begin with a line and mark off three points on it so that B is between A and C, with AB = x and BC = y. This is possible by the Point-Line-Plane Postulate (the Ruler Postulate). By the forward Betweenness Theorem, AC = x + y.

Now we want to show uniqueness -- that is, that B is the only point that is exactly x units from A and y units from C. We let D be another point that is x units from A, other than B. By the Ruler Postulate again, D can't lie on AC -- it can only be on the same line but the opposite side of A (so that A is between D and C), or else off the line entirely (so that ACD is a triangle).

In the former case, the forward Betweenness Theorem gives us AC + AD = DC. Then the Substitution and Property of Equality give us DC = (x + y) + x or 2x + y, which isn't equal to y (unless 2x = 0 or x = 0, making A and D the same point when they're supposed to be distinct).

In the latter case, with ACD a triangle, we use the Triangle Inequality to derive AD + DC > AC. Then the Substitution Property gives us x + CD > x + y, and then the Subtraction Property of Inequality gives us the statement CD > y, so DC still is not equal to y.

So B is the only point that makes BC equal to y -- and it lies on AC. QED

(The explanation in the book is similar, but this is more formal.) Later on, we're able to use this trick to prove converses of other theorems. So the converse of the Pythagorean Theorem will be proved using the forward Pythagorean Theorem plus a uniqueness statement -- and that statement is SSS, which tells us that there is at most one unique triangle with side lengths a, b, and c up to isometry. And, more importantly, the converse to the what the text calls the Corresponding Angles Postulate (the forward postulate is a Parallel Test, while the converse is a Parallel Consequence) can be proved using forward postulate plus a uniqueness statement. The uniqueness statement turns out to be the uniqueness of a line through a point parallel to a line -- in other words, Playfair's Parallel Postulate. This explains why the forward statement can be proved without Playfair, yet the converse requires it.

In the end, I decided to throw out Question 9 -- I don't want to confuse students by having a statement whose converse requires the Triangle Inequality that we skipped (and the students would likely just assume that the converse is true without proving it) -- and used Question 10 instead.

But now we get to Question 13 -- and here's where the controversy begins. I like to include examples in this section that aren't mathematical -- doing so might engage students who are turned off by doing nothing but lifeless math the entire period. This chapter on mathematical logic naturally lends itself to using example outside of mathematics. But the problem is that Question 13 is highly political. Written in conditional form, Question 13 is:

-- If a country has communist, then it has socialized medicine.

and its converse is:

-- If a country has socialized medicine, then it is communist.

Just like the article at the beginning of the chapter (with discussion of "terrorists" before 9/11), the change in political climate since the publishing of this text has rendered the question controversial. For during the past few years, this country has moved in the direction of socialized medicine with the passage of the Affordable Care Act -- better known by its nickname, "Obamacare."

Now some opponents of Obamacare would argue that moving towards socialized medicine and away from a free-market solution is indeed a step towards socialism or communism. But the intended answer of this question is that this reasoning is incorrect, since it assumes that the converse of a statement is true just because the original conditional is true. But some opponents of Obamacare might argue that the converse is true, anyway. Since defenders of Obamacare tend to be Democrats -- that is, of the same party as the administration that passed the law -- and opponents of the law tend to be Republican, the question may be viewed as having a pro-Democratic slant.

Now I wish this blog to be politically neutral. But unfortunately, the problem is that this is a Common Core blog, and Common Core is itself politically charged. Once again, it was a Democratic administration that passed Common Core, and so once again, many Republicans oppose the standards (indeed, one nickname for the standards is "Obamacore"). One argument against Common Core is that it promotes slanted viewpoints that favor the Democrats -- and by including Question 13 on a blog that purports to be a Common Core blog, I'm lending credence to that argument!

Notice that I have a mixed opinion of the Common Core Standards. I write about CCSS not because I wholeheartedly endorse them, but because I want teachers to know know to teach to them. But I do defend Common Core against invalid arguments. For example, the argument that Common Core endorses a slanted view of history is invalid because the standards cover only English and math (and with the upcoming Next Generation Science Standards, science as well), not history.

Now I don't want Question 13 to convince readers that the Common Core encourages a political view slanted towards the Democrats. (Recall that the question comes from a book that was written years before there even was a Common Core, and that the question has a Chinese immigrant -- not an American Republican -- making the fallacious argument.) It is argued by critics that math classes should be teaching nothing but math, not non-mathematical politics. But I want to include non-mathematical topics because these are more likely to engage the students than teaching nothing but math.

In the end, I will include Question 13 on the worksheet. But I left a space so that if a teacher feels that the question is politically slanted, then he or she could add another question to balance it. For example, that teacher can add a fallacious argument often made by Democrats, for example:

-- If a white person is racist, then he or she opposes Obama.
-- If a white person opposes Obama, then he or she is racist.

Of course, this question adds a new layer of controversy (race) to the mix. Teachers who want to add a balancing question should just write in their own question, or just throw out the question about socialized medicine altogether.

In the review section, I'd have loved to include Question 15, a review of the last section on programming (and of course changed it to TI-BASIC). But in deference to those classes that skipped the section because not every student has a graphing calculator, I've thrown it out and included only questions from the fully covered Sections 2-2, 1-8, and 1-6.

Wednesday, August 20, 2014

Section 2-3: If-then Statements in Computer Programs (Day 10)

Section 2-3 of the U of Chicago text discusses computer programs written in BASIC. We must remind ourselves that this book was written over 20 years ago, back when BASIC was a popular language. But now, BASIC has been derided as "spaghetti code" and isn't as widely used anymore.

Of course, I asked myself whether I should even include this lesson on this blog. After all, I could have skipped it just as I omitted parts of Chapter 1 -- I can't expect the students in the classroom to have access to a computer at all, much less one that can be programmed in BASIC. But many students do carry around something that can be programmed -- a graphing calculator.

Now I'm aware that many students, especially below Algebra II, don't have graphing calculators. And even those who do have such a calculator usually don't know how to program it. I have an old TI-83 that I've owned ever since I was an AP Calculus student. Nowadays the standard is a TI-84. But still, I want to discuss how to program the TI graphing calculator. If the students have graphing calculators, then they might find this to be an interesting lesson -- especially since one of the reasons given for studying higher math is its use in computer science and video games. But if the students don't have access to calculators, or if the teacher doesn't wish to teach programming, then there can be a second day of Section 2-2.

The first BASIC program in the U of Chicago text gives the number of diagonals in a polygon:

50 END

Now let's convert this program to the language of the TI. This language is often called TI-BASIC, even though it's not quite the same as usual BASIC. The first thing that we notice is that unlike BASIC programs, TI-BASIC programs have names. Also, BASIC lines have numbers, while TI-BASIC lines simply begin with a colon. Let's give this program the name DIAGONAL:

:Disp "OF SIDES"

:Input N
:If N>3

We notice that the command PRINT in BASIC corresponds to Disp in TI-BASIC. The problem is that unlike a computer screen, the TI screen is only 16 characters across. So I had to divide up the lines of text into several Disp lines on the TI. The command Input is the same for both languages.

But, considering the title of this lesson, the emphasis is on the If line. We observe several differences between BASIC and TI-BASIC. First, in BASIC we write "greater than or equal to" as >=, but on the TI it has its own symbol, found on the TEST menu (2nd-MATH). Next, we see that Then has its own line on the TI, where in BASIC it is in the middle of the line. When displaying the number of diagonals, we notice that TI-BASIC uses a comma where BASIC uses a semicolon. Finally, on the TI we may use juxtaposition for multiplication, where in BASIC we must use an explicit multiplication system. Notice how the TI displays multiplication and division with an asterisk and a slash, respectively. These symbols actually date back to computer BASIC and other languages.

When I wrote this program in the exercises, I decided to include less text to display, since all the text printed by BASIC in the U of Chicago book was intended for widescreen computer monitors. Notice that my new if-then statement looks like this:

:If N>3

A TI-BASIC programmer wouldn't usually write this. As it turns out, if the body of the Then section contains only a single statement, then the line Then can be left out. But I decided to include it anyway in order for this program to look more like the BASIC program -- and to emphasis the word Then because of its importance in mathematical logic and geometry. Actually, if the word Then is omitted, then so should the word End. In TI-BASIC, End denotes the end of the if-then statement, whereas in BASIC, the word END denotes the end of the entire program. Actually, when I used to program in BASIC, I didn't need to include Line 50 in my program, but some computers require that line.

Here are some of the other programs from the questions, converted into TI-BASIC. First, here's the program from Question 12:

:Disp "HOW MANY 
:Input N
:Disp "PROFIT",5
:If N<300

Sorry about the bad grammar there. The sentence "Order more robots" contains 17 characters when the maximum is 16.

Notice that the program in Question 13 uses 3.14159 for pi and multiplies by r twice. But the TI has both a pi key and a squared key. The proper way to enter the line corresponding to 20 is: start by entering pi (which is 2nd-^), then the letter R, then the squared key (left side of the calculator), then the STO-> key (just above ON -- this corresponds to the BASIC LET), and finally the letter A.

Question 25, in the Exploration section, contains a FOR loop in Line 20. As it turns out, TI-BASIC also has For loops. Here is the entire program:

:Disp "SIDES",N

I decided to add a Pause at the end of the loop. Once again, this is because not all the lines of output can be displayed at the same time on the tiny calculator screen. In order to get the calculator to resume after pausing, simply press the ENTER key. Notice that the End line at the end of this program actually ends the For loop, so it corresponds to the line 50 NEXT N in BASIC, not the line 60 END. So this End line cannot be omitted.

Finally, question 26 asks the student to print an appropriate message if the input is less than 3. I assume that the book intends the student to add a second if-then statement:

:If N>3
:If N<3

But no professional programmer would do this. Instead, a programmer in both BASIC and TI-BASIC would use the Else command:

:If N>3

But Else, unfortunately, is not mentioned in the U of Chicago text. In the end, since there isn't enough room (nor likely enough time, unless this becomes a two-day lesson) I decided to include the example program and the program in Question 12, but not the one in Question 13 (in order to avoid explaining the pi key or the STO-> key). Question 25 is included (since it's only a bonus question anyway), but not 26 (since it would be better written with Else).

If the condition in an If statement is false, then the Then statement is not executed. The book uses this fact to segue into what happens in mathematical logic if the hypothesis is false. Then as it turns out, the entire conditional is automatically true. This is called vacuous truth. The concept of vacuous truth can be confusing to many students. For example, the statement:

All unicorns are white.

is actually true -- after all, we have never seen a unicorn that isn't white (precisely because there exists no unicorns at all, much less ones that aren't white). Another way of thinking about this is that there are zero unicorns in this world, and all zero of them are white! In if-then form this statement becomes:

If an animal is a unicorn, then it is white.

The hypothesis is false (since there are no unicorns), so the entire conditional is true. This statement has no counterexamples (unicorns that aren't white), and conditionals without counterexamples are normally called true.

The book then derives, from the statement 1=2, the statement 131=177. There is a famous example of a derivation of a false conclusion from a false hypothesis, often attributed to the British mathematician Bertrand Russell, about a hundred years ago. From the statement 1=2, Russell proved that he was the Pope:

The Pope and I are two, therefore the Pope and I are one.

that is, he used the the Substitution Property of Equality from the hypothesis 1=2.

Tuesday, August 19, 2014

Section 2-2: "If-then" Statements (Day 9)

Section 2-2 of the U of Chicago text continues the study of logic by focusing on "if-then" statements. I certainly agree with the text when it writes:

"The small word 'if' is among the most important words in the language of logic and reasoning."

There are a few changes that I will make to the text. First of all, the text refers to the two parts of a conditional statement as the antecedent and the consequent -- although it does mention hypothesis and conclusion as acceptable alternatives. I'm going to follow what the majority of texts do and just use the words hypothesis and conclusion. Actually, Dr. Franklin Mason doesn't even use the word hypothesis -- he simply uses the word given -- since after all, the hypothesis of a theorem corresponds to the "given" statement in a two-column proof.

When I teach or tutor students in geometry, one of my favorite examples is "if a pencil is in my right hand, then it is yellow." So I pick up three yellow pencils, and we observe that the conditional is true. But let's suppose that I pick up a blue pencil in addition to the three yellow pencils. Now the conditional is false, since we can find a counterexample -- the blue pencil, since that's a pencil in my right hand yet isn't yellow.

Notice that I decided to replace the word instance with the word example -- so that the connection between examples and counterexamples becomes evident.

The text has to go back to an example from that dreaded algebra again. Of course, it's an important example, since students often forget that 9 has two square roots, 3 and -3. But I decided to include it anyway since it's simple -- it's not as if I'm making students use the quadratic formula or anything like that.

Then the book moves on to a famous mathematical statement: Goldbach's conjecture, named after the German mathematician Christian Goldbach who lived 300 years ago:

If n is an even number greater than 2, then there are always two primes whose sum is n.

At the time the book was written, the conjecture had been verified up to 100 million, but the conjecture had yet to be proved. But what about now -- has anyone proved Goldbach's conjecture yet? As it turns out, the answer is still no -- but now the conjecture has been verified up to four quintillion -- that is, the number 4 followed by 18 zeros.

But there has been work on a similar statement, called Goldbach's weak conjecture:

If n is an odd number greater than 5, then there are always three primes whose sum is n.

This is called weak because if the better-known (or strong) conjecture is true, the weak is automatically true because we can always let the third prime just be 3. Ironically, when Goldbach himself actually stated his conjecture, he stated the weak version of the conjecture. It was a letter from Euler -- you know, the same Euler who solved the bridge problem that we discussed as an Opening Activity -- that convinced Goldbach to state the strong conjecture instead.

Now as it turns out, someone has claimed a proof of Goldbach's weak conjecture -- namely the Peruvian mathematician Harald Helfgott. But Helfgott's proof is still being peer-reviewed -- that is, checked by other mathematicians to find out whether the proof is correct. Perhaps by this time next year, Helfgott's proof will have finally been verified (or dismissed as incorrect).

Dr. M also mentions Goldbach's conjecture, on a worksheet for his Section 2-1. Often students are fascinated when they hear about conjectures that take centuries to prove, such as Goldbach's conjecture or Fermat's Last Theorem. I often use these examples to motivate students to be persistent when trying to come up with proofs in geometry -- if mathematicians didn't give up even after centuries of trying to prove these conjectures, then why should they give up after minutes?

The final example in this section has students rewrite statements into if-then form. I've found that oftentimes, English learners struggle with this part of the lesson. The teacher must point out why, for example, the "something" in the example "all triangles have three sides" must be a figure: "if a figure is a triangle, then it has three sides." So not only must we appease algebra haters when we include algebra in the geometry lesson, but we must also consider English learners when including English in the geometry lesson.

Once again, I decided to include some review questions. Notice that the most of the review questions in this section are from yesterday's lesson, Section 2-1. We skipped Section 1-9 so I threw out the Triangle Inequality question. Once again, that question marked Previous course is an Algebra I question, and so once again, I rewrote it so that the solution is a whole number. Finally, I decided to avoid that inequality question completely.

Monday, August 18, 2014

Section 2-1: The Need for Definitions (Day 8)

The second chapter of nearly any high school geometry text discusses the logical structure of geometry -- to prepare students for proofs. This includes the U of Chicago text, as well as Dr. Franklin Mason's text, and many others.

Section 2-1 of the U of Chicago text deals with definitions. But the introduction to the chapter mentions a 1986 USA Today article concerning a non-mathematical definition: cookie. Normally, as teachers we'd ignore this page and skip directly to the first section, except that this article is mentioned all throughout 2-1, even including the questions!

Now, of course, a teacher could have the students discuss the article as an introduction to the importance of precise definitions. Such an introduction is often called an anticipatory set -- a concept that apparently goes back to the education theorist Madeline Hunter.

A teacher could present the article as an anticipatory set, but I should point out that the article is over a quarter of a century old -- after all, my text itself is nearly that old. The article points out that the word terrorist was controversial even back then. As we already know, a decade after the book was written, the 9/11 attacks occurred -- and since then, that word terrorist has been thrown around so much more, with very strong political implications.

And, of course, there was another definition that led to a politically charged debate -- one that occurred just a few years after the publishing of the text. During the investigation during the impeachment of Bill Clinton, the former president questioned the definition of the word is. So we see that there are two fields where precise definitions matter greatly: law and mathematics.

To me, it might be fun to discuss these examples in class. But it may be tough for the teacher to remain politically neutral during such a discussion, so we must proceed with caution.

The images at the end of this post do not mention the article -- I threw out any part of the section that refers to the article. I preserve the discussion about what a rectangle is, and the one definition given in the lesson -- that of convex set.

When approaching the questions, I first threw out Questions 1 through 3, since these questions go back to the article. I kept all of the questions about convex sets, since that's the term defined in the lesson, then kept the question where students guess the definition of midpoint -- a preview of Section 2-5.

Now I want to consider including the review questions as well. As any teacher knows, students have trouble retaining what they've learned, so we give review questions to make them remember. I avoided review questions during Chapter 1 since most of them were review of the skipped Sections 1-1 through 1-5. But most of the review questions in this section are labeled Previous course. I must be careful about these, since it all depends on which previous course is being mentioned here.

Question 20 in the text discusses the definition of words like pentagon and octagon. Like midpoint, this book will define these terms later in the chapter (Section 2-7), but this is labeled Previous course. I assume that the intended previous course is probably a middle school course. But -- remembering that this is a Common Core blog -- I decided to look up the Common Core Standards. The only standard mentioning the word pentagon is a 2nd grade standard!

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces. Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

So in theory, it might have been nearly a decade since the students saw the word pentagon. (The word octagon doesn't appear in the standards at all.) But I figure that upon seeing the question, the students will remember vaguely that these words all refer to shapes with different numbers of sides -- and at least know that a triangle has three sides, even if they must guess on all the rest. This is a good preview of Section 2-7.

In the other questions marked Previous course, the course referred to is clearly Algebra I. Once again, I don't want to intimidate the students with Algebra I questions in a Geometry class. Of course, we can see how Questions 21 and 22 came about -- they are clearly translations of the word problems "23 degrees less than the measure of an angle is the measure of its supplement" and "the measure of an angle is six times the the measure of its complement," respectively. I'm torn whether to include such problems. One thing that I definitely want to avoid is algebra problems masquerading as geometry problems -- for example, we take a linear equation from algebra and write its two sides as the measures of vertical angles (provided the two sides equal valid angle measures). The geometry in such a question is trivial -- just set the two sides equal to each other since vertical angles are congruent, then the rest is all algebra. The geometry in a question about complementary and supplementary angles is less trivial, but then -- so is the algebra, since a typical question will often have variables on both sides, and many students struggle with these.

In the end, I decided to keep Questions 21 and 22 but at least give the students a break by making the solutions whole numbers -- notice that as written, the solutions to both contain fractions. Question 23 seems to serve no geometric purpose at all. I decided to drop the second variable z and change the number 225 to 360, since students will often need to divide 360 degrees by various numbers -- for example, when finding the exterior angle measures of a 15-gon. This is the most difficult algebra/arithmetic that I want appearing in the first semester of a geometry course -- nothing beyond this is acceptable.

Finally, we reach Question 24. This is an Exploration question, asking the students to define the words cookie and terrorist. Once again, this makes a lot more sense if the article is mentioned in class. I decided that I'll include this and other Exploration questions, but label them as Bonus questions to emphasize that these questions are optional for the students. Of course, it can be thrown out completely if a teacher wants to avoid politically charged debates over the word terrorist.

Friday, August 15, 2014

Chapter 1 Quiz (Day 7)

This concludes Chapter 1 of the U of Chicago text. It is traditional to give some sort of test at the end of the chapter, but since we formally covered only three sections, a quiz is more appropriate here.

Now for me, designing this quiz is rather tricky. As everything else in this course, I wanted my quiz to based on the U of Chicago text. There is a Progress Self-Test included in the book. But even if I threw out the questions based on sections 1-6 through 1-8, there are some questions that I chose not to include.

For example, the first question on the Self-Test asks the students to find AB using a number line. This is very similar to some of the questions that I gave on the Wednesday and Thursday worksheets. But there is one crucial difference -- this one is the first in which both A and B have negative coefficients.

Now I know what the test writers are thinking here. The test writers want to know whether the students understand a concept. There's not enough room on the test to give both easier and harder questions. If a student gets a harder question correct, we can be sure that the student will probably get a much easier question right as well. But if the student only answers an easier question correctly, we can never be sure whether the student understands the more difficult question. Therefore, the test should contain only harder questions, since anyone who gets these right understands the simpler concepts too.

But now let's think about this from the perspective of the test taker, not the test maker. Let's consider the following sequence of hypothetical conversations:

Student: The distance between 4 and 5 is 9.
Teacher: Wrong. You're supposed to subtract the coordinates, not add them. The distance is 5 - 4 = 1.
Student: Oh.

Student: The distance between -4 and 2 is 2.
Teacher: Wrong. When subtracting, change the sign. The distance is 2 - (-4) = 6.
Student: Oh.

Student: The distance between -8 and -4 is 12.
Teacher: Wrong. You forgot the negative in front of the 4. The distance is -4 - (-8) = 4.
Student: Oh.

And we can see the problem here. The teacher wants the student to be able to find the distance no matter what the sign of the coordinates are -- not just when they're positive. But the problem is that the instant that a student finally understands how to solve the first problem, the teacher suddenly makes the problem slightly harder, and the student becomes confused.

Of course, you might be asking, why only give one problem on Wednesday? Why can't we give more problems to check for student understanding of the all-positive case, then move on to negatives? But you see, I'm imagining the above hypothetical conversations as occurring during, say, a warm-up given during the first few minutes of class -- and warm-ups typically contain no more than one or two questions. The student is never allowed to taste success, because each day a little something is added to the problem (like a negative sign) that's preventing the student's answer from being completely correct. The student never hears the words "You're right." And that's just with negative signs -- the U of Chicago text includes questions with decimals as well. I immediately threw all decimals out of my problems -- since decimals confuse the students even more, most notably when we draw number lines and mark only the integers.

Well, I don't want this to happen, especially not on the quiz or test where most of the points are earned. I want the student to taste success -- and this includes the student who's coming off of a tough second semester of Algebra I and is now in Geometry. Sure, if you feel that some students need to be challenged, then challenge them with all the negatives and decimals you want. But I don't want to dangle the carrot of success in front of a student (making them think that they've understood a concept and will get the next quiz question right), only to jerk it away at the last moment (by adding extra negative signs that will make the student get the next quiz question wrong), all in the name of challenging the students.

And so my quiz questions are basically review questions rewritten with different numbers. My rule of thumb is that the quiz contains exactly the same number of negative signs as the review. Some teachers may see this as spoon-feeding, but I see it as setting the students up for success. Any student who works hard to prepare for the quiz by studying the review will get the corresponding questions correct on the quiz.

Of course, some questions about the properties are hard to rewrite. I considered using the question from the U of Chicago text, to get from "3x > 11" to "3x + 6 > 17." But notice that the correct answer -- Addition Property of Inequality -- is difficult to remember and will result in many students getting it wrong. So even here I changed it to the Addition Property of Equality. After all, the whole point of learning the properties is to be able to use them in proofs. The Addition Property of Equality is much more likely to appear than the corresponding Property of Inequality. All including Inequality on the quiz accomplishes is increasing student frustration over a property that rarely even appears in proofs.

And so this is what I came up with:

Thursday, August 14, 2014

Chapter 1 Review (Day 6)

The first thing you might be wondering is, what happened to Section 1-9? In this section, the U of Chicago text presents the Triangle Inequality, which seems to be an important concept for students to learn. But here's where the problem lies:

"In this book we treat the Triangle Inequality as a postulate. That is, we do not prove it from other postulates."

The problem is that the Triangle Inequality is actually a theorem. It is, contrary to what's written in the book, provable from other postulates. And so, following David Joyce, we shouldn't label it as a postulate.

Franklin Mason ("Dr. M"), meanwhile, indeed gives a proof of the Triangle Inequality -- and his proof goes all the way back to Euclid. It is Euclid's Proposition 20, where the ancient geometer proves that in triangle ABC, AB + AC > BC:

Euclid, Proposition 20:
"Draw BA through to the point D, and make DA equal to CA. Join DC.
Since DA equals AC, therefore the angle ADC also equals the angle ACD.
Therefore the angle BCD is greater than the angle ADC.
Since DCB is a triangle having the angle BCD greater than the angle BDC, and the side opposite the greater angle is greater, therefore DB is greater than BC.
But DA equals AC, therefore the sum of BA and AC is greater than BC.
Similarly we can prove that the sum of AB and BC is also greater than CA, and the sum of BC and CA is greater than AB.
Therefore in any triangle the sum of any two sides is greater than the remaining one."

We can follow the proof to see where Euclid's reasoning comes from.

-- The first line "Draw BA..." is essentially the Point-Line-Plane Postulate (with DA = CA coming from the Ruler Postulate part of the P-L-P Postulate).
-- The second line "Since DA..." is the Isosceles Triangle Theorem (since triangle ACD is isosceles). This is in Section 5-1 of the U of Chicago text.
-- The third line "Therefore the angle..." comes from the Angle Addition Postulate (Section 3-1), the Equation to Inequality Property, and Substitution Property of Equality.
-- The fourth line "Since DCB..." comes from the Unequal Angles Theorem, which isn't given in the U of Chicago text until Section 13-7 as it depends on indirect proof.

So the U of Chicago text does provide all of the results needed to prove the theorem, but not until we reach Chapter 13. (Here in Chapter 1, we haven't even defined angle yet!) Dr. M includes a two-column proof with eight steps that the students are to fill in. This result is part of his Chapter 5, which gives some of the theorems that U of Chicago provides in 13-7, including the Unequal Angles Theorem (which Dr. M calls the "Triangle Angle Side Inequality").

Now both Dr. M and U of Chicago derive the Unequal Angles Theorem (that is, TASI) from the (Triangle) Exterior Angle Inequality. But then again, the paths diverge. In Dr. M the TEAI is a postulate (but recall that an old blog post of his derives this from SAS) while in U of Chicago, it is derived from the Exterior Angle (Equality) Theorem, which itself goes back to the sum of the angles of a triangle being 180. (Actually, I've seen other high school geometry texts that do derive the Triangle Inequality from the Unequal Angles Theorem but then proceeds back to the Exterior Angle Equality as in U of Chicago.)

But here's the thing -- this is Euclid's Proposition 20, and recall that all of his propositions up to 28 don't require the Parallel Postulate. Thus even though the sum of the angles of the triangle requires Playfair, the Triangle Inequality doesn't require Playfair. (In particular, that an exterior angle of a triangle is the sum of the two remote interior angles requires Playfair, but that the exterior angle is merely greater than the two remote interiors doesn't require Playfair.)

What, then, should I do about the Triangle Inequality? I could follow the Joyce ideal -- never present a statement as a postulate when it can be proved as a theorem, prove every theorem, and never use a statement requiring Playfair to derive a statement that is provable without Playfair. This is what most professional mathematicians do -- it's considered more elegant to have as few postulates and as many proved statements as possible.

But high school students are not professional mathematicians. Student understanding has priority over mathematical elegance. Recall that last month, I distinguished low-, medium-, and high-level proofs -- with high-level proofs being the ones that students are usually asked to complete. Medium-level proofs of theorems are used to prove high-level results, and low-level proofs of theorems are used to prove both medium- and high-level results. Surely the Triangle Inequality is a low- or medium-level result -- students are more likely to use the Inequality than to prove it. Since the U of Chicago text doesn't expect students to prove it, the text declares the Inequality to be a postulate. Dr. M expects students to prove it, with the prove involving the Exterior Angle Inequality. But he doesn't expect students to prove the TEAI, so he declares the TEAI to be a postulate.

Also having priority over mathematical elegance are the Common Core Standards. I wish to focus on proving the results that the standards explicitly ask students to prove. And the standards currently don't mention the Triangle Inequality at all. But, when I performed a Google search just to double check, I discovered that an older version of the standards do mention both the Triangle Inequality and TEAI as results to prove! (Of course, the standards didn't mention whether TEAI should be derived from Playfair or SAS.)

I've decided to delay the Triangle Inequality until it can be proved. I could try to prove it before giving the Playfair Postulate in the first semester (since the Inequality doesn't derive from Playfair), but I've already mentioned that I'll wait until second semester to give indirect proofs -- once again, in order to avoid confusing students with indirect proofs. And so I don't have to decide whether to give the Playfair or the SAS proof of the TEAI until then.

Instead, I now give a quick review of the sections of Chapter 1 that I did cover. This is mostly derived from the Vocabulary and Questions on SPUR Objectives that appear in the text, except that I skip over sections that I didn't cover. Notice that the text gives vocabulary and questions from Sections 1-4 and 1-5, and I did cover these sections, but only as a possible Opening Activity (and teachers may have come up with their own Opening Activity). So my review worksheet only covers 1-6 through 1-8. (If you prefer, you may think of these exercises as Section 1-8 Part 2, since that's where most are from.)

Wednesday, August 13, 2014

Section 1-8: One-Dimensional Figures (Day 5)

Section 1-8 of the U of Chicago text deals with segments and rays. The text begins by introducing the simple idea of betweenness. In Common Core Geometry, betweenness is an important concept, because it's one of the four properties preserved by isometries (the "B" of "A-B-C-D").

As I mentioned a few days ago, for Hilbert, betweenness is a primitive notion -- an undefined term, just as point, line, and plane are undefined. Yet the U of Chicago goes on to define it! It begins by defining betweenness for real numbers:

"A number is between two others if it is greater than one of them and less than the other."

Then the text can define betweenness for points:

"A point is between two other points on the same line if its coordinate is between their coordinates."

But Hilbert couldn't do this, because his points don't have coordinates. Recall that it was Birkhoff, not Hilbert, who came up with the Ruler Postulate assigning real numbers to points. Instead, Hilbert's axioms contain statements about order (Axioms II.1 through II.4), such as:

"II.2. If A and C are two points of a line, then there exists at least one point B lying between A and C."

Since we have a Ruler Postulate (part of the Point-Line-Plane Postulate), this statement is obvious, since points have coordinates and the same is true for real numbers -- between reals a and c is another real b.

I've seen some modern geometry texts mention a Ruler Postulate, but nonetheless leave the term betweenness undefined. Now as we mentioned earlier with point, line, and plane, if a term such as betweenness is undefined, then we need a postulate to describe what betweenness is. This postulate is often called the Segment Addition Postulate:

"If B is between A and C, then AB + BC = AC."

Notice that this statement does appear in the U of Chicago text. But the text doesn't call it the Segment Addition Postulate, but rather the Betweenness Theorem. As a theorem, we should be able to prove it -- and since after all, the text defines betweenness in terms of real numbers, we should be able to use real numbers to prove the theorem. Indeed, the text states that we can use algebra to prove the theorem, but the proof is omitted.

Following David Joyce's admonition that we avoid stating a theorem without giving its proof, let's attempt a proof of the Betweenness Theorem. We are given that B is between A and C. Now let us assign coordinates to these points. To make it easy to remember, we simply use lowercase letters, so point A has coordinate a, point B has coordinate b, and point C has coordinate c.

We are given that B is between A and C, so by definition of betweenness, we have either a < b < c, or the reverse of this, a > b > c. Without loss of generality, let us assume a < b < c (especially since the example in the book has a < b < c). Now by the Ruler Postulate (the Distance Assumption in the Point-Line-Plane Postulate), the distance between A and B (in other words, AB) is |a - b|. Since a < b, a - b must be negative, and so its absolute value is its opposite b - a. (To avoid confusing students, we emphasize that to find AB, we just subtract the right coordinate minus the left coordinate, so that AB isn't negative. This helps us to avoid mentioning absolute value.) Similarly BC = c - b and AC = c - a. And so we calculate:

AB + BC = (b - a) + (c - b) (Substitution Property of Equality)
               = c - a (simplification -- cancelling terms b and -b)
               = AC

The case where a > b > c is similar, except that AB is now a - b rather than b - a. All the signs are reversed and the same result AB + BC = AC appears. QED

Don't forget that I want to avoid torturing geometry students with algebra. And so I simply give the example with numerical values, with the variables off to the side for those who wish to see the proof.

The text proceeds to define segments, rays, and opposite rays in terms of betweenness. Notice that these definition are somewhat more formal than those given in other texts. A typical text, for example, might define a segment as "a portion of a line from one endpoint to another." But the U of Chicago text writes:

"The segment (or line segment) with endpoints A and B is the set consisting of the distinct points A and B and all points between A and B."

The definitions of ray and opposite ray are similarly defined in terms of betweenness.

The section concludes with the notation for line AB, ray AB, segment AB, and distance AB. But although every textbook distinguishes between segment AB and distance AB, many students -- and admittedly, many teachers as well -- do not. The former has an overline, but the latter doesn't. Unfortunately, Blogger allows me to underline AB and strikethrough AB, but not overline. For the purpose of the rest of this post, let's pretend that the strikethrough AB is really the overline for segment AB.

Now if AB and CD are both of, say, unit length, then AB = CD, but we can't write AB = CD. After all, AB and CD are real numbers -- both are 1 -- and those numbers are equal. But the segments AB and CD can't be equal unless they have the same endpoints (that is, A and C are the same point, as are B and D, or vice versa). The numbers (the lengths) are equal, while the segments are congruent. But students and teachers alike confuse a segment with its length, and confuse equality with congruence.

To avoid confusion, in the following images I threw out Question 8 from the text, a multiple choice question which states that AB literally equals BA (but ray AB is not the same ray as BA). Notice that I tried to draw all of the one-dimensional figures in red.