Friday, February 27, 2015

Chapter 14 Test (Day 121)

Today is the Chapter 14 Test. Here are the answers to my posted test:

1. DE = 32, EF = 16sqrt(3).

2. TU = 16, US = 8sqrt(3), SK = 8, TK = 8sqrt(2).

3. 3/4

4. 3/5

5. 0.309

6. 0.625

7. 1/2

8. sqrt(3) (Some people may consider this question unfair, since the above question and both corresponding questions on the practice had rational answers, leading students to believe that they can just use a calculator to find the exact value rather than use 30-60-90 triangles.)

9-10. These are vectors that I can't reproduce easily here.

11. BC/AC (or a/b, if the students learned it that way).

12. AB and AD

13. ACD and CBD

14. This is a vector that I can't reproduce easily here.

15. (9, 6)

16. 115 feet, to the nearest foot.

17. (1, 4)

18. (3, -3)

19. (3, 2). (I hope students don't get confused here and solve these three backwards!)

20. This is a vector that I can't reproduce easily here.

Thus concludes Chapter 14. Stay tuned -- we're jumping back to Chapter 8 on Area next week!

Thursday, February 26, 2015

Review for Chapter 14 Test (Day 120)

Today is the 120th day of school, according to the blog day count. Therefore today marks the end of the second trimester, but this is relevant mostly for elementary school students (including first graders, since Common Core requires that they learn how to count to 120).

Last night I tutored my geometry student for the second time this week. Section 6-6 of the Glencoe text is on -- and don't say I didn't warn you, since I mentioned it earlier -- trapezoids. And my student already knew about trapezoids, since I showed them to him during the first tutoring session. Indeed, I didn't given my student another worksheet, since our worksheet from earlier was on trapezoids.

My student had two proofs to complete last night. Naturally, both of them were on isosceles trapezoids, since, as the mathematician Conway pointed out, there's hardly anything to prove about a trapezoid that isn't isosceles.

Given: HJ | | GK, Triangles HGK and JKG are congruent,
          HG is not parallel to JK
Prove: GHJK is an isosceles trapezoid

Glencoe asks the students to write flow proofs for this assignment. I won't post any proof, since this has nothing to do with the current U of Chicago chapter (trigonometry or vectors). But notice out the exclusive definition of trapezoid led necessitated the statement "HG is not parallel to JK" being included in the list of Given statements -- otherwise the figure could be a rectangle. Under the inclusive definition, this statement would not be necessary.

Given: ZYXP is an isosceles trapezoid.
Prove: PWX is an isosceles triangle.
(The diagram shows that lines PZ and XY intersect at W.)

This is a straightforward proof -- albeit long since Glencoe requires the students to use the definition of congruence to convert between the congruence of segments and the equality of their lengths before one can apply the Segment Addition Postulate.

Also, today I subbed in another math class. This time, it was a sixth grade class. The text that this class is using for Common Core was published by Carnegie Learning. Here is the course layout:

Chapter 1. Factors, Multiples, Primes, and Composites
Chapter 2. Prime Factorization and the Fundamental Theorem of Arithmetic
Chapter 3. Fractions
Chapter 4. Decimals
Chapter 5. Ratios
Chapter 6. Percents
Chapter 7. Introduction to Expressions
Chapter 8. Algebraic Expressions

The class was on the final section of the sixth chapter -- Section 6.5, "Practical Percents Practice." This is just about the right pace for the class to finish Chapter 8 before the SBAC exam. In this section, the students had to deal with the common misconception that a 25% discount followed by a 40% discount equals a 65% discount. I even told them about some bookstores that once had to explain that if a book can be preordered at 40% and members get a 10% discount, the book is only 46% off. The main sixth grade Common Core standard for percents is:

Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.

Today I finally post the vector activity that I've been planning this week. But today is supposed to be the review for the Chapter 14 Test. Well, that's no problem -- technically this activity counts as part of the test review.

Even then, I still have problem making this activity organized. I quickly created my worksheets for both Partner A and Partner B, but I didn't have time to scan them before posting them here. Since I can't keep delaying the activity, I will cut and paste what I wanted the worksheet to say right here:

Name _________________________ Date _______________ Period _____
Partner’s Name _________________________

Partner A: Vector Activity

Task 1: What vector did you select?                             (_____, _____)

Add (2, 1) to this vector. Answer:                                 (_____, _____)

Tell this answer to Partner B. Did Partner B guess your original vector?

Task 2: What vector did Partner B tell you?                   (_____, _____)

Partner B multiplied the original vector by the scalar 5 to obtain this vector. What is the original vector?                                           (_____, _____)

Tell your guess to Partner B. Did you guess correctly?

Task 3: What vector did you select?                             (_____, _____)

Multiply this vector by the scalar 4. Answer:                           (_____, _____)

Add (-2, -2) to this vector. Answer:                              (_____, _____)

Tell this answer to Partner B. Did Partner B guess your original vector?

Task 4 (Optional): Graph the original and final vectors from Task 3.

Name _________________________ Date _______________ Period _____
Partner’s Name _________________________

Partner B: Vector Activity

Task 1: What vector did Partner A tell you?                   (_____, _____)

Partner A has added (2, 1) to the original vector to obtain this vector. What is the original vector?                                                   (_____, _____)

Tell your guess to Partner A. Did you guess correctly?

Task 2: What vector did you select?                             (_____, _____)

Multiply this vector by the scalar 5. Answer:                           (_____, _____)

Tell this answer to Partner A. Did Partner A guess your original vector?

Task 3: What vector did Partner A tell you?                   (_____, _____)

Partner A multiplied the original vector by 4, then added (-2, -2) to obtain this vector. What is the original vector?                    (_____, _____)

Tell your guess to Partner A. Did you guess correctly?

Task 4 (Optional): Graph the original and final vectors from Task 3.

As I've mentioned earlier, this activity is based on the one that the class I subbed for earlier this week were working on, except that this one involves vectors. Also, I pointed out that I designed this activity to run more smoothly than the one in that class, by making sure that Partner A and Partner B have clearly defined roles.

In fact, this one should run even better than my improved 6th period version ran. In that class, I had to tell the Partner A students to cross out the boxes since only the Partner B students need them. But that wasted time, and students on the other side of the classroom were goofing off while I was checking to see that the students on my side were crossing out the boxes. So for my lesson, Partner A and Partner B already have separate worksheets, so there is nothing for anyone to cross out -- so no time is wasted crossing out boxes.

There is also a page to cut out with 36 given vectors -- since as I mentioned earlier, I don't want the students to choose the vectors. Of course, cutting out the vectors is time consuming -- even if the teacher does it before the class -- and those tiny slips of paper are easily lost. Then teachers will have to cut them out several times throughout the day.

Another way to have the students choose vectors would be to have something larger represent the vectors, such as playing cards. The playing cards can be converted to vectors, as follows:

For the horizontal component:
Ace through 10 -- valued 1 through 10
Jack -- valued -2
Queen -- valued -1
King -- valued 0

For the vertical component, use the suit:
Clubs -- valued -1
Diamonds -- valued 0
Hearts -- valued 1
Spades -- valued 2

Eight of Diamonds -- (8, 0)
Ace of Clubs -- (1, -1)
Jack of Hearts -- (-2, 1)

Because they are larger, playing cards are less likely to be lost than the little slips of paper that I provide for this activity. But there are problems with using playing cards. First of all, the conversion from playing card to vector is another step that the first partner can get wrong -- and once again this may frustrate the second partner. (I've heard that some people don't even know the difference between clubs and spades!) Furthermore, vectors with large components, such as (8, 0) for the eight of diamonds above, become (30, -2) after performing the steps in Task Three -- and then they are asked to graph that vector (30, -2) in Task Four. So I leave it up to individual teachers whether to use playing cards or the slips of paper that I provide.

Wednesday, February 25, 2015

Section 14-7: Adding Vectors Using Trigonometry (Day 119)

I have prepared the vector activity lesson that I mentioned yesterday. But the problem is that I don't know whether there is any time for me to post it. This is because Section 14-7 of the U of Chicago text is on adding vectors using trigonometry -- and we can't skip it because it appears in the following Common Core standard:

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Notice that we are essentially converting the vectors from polar to rectangular form, adding their components, and then converting the sum back to polar form. And all of this is done without the students' even knowing what polar coordinates are.

In an actual trig course (as part of precalculus), we find out that we can avoid thinking about angles larger than 90 degrees by considering the reference angle -- the angle formed by the x-axis and the terminal side of the original angle. We see that even though the U of Chicago's Geometry text doesn't teach reference angles, the angles shown in the text are always formed using the x-axis -- that is, the west-east axis -- and never using the y-axis. So the U of Chicago, while not explicitly teaching reference angles, clearly has these reference angles in mind when writing this chapter.

The text states that one of the two components is found using the sine of the angle, while the other is using the cosine. But because of the way the angles are drawn, the horizontal component will always use the cosine and the vertical component will always use the sine -- just as they would be for polar coordinates in a trig class.

To convert from rectangular back to polar coordinates, we use the distance formula and the inverse tangent function. This is the only time that an inverse trigonometric function appears in the U of Chicago text -- although many other geometry texts discuss inverse trig in more detail.

Tuesday, February 24, 2015

Section 14-6: Properties of Vectors (Day 118)

Last night I tutored my geometry student. He is now in Chapter 6 of the Glencoe text, which covers various polygons. Let's see how this chapter is organized:

Section 6-1: Angles of Polygons
Section 6-2: Parallelograms
Section 6-3: Tests for Parallelograms
Section 6-4: Rectangles
Section 6-5: Rhombi and Squares
Section 6-6: Trapezoids and Kites

Clearly, Chapter 6 of the Glencoe text corresponds best to Chapter 5 of the U of Chicago. Angles of polygons (most notably, their sum) is in U of Chicago's Section 5-7. Notice that the parallelogram consequences in Glencoe's 6-2 and 6-3 wait until U of Chicago's 7-6 and 7-7, respectively. The rest of the Glencoe chapter corresponds to U of Chicago's 5-4 and 5-5.

My student had just finished Section 6-4 in Glencoe, on rectangles, and he asked me whether I had a worksheet prepared for rectangles. But there are two main differences between the U of Chicago text and a more traditionalist text such as Glencoe. The first is the U of Chicago's dependence on symmetry (reflected in the Common Core standards), while Glencoe uses triangle congruence to derive the parallelogram properties. The other is the U of Chicago's inclusive definitions of the various quadrilaterals. Recall that in the U of Chicago, a parallelogram is a trapezoid, and a rectangle is an isosceles trapezoid. Glencoe, like most texts of its era, defines trapezoid exclusively.

My worksheets are based on the U of Chicago. In this text, rectangles are covered in Section 5-5, "Properties of Trapezoids," because a rectangle is a trapezoid -- indeed, an isosceles trapezoid. In fact, the properties of the rectangle derive trivially from those of the isosceles trapezoid. This would only confuse my student since he is only in Section 6-4, while trapezoids don't occur until Section 6-6 of his text. So my worksheet shows how the properties of the rectangle and isosceles trapezoid are related, while he doesn't even know what a trapezoid is yet! Glencoe can define rectangle before defining trapezoid, since a rectangle isn't a trapezoid in that text. On the other hand, notice that rectangle is defined after parallelogram, since a rectangle is still a parallelogram in that text.

Now my student wants to see a proof involving rectangles. I decided to prove the least obvious properties of rectangles -- namely that their diagonals are congruent. Notice that my Section 5-5 worksheet contains a proof that the diagonals of isosceles trapezoids are congruent. And so I was forced to show him the Quadrilateral Hierarchy and explain what a trapezoid is. When I finally showed him a proof, I used a traditional one based on SAS -- since that is what he's expected to learn in his Glencoe class -- rather than use the Rectangle Symmetry Theorem.

I like the idea of showing that, since rectangles are isosceles trapezoids and isosceles trapezoids have congruent diagonals, so do rectangles. Rectangles inherit all the properties of isosceles trapezoids -- just as they inherit all the properties of parallelograms. The inclusive Quadrilateral Hierarchy means that there's less to memorize, and less to prove. But the benefits are lost if we suddenly spring the inclusive hierarchy on someone who learned the exclusive hierarchy!

Meanwhile, today I subbed in a math class. The regular teacher has three sections of Calculus (periods 3-5) and two sections of 9th grade Integrated Math (per. 1 and 6). I point out that to opponents of Integrated Math, this is ironic, since it's because of Integrated Math that there might not be any Calculus students in three years -- and this is especially true since "9th grade" Integrated Math is based on 8th grade packets.

The "9th" (8th) grade MathLinks packet for today's class is "Slope and Slope-Intercept Form of a Line." I noticed that once again, today's lesson is geared more towards algebra than geometry, even though the 8th grade Common Core Standards has both algebra and geometry. I was hoping that there'd be more geometry now that we're in the second semester, but so far this has not turned out to be the case.

Here are the sections of the current Student Packet 8-8:

8.1 Introduction to Slope
8.2 Input-Output Investigation
8.3 Slope-Intercept Form
8.4 Skill Builders, Vocabulary, and Review

The students are now in Section 8.2, "Input-Output Investigation." Today, the packet directs the students to play a game: the Input-Output Game. Here are the rules of the game, as written by the teacher himself:

One person thinks of a combination of cups & counters -- the other person then asks two questions (numbers) to determine the equation, the value of y when x = 0, and the slope (also graph the equation). After this, the players switch responsibilities and repeat the process.

Notice that this defines a function whose slope is the number of cups, and whose intercept is the number of counters. That is, the input x represents the number of counters in each cup.

But this game turned out to be highly problematic. Here are some of the problems that I had when I played this game in first period:

-- Many students couldn't choose a good number of cups and counters.
-- When they did choose a number of cups and counters, they wrote the number of cups and counters under "Input" and "Output." (In other words, they wrote the values of m and b where the values of x and y are supposed to go!)
-- Because this is a partner activity, if the first student doesn't perform his/her task correctly, then the second partner can't perform his/her task at all.
-- When I tried to correct them, many students were talking and not paying attention to me at all.

And all of this is despite the teacher having shown them how to play the game during yesterday's class! (Then again, the teacher showed them a specific number of cups -- three -- and counters -- two -- and input values x = 2 and x = 4. It's easier for the students when they don't have to choose the numbers!)

I remember back when I was student teaching in a traditionalist Algebra I course, there was a similar activity near the beginning of the school year. The students were to divide into pairs. One student would choose a value of x -- say x = 4 -- and then perform the following tasks based on the number they chose:

Multiply both sides by 4: 4x = 16
Add 10 to both sides: 4x + 10 = 26
Divide both sides by 2: (4x + 10)/2 = 13
Subtract 1 from both sides: (4x + 10)/2 - 1 = 12

The second partner now receives the equation (4x + 10)/2 - 1 = 12 and now must solve that equation. The whole point of this activity is for the students to see that they must perform inverse operations in order to solve the equation -- instead of subtracting 1, they should add 1 to both sides. Instead of dividing by 2, they should multiply by 2 on both sides, and so on.

But once again, the problem was that the first student didn't do his/her task right -- and the second partner's success is contingent on the first student's. Both that day and today, it was often the second partner who was feeling frustrated, since the second partner couldn't do anything if the first student makes a mistake.

Now this is often the point where traditionalists point out the flaws with a more progressive curriculum. The partner activity failed because when one partner couldn't perform the task, the second can't even begin. Instead, the students need to be led not by a fallible student partner, but by the one person who's supposed to know already what he/she is doing -- the "sage on the stage," the teacher. My problem with pure traditionalism at this age is that it presumes that the students are even willing to listen to the teacher. I can easily see a teacher giving a traditionalist lecture on this material and the students simply tuning him/her out -- especially when it's time to graph.

(By the way, just as another word for the progressive philosophy is constructivism, there is a similar-sounding word for the traditionalist philosophy -- instructivism.)

To me, I could improve the lesson not simply by tossing it out completely to replace it with a traditionalist lecture. Instead, I intervene at the points where students are likely to make a mistake. Here are the changes that I implemented by the time I reached the other 9th grade class -- 6th period:

-- In both the student teaching class and today's subbing class, the students had trouble choosing numbers. I conclude that it's a bad idea to have students choose anything in this sort of activity, except to choose a partner. (And some teachers oppose having the students choose their partners as well!) Instead of having the students choose the numbers, I handed one partner a post-it note telling them how many cups and counters there were. I prepared the post-it notes during the teacher's conference period (2nd).

-- I told the student to whom I gave each post-it note to cross out the chart showing the input and output. That chart is only for the second partner, so the first student should cross it out rather than put numbers in the wrong places. In theory, the students switch roles after each game, so that the first partner becomes the second. This means that I can only play half as many games as the teacher intended -- but I'd much rather play fewer games correctly than more games incorrectly.

-- After that, instead of having the second partner choose the input values, I, the teacher, make that choice. I chose x = 3 for the first input, and x = 1 for the second. So the second partner writes these values under the "Input" column. The first person multiplies the input by the number of cups, then adds the number of counters, and then reports this number as the "Output" for the second person to write down. The second person's job is still to guess how many cups and counters there are.

-- The second partner gives the equation, the value of y if x = 0 (that is, the y-intercept), and the slope. Finally, both partners work together to graph the line. My job is to make sure that the lines are graphed correctly.

Well, with these changes, 6th period fared somewhat better than 1st. A few students were still confused, especially with the graphing. But at least some were less confused than period 1 was with everything before the graphing.

In case you're curious, the Calculus classes were working on a project of their own -- an "Optimal Can Project." The students were to divide into groups and determine what dimensions maximize the volume of a cylindrical can for a given surface area. The connection to geometry, the topic of this blog, is obvious -- we need to formulas for the volume and surface area of a cylinder. We will reach Chapter 10 of the U of Chicago text, on the volume and surface area formulas, some time in late March or early April.

In some ways, the cylinder optimization project is a bit difficult. One only has to do a few rectangle optimization questions -- that is, give the dimensions of a rectangle to maximize the area given a fixed perimeter -- before one conjectures that the optimal rectangle is the square. The teacher has to modify the question -- such as stating that the rectangle needs fencing on only three sides (because the fourth is either a river or a wall) -- in order to prevent the answer from being a square. But with a cylinder, the answer is less predictable.

Sometimes I like to post on the blog based on what I tutor or teach. I'm considering posting an activity similar to the one that I taught today in class, except that it would be on vectors, since that's what we covered today. I didn't want to create the activity before trying it out in 6th period today, and by then it's too late for me to create the activity since I want to make sure that students don't have to choose the vectors, so I'll have to make a cut-out page with all of the vectors on it. Instead, I post my originally planned lesson for Section 14-6, which contains many of those properties of vector addition from the Common Core Standards that I mentioned yesterday.

Monday, February 23, 2015

Section 14-5: Vectors (Day 117)

Let me congratulate the two scientist movies that I described last week for the Oscars they won. In particular, Graham Moore won Best Adapted Screenplay for Imitation Game, and Eddie Redmayne won Best Actor for his portrayal of Stephen Hawking in Theory of Everything.

Section 14-5 of the U of Chicago text is on vectors. As I mentioned last week, much of physics deals with vectors, and indeed the Theory of Everything that Stephen Hawking sought involves unifying the four forces. Force is a vector quantity.

I remember back when I was a high school senior taking AP Physics C, and our teacher wanted us to remember one thing about vectors:

Vectors operating at right angles are independent.

This means that if two vectors are perpendicular to each other -- most notably if one is parallel to the x-axis and the other to the y-axis -- then they are linearly independent. Motion along one vector has nothing to do with motion along the other. We notice this most clearly when one is throwing a ball, and we resolve the velocity into its horizontal and vertical components. The horizontal component of velocity is slowed down slightly by the force of air resistance, while the vertical component is slowed down -- and its direction is ultimately reversed -- by the force of gravity.

In fact, I remember learning about acceleration so many years ago, and I was confused as to why an object that is turning or slowing down is said to be "accelerating." After all, I thought that an object slowing down was decelerating. To find out the answer, we must look at vectors. If the velocity and acceleration vectors are perpendicular, then the object is turning. (They must be perpendicular because of the theorem from Chapter 13 (Section 13-5) that the tangent and radius of a circle are perpendicular.) So if the velocity and acceleration vectors are in opposite directions, then the object is slowing down. I was confused by the use of the word "acceleration" by non-technical English speakers, where it usually refers to an increase in speed without considering vectors at all.

As it turns out, we prove in Linear Algebra -- a college course beyond Calculus -- that any two vectors that aren't parallel are linearly independent. But the linear independence that we consider the most in physics involve right angles. We don't discuss this idea in Section 14-5, but we will look at both velocity and force vectors on the posted worksheet.

Vectors appear in the Common Core Standards, but not in the geometry section. Indeed, they appear in the "Vector & Matrix Quantities" section:

(+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).

(The U of Chicago text uses the first option, a single boldface roman letter. So v is a vector.) The point is that the various domains of the standards don't line up exactly with courses -- not even for the traditional (as opposed to the integrated) pathway. So there are standards that appear in the geometry section that don't appear in a geometry course (i.e., conic sections), as well as vice versa (vectors).

The various Common Core Standards for vectors are spread out among the last three sections of the chapter, 14-5 through 14-7. One standard that appears in today's Section 14-5 is:

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

This is only partly realized in Section 14-5 -- component-wise addition will appear in 14-6. But adding vectors end-to-end and by the parallelogram rule do appear in 14-5, and the U of Chicago does give an example where the magnitude of a sum isn't the sum of the magnitudes -- indeed, there's an example where the sum is shorter than one of the vectors being added! The text explains that this is to be expected since one vector is the velocity of the boat and the other is the velocity of the current (and the boat is moving nearly upstream).

So in a way, we are beginning this standard today as well:

(+) Solve problems involving velocity and other quantities that can be represented by vectors.

The text points out that vectors are closely related to translations. Many texts tend to define the translation in terms of its vector. The U of Chicago does the opposite. Instead, translations are defined as compositions of reflections. The Two Reflection Theorem of Section 6-2 tells us how to find the magnitude and direction of a translation in terms of the two mirrors. Of course, some would say that the magnitude and direction of the translation really means the magnitude and direction of the translation vector. But the word "vector" can't appear in Section 6-2, since it isn't even defined until Section 14-5. Instead, we see the following theorem:

Two vectors are equal if and only if their initial and terminal points are preimages and images under the same translation.

This theorem, therefore, shows how to connect the U of Chicago definition of translation (two reflections) with the definition found in other texts (vectors). Notice that in both cases, "equal vectors" are defined to be vectors with the same magnitude and direction.

Finally, the text defines vector addition:

The sum or resultant of two vectors AB and BC, written AB + BC, is the vector AC.

David Joyce criticizes the use of the word "resultant" to refer to vector sum:

The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?)

But then again, U of Chicago rarely uses the word "resultant" and mainly uses the word "sum" -- the word that we normally use when referring to addition. Joyce would prefer to omit the section on vectors, but they are included in the Common Core Standards -- and as I mentioned, the U of Chicago devotes three sections to vectors, not just one!

Friday, February 20, 2015

Activity: The Sine, Cosine, and Tangent Ratios (Day 116)

I'm still thinking about the Chinese New Year. As it turns out, 2015 is an exceptional year in the calculation of the date of the new year. As I mentioned yesterday, Chinese New Year falls near Lichun (or Imbolc), the midpoint between the winter solstice and the spring equinox, and that it falls at a dark (or new) moon. Indeed, Chinese New Year is almost always the dark moon that's the closest to Lichun. But one exception is 2015. According to the link I mentioned yesterday, this is because Lichun falls almost exactly halfway between two dark moons (i.e., at a full moon). The rule is quite complicated and is only explained fully at the link from yesterday.

The following link mentions that the past Year of the Horse contained a Leap Month -- that is a 13th month, since the solar year can't be divided evenly into lunar months. Unlike our solar calendar, where the Leap Day is always February 29th, or even the Hebrew lunisolar calendar, where the Leap Month is always in winter (the month "Adar" is repeated), in the Chinese calendar, any month can be the repeated month. According to the following link, the ninth month was the repeated month:

Finally, here's a link to the Archetypes Calendar, a calendar with 10 days per week. It is a rule-based calendar, rather than astronomical, but it turns out to match the Chinese calendar almost perfectly.

I could go on and on about the mathematics of the Chinese calendar, but that's enough for now. This is a geometry blog, not a Chinese calendar blog!

Here are the activities that I've decided upon for this activity day. All of them are are discovery activities found in the Exploration Questions in the U of Chicago.

(From Section 14-3: The Tangent Ratio)
Choose three angle measures (other than 90) whose sum is 180. (For example, you could choose 25, 97, and 58.)
a. Using a calculator, find the sum of the tangents of the numbers you have chosen.
b. Calculate the product of the tangents of the numbers you have chosen.
c. Repeat parts a and b with a different three numbers.
d. Make a conjecture based on what you find.

The goal of this activity is for the students to discover the surprising identity: if A, B and C are three angles whose measures add up to 180, then tan A + tan B + tan C = tan A tan B tan C. A full proof of this fact requires the addition formula for tangent, which one doesn't learn until a full trigonometry course later on. Of course, none of the angles can be 90, since the tangent of 90 degrees is undefined.

(From Section 14-4: The Sine and Cosine Ratios)
a. Fill in this table of values of the sine and cosine using your calculator.
(In the given table, all multiples of 5 degrees, from 0 to 90 degrees, are given.)
b. For which values of x does (sin x)^2 + (cos x)^2 = 1?

Obviously, the goal is for the students to discover the Pythagorean identity.

(From Section 14-5: Vectors)
a. Find out how long it takes by airplane to go from a nearby airport on a nonstop flight to some other location, and how long the return flight takes.
b. Allowing some time for takeoff and landing (from 5 minutes at a smaller airport to 20 minutes at the largest airports), about how fast does the schedule assume the plane can travel? Is there any assumption about wind?

As it turns out, a round trip flight can be shorter going one way than returning. Flying from California to New York is about 30 minutes shorter than flying from New York back to California. Of course there's an assumption about wind -- one is flying with the wind when going east and against the wind when flying west. This is what the students are to discover, and this is an introduction to the concept of vector.

This question requires research, but that should be easier now in the age of the Internet than back when the U of Chicago text was written. All three problems require calculators -- obviously for the trig questions, and likely for the last question as well, since students will have to divide numbers that almost certainly won't come out even to calculate the speed.

Thursday, February 19, 2015

Sections 14-3 and 14-4: The Sine, Cosine, and Tangent Ratios (Day 115)

Today is Chinese New Year, one of the most celebrated holidays in the world. So now is a good question to ask, how is Chinese New Year calculated?

Recall what I wrote at the Western New Year regarding calendars. We can describe them in terms of which boxes from that calendar checklist would be checked. For the Chinese Calendar, we'd have:

(x) the lunar month cannot be evenly divided into solar days
(x) the solar year cannot be evenly divided into lunar months
(x) having months of different lengths is irritating
(x) having months which vary in length from year to year is maddening

The Chinese Calendar is a lunisolar calendar -- indeed Chinese New Year is often called "Lunar New Year," in both China and other nearby Asian nations that celebrate it.

The following link describes the Chinese Calendar in more detail:

Here are a few comments about this calendar, as mentioned at the link:

The Chinese Calendar assumes a prime meridian of 120 degrees East (120°E). This means that a day (or rather, a nychthemeron, a day and a night) is taken to run from midnight Beijing standard time (BST = CCT = GMT+8) to the next midnight BST.

Oops, I forgot to check one of the boxes:

(x) Greenwich is not unambiguously inferior to any other possible prime meridian

Of course, the Chinese calendar has existed for thousands of years, before anyone from China had ever heard of Greenwich, so to them Greenwich was inferior to a prime meridian through China.

As we read the link above, we see that not only New Year, but every month is supposed to begin at an astronomical dark (or "new") moon, no matter what. This means that not only do months have different lengths, but the same month has can have different lengths in different years (although it's always either 29 or 30 days). Here astronomical exactness has priority over simplicity of month or year lengths, as there is always a trade-off between the two.

Why is Chinese New Year always in January or February? Let's see:

New Year's Day in the Gregorian Calendar always occurs about a week after the northern winter solstice, whereas on average New Year's Day in the Chinese Calendar occurs approximately midway between that solstice and the northern vernal equinox.

The day that is halfway between the winter solstice and spring equinox is called Lichun in Chinese -- this means the same thing as Imbolc to the pagan Wiccans. (Westen Christians often refer to this as Candlemas or Groundhog Day.) There is often a debate as to when the seasons begin. In China, the seasons run as follows:

Spring: February through April
Summer: May through July
Fall: August through October
Winter: November through January

The winter solstice is the darkest day of the year, while the coldest day of the year often doesn't occur until almost Lichun/Imbolc. Likewise, the lightest day of the year is the summer solstice, but the hottest day isn't until July or even August. The temperature trails the light by about a month or two, especially near the oceans due to the high specific heat of water. This means that using the Chinese definition of season, the hottest season of the year is often the fall and the coldest is the spring.

Nonetheless, to the Chinese, Lichun is the luckiest day of the year, and so a year containing two Lichuns -- a year containing a Leap Month, a 13th lunar month -- is also lucky. The year that just ended, the Year of the Horse, was a lucky year containing two Lichuns, while the new year, the Year of the Sheep, is an unlucky year with no Lichun.

Here's another webpage, from Singapore, about the Mathematics of the Chinese Calendar. It refers to the current unlucky year as a "Double Blind Year":

Section 13-3 of the U of Chicago text is on the tangent ratio, and Section 13-4 of the U of Chicago text is on the sine and cosine ratios. I have decided to combine all three trig ratios into one lesson.

David Joyce was not too thrilled to have trig in the geometry course. He wrote:

Chapter 11 [of the Prentice-Hall text -- dw] covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.
As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.

Yet most geometry books include trig because most state standards require it. And this most certainly includes the Common Core Standards:

Define trigonometric ratios and solve problems involving right triangles

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.
Explain and use the relationship between the sine and cosine of complementary angles.
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

And all three of these standards appear in this lesson.

Some people may wonder, why do we use the same name "tangent" to refer to the "tangent" of a circle and the "tangent" in trigonometry? A Michigan math teacher, Mike Shelly, discusses the reasons at the following link:

On the other hand, the reason that "sine" and "cosine" have the same name is less of a mystery. In fact, the U of Chicago tells us that "cosine" actually means complement's sine -- since the cosine of an angle is the sine of its complement. This is Common Core Standard C.7 above.

OK, let me post the worksheet, and wish everyone a very happy Chinese New Year!

Wednesday, February 18, 2015

Section 14-2: Lengths in Right Triangles (Day 114)

Well, last night I was able to watch The Theory of Everything. The subject of this movie is the 20th (and 21st) century physicist, Stephen Hawking.

Here is a link to a biography of Stephen Hawking:

In some ways, Hawking is to physics as Turing was to computer science. Just as Turing sought out a Universal Turing Machine that could solve any problem, Hawking sought a Theory of Everything, a single equation that could describe any force. As the scientist describes in the movie, there are two major theories -- Einstein's Theory of Relavity (which models the very fast) and Quantum Mechanics (which models the very small). Before those theories, Isaac Newton's mechanical laws ruled. But what Hawking wanted was a single equation that would subsume all of the currently known theories, but as of today, a Theory of Everything remains elusive.

Notice that this marks the difference between physics and mathematics. We can mathematically prove that a Universal Turing Machine is impossible. But we can't actually prove things impossible in physics, like a Theory of Everything.

A Theory of Everything would describe all forces. There are four known forces -- electromagnetic, gravity, strong, and weak. Forces are usually described mathematically as vectors, since they have a magnitude and a direction. For example, the force that gravity exerts on us is called our weight. It has a magniude -- our weight in pounds -- and a direction -- down, towards the center of the earth. Recall that vectors (second half of Chapter 14) will be our main topic here on the blog next week.

SPOILER ALERT: Hawking is best known for astrophysics, and the movie discusses how he wrote his first book, A Brief History of Time, in the late 1980's. END SPOILER. But Hawking also wrote a book about mathematics -- actually, he compiled it. God Created the Integers is an anthology of 31 of the most famous mathematicians of all time, from Euclid up to -- surprise, surprise -- Alan Turing! I point out that the title God Created the Integers refers to a quote by the 19th century German mathematician Leopold Kronecker: "God created the integers; all else is the work of man." Like many students today, Kronecker preferred integers to rational and especially irrational numbers -- he was what we call a constructivist. Mathematical constructivists also reject some use of indirect proof.

SPOILER ALERT: It seems interesting that the first word of a book written by Hawking would be "God," considering how his atheism was at odds with his wife's Anglicanism during the movie. This is why I point out that the word "God" refers to that Kronecker quote. END SPOILER.

I hope that the two movies about famous scientists, Imitation Game and Theory of Everything, clean up at the Oscars this weekend. I believe that one way that we can convince our students to be more interested in math and science is to show them these famous scientists whenever we can. Even today's Google Doodle is an opportunity to tell our students about Alessandro Volta, the Italian physicist who invented the first battery at the end of the 18th century. (So now you know why we refer to batteries as having nine volts!)

Section 14-2 of the U of Chicago text is on lengths in right triangles -- specifically, those lengths that are related to the altitude and involve the geometric mean.

Geometric Mean Theorem:
The geometric mean of the positive numbers a and b is sqrt(ab).
(Note: This may sound like a definition, but actually the U of Chicago defines geometric mean to be the number x such that a/x = x/b, so we need a theorem to get the geometric mean as sqrt(ab).)

Right Triangle Altitude Theorem:
In a right triangle:
a. The altitude of the hypotenuse is the geometric mean of the segments dividing the hypotenuse.
b. Each leg is the geometric mean of the hypotenuse and the segment adjacent to the leg.

In this lesson, I give the proof of the Pythagorean Theorem based on similarity, but this time I gave the proof in the book, which mentions the geometric mean. Let's look at the proof -- as usual, with an extra step for the Given:

Given: Right triangle
Prove: a^2 + b^2 = c^2

Statements                                Reasons
1. Right triangle                       1. Given
2. a geometric mean of c & x,  2. Right Triangle Altitude Theorem
    b geometric mean of c & y
3. a = sqrt(cx), b = sqrt(cy)       3. Geometric Mean Theorem
4. a^2 = cx, b^2 = cy                 4. Multiplication Property of Equality
5. a^2 + b^2 = cx + cy               5. Addition Property of Equality
6. a^2 + b^2 = c(x + y)              6. Distributive Property
7. x + y = c                                7. Betweenness Theorem (Segment Addition Postulate)
8. a^2 + b^2 = c^2                    8. Substitution (step 6 into step 7)

It is uncertain whether this is the proof that Common Core intends the students to learn, or whether my earlier proof that avoids geometric means suffices. We'll find out soon enough.

Notice that one of the questions that I included from the text involves the Girl Scouts -- in particular, a girl scout troop leader who calculates the height of a tower using notebook paper (the sole purpose of which is to ensure that the angle is actually 90 degrees). This time of year is Girl Scout Cookie season, so of course I had to include a Girl Scout problem.