My Favorite One-Liners: Part 40

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

In some classes, the Greek letter \phi or \Phi naturally appears. Sometimes, it’s an angle in a triangle or a displacement when graphing a sinusoidal function. Other times, it represents the cumulative distribution function of a standard normal distribution.

Which begs the question, how should a student pronounce this symbol?

I tell my students that this is the Greek letter “phi,” pronounced “fee”. However, other mathematicians may pronounce it as “fie,” rhyming with “high”. Continuing,

Other mathematicians pronounce it as “foe.” Others, as “fum.”

My Favorite One-Liners: Part 8

In this series, I’m compiling some of the quips and one-liners that I’ll use with my students to hopefully make my lessons more memorable for them.

At many layers of the mathematics curriculum, students learn about that various functions can essentially commute with each other. In other words, the order in which the operations is performed doesn’t affect the final answer. Here’s a partial list off the top of my head:

  1. Arithmetic/Algebra: a \cdot (b + c) = a \cdot b + a \cdot c. This of course is commonly called the distributive property (and not the commutative property), but the essential idea is that the same answer is obtained whether the multiplications are performed first or if the addition is performed first.
  2. Algebra: If a,b > 0, then \sqrt{ab} = \sqrt{a} \sqrt{b}.
  3. Algebra: If a,b > 0 and x is any real number, then (ab)^x = a^x b^x.
  4. Precalculus: \displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i.
  5. Precalculus: \displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i.
  6. Calculus: If f is continuous at an interior point c, then \displaystyle \lim_{x \to c} f(x) = f(c).
  7. Calculus: If f and g are differentiable, then (f+g)' = f' + g'.
  8. Calculus: If f is differentiable and c is a constant, then (cf)' = cf'.
  9. Calculus: If f and g are integrable, then \int (f+g) = \int f + \int g.
  10. Calculus: If f is integrable and c is a constant, then \int cf = c \int f.
  11. Calculus: If f: \mathbb{R}^2 \to \mathbb{R} is integrable, \iint f(x,y) dx dy = \iint f(x,y) dy dx.
  12. Calculus: For most differentiable function f: \mathbb{R}^2 \to \mathbb{R} that arise in practice, \displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}.
  13. Probability: If X and Y are random variables, then E(X+Y) = E(X) + E(Y).
  14. Probability: If X is a random variable and c is a constant, then E(cX) = c E(X).
  15. Probability: If X and Y are independent random variables, then E(XY) = E(X) E(Y).
  16. Probability: If X and Y are independent random variables, then \hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y).
  17. Set theory: If A, B, and C are sets, then A \cup (B \cap C) = (A \cup B) \cap (A \cup C).
  18. Set theory: If A, B, and C are sets, then A \cap (B \cup C) = (A \cap B) \cup (A \cap C).

However, there are plenty of instances when two functions do not commute. Most of these, of course, are common mistakes that students make when they first encounter these concepts. Here’s a partial list off the top of my head. (For all of these, the inequality sign means that the two sides do not have to be equal… though there may be special cases when equality happens to happen.)

  1. Algebra: (a+b)^x \ne a^x + b^x if x \ne 1. Important special cases are x = 2, x = 1/2, and x = -1.
  2. Algebra/Precalculus: \log_b(x+y) = \log_b x + \log_b y. I call this the third classic blunder.
  3. Precalculus: (f \circ g)(x) \ne (g \circ f)(x).
  4. Precalculus: \sin(x+y) \ne \sin x + \sin y, \cos(x+y) \ne \cos x + \cos y, etc.
  5. Precalculus: \displaystyle \sum_{i=1}^n (a_i b_i) \ne \displaystyle \left(\sum_{i=1}^n a_i \right) \left( \sum_{i=1}^n b_i \right).
  6. Calculus: (fg)' \ne f' \cdot g'.
  7. Calculus \left( \displaystyle \frac{f}{g} \right)' \ne \displaystyle \frac{f'}{g'}
  8. Calculus: \int fg \ne \left( \int f \right) \left( \int g \right).
  9. Probability: If X and Y are dependent random variables, then E(XY) \ne E(X) E(Y).
  10. Probability: If X and Y are dependent random variables, then \hbox{Var}(X+Y) \ne \hbox{Var}(X) + \hbox{Var}(Y).

All this to say, it’s a big deal when two functions commute, because this doesn’t happen all the time.

green lineI wish I could remember the speaker’s name, but I heard the following one-liner at a state mathematics conference many years ago, and I’ve used it to great effect in my classes ever since. Whenever I present a property where two functions commute, I’ll say, “In other words, the order of operations does not matter. This is a big deal, because, in real life, the order of operations usually is important. For example, this morning, you probably got dressed and then went outside. The order was important.”

 

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 18

The Riemann Hypothesis (see here, here, and here) is perhaps the most famous (and also most important) unsolved problems in mathematics. Gamma (page 207) provides a way of writing down this conjecture in a form that only uses notation that is commonly taught in high school:

If \displaystyle \sum_{r=1}^\infty \frac{(-1)^r}{r^a} \cos(b \ln r) = 0 and \displaystyle \sum_{r=1}^\infty \frac{(-1)^r}{r^a} \sin(b \ln r) = 0 for some pair of real numbers a and b, then a = \frac{1}{2}.

As noted in the book, “It seems extraordinary that the most famous unsolved problem in the whole of mathematics can be phrased so that it involves the simplest of mathematical ideas: summation, trigonometry, logarithms, and [square roots].”

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When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

What I Learned from Reading “Gamma: Exploring Euler’s Constant” by Julian Havil: Part 15

I did not know — until I read Gamma (page 168) — that there actually is a formula for generating nth prime number by directly plugging in n. The catch is that it’s a mess:

p_n = 1 + \displaystyle \sum_{m=1}^{2^n} \left[ n^{1/n} \left( \sum_{i=1}^m \cos^2 \left( \pi \frac{(i-1)!+1}{i} \right) \right)^{-1/n} \right],

where the outer brackets [~ ] represent the floor function.

This mathematical curiosity has no practical value, as determining the 10th prime number would require computing 1 + 2 + 3 + \dots + 2^{10} = 524,800 different terms!

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When I researching for my series of posts on conditional convergence, especially examples related to the constant \gamma, the reference Gamma: Exploring Euler’s Constant by Julian Havil kept popping up. Finally, I decided to splurge for the book, expecting a decent popular account of this number. After all, I’m a professional mathematician, and I took a graduate level class in analytic number theory. In short, I don’t expect to learn a whole lot when reading a popular science book other than perhaps some new pedagogical insights.

Boy, was I wrong. As I turned every page, it seemed I hit a new factoid that I had not known before.

In this series, I’d like to compile some of my favorites — while giving the book a very high recommendation.

A Visual Proof of a Remarkable Trig Identity

Strange but true (try it on a calculator):

\displaystyle \cos \left( \frac{\pi}{9} \right) \cos \left( \frac{2\pi}{9} \right) \cos \left( \frac{4\pi}{9} \right) = \displaystyle \frac{1}{8}.

Richard Feynman learned this from a friend when he was young, and it stuck with him his whole life.

Recently, the American Mathematical Monthly published a visual proof of this identity using a regular 9-gon:

Feynman identity

Source: https://www.facebook.com/AmerMathMonthly/photos/a.250425975006394.53155.241224542593204/1045091252206525/?type=3&theater

This same argument would work for any 2^n+1-gon. For example, a regular pentagon can be used to show that

\displaystyle \cos \left( \frac{\pi}{5} \right)  \cos \left( \frac{2\pi}{5} \right) = \displaystyle \frac{1}{4},

and a regular 17-gon can be used to show that

\displaystyle \cos \left( \frac{\pi}{17} \right) \cos \left( \frac{2\pi}{17} \right) \cos \left( \frac{4\pi}{17} \right) \cos \left( \frac{8\pi}{17} \right) = \displaystyle \frac{1}{16}.

A natural function with discontinuities (Part 2)

Yesterday, I began a short series motivated by the following article from the American Mathematical Monthly.

discontinuous

Today, I’d like to talk about the how this function was obtained.

If 180^\circ \le latex \theta \le 360^\circ, then clearly r = R. The original circle of radius R clearly works. Furthermore, any circle that inscribes the grey circular region (centered at the origin) must include the points (-R,0) and (R,0), and the distance between these two points is 2R. Therefore, the diameter of any circle that works must be at least 2R, so a smaller circle can’t work.

reflexangle

The other extreme is also easy: if \theta =0^\circ, then the “circular region” is really just a single point.

Let’s now take a look at the case 0 < \theta \le 90^\circ. The smallest circle that encloses the grey region must have the points (0,0), (R,0), and (R \cos \theta, R \sin \theta) on its circumference, and so the center of the circle will be equidistant from these three points.

acuteangle

The center must be on the angle bisector (the dashed line depicted in the figure) since the bisector is the locus of points equidistant from (R,0) and (R \cos \theta, R \sin \theta). Therefore, we must find the point on the bisector that is equidistant from (0,0) and (R,0). This point forms an isosceles triangle, and so the distance r can be found using trigonometry:

\cos \displaystyle \frac{\theta}{2} = \displaystyle \frac{R/2}{r},

or

r = \displaystyle \frac{R}{2} \sec \frac{\theta}{2}.

This logic works up until \theta = 90^\circ, when the isosceles triangle will be a 45-45-90 triangle. However, when \theta > 90^\circ, a different picture will be needed. I’ll consider this in tomorrow’s post.

Engaging students: Defining sine, cosine and tangent in a right triangle

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Loc Nguyen. His topic, from Geometry: defining sine, cosine and tangent in a right triangle.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

There are many real world applications that involve in this topic and I will incorporate some problems in real life to engage the students.  Suppose I have a classroom that has the shape of rectangular prism.  I will begin my lesson by challenging the students to find the height of the classroom and of course I will award them with something cool.  I believe this will ignite students’ curiosity and excitement to participate into the problem.  In the process of finding the height, I will gradually introduce the concept of right triangle trigonometry.  The students will learn the relationship of ratios of the sides in the triangle.  Eventually, the students will realize that they need this concept for finding the height of the classroom.  I will pose some guiding questions to drive them toward the solution.  Such questions could be: what can I measure? Can we measure the angle from our eyes to the opposite corner of the ceiling point?   What formula will help me to find the height?
trig1

 

After this problem I will provide them many different real world problems to practice such as:

trig2 trig3

 

 

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How can this topic be used in your students’ future courses in mathematics or science?

Knowing how to compute sine, cosine or tangent in the right triangle will help students a lot when they get to higher level math or other science class, especially Physics.  In higher level math, students will always have the chance to encounter this concept.  For example, in Pre-Calculus, the students will likely learn about polar system.  This requires students to have the strong fundamental understandings of sine, cosine and tangent in a right triangle.  Students will be asked to convert from the Cartesian system to polar system, or vice versa.  If they do not grasp the ideas of this topic, they will eventually encounter huge obstacles in future.  In science, especially physics, the students will learn a lot about the motions of an objects.  This will involve concepts of force, velocity, speed, momentum.  The students will need to understand the how to compute sine, cosine and tangent in the right triangle so that they can easily know how to approach the problems in physics.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

This website, https://www.geogebra.org/material/simple/id/48148 , can be a great tool for the students to understand the relationships of the sides in the right triangle.  The website creates an activity for students to explore the ratios of the sides such as AC/BC, AC/AB, and BC/AB.  The students will observe the changes of the ratios based on the changes of theta and side BC which is the hypotenuse.  At this point, the students will be introduced the name of each side of the right triangle which corresponds to theta such as opposite, adjacent and hypotenuse.  This activity allows the students to visualize what happens to the triangle when we change the angle or its side lengths.  The students will then explore the activity to find interesting facts about the side ratios.  I will pose some questions to help the students understand the relationships of side ratios. Such questions could be:  What type of triangle is it?  Tell me how the triangle changes as we change the hypotenuse or angle.  If we know one side length and the angle, how can we find the other side lengths?  Those questions allow me to introduce the terms sine, cosine, and tangent in the right triangle.

trig4

References

https://en.wikibooks.org/wiki/High_School_Trigonometry/Applications_of_Right_Triangle_Trigonometry

https://www.geogebra.org/material/simple/id/48148

 

 

Sign and Cosign

Math_Cartoon_Trig_signatures

SOHCAHTOA

Years ago, when I first taught Precalculus at the college level, I was starting a section on trigonometry by reminding my students of the acronym SOHCAHTOA for keeping the trig functions straight:

\sin \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Hypotenuse}},

\cos \theta = \displaystyle \frac{\hbox{Adjacent}}{\hbox{Hypotenuse}},

\tan \theta = \displaystyle \frac{\hbox{Opposite}}{\hbox{Adjacent}}.

At this point, one of my students volunteered that a previous math teacher had taught her an acrostic to keep these straight: Some Old Hippie Caught Another Hippie Tripping On Acid.

Needless to say, I’ve been passing this pearl of wisdom on to my students ever since.

10 Secret Trig Functions Your Math Teachers Never Taught You

Students in trigonometry are usually taught about six functions:

\sin \theta, \cos \theta, \tan \theta, \cot \theta, \sec \theta, \csc \theta

I really enjoyed this article about trigonometric functions that were used in previous generations but are no longer taught today, like \hbox{versin} \theta and \hbox{havercosin} \theta:

http://blogs.scientificamerican.com/roots-of-unity/10-secret-trig-functions-your-math-teachers-never-taught-you/

Naturally, Math With Bad Drawings had a unique take on this by adding a few more suggested functions to the list. My favorites: