Engaging students: Law of Cosines

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 Allison Metlzler. Her topic, from Precalculus: the Law of Cosines.

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

Real world word problems are an effective engagement because the students can actually relate to the events occurring in the problem. Below are two word problems where one deals with animal footprints and the other talks about trapeze artists.
1. Scientists can use a set of footprints to calculate an organism’s step angle, which is a measure of walking efficiency. The closer the step angle is to 180 degrees, the more efficiently the organism walked. Based on the diagram of dinosaur footprints, find the step angle B.
2. The diagram shows the paths of two trapeze artists who are both 5 feet tall when hanging by their knees. The “flyer” on the left bar is preparing to make hand-to-hand contact with the “catcher” on the right bar. At what angle (theta) will the two meet?
The problems were obtained from http://www.muhsd.k12.ca.us/cms/lib5/CA01001051/Centricity/Domain/547/Trig/13-6%20Law%20of%20Cosines.pdf.

How could you as a teacher create an activity or project that involves your topic?

Activities are a great way to engage students. They require the students to explore the topic and make new discoveries. It can also benefit students who learn best by doing hands-on work. The activity, http://hilbertshotel.wordpress.com/2013/01/10/law-of-sinescosines-mapquest/ involves the law of sines, the law of cosines, and MapQuest. You will need a map of your school or just one of your school’s buildings. The students will then create triangles to figure out the length of different parts of the school. In order to do this, the students will have to use the law of cosines and sines. They will be able to measure the angles of the triangles using protractors. Then they can calculate the lengths of the sides of the triangles. You can then relate this activity to the real world job of surveyors. You would also need to point out to the students that because they are rounding their calculations of the distances and angles, there is a loss of accuracy. Also, you should note that in real life, surveyors would compute the distances using a different method in order to be completely accurate. This activity is very interesting and helps the students get a good understanding of the law of cosines.

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

A video is a great way to engage students because it’s visual and auditory which helps student understand concepts better. The video below uses Vanilla Ice’s song, Ice, Ice Baby, to introduce the law of cosines. I would play it from the start until1:51. At 1:51, the video starts introducing the idea of the law of sine. Besides just introducing the general idea of the law of cosines, it also shows how it’s derived from the Pythagorean Theorem. The video also clearly states that the Pythagorean Theorem only works with right triangles so that’s why we need the law of cosines- to help solve all triangles. It points out that you cannot only solve for a side of the triangle, but also the angles of the triangle. Another reason this video is engaging is that it is a well-known song that is catchy. Thus, the students will be able to remember the connection between the video and the concept of the law of cosines.

References:

Apply the Law of Cosines (n.d.). In MUHSD.k12. Retrieved April 4, 2014, from http://www.muhsd.k12.ca.us/cms/lib5/CA01001051/Centricity/Domain/547/Trig/13-6%20Law%20of%20Cosines.pdf

Dahl, M. (Producer). (2009). Law of Cosines Rap- Vanilla Cosines [Online video]. YouTube. Retrieved April 4, 2014, from http://www.youtube.com/watch?v=-wsf88ELFkk

Newman, J. (2013, January 10). Law of Sines/Cosines “Mapquest”. In Word Press. Retrieved April 4, 2014, from http://hilbertshotel.wordpress.com/2013/01/10/law-of-sinescosines-mapquest/

Statistical Inference for the General Education student

From the opening and closing paragraphs:

Many mathematics departments around the country offer an introductory statistics course for the general education student. Typically these students come to the mathematics classroom with minimal skills in arithmetic and algebra. In addition it is not unusual for these students to have very poor attitudes toward mathematics.

With this target population in mind one can design courses of study, called statistics, that will differ radically depending on what priorities are held. Many people choose to teach arithmetic through statistics and thereby build most of the course around descriptive statistics with some combinatorics. Others build most of the course around combinatorics and probabilities with some descriptive statistics. Few courses offered at this level spend much time or effort on statistical inference.

We believe that for the general education student the ideas of statistical inference and the resulting decision rules are of prime importance. This belief is based on the assumption that general education courses are included in the curriculum in order to help students to gain an understanding of their own essence, of their relationship to others, of the world around them, and of how man goes about knowing.

If you inspect most of the texts on the market today, you will find that they generally require that a student spend approximately a semester of study of descriptive statistics and probability theory before attempting statistical inference. This makes it very difficult to get to the general education portion of the subject in the time allotted most general education courses. If you agree with the analysis of the problem to this point the logical question is ‘Is there a way to teach statistical inference without the traditional work in descriptive statistics and probability?’. The remainder of this article describes an approach that allows one to answer this question with a yes…

It should be pointed out that there are some unusual difficulties in this approach to statistics [since] one trades traditional weakness in arithmetic and algebra for deficiencies in writing since the write-ups of the simulations demand clear and logical exposition on the part of the student. However, if you feel that the importance of ‘statistics for the general education student’ lies in the areas of inference and decision rules, then you should try this approach. You will like it.

This article won the 1978 George Polya award for expository excellence. Several techniques described this article probably would be modified with modern computer simulation today, but are still worthy of reading.

Click to access 00494925.di020678.02p03892.pdf

Introduction to the Catalan numbers

From the YouTube description:

Alissa S. Crans, Associate Professor of Mathematics at Loyola Marymount University, introduces viewers to the Catalan numbers, which take on a variety of different guises as they provide the solution to numerous problems throughout mathematics.

More on the Catalan numbers can be found at MathWorld and at Wikipedia and at http://www-math.mit.edu/~rstan/ec/. This video is accessible to the general public, including gifted elementary school students.

From the European Girls Mathematical Olympiad

Source: https://www.facebook.com/maanews/photos/a.170415820418.120984.153302905418/10152497151470419/?type=1&theater

Pythagorean quadruplets

Source: https://www.facebook.com/photo.php?fbid=494942050639298&set=a.416585131808324.1073741827.416199381846899&type=1&theater

Proofs that the square root of 2 is irrational

Here’s a resource with over 25 different proofs for demonstrating that $\sqrt{2}$ is irrational: http://www.cut-the-knot.org/proofs/sq_root.shtml

Mathematician maze

Source: https://www.facebook.com/maanews/photos/a.170415820418.120984.153302905418/10152497214445419/?type=1&theater

Approximations

Source: http://www.xkcd.com/1047/

An infinite number of mathematicians walk into a bar

An infinite number of mathematicians walk into a bar. The first one orders a beer, the second one orders a half a beer, the third orders a quarter of a beer, and this trend continues on for some time.

After a while, the bartender gets fed up and hands them 2 beers, shakes his head and says, “You mathematicians just don’t know your limits.”

Functions that commute

At the bottom of this post is a one-liner that I use in my classes the first time I present a theorem where two functions are permitted to commute. 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:

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.
Algebra: If $a,b > 0$ , then $\sqrt{ab} = \sqrt{a} \sqrt{b}$ .
Algebra: If $a,b > 0$ and $x$ is any real number, then $(ab)^x = a^x b^x$ .
Precalculus: $\displaystyle \sum_{i=1}^n (a_i+b_i) = \displaystyle \sum_{i=1}^n a_i + \sum_{i=1}^n b_i$ .
Precalculus: $\displaystyle \sum_{i=1}^n c a_i = c \displaystyle \sum_{i=1}^n a_i$ .
Calculus: If $f$ is continuous at an interior point $c$ , then $\displaystyle \lim_{x \to c} f(x) = f(c)$ .
Calculus: If $f$ and $g$ are differentiable, then $(f+g)' = f' + g'$ .
Calculus: If $f$ is differentiable and $c$ is a constant, then $(cf)' = cf'$ .
Calculus: If $f$ and $g$ are integrable, then $\int (f+g) = \int f + \int g$ .
Calculus: If $f$ is integrable and $c$ is a constant, then $\int cf = c \int f$ .
Calculus: If $f: \mathbb{R}^2 \to \mathbb{R}$ is integrable, $\iint f(x,y) dx dy = \iint f(x,y) dy dx$ .
Calculus: For most differentiable function $f: \mathbb{R}^2 \to \mathbb{R}$ that arise in practice, $\displaystyle \frac{\partial^2 f}{\partial x \partial y} = \displaystyle \frac{\partial^2 f}{\partial y \partial x}$ .
Probability: If $X$ and $Y$ are random variables, then $E(X+Y) = E(X) + E(Y)$ .
Probability: If $X$ is a random variable and $c$ is a constant, then $E(cX) = c E(X)$ .
Probability: If $X$ and $Y$ are independent random variables, then $E(XY) = E(X) E(Y)$ .
Probability: If $X$ and $Y$ are independent random variables, then $\hbox{Var}(X+Y) = \hbox{Var}(X) + \hbox{Var}(Y)$ .
Set theory: If $A$ , $B$ , and $C$ are sets, then $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ .
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.)

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

I 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.”