# AIM math

I ran across the AIM math website recently by stumbling onto their section for free math textbooks which, according to their evaluation criteria: "...must be able to serve as the primary text in a mainstream mathematics course at the undergraduate level in U.S. colleges and universities....Furthermore, we expect books of the quality we seek to be class-tested. They should have been used (and be in current use) by faculty other than the author....We are impressed with books that have a support system, signs of which are a website that is maintained, a community of users, a means for submitting support material from faculty such as test questions and project ideas, and a mechanism for improving the book by correcting errors, publishing errata lists, and displaying user comments."

These are some of the best free textbooks out on the internet; you should take a look. While looking around the website I found the Math Teacher's Circle Network had some resources that you might find useful as well. The latest newsletter (Summer 2013) focuses on Common Core math and the Resources section has some PDFs you might be able to integrate into your classroom. For example, the "Introduction to Problem Solving", "Mathematical Games" and "Introduction to Matrices" PDFs stand out.

# Sagetex: trig problem 1 (radians)

In an earlier post sagetex was used to create a problem that said, "Use reference triangles to find the sine/cosine/tangent/etc of the angle x (in degrees)". As I mentioned there, Sage prints out radian angles such as $\frac{3\pi}{4}$ so that $\pi$ is multiplied by $\frac{3}{4}$. You can't just code in $\LaTeX$ and insert the numerator or denominator via a Sage command because $0$ and $\pi$ aren't fractions. I opted to crunch things out case by case; simple and quick to implement. It gets the job done by referring to the numerators and denominators:

anglesR = [0, pi/6, pi/4, pi/3, pi/2, 2*pi/3, 3*pi/4, 5*pi/6, pi, 7*pi/6, 5*pi/4, 4*pi/3, 3*pi/2,5*pi/3,7*pi/8,11*pi/6]
#Numerators and Denominators of the radian angles
anglesRnum = [0,1, 1, 1, 1, 2, 3, 5, 1, 7, 5, 4, 3,5,7,11]
anglesRden = [1, 6, 4, 3, 2, 3, 4, 6, 1, 6, 4, 3, 2,3,8,6]

Regardless of the angle, it's passed to a function GetTheta that will take the index and return the $\LaTeX$ string formatted as a single ratio: $\sin\left(\sage{theta1}\right)$. Here's the GetTheta function; note that the code isn't formatted properly:

def GetTheta(Index):
if Index == 0:
theta = '0'
elif Index == 8:
theta = '\\pi'
elif (Index == 1) or (Index == 2) or (Index == 3) or (Index == 3) or (Index == 8):
theta = '\\frac{\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 5):
theta = '\\frac{2\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 6) or (Index == 12):
theta = '\\frac{3\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 7) or (Index == 10):
theta = '\\frac{5\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 9) or (Index == 14):
theta = '\\frac{7\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 11):
theta = '\\frac{4\\pi}{'+ str(anglesRden[Index])+'}'
elif (Index == 15):
theta = '\\frac{11\\pi}{'+ str(anglesRden[Index])+'}'

return theta

That's some long, ugly code but as you can see from the screenshot, it works. Note that backslash has to be typed twice to be picked recognized in a regular string (but not a raw string). The full code is posted on the SageTeX: Trigonometry page.

# Sage Interact: Triangle creator

I found myself looking for a triangle to insert into a test. It was going to be used as in a solve-the-triangle type of problem. The answer to the problem, though, could not have been the triangle graphic I wanted to use because the angles were too far off. It would be nice, I thought, to have a quick and easy was of creating triangles that would look like they should. The Triangle Creator fills that role. Specify 2 angles and this simple manipulative will create a triangle with those specific angles. You have the option to change the color and thickness of the lines and even add labels A, B, C as you can see in the screenshot above.

The math behind it is basic: start with a straight line segment from $(0,0)$ to $(2,0)$ and the user input of angles $\theta_1$ and $\theta_2$. To find where to place the 3rd point of the triangle note that the line to that 3rd point through $(0,0)$ is given by $y=\tan(\theta_1)$ and the equation of the line through $(2,0)$ and including the 3rd point is $(y-0)=\tan(180^{\circ}-\theta_2)(x-2)$. Solving the 2 equations simultaneously gives us $x=\frac{\tan(\theta_1)}{\tan(180^{\circ}-\theta_2)}(x-2)+2$ hence $y=\tan(\theta_1)\left(\frac{\tan(\theta_1)}{\tan(180^{\circ}-\theta_2)}\right)$. With that 3rd point we create the triangle. You can find the code for Triangle Creator on the Python/Sage page.

# Sagetex: Composition of Functions

For those of you using Sagetex in creating randomized test/quizzes/worksheets/etc, I've added an other problem. I started off creating a list of functions that I wanted to use and put allowed the coefficients of each function to be determined at random. I've included some basic functions that all precalculus students should be familiar with at the beginning of the year. Of course you can easily modify this to suit your need

The resulting problem is shown above; click on the image for a larger version. The student must calculate the composition of the two functions. Sage calculates the answer for us, eliminating the time needed to make an answer key (and preventing any errors by us in making the key). The code is posted on the Sagetex: Functions page.

# Sagetex: Trig Problem 2

I've added a problem to the Sagetex: Trigonometry page. You need to "...plot the function $y=a\sin(bx+c)+d$ for at least 1 period. Find the amplitude, midline, period, and horizontal shift.". Although the problem uses sine, it's trivial to modify the problem to handle other trig functions as well. In generating the graph for the solution I've made the output look a little bit nicer as follows:

plt1 = plot(p1,(x,-(((pi/c)/b)+2*pi/b),((pi/c)/b)+2*pi/b))

This command uses the fact that 1 period for y=asin(bx-c) goes from c/b to c/b+2pi/b.
By starting the graph at -(((pi/c)/b)+2*pi/b) then about 2 periods get graphed and the graph doesn't looked as cramped as it would if, say, 8 periods were plotted.

# Teacher shortages, revisited

As bad as public education is when seen from the outside, it's even worse on the inside; you see the problems and the stupidity up close and personal. The system is broken and it's not going to be fixed; it's massive bureaucracy and entrenched interest groups make meaningful change extremely difficult to achieve.

One of the many issues that deeply disturbs me are the numerous barriers that are placed in front of potential teachers. I've mentioned it before, and I had numerous issues getting into the system myself. I still remember being told I'd need to demonstrate that I was strong enough in math to teach at the high school level. When I reminded the bureaucrat I had 3 degrees in math from accredited schools I was rebuked, "You're not qualified unless I say you're qualified.". That meant, among other requirements, taking a multiple choice test which required the purchase of a graphing calculator. The level of the questions was below what I had experienced in high school and yet it is the only acceptable  measure of my math qualifications. Why aren't degrees from accredited schools recognized? Whatever the reason, the result is more time, effort, and money to become certified. And remember, top private schools don't value a teaching certificate, they care about the having teachers with advanced degrees.

The accumulation of barriers typically adds about 2 years of extra garbage to go through if you want to enter the public school system. I was reminded of that yet again with 2 recent articles. The first is article, "The Repurposed Ph.D." gives you a look into the poverty of a group: "Most held doctorates; a few were either close to completion or had left before finishing"..."One attendee recalled scraping by on $9,000 a year.". They've had to abandon academics because they can't make a living wage. But more than just the personal, anecdotal information, the statistics are staggering: • "According to a 2011 National Science Foundation survey, 35 percent of doctorate recipients — and 43 percent of those in the humanities — had no commitment for employment at the time of completion." • "Fewer than half of Ph.D.’s are expected to land tenure-track jobs." • "In this view, Ph.D. programs, with their false promises, lure students to serve as cheap labor, first as teaching assistants, then as poorly paid adjuncts when tenure-track jobs elude them." What's not mentioned in the article are these academics going on to the high school level. That's almost certainly due to the fact they can't because of the various barriers I've experienced first hand myself; barriers that take time and money to overcome for people who, in not having a job, lack the financial means to pursue. Qualified to teach freshman at college for slave wages with no benefits they are completely unable to teach students one year younger in the public school system because of the certification requirements. The second article is "From Welfare to Tenure Track" on the Chronicle of Education's website. The person at the center of the article had a Ph.D. in medieval history and was unable to get more than an adjunct position where "Her adjuncting brought in$900 a month, $750 of which went immediately toward rent. To make the remaining$150 last longer, she learned tricks familiar to those with little room for financial maneuvering: stretching two pounds of hamburger meat over six meals, reminding her daughter to use a washable rag instead of paper towels, asking friends if they had an extra roll of toilet paper when she ran out. At times she borrowed her mother’s car to drive to campus—about 100 miles round trip—because she couldn’t afford to fill up her own gas tank"...."[she] ended up relying on food stamps and Medicaid, barely scratching out a living for herself and her 17-year-old daughter. When she wasn’t grading papers, or worrying about keeping the lights on and the hot water running, she was trawling the Web in search of articles about the brutal academic job market and colleges’ use of adjuncts.".

Again, the anecdotal information is buttressed with statistics: "She had a point, and she wasn’t speaking just for herself. Between 2007 and 2010 the numbers of Ph.D.-holders receiving aid more than tripled, from just shy of 10,000 to 33,000.".

If the public schools are interested in getting "experts in the field" to teach at the high school level, there is a huge supply of people who would be willing and able to teach, if given the chance. But the numerous nuisance barriers (such as making someone with a graduate degree prove they have mastered their subject, or passing a test on the US Constitution to teach math) on top of the legitimate barriers (background check) make it onerous for people with limited financial means to pursue. Making it easier for these people who are passionate and knowledgeable in their field to teach in a public school without the nuisance barriers would tighten the supply of people qualified to teach at the college level and ultimately force colleges to pay a more reasonable wage. During these tough economic times, it's even more imperative. But nobody in the system wants that; that would make a teaching certificate less important if it could be replaced by actually being knowledgeable in the field.

# Problems: Trigonometry

The website TeXample.net has a diagram which I modified to produce the image above. The problem is to find the length of the belt in terms of $\cot(\theta)$. I've added this to the Problems page.

You'll need to recall a little geometry: the belts between the two pulleys are tangent to each circle. In addition, there's some symmetry which gives us a total of 4 angles with degree $\theta$.

# Why do we need radians?

When you teach unit circle trigonometry you've got to introduce radians, and inevitably that leads to problems. Students aren't initially comfortable with radians and you'll almost certainly have someone ask why they need to use radians.

To be sure, the degrees/minutes/seconds system only described a finite number of angles but with decimal degrees, there is the required infinite number of angles needed to map each degree to a point on the unit circle. For any angle in radians we can simply convert to degrees using the conversion factor $\frac{180^{\circ}}{\pi}$; why then do we need radians?

You can address this natural question by including a simple math model into your lesson. The displacement of a weight attached to a spring moves up and down over time (non damping) according to some equation like $y=\frac{1}{5}\cos(4t)$. Physclips is a good site for animations; check out this page to see spring movement juxtaposed with movement along the sine curve. You can download clips to your computer, the harmonic motion example is found here. After seeing how the equation models reality you're ready for the teaching point: Radians are necessary because mathematical models use trig functions where the variable is time, which is not measured in degrees.

# Sagetex: Polynomials 2

I've added a problem to the Polynomials page for use in creating your own randomized tests (with answer key). By choosing the the linear factors of the polynomial and then expanding to create the polynomial you have control over how difficult it will be to find the real roots. You can see the output running in Sagemath Cloud.

# Odds and Ends: September 30, 2013

Some odds and ends:

• The WebEquations link no longer works and was removed. The WriteLaTex link was moved as Sagemath Cloud, with support for Sage, is now the only tool you need.
• A helpful reader found some mistakes in the Trig formulas PDF posted on the Handouts page. The PDF and LaTeX code were fixed and I've updated those files.
• I've run across a nice post at Un peu de math which has a lot of interesting links to explore: a wiki with "...a gallery of all images — illustrations, diagrams and animations"  created by LucasVB. I've already downloaded 2 animated gifs on radians that will be useful when I get to trig and one on the transpose of a matrix. A nice article on the use of computers in mathematical proofs from Quanta Magazine. Another link about computers in math and the  "curse of computing". Of course the Sage Interacts on graph theory were nice, too.

An important point worth emphasizing from the above articles. Sage, being open source, can be checked for the accuracy of the code. Closed source mathematical programs are "black box"; the underlying code might be (and sometimes is) wrong, but how do you know until you spot the error.

Trusting closed source math programs violates the very spirit of math and science.