This is a page of scratch work and notes related to the delivery of MS 101 Algebra and Trigonometry summer 2001 at the College of Micronesia-FSM. These are essentially notes for the instructor, not necessarily for the students, hence the rambling, discombobulated, random nature of these notes.
Technical note: these notes do not validate because the URLs that are produced by WolframAlpha contain bad values for attribute href on element a: Illegal character in query component. The bad values, or "unsafe" characters, are unsafe "because gateways and other transport agents are known to sometimes modify such characters. These characters are "{", "}", "|", "\", "^", "~", "[", "]", and "`"." (Berners-Lee, Masinter, McCahill: RFC1738). As these characters are a) fully functional for my non-transport/gateway use, b) an extra step inconvenient step to convert for notes being built in real time on the fly, I simply am not going to fix this issue. WolframAlpha ought to encode those characters when they appear in a query.
Some of the notes refer to homework in Algebra and Trigonometry, 8th Edition, by Ron Larson.
The stucture of the course is to chase a particular model or system each day, and let the math flow from the demands of the model. Models come from a diverse array of fields including genealogy, physical science, health, and web page programming.
Go to Geni.com and start your family tree.
On the home screen, use the drop-down menu to switch to the tree view.
In the lower right hand corner click on the five-generation horizontal view.
Fill in the five generation chart as completely as you can.
If the chart extended to generation six, how many great-great-great-grandparents would you have?
Is there a function that can calculate the number of ancestors you have in any given generation?
Obtain historic population numbers
| State | Year | Population |
|---|---|---|
| Chuuk | 1925 | 10000 |
| Kosrae | 1880 | 274 |
| Pohnpei | 1920 | 5000 |
| Yap | 1920 | 8000 |
If you have two children and each of them have two children, how many grandchildren will you have?
What function describes the number of children in any generation?
What if you have three children who each have three children who each have three children... and so forth. What function would calculate the number of children in generation x?
Graph the functions above using
WolframAlpha
.
plot f(x)=2^x
To set a specific domain:
plot f(x)=2^x for x=0 to 5
To add gridlines:
plot 2^x from x=0 to 5 with gridlines.
There is a good note on the focus of WolframAlpha on natural language processing and the logic to the limitations of WolframAlpha versus the full Mathematica platform. That said, many of the Mathematica functions also work in WolframAlpha. There is, however, no intent in the course to teach Mathematica commands.
Substitution and evaluation for a single value:
evaluate f(x)=0.9^x at x=1.4
ReplaceAll[0.9^x, x -> 1.4]
0.862858
(1/2)^x, x=2
Mathematica:{(1/2)^x, x == 2}
evaluate f(x)=2*e^(-5*x) at x=10 yields
ReplaceAll[2/E^(5 x), x -> 10]
2/e^50~~3.8575x10^-22
Thus the new skill to teach is where to put * and parentheses to faithfully reproduce the function in WA. This is not trivial.
As of early May 2011, finding natural language inputs for evaluating a function across a domain of values has eluded me. In this case one has to either fall back on Mathematica functions or utilize a widget. There are many widgets already in existence that are custom designed to support a math instructor in the classroom, such as those by RWLMath
Generating a table of values for 0.5x from -6 to 6:
Table[0.5^x, {x, -6, 6}]. The format is Table[function, {variable, start, stop}]
To generate a table in specific steps:
Table[e^x, {x, -3, 4, 0.5}]
The HPE - Table of Values Calculator widget generates tables of values for a function. The author of that widget, RWLMath , has created a variety of widgets that appear to be directly useful in the classroom.
One can embed already developed widgets into a page or one can develop one's own widgets. Embedded widgets, however, stall the building of the web page beyond the widget in FireFox 4.
WolframAlpha's natural language is quite capable of parsing the complex evaluation:
Evaluate p*(1+(r/n))^(n*t) for p=1500,r=0.02,n=365,t=10
Note that the when typed into WolframAlpha, WolframAlpha will ask, "Wolfram|Alpha needs more time to respond to your query... Go on computing »". Click on Go on Computing to obtain the solution. Note that the above URL include the incParTime=True.
Mathematica: ReplaceAll[p (1 + r/n)^(n t), p -> 1500, r -> 0.02, n -> 365, t -> 10]
1832.09
WolframAlpha does NOT always follow the PEMDAS order of operations. (parentheses, exponents, multiplication, division, addition, and subtraction). For example, following strict PEMDAS rules, 81^1/4 should be equal to 20.25. Under PEMDAS, the exponent should be evaluated prior to the division. WolframAlpha, however, presumes that one meant 81^0.25 with a result of 3. WolframAlpha has implicitly added parentheses: 81^(1/4) To force the 20.25 result to appear, one has to enter (81^1)/4.
In other words, one should always use parentheses to ensure the WolframAlpha is interpreting an input as intended. One should not rely on PEMDAS.
My blog details the difference between the best fit functions generated by Excel and Calc.
Facebook: spread of viral messages, spam. Growth rates at FB average of 130 friends. Growth rate at experimentally determined class average.
Logarithmic fitting is easier from a spreadsheet than WolframAlpha. One of the intents of the course is to help students understand that different software tools have different strengths. Choosing the correct tool for the job is important.
Data table
| Age in months since conception | Actual mass |
|---|---|
| 9 | 3 |
| 9.5 | 3.4 |
| 11.5 | 5.2 |
| 13.3 | 6.1 |
| 14.2 | 6.4 |
| 15 | 6.6 |
| 17.5 | 7 |
| 21 | 9.1 |
| 25 | 8.2 |
| 26 | 8 |
| 27 | 8.8 |
| 37 | 10.5 |
| 57 | 13.2 |
The data is for my son. The dip at 26 and 27 months post-conception was a week long bout of diarrhea that required hospitalization. Note that "post-conception" is necessary because ln(0) is undefined. Mathematically the age of the baby cannot be zero. The choice of nine months is effectively abitrary.
WolframAlpha again behaves in a predictable manner with regard to natural language input:
Evaluate f(x)=log2(x) for x=32 yields x=32. Code for this:
Mathematica: ReplaceAll[Log[x]/Log[2], x -> 32]
Note the change of base structure in the Mathematica format. Again, there is no intent to teach the Mathematica form, although awareness of its existence is likely to be mentioned. The forms are useful as they can be reverse engineered if the need arose to handle a more complex situation.
log(x) is the natural log of x for WolframAlpha/Mathematica. The terms plot and graph are interchangeable.
log[b,z] is Mathematica format logarithm base b.
Graphing the natural logarithmic function:
Graph f(x)=ln(x)
Omitting the f(x)= or y= is useful as this permits extension to multiple plots on one chart.
graph x,e^x,log(x)
The above does is not aesthetically pleasing to a 20th century chalkboarder as the y = x reflection axis is not at a 45° angle. As of May 2011 attempts to either directly use a Mathematica command or a natural language input to control either the AspectRatio or the range have failed. WolframAlpha usually reverts to plotting only a single function when anything else is added to the multiple function list.
Change of base: log5(25)
Mathematica:Log[5, 25]
Result: 2
57.
expand log2(sqrt(a-1)/9) is interpreted correctly:
Plaintext input equivalent:expand | log_2(sqrt(a-1)/9)
Note the underscore above after the log and before the 2.
Mathematica input: Expand[PowerExpand[Log[Sqrt[-1 + a]/9]/Log[2]]]
The underscore is optional, however WolframAlpha "suggests" the use of the underscore in the following example. The input was in Mathematica format, the plaintext equivalent includes the underscore character.
Another expansion
Plaintext output equivalent: (log(a-1))/(2 log(2))-(2 log(3))/(log(2))
Mathematica output: (-2 Log[3])/Log[2] + Log[-1 + a]/(2 Log[2])
5.3 #95 Galloping speed opendocument spreadsheet
Galloping speed WolframAlpha input.
Output: -20.8056 log(4.47779x10^-6 x)
Where x is the weight of the animal in pounds, y is the galloping strides per minute.
One can then expand that result to obtain:
256.25-20.8056 log(x)
Failed example:
Planetary distances from sun
Copy the plain text. Use paste special into LibreOffice.org and merge the space delimiters. Getting fancy: include the | delimiter. Select and sort on AU distance. Dead end. Leads to exponential. Can use log to linearize, but text fudges AU distances to improve fit. Plus the text does not return to the logarithm to a mathematical explanation.
WolframAlpha will expand logarithms using the expand command, but will not "condense" (textbooks word) logarithms. Attempts to use simplify and combine also failed to produce the "condensed" result.
solve e^(-x^2)=e^(-3*x-4) for x
Yields both the wave results and the real results: a need to learn to read answers.
solve log_10(5x)+log_10(x-1)=2 for x
Provides only the x = 5 solution, not the spurious x = -4 that hand working generates. WolframAlpha reinforces the concept of the crossing of the graphs with a graphical solution presented as well. Getting these typed in correctly will clearly take time.
8th edition p417 #137. The automobiles problem, part a: One way to get answers for part a is by substituting the table values into the equation: -3+11.88ln(.2)+(36.94/.2).
There, however, a way to have WolramAlpha fill out a table of evenly spaced values. The way is a little complicated and I do NOT expect you to use it. But you should know that a way exists to fill out a table in Wolfram Alpha. The following link will display the function and the result:
Table[-3+11.88ln(x)+(36.94/x), {x, 0.2, 1.0, 0.2}]
The format of that function is Table[function, {variable, startValue, endValue, increment}]
Chose to focus on Usain Bolt's 2008 100m world record. At 8:00 in the morning there is still enough bandwidth to show an on line video - sort of. So I found and showed a video of Usain Bolts run, not a very good one, but just to give the class an idea of what happened.
Then I had the students enter Usain distance versus speed data into a spreadsheet and then try the four regression models in LibreOffice.org to find a fit.
| Distance (m) | Speed (m/s) |
|---|---|
| 0 | 0.0 |
| 10 | 5.4 |
| 20 | 9.8 |
| 30 | 11.0 |
| 40 | 11.5 |
| 50 | 11.8 |
| 60 | 12.2 |
| 70 | 12.2 |
| 80 | 12.2 |
| 90 | 12.1 |
| 10 | 11.1 |
None of the four models fits well, although the students favored the logarithmic fit. The logarithmic model may be a best fit, but the function cannot handle x = 0, starts too fast, and, possibly most problematic, the speed continues to rise without limit. Sprinters hit a maximum speed and then attempt to hold that speed into the finish line. The logarithmic model simply behaves incorrectly.
I then introduced a logistic model. plot -12.2+(24.4/(1+e^(-x/9))) from x=0 to 100. I first displayed the model using WolframAlpha to show the shape. Then I instructed the students how to enter the function into LibreOffice.org. Spreadsheets use the EXP() function for the base e. The EXP function does not use a exponentiation symbol, so there is no "^". Spreadsheets also do not use the variable x, a cell address must be used instead. If the Distance data is in column A, then the function in cell C2 would be =-12.2+(24.4/(1+EXP(-A2/9))).
I wrapped with some homework including Usain's speed at 15 meters and the distance at which Usain reaches my running speed, 2.5 m/s.
The pedagogical intent was again a focus on the 60% of the class that are elementary education majors. While a first grade teacher may never again see a logistic function, they will see a spreadsheet and likely use one at some point. Familiarity with spreadsheets and entering functions into spreadsheets is a career-useful skill for an educator. Learning to team up tools such as WolframAlpha and a spreadsheet, using each where it is strongest, is useful.
Note too I neglected the other models, in part because I have been focusing on models from day one. The one model I neglected, the Guassian curve, is the statistics normal curve. Yet even in statistics that curve is never tackled algebraically.
FWIW, WolframAlpha will plot the Usain data:
ListPlot[{{0,0},{10,5.39},{20,9.806},{30,11},{40,11.5},{50,11.8},{60,12.19},{70,12.19},{80,12.19},{90,12.06},{100,11.1}}]
.
To date, however, I cannot get WolframAlpha to plot the data and the function together on one plot.
Data from the banana leaf marble ramp exercise was used on test two.
| Height h (cm) | Velocity v (cm/s) |
|---|---|
| 0 | 0 |
| 10 | 97 |
| 20 | 142 |
| 30 | 182 |
| 40 | 208 |
Despite playing with options, I could not find a way to have WolframAlpha run a power regression on the
data
. A power regression fits better than a logarithmic. As noted above, finding a way of getting points and functions plotted together in WolframAlpha has eluded me. The students used LibreOffice.org Calc to plot the data and find the best fitting regression function.
Concept is to hope for sun and run three radii of yarn out. Get angle in degrees. Use surveyor's wheel to measure actual arc length. Calculate radian measure.
Vertex, angles, acute 30°, 45°, 60°, right 90°, complementary, obtuse, straight 180°, supplementary. θ.
s= rθ.
Rotational velocity ω
Character map (charmap.exe) can be used to obtain a degree symbol. convert 30° to radians
Alternatively one can type the word degrees. convert 30 degrees to radians. Note that for both of the above, WolframAlpha will provide an "instant answer" via JavaScript.
I found that WolframAlpha could solve the arclength problems with an angle input as degrees if set up as:
solve for s: s=14*(45 degrees)
This input is faulty as the angle should be in radians, but WolframAlpha obliges, dumps out two options for the answer, the second of which is the desired answer:
s = 630 ° ~~ 10.9956
Over the next couples of days the models focus on web page creation using the new Scalable Vector Graphic (SVG) extensions to HTML. You will be working laying out some basic shapes on a 200 by 200 grid.
The SVG grid is like an xy graph EXCEPT that the y-axis is UPSIDE-DOWN. To draw shapes on the grid you will use a text editor. This will be demonstrated in class. Then the pages will be opened and viewed using FireFox.
HTML code is entered into a text editor. The text editor in Ubuntu has to be opened from the activities button in the top left corner of the screen to open the launchers menu.
Click on the More apps button and start typing "text editor".
Click on the text editor. To help check for errors, set the highlighting mode to HTML.
The following HTML code will draw a circle on a web page.
<!DOCTYPE html> <html> <head> <title>SVG test bed</title> </head> <body> <svg width="220" height="220"> <circle fill="cyan" stroke="teal" cx="100" cy="100" r="100" /> </svg> </body> </html>
HTML is very very fussy - the slightest typing error will cause the web page to fail.
The next task is where trigonometry comes into play: we are going to draw a square in the circle. To draw the square exactly inside the circle ("inscribed") we need to determine where the corners of the circle are located. The circle has a radius r of 100. For those who are curious, the units are "pixels" which is short for "picture elements."
To draw a square in a circle we will need to locate the corners of the square. To do this we will need to work out the length of the adjacent and opposite sides of a 45° triangle. This will help us locate the corners of the 45-45-90 right triangle that is the building block of the square.
The SVG code to draw this square will have the form of...
<path fill="none" stroke="silver" d="M x1,y1 L x2,y1 x2,y2 x1,y2 x1,y1" />
where the x1,y1 elements will be replaced by numbers, by coordinate pairs.
Calculating the coordinates requires determining the length of the bases of the 45-45-90 triangle. This calculation requires using sine and cosine. The sine function, abbreviated sin, can be used to calculate the length of the side opposite the angle theta θ.
Cross multiply and one gets y = r sin(θ).
r = 100 and θ = 45°
Thus y = 100*sin(45°)
To calculate this use WolframAlpha:
100*sin(45 degrees)
The cosine function, abbreviated cos, can be used to calculate the length of the side adjacent the angle theta θ.
Cross multiply and one gets y = r cos(θ). r = 100 and θ = 45° Thus y = 100*cos(45°) To calculate this use WolframAlpha: 100*cos(45 degrees)
For both of the above results, we will have to round off the decimal portion of the number. SVG does not use fractional pixel values. Both calculations have the same answer: 71 pixels.
The center of the circle is (100,100). This is our "starting" place. Moving to the right 71 pixels adds 71 to x-value in the (x, y) coordinate. This puts us at the right-angle corner of the triangle.
Moving up 71 pixels is moving in the negative y direction. This means we have to subtract 71 from the y value at the right-angle corner.
Thus the coordinate for the upper right hand corner is (171,29). This will be the first coordinate in the path command that will draw our square
<path fill="none" stroke="black" d="M 171,29 L , , , , " />
The "M" in the path attribute means "Move to..." This tells the computer where to start our SVG diagram. The "L" means "Line" as in "draw a Line to". The next step is to tell the computer where the line should go.
Let us start by going down to the lower right corner of the square. Using the same basic argument as above, the lower right corner of the square must be 100*sin(45 degrees) below the center - that is, 71 pixels down from 100. Or 142 pixels down from the upper right corner. Each side of the square is two times 71 pixels long.
<path fill="none" stroke="black" d="M 171,29 L 171,171 , , , " />
Using the above approach, we can work our way around the square, corner to corner, returning eventually to the upper right corner at (171,29).
<path fill="none" stroke="black" d="M 171,29 L 171,171 29,171 29,29 171,29" />
Next we will tackle an equilaterial triangle. The key to solving this will be using the 30-60-90 triangle in the diagram below.
The angle θ is 60°. Your text book notes that a 30-60-90 triangle is another "special" triangle. They are special because they build equilaterial triangles and hexagons, among other reasons.
The hypotenuse r is 100 pixels long. The angle θ is 60°. Look carefully at the diagram. The adjacent side generates a vertical value - a y value. The opposite side generates a horizontal value - an x value. Make a diagram when in doubt.
The adjacent side still uses the cosine function:
adjacent = r cos(θ). r = 100 and θ = 60° Thus adjacent = 100*cos(60°) To calculate this use WolframAlpha: 100*cos(60 degrees)
The opposite side still uses the sine function:
opposite = r sin(θ). r = 100 and θ = 60° Thus opposite = 100*sin(60°) To calculate this use WolframAlpha: 100*sin(60 degrees)
The adjacent side is 50 pixels. This tells us that the y-coordinate of the bottom corners is at y = 150.
The opposite side is 87 pixels (rounded off to the nearest whole number. This tells us to go +87 from the center line (x = 100) to get to the lower right corner.
The center of the circle is (100, 100), therefore the lower right corner is (100 + 87, 100 + 50) or (187, 150).
The lower left hand corner can be worked out in a similar fashion. The y-coordinate is y = 150. The x-coordinate is y =(100 − 87) or y = 13. The coordinate for the lower left corner is (13, 87).
The top of the triangle is straight up, − pixels up from the center at (100,0).
<path fill="none" stroke="black" d="M 100,0 L 187,150 13,150 100,00" />
Homework: Draw an equilateral triangle with one vertex at the bottom of the square.
The relationship between x, y and the angle theta θ is given by a third trigonometric function, the tangent function. The tangent is also the slope of a line if the adjacent side is horizontal.
The following will display in FireFox 6.0a1 (Mozilla/5.0 (Windows NT 6.1; Win64; x64; rv:6.0a1) Gecko/20110523 Firefox/6.0a1). The html will not pass validation - no doctype is the primary error.
<html> <head> <title>SVG test bed</title> </head> <body> <svg fill="turquoise" stroke="teal" > <circle cx="20" cy="20" r="10" /> </svg> </body> </html>
But oddly enough the following actually validates!
<!DOCTYPE html> <html> <head> <title>SVG test bed</title> </head> <body> <svg fill="turquoise" stroke="teal" > <circle cx="20" cy="20" r="10" /> </svg> </body> </html>
The above, however, is limited in FireFox to some sort of default vertical height of 150. SVG elements must explicitly close or subsequent elements fail.
6.3 was integrated into the above SVG material.
I used my RipStik to introduce section 6.4.
Introducing Sine Waves in Trigonometry with a RipStik Using this approach I also introduced the wavelength - something the text omits in favor of focusing on the period. I also introduced frequency, something the text leaves for a homework problem (77 in the seventh, 91 in the eighth). I brought in a tuning fork and showed the class that Larson has the problem the wrong-end around: tuning forks always have their frequency stamped on the them, not their period. One can then work out the period from the frequency.
I also worked the wavelength from the relation wave velocity = frequency * wavelength and the speed of sound (http://www.wolframalpha.com/input/?i=sound+speed+32+Celsius+74%25+relative+humidity yields about 350 m/s. Data is from my physical science class.)
plot 4 sin(2*pi*x/32) from 0 to 64
In 6.5 I had the students using WolframAlpha to produce graphs, and I connected tangent to the slope of a line when the angle theta is in the "standard" position.
WolframAlpha accepts either arcsin() or sin^-1() as input notation.
I wanted to build finding the arc functions on top of finding the angles for Pythagorean triples. I asked the class if they knew of whole numbers which satisfied a² + b² = c². I also noted that equation can also be written as x² + y² = r² or adj² + opp² = hyp². One student eventually answered 3-4-5. After showing this worked, I showed that 3-4-5 is the first triple in a family of similar triangles that form triples.
I then noted that the next step was one that was focused on the CIS majors: using a mixed address formula in a spreadsheet to generate a map of Pythagorean triples.
Taking a cue from the text, I focused on harmonic motion. I brought in a spring and masses, timed the period, and presented the physics forms of the equations. I noted that Larson seemed bent and determined on confusing the students by using omega in chapter 6.7 while in an earlier chapter Larson had used b. And in neither instance did he start with nor explicity state the physically meaningful versions of the sine functions with the period, frequency, or wavelength specifically stated. So I put masses on the spring and used a stopwatch to time the period. After a demo at 500 grams, I had the students try one at 700 grams. The bounce rate of the spring, by the way, varies with the mass.
ln|cos x|-ln|sin x|
ln|cot (theta)| + ln|1+tan^2(theta)|
solve for mu: mu*W*cos(theta)=W*sin(theta)
tan t cot t=1
tan^2(theta)/sec(theta)=sin(theta)*tan(theta)
(1+cot^2(x))*(cos^2(x))=cot^2(x)
tan^4(x)+tan^2(x)-3=sec^2(x)*(4*tan^2(x)-3)
The gnomon problem reveals most clearly that Larson does not come at true physical problems from a physical perspective. One would never ever wind up with:
h sin(pi/2-theta)/sin(theta)
as a starting point. One would most like soh cah toa and then use tan(theta)=opposite h/adjacent s.
Solving for s yields s = h/tan(theta) or s=cot(theta) - the solution to the homework problem above (8th edition #70 p546).
solve 1/12*(cos(8*t)-3*sin(8*t))=0 for 0<t<1 is the only form that generates decimal equivalents.
Using domain limits is tricky business at best. WolframAlpha sometimes balks. And WA seems to prefer < to <=.
WolframAlpha by default quotes angular solutions and being within the range negative pi radians to pi radians. The text quotes solutions in quadrant III and IV as being angular measures between positive pi and positive two pi. I was able to force solutions to land between 0 and two pi in the following instance:
solve Cos[2 x] - Cos[x] = 0 for -1 < x < 7.
This was based on the following Mathematica construct:
Reduce[{(Cos[2 x] - Cos[x]) == 0, 0 < x < 7}, x]
Note the intentional omission of the N[Reduce which generates the two pi n repeat cycle.
WolframAlpha behaves very differently depending on what one enters. Run the following sequence:
4 sin(x) cos(x)=1
yields a series of unworked out decimal solutions with an n-repeat.
4 sin(x) cos(x)=1 for 0<x<7
yields a graph only, no solutions.
solve 4 sin(x) cos(x)=1 for 0<x<7
yields worked out decimal solutions between 0 and two pi.
vector {80, 80}
vector {-86.16, 50}
vector {-40,-30}
vector {95,0}
After a very brief introduction of magnitude and direction angle, I moved on day two into dot and cross products. Using the physics definitions of the dot product being F d cos(θ) and the cross product being F d sin(θ) where F is the force, d is either the distance the object moves or the lever arm length, and θ is the angle between the two vectors. I used the toy pull cart wagon in my office to illustrate the effect of pulling the cart versus applying a torque and thus lifting the front wheels of the toy wagon up.
I wanted the students to see some mathematical operators beyond what they had seen thus far in life, to lift them higher up on the mountain of functions.
This also allowed me to introduce a reason why I use asterisk for multiplication, besides its default use spreadsheets, programming languages, and WolframAlpha this way. I save dots and crosses for dot and cross products. Larson covers dot products but in a horribly obtuse and obscure fashion with the throttle set to full theoretic. Then, as always, "real world" problems are glued on during the homework.
Three equations in two unknowns...
solve x^2+y^2=144,(x-3)^2+(y-3)^2=144,y=-x+3
Note that the red solution marking spots, which can be hovered to reveal the decimal answer, have disappeared. The answer is still quoted.
A sphere at the origin, radius 5: Sphere[{0, 0, 0}, 5]
Sphere, defined mathematically: Sphere
Sphere, defined mathematically: Sphere
Cylinder, defined mathematically: cylinder
Platonic solids: Platonic solids
Archimedean solids: Archimedean solids