Thursday, March 29, 2007

Grey-M's DEV Project

Hello everyone and welcome to my Developing Expert Voices project! Why don't you give the questions below a try, I dare ya. If you can't get them then take a look at the answers that I have come up with because I believe that they are right. The sound adds a little bit to the questions at times but it is kind of eradic volume-wise (and I really don't sound too thrilled in them).

Question 1 – The Velodrome

Two bicyclists are going around the track in a bike race. One bicyclist decides to pull away when he starts going down the slope at one end of the track improving his lap time to twenty seconds. Their speeds (m/s) are represented by the following sinusoidal functions where x is twentieths of the track and y is the speed in m/s:

Racer 1: f(x) = 10sin((pi/5)(x))+40

Racer 2: g(x) = 5sin((pi/5)(x))+35

How many laps will it take for racer 1’s speed to get to double racer 2's speed if racer 1 is picking up speed at a rate of x for every x twentieths of the track when he started to pull away and the range of racer 2’s speed is now half that it was before?
See the larger version of the slideshow here

Question 2

Solve for x, y and z:

eiπ(4(cos2(y-(pi/2)) + cos2(y))) – z = 4x(cos2((pi/5)(y –2.5) + cos2((pi/5)(y))y ^ y

3(sin2(y)+cos2(y))y! + x = 2z(csc2(y) - cot2(y))xyz

You may look at this and say to yoursel
f "To solve for three variables you need three equations" if you remember back to systems. But if you look carefully this is an easy problem and a solvable one.
eiπ(4(cos2(y-(pi/2)) + cos2(y))) – z = 4x(cos2((pi/5)(y –2.5) + cos2((pi/5)(y))y ^ y
First, simplify what is in the brackets i
n the first term. The first term within the brackets is a shifted squared cosine function. Well with the shift it has it is a squared sine function.
eiπ(4(sin2(y) + cos2(y))) – z = 4x(cos2((pi/5)(y –2.5) + cos2((pi/5)(y))y ^ y)
sin2(y) + cos2(y) you should recognize as the rudimentary trig identity, so it is equal to 1 and 4 times 1 is 4.
eiπ(4) – z = 4x(cos2((pi/5)(y –2.5) + cos2((pi/5)(y))y ^ y
The last thing you have to do to this is recognize Euler's identity(
eiπ+1 = 0).
-4 – z = 4x(cos2((pi/5)(y –2.5) + cos2((pi/5)(y))y ^ y
That side is really easy now so we'll go to the other side to the first term within the brackets. It is a squared cosine function that has a parameter B of (pi/5) which means that the period of the function is going to be 10 ( 2pi/B --> 5*2pi/pi --> 10). It has a phase shift of 2.5 to the right. Hmmm... well that's just perfect. That makes is a squared sine function with the same parameter B and no phase shift. Once again this is the rudimentary trig identity so it is equal to 1.
-4 – z = 4x(1)y ^ y
Now 1 to any exponent is still 1.
-4 – z = 4x * 1
Anything times 1...

-4 – z = 4x

So the first equation is now easy as pi (couldn't resist, I'm so corny...). What we do to the next equation is the same thing...
3(sin2(y)+cos2(y))y! + x = 2z(csc2(y) - cot2(y))xyz
Identify the trig identities.
1)y! + x = 2z(1)xyz
One to any exponent is 1 (y! is a factorial).

3(1) + x = 2z(1)
Anything times 1 is itself.
3 + x = 2z

Now you have two easy equations. Y has been completely eliminated meaning that the value of y is an element of all numbers real or imaginary (Since it doesn't effect the equations at all it can be anything).

Then just solve the system. (The quality of the picture really is downgraded for some reason, if you want to look at the good quality image go here)

So your answer would be -11/9 = x, 8/9 = z and y is an element of the real or imaginary numbers.

Question 3

Solve for x: sin(x)sin(x) = sin2(x)


One way you could solve this is to let sin(x) = a. Then you have a^a = a^2. The only values that will work for this are 1 and 2 (2^2 = 2^2, 1^1 = 1^2) because any other numbers and they are inequalities (0.001^0.001 <> pi^2). If you ask "Why not 0?" Then my answer is I don't know because not even the experts are certain, or atleast agree, what the value of 0 to the 0 is, so I cannot say that 0^0 = 0^2. Then we reject the answer of 2 because sin(x) will never give us an answer greater than 1. So within one period of the sine function sin(x)
sin(x) the only possible value for sin(x) will be 1. The value in which sin(x) = 1 is pi/2. Since this is a sine function this will repeat. But how often? It will not repeat every period because when the sine function enters the negative outputs we'll begin to get imaginary answers because you cannot have negatives under the radical sign (the only one that will still work would be -1, any others are decimals and when changed from exponential to radical form they will have the negative base under the radical). Since both are sinusoidal functions this will repeat every 2kpi where k is an element of integers because we eliminated it repeating every period or kpi where k is an element of the integers.

So our answer formally is x = (pi/2) + 2kpi
; kEI.

Graphically to support my answer...

Question 4

How many equations are there that at infinite points touches/intersects the functions:

f(x)= x/10 and g(x) = -x/10.

Name atleast 3 of them and you cannot use tangent.


First visualize what your trying to do.
f(x)= x/10 g(x) = -x/10

If you look you may see that these graphs are the same except for the fact that the outputs of the second graph are negative that of the first. So if you had a function that had infinite amount of points equal to that of the first graph in the positive inputs then when it crosses the y-axis and starts getting negative inputs makes those outputs opposite sign then you would have an infinite amount of points on both graphs. It just so happens that h(x)= |x/10| does just that.

If that works then the negative of that will work also just the positive inputs will go along our second function while our negative inputs will go along our first function, j(x)= -|x/10|.

Now the answer that is the point of this question. We know that sine, cosine and tangent graphs will get positive and negative values periodically.
k(x) = sin(x) f(x)= x/10 g(x) = -x/10

This graph, k(x) = sin(x), doesn't touch these lines at infinite points. But we do know some things about this graph. For one we know that the maximum and minimum values are 1 and -1 respectively. Those are easy numbers to work with. But they occur in relation to pi which is not as easy. So since we can manipulate the sine function to suit our needs we can set the period to be an easy number like 2. Our sine function will then be m(x) = sin(pi(x)) because...
period = 2pi/B
2 = 2pi/B
B = pi

Now we have our outputs of 1 and -1 alternately at every input of a+0.5 ;aEI. Now how do you change 1 to a number of your choice? You multiply by that number. So if the outputs of m(x) = sin((pi)(x)) are 1 then you just multiply whatever sin((pi)(x) )gives you by (x/10) and when sin((pi)(x)) gives an output of 1 it will be the same as the output of f(x) = (x/10). The new function you would now get is n(x) = m(x)f(x) or n(x) = (x/10)(sin((pi)(x)). Now since the bottom line is the same equation just with negative outputs of f(x) = x/10, or in other words f(-x) = g(x), when (sin((pi)(x)) equals -1 then n(x) will equal f(-x) or g(x) ((-x)/10 = -x/10). So n(x) = (x/10)(sin((pi)(x)) will touch both lines infinitely because (sin((pi)(x)) will infinitely give outputs of 1 and -1.

Now n(x) = (x/10)(sin((pi)(x)) can be made simpler and in making it simpler we also find that there is an infinite amount of functions that will infinitely intersect the functions f(x) and g(x). We said that when (sin((pi)(x)) in our function n(x) = 1 or -1 that we'll get the outputs of f(x) and g(x) at those points respectively. Well does parameter B in a sine function affect the maximum and minimum values of 1 and -1 in a regular sine function? No. So our function could be q(x) = (x/10)(sin(z)(x)); zER. Now there is also another way to see that there is an infinite amount of possibilities. If you had any set of lines with a steeper slope than f(x) = x/10 then for our sine function to touch that line then it would have to cross f(x) and g(x). For a line to have a steeper slope than f(x)= x/10 it would have to have x divided by a number less than 10 but larger than -10. An example: r(x) = (x/5)(sin(x))

So since this is true a function that will infinitely cross the functions f(x), g(x) is:
s(x) = (x/t)(sin(z)(x));-10<=t<=10; tER; zER This same answer will apply to cosine with any phase shift because phase shift makes no difference in this and cosine just like sine will infinitely get outputs of 1 and -1. u(x) = (x/t)(cos(z)(x));-10<=t<=10; tER; zER Summary • Why did you choose the concepts you did to create your problem set? I chose the concepts that I did because I wanted to cover a lot of what we've done in grade 12 so far no matter how comfortable or uncomfortable I was with them. • How do these problems provide an overview of your best mathematical understanding of what you have learned so far? Well I found them rather hard so I guess that it would show the upper level of my skill which would be what I have learned so far. • Did you learn anything from this assignment? Was it educationally valuable to you? I learned lots from this. For one I learned that making problems that are hard but not impossible for yourself is *really* hard. Second I learned that when making these you should really work backwards from the answer because solving a problem that is on the brink of what you can do with no clue as to what the answer is will make you end up at the wrong answer a high percentage of the time. That first question was actually a five part question with the one I used being the last one. But I made the questions and then tried to get the answers. So I found out one was wrong and it made all the others wrong except for the last question (two days of spring break wasted). Then in my first answer I created for question 3 I found that I completely screwed that up. Overall I think that this was valuable because I don't know a time when I have thought so hard (and I truly hope I didn't get any of those wrong). One thing I must say though is that being able to do the problems is one thing but explaining them is another. When I did the problems I knew exactly what I was doing but when explaining them the words escaped me. I almost think that it would have been worth doing journeyman level questions for the boost in the mark of annotation because I don't know if my answers are going to get through to everyone. I know what I mean but others may not. With an easier question that you have a stronger foundation in you'll be able to explain it much better. Give me feedback. If you don't get how I solved the questions or if you think I got them wrong then tell me in the comments. Maybe I'll be able to explain them in the comments and Mr.K will allow another modifier in the annotation/solution mark for ongoing annotation/solutions.

Tuesday, March 27, 2007

The Assessment Rubric

Here is the updated version 1.1 of our rubric ...

Teaching mathematical concepts is the main focus of this project; so we can teach other people and learn at the same time.

Achievement Descriptors
Instead of levels 1-4 (lowest to highest) we use these descriptors. They better describe what this project is all about.

Novice: A person who is new to the circumstances, work, etc., in which he or she is placed; a beginner.
Apprentice: One who works for an expert for instruction or to learn a skill or trade.
Journeyperson: Any experienced, competent but routine worker or performer.
Expert: One who possesses special skill or knowledge; trained by practice; skillful and skilled.

Mathematical Challenge (25%)
Annotation (40%)
Solutions (15%)
Presentation (20%)
Novice Problems illustrate only an introductory knowledge of the subject. They may be unsolvable or the solutions to the problems are obvious and/or easy to find. They do not demonstrate mastery of the subject matter. Explanation does not "flow," may not be in sequential order and does not adequately explain the problem(s). May also have improper mathematical notation. One or more solutions contain several errors with insufficient detail to understand what's going on. Presentation may or may not include visual or other digital enhancements. Overall, a rather uninspired presentation. Doesn't really stand out. It is clear that the student has invested little effort into planning their presentation.
Apprentice Problems are routine, requiring only modest effort or knowledge. The scope of the problems does not demonstrate the breadth of knowledge the student should have acquired at this stage of their learning. Explanation may "flow" well but only vaguely explains one or more problems. Some parts of one or more solutions are difficult to follow. May include improper use of mathematical notation. One or more solutions have a few errors but are understandable. The presentation style is attractive but doesn't enhance the content; more flashy than functional. It is clear that the student has invested some effort into planning their presentation.
Journeyperson Problems showcase the writer's skill in solving routine mathematical problems. They span an appropriate breadth of material. One or more problems may require careful thought such as consideration of a special case or combine concepts from more than one unit but not necessarily. Explanation "flows" well and explains the problems step by step. Solution is broken down well and explained in a way that makes it easy to follow. May have minor use of improper mathematical notation. May point out other ways of solving one or more problems as well. All solutions are correct and easy to understand. Very few or no minor errors. The presentation may use multiple media tools. The presentation style is attractive and maintains interest. Some of the underlying message may be lost by some aspects that are more flashy than functional. It is clear that the student has given some forethought and planning to their presentation.
Expert Problems span more than one unit worth of material. All problems are non-routine. Every problem includes content from at least two different units. Problems created demonstrate mastery of the subject matter. Showcases the writer's skill in solving challenging mathematical problems. Explanation "flows" well, explains the problems thoroughly and points out other ways of solving at least two of them. All solutions correct, understandable and highly detailed. No errors. The presentation displays use of multiple media tools. The presentation style grabs the viewer's or reader's attention and compliments the content in a way that aids understanding and maintains interest. An "eye opening" display from which it is evident the student invested significant effort.

Creativity (up to 5% bonus)
The maximum possible mark for this assignment is 105%. You can earn up to 5% bonus marks for being creative in the way you approach this assignment. This is not a rigidly defined category and is open to interpretation. You can earn this bonus if your work can be described in one or more of these ways:
  • unique and creative way of sharing student's expertise, not something you'd usually think of;

  • work as a whole makes unexpected connections to real world applications;

  • original and expressive;

  • imaginative;

  • fresh and unusual;

  • a truly original approach; presentation method is unique, presented in a way no one would expect, e.g. song, movie, etc.

Monday, March 26, 2007

The Assignment

The Assignment
Think back on all the things you have learned so far this semester and create (not copy) four problems that are representative of what you have learned. Provide annotated solutions to the problems; they should be annotated well enough for an interested learner to understand and learn from you. Your problems should demonstrate the upper limit of your understanding of the concepts. (I expect more complex problems from a student with a sophisticated understanding than from a student with just a basic grasp of concepts.) You must also include a brief summary reflection (250 words max) on this process and also a comment on what you have learned so far.

You will choose your own due date based on your personal schedule and working habits. The absolute final deadline is May 31, 2007. You shouldn't really choose this date. On the sidebar of the blog is our class Google Calendar. You will choose your deadline and we will add it to the calendar in class. Once the deadline is chosen it is final. You may make it earlier but not later.

Your work must be published as an online presentation. You may do so in any format that you wish using any digital tool(s) that you wish. It may be as simple as an extended scribe post, it may be a video uploaded to YouTube or Google Video, it may be a SlideShare or BubbleShare presentation or even a podcast. The sky is the limit with this. You can find a list of free online tools you can use here (a wiki put together by Mr. Harbeck and myself specifically for this purpose). Feel free to mix and match the tools to create something original if you like.

So, when you are done your presentation should contain:
(a) 4 problems you created. Concepts included should span the content of at least one full unit. The idea is for this to be a mathematical sampler of your expertise in mathematics.

(b) Each problem must include a solution with a detailed annotation. The annotation should be written so that an interested learner can learn from you. This is where you take on the role of teacher.

(c) At the end write a brief reflection that includes comments on:

• Why did you choose the concepts you did to create your problem set?
• How do these problems provide an overview of your best mathematical understanding of what you have learned so far?
• Did you learn anything from this assignment? Was it educationally valuable to you? (Be honest with this. If you got nothing out of this assignment then say that, but be specific about what you didn't like and offer a suggestion to improve it in the future.)

Experts always look back at where they have been to improve in the future.

(d) Your presentation must be published online in any format of your choosing on the Developing Expert Voices blog. url: tba.
Experts are recognized not just for what they know but for how they demonstrate their expertise in a public forum.

Levels of Achievement
Instead of levels 1-4 (lowest to highest) we will use these descriptors. They better describe what this project is all about.

Novice: A person who is new to the circumstances, work, etc., in which he or she is placed.

Apprentice: To work for an expert to learn a skill or trade.

Journeyperson: Any experienced, competent but routine worker or performer.

Expert: Possessing special skill or knowledge; trained by practice; skillful and skilled.