D.E.V.

D.E.V. (Developing Expert Voices)

Once upon a time in a land far.. Far away, there lived a man named Isosceles. He lived in a town called Phi. It was a very mathematical place. Where numbers were free to divide and multiply. Where the all trees were left to grow accordingly to the Fibonacci principle.

One day Isosceles was invited to go to a “rave”. After riding his Pi mobile there he got off and entered the building.

Problem #1 Isosceles noticed that there were 4 girls and 4 boys dancing in a circle at a party. Isosceles being such a great mathematician wondered: “How many different ways can I rearrange them?”

Solution 5040 ways Isosceles took out his “handy dandy notebook” and started to figure it out. “ Hmm.. Well since there are 8 people I should be able to arrange them 8 factorial ways. However since they’re dancing in a circle we must first point out a reference point. Thus we use (n-1)! - (8-1)! 7! = 5040 ways to arrange the dancers

Problem 1-2 He saw that the girls preferred to be with the girls and the boys with the boys. “ Hmm.. How many different ways can I rearrange them so that all the girls stay with the girls and the boys stay with the boys?..”

Solution 576 ways “Well if I take only the girls that would be 4!. Then the boys would be 4!. This gives us 4! And 4!.” => (4!)(4!) 576 ways to arrange them so that each gender stays with only that gender.

Problem 1-3 Suddenly a slow song played. Now the girls and the boys would each need to have their own partners of the opposite gender in a circle. “ How many different ways can I rearrange them so that the boys and girls are alternating?”

Solution 144 ways “ Well first we need a reference point. We take one girl and use her as a reference point and place the other girls along to form a circle. This gives us 3! Ways for girls. Then to place their boy partners we have 4!.” => (3!)(4!) 144 different ways to mix and match the boys with girls.

Problem 2 Isosceles found a hula hoop on the dance floor. He grabbed it and measured the circumference. Who knows why? It turned out to be 60 meters. He began to play with it and started to roll it. He rolled it 10 meters. “ Hmm.. I wonder how many radians I turned it?”

Solution Q=Theta n= pi(3.14) “ Luckily in my handy dandy notebook I have the formula (C=2nr). I have the circumference I can use the to find the radius. Then I can use the equation S=Qr. To find out how many radians it has moved.” *note that S equals the arc length. In this case its 10 meters => C=2nr => S=Qr 60=2nr 10=Q(30/n) r=30/n Q=n/3 radians “ I see.. So I’ve turned the hula hoop n/3 radians.”

Isosceles exits the dance to get a breather and sits on the stairs of the entrance to the school. As he looked around he noticed a bicycle passing by. As he looked at the bicycle he suddenly noticed the math within the bicycle. He then grabbed a notebook out of his bum pocket and started to write information down.

Problems 3 Isosceles concluded that the maximum height of the bicycle wheel is 70cm as it rolled, and the minimum height is at 0cm touching the ground. He also noticed that the wheel would complete a full spin in 4 seconds. He then began to think of questions for himself. a) Create two equations in terms of sine and cosine. b) Make a graph showing the max, min, average value, and period.

Solutions a) cos(x) = 35cos(2π/4(x))+35, 35sin(2π/4(x+1))+35 This questions roughly pretty easy you’ll need to first make a graph to show all the information required to create an equation. It’s actually more easier if you had a graph. The graph to the left will eventually give you the equations for sin and cos. Cos(x)= 35cos(2 π/4(x))+35 Sin(x)= 35sin(2 π/4(x+1))+35

Solutions b) This one was actually solved already. But here it is again!

Isosceles then heads home, from all the dancing he did and calculating math problems he was really exhausted. He quickly got changed into his home clothing and started playing his play station 2. He played a game called Marvel Vs Capcom 2. He then noticed, this certain characters attack resemble a hyperbola. He took a closer look and paused it, and it turns out that it was a hyperbola! He then creates another mathematical problem with his inspiration.

Problems 4 The center of the Hyperbola is (0,0) The equation of the asymptote is y= 7/11x. Find the equation of the hyperbola.

Solution x^2/121 – y^2/49 = 1 The answer to this question is already solved as well. By looking at it you see that the value of “ b ” is 7^2 which is 49. The value of a is 11^2 which is 121. The result is:

Problem 5 After that short little segment he experiences with math in video games, he then gets a phone call. For some odd reason his friend Maris, calls him dealing with his other friend Paris. She told him that Paris invested $700 somewhere, and the interest was 7% and compounded semi-annually. She then asked him how long will it take for his investment to quadruple.

Solution 20 years The solution to this question is quite easy. With the help of this equation you’ve got what you need. A = P(1 + r/n)^(t)(n) Now simply plug your information in and it should look like this: 2800 = 700 ( 1 + 0.07/4 )^4t ( equation plugged in ) 4 = (1.0175)^4t ( a more simplified step ) ln4 = 4tln(1.0175) ( take the “ln” of both sides ) ¼((ln4/ln(1.0175)) = t 19.97 = t t = 20 years The time it takes the investment to quadruple is 20 years.

RFLECTION Reflection- Oliver Why did you choose the concepts you did to create your problem set? I believe that these questions were the questions that I have been experiencing some difficulties in. It’s also a refresher to my mind. I liked all the units and I thought that these problems would be quite fun to revisit, and of course we’ll all be revisiting them on the exam. So I thought why not do them anyway. How do these problems provide an overview of your best mathematical understanding of what you have learned so far? Since these were problems that I was not able solve before I believe that being now able to solve it shows that I have learned how to answer these types of questions properly. The ''half life'' question is the one in which I am most proud of. I’ve only recently learned how to properly solve these types of questions. I know now that in the future if this type of question were to come up I would be able to solve it. Did you learn anything from this assignment? Was it educationally valuable to you? I think that I did learn something doing this assignment. More than one thing actually. In each and every problem that we had created I wasn't so sure of how to go about solving the question. After doing the assignment I feel that I am now able to break down and solve these certain types of problems. It also gave me a chance to remind myself of past formula and concepts that we had been taught.

REFLECTION 2 Reflection- Miller Why did you choose the concepts you did to create your problem set? i chose to create these questions because they resemble some of the questions that I've been experiencing difficulty on these past few units. Creating and solving these questions will hopefully allow myself to fully understand the concepts and ideas going on in these sections. Which are Conics, Permutations, unit Circle, trigonometry and Logarithms. How do these problems provide an overview of your best mathematical understanding of what you have learned so far? these questions have proven that i am now able to answer these types of questions. Before i may not have been successfully able to find the equation of a hyperbola by just being given the center and equation of the asymptote. Now i am fully confident that if this question arises somewhere in the future i would be able to solve it. Did you learn anything from this assignment? Was it educationally valuable to you? to be totally honest after doing this assignment i feel as if i have only scratched the surface of these units. i feel that if i had chosen to dig deeper and create more challenging question this assignment would have been as good as gold for me. Needless to say that i still did learn a lot from doing this and was able to brushing up my mathematical skills.

D.E.V.

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D.E.V.