163 – Calculus Project Presentations Day 3 (LAST DAY OF CLASSES!)

(these are my quick notes from Day 3 of presentations)

Python and Calculus

Described the python, vpython, sympy programming languages/libraries. Next they are showing how they used python and the sympy libraries to calculate the derivative of a given function. It graphs the function and asks where they want to draw the tangent line.

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Parachutes

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They were trying to make a parachute that fell the slowest. They changed the diameters of the parachutes, and the wire placement. They also varied the amount of weight on the parachute. Used tracker analysis to see position, velocity, and acceleration. Showed the tracker info to find the velocity of the parachute. Used a t=1 to compare the speeds of all the parachutes. The “best” parachute has the least velocity. Conclusion – bigger parachutes with cross wires worked the best.

How to count cards and betting strategy

Explained the rules of Blackjack. Went through basic strategy, and then went through how to count cards. Cards 2-6 add 1, 10 and face subtract 1. If the count is positive, then you are more likely to get a high card. If you can count cards, you can win around 52% of the time instead of 49% of the time. If you have a high count, then you increase your betting amount because you are more likely to win. Flat betting is betting the same amount of money each hand. Martingale betting system has the person double their previous losing bet. Cancellation Betting system is where you write down a series of numbers, you bet the sum of the first and last numbers etc. Mimic the Dealer is a bad gambling strategy. Never Bust is where the house has a advantage of 4%. Assume ten in the hole gives the house 10% advantage. Walked through the python program and showed the simulations

Relationship between calculus, projectile motion and discus

(lost the description – darn you google docs)

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Pendulum Wave

Showed the blacklight pendulum wave machine. Each string is a different length so they have a different period. The longest pendulum has 45 oscillations in 60 sec, 46 oscillations in 60 sec, 47 oscillations in 60 sec. etc. You can choose what period you want by using a formula that will find the pendulum length. Did a full demo and then showed their full video. Used desmos to show the sine waves for the 5 different masses.

Their video!

String Art

Researched how to create string art. String art uses tangent lines to create a curve. String art can create all differentiable functions. (unfortunately her presentation was rushed due to the lack of time in the block).

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Baking Soda and Vinegar Rockets

Launched the rockets at 45 degree angle to get the maximum distance. Measured the distances of the rockets. They found the ratio 12g vinegar to 1g baking soda to maximize the distance of the rocket. Went over the horizontal distance equation and how you could use the time of flight to find the distance of the rocket (or vice versa).

 

 

161 – Calculus Project Presentations Day 2

(these are my quick notes from Day 2 of presentations)

Limits of Human Speed

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What is the theoretical limit of human speed? Showed world records in 100, 200, 400, 800, 1600, and marathon.
mv = F – D(v).
t-t_0 = m integral of dv/(F_0 – D(v’)) from v_0 to v.
Can find the force of Usain Bolt by knowing his mass and velocity. Acceleration 9.49 m/s^2. 800+ Newtons. Showed graph of his position, velocity, and acceleration. Talked about what the graphs mean. Shows slope of secant lines for average velocity over each 10m interval. Possible reasons why humans have increased their speed (spikes, starting blocks, better tracks, weather patterns, performance enhancing drugs, better timing). Showed his power output throughout the sprint. Peaks at over 2500+ watts (3.51HP). The theoretical limit for the 100m dash is around 9.48 seconds. Talked about how the fast humans can run over distance. For every doubling in distance, multiply the time by 2.3 – 2.1. Showed NYTimes videos on the 100m records over time, and then showed usain bolts world record.

Water Bottle Rocket

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Video – 1 liter of water in a 2L bottle. Goes up HIGH (100ft, 300ft/s, 50-60gs of force) 90 to 120psi. Second launch 100g’s of force. 1/2 liter of water, 60ft high 250ft/sec 87g’s. Second launch 130g’s. Reaches top velocity right after the water has been expelled (no more acceleration). Tried to put more than 1L of water in, but it was far more difficult to put air in.

Knot theory

Describing the basics of Knot Theory. Most basic knot: unknot. How you can “untangle” a knot using the Reidemeister moves. Different types of knots, to classify knots, unknot, trefoil, etc. Applications of Knot Theory, untangling knotted DNA, Protein Strutures, Statistical Mechanics, etc. Moving on to the Borromean Rings (any two of the rings are not connected, but all three are inseparable). Talked about the Human Knot game – how sometimes it’s unsolvable. Students demonstrated the human knot, in this case it was unsolvable.

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Encryption and Hashing

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The basics of encryption. How passwords work, and how passwords are hashed. Why computers use hashing to store passwords. Went into the basics of why computers use passwords. Talked about the connection to LARGE prime numbers and how to verify if a number is prime or not. Hashing is easy to go from a word to the hash, but VERY difficult to go the reverse route. Salting the hash allows each username to have unique hash possibility.

Tilings

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Talked about the different types of tiling. Domino Tilings, related to fibonacci numbers. Aztec Diamond tiling, harder to tile than the checkerboard pattern. Hexagon tiling, must think about the area taken by the tiling. Penrose dart kites and rhombi. Formed by the golden ratio to make the angles. phi = 0.5 *  5^0.5 + 0.5 ! Golden triangle, angles of 36, 72, and 72, and the Golden gnomon 180, 36, and 36 degrees. Arc and Edge matching rules. Penrose pentacles and kosh curve. Applications: flooring, etc. This student’s work has already been featured on this blog.

 

160 – Calculus Project Presentations Day 1

(from my notes on the calculus projects so far)

Farming Calculus Game – a Bolivian farming themed board game

With powerpoint: explaining the rules of the game. Spin the spinner and solve an area underneath the curve example. Teams solve a calc program and get “profit units”. There are integral problems, max/min problems, arc length problems, etc. Review game with “real life” problems. Alpacas!!!2013-06-04_09h53_46

Related Rates Programming “Falling ladder problem, wrote code in BlueJay (Java)

Showed and explained the variables in the code. Showed how the code reads in and interprets what to solve for. Shows pythagorean theorem in code, shows the code to find the dx,dy, and dz using the calculus.
Now going to a website to get an example and they are going to use their program to confirm the answer. Enters a sample problem in their code, and they get the answer correct! Nice code.

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159 – blank

Plenty of things were going on in class but no pictures. Calculus is finishing up projects, and Consumer math played a review game and continues to review. The final question in the review game was: “Name as many videos as you can in the top 10 viewed videos on Youtube. 10 points per video.” They’re surprisingly bad at it! Most groups could only get one or two.

 

155 – Twin Primes

Today is a senior skip day so we talked Twin Prime conjecture and the new discovery. Used the Slate article and Mr. Honner’s writeup for basis. Fantastic. Also wrote up a quick python script to print twin primes. Runs pretty quick, finds the first half a million in around 10 minutes. So no picture today, but here’s some code 🙂

from math import sqrt

def prime(n):
   for i in range(2,int(sqrt(n))+1):
      if (n%i == 0):
         return False
   return True

count = 1
numberfound = 0
previous = 0
while True:
   if count % 5 != 0:
      if prime(count):
         if prime(count + 2):
            numberfound += 1
            print "number found ",
            print numberfound,
            print "twin primes",
            print count,
            print count + 2,
            print " the gap is ",
            print count - previous
            previous = count
   count = count + 2