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Course Topic: Multiplication Principle for Probabilities Video Link: https://www.youtube.com/watch?v=CNr_7gPhYtY&list=PL2CD836B66D3CEBED&index=17 Time: 45:45 - 47:49 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: John Huelsenbeck Teaching Ideas: This video uses the multiplication principle for independent probabilities to determine the probability of a genetic match with two alleles. The professor goes on to talk about how this is used in criminal trials. This can be used as a nice introduction to probability theory.
Course Topic: Independence of Two Events Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-10 Time: 45:15 - 48:02 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video applies the uses coins stuck together to demonstrate that two events are not always independent. The professor uses it to demonstrate that the orbitals of multiple electrons are dependent since electrons repel one another. This is nice lesson in resisting the temptation to always multiply probabilities together. Course Topic: Expected Value and Standard Deviation Video Link: http://oyc.yale.edu/economics/econ-251/lecture-13 Time: 67:52 to 71:09 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video goes over the expected value (he calls it expected payoff) and the variance and standard deviation. The professor describes these just as they do in math classes. This is a nice video that shows the most important topic from statistics is used in finance.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-14 Time: 1:58 to 3:24 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows the professor calculating the expected value, variance and standard deviation by hand for the situation where the sample space is {-1,1} each with probability 1/2. He refers to an investment where you either win or lose $1. This is a simple example that shows the hand computation.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-20 Time: 7:15 to 7:48 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows the professor calculating the expected value for a bet on the Yankees winning a game. This is quickly done and gives a fun example of the topic of expected values for a discrete probability distribution function.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-9 Time: 6:10 to 9:59 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses looks at the expected payoff (value) for a game based on a payoff matrix and probabilities of each strategy. The computations are done by hand and are easy to follow.
Course Topic: Discrete Probability Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-5 Time: 28:55 - 35:13 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video is an application of discrete probability to make genetic counseling decisions. The specific mathematics is all at the level of elementary statistics class, but it uses several ideas and formulas. Only the very best students will be able to solve the problem without plenty of explanation from their instructor. Students will need help to follow the reasoning, but the application if incredibly relevant: There is a deadly disease caused by a homozygous gene. 1% of the population dies of this disease. John's brother died of the disease and Jill knows nothing about her family history. If John and Jill have a child, what is the probability that the child will die of the disease? Course Topic: The Z-Score Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-4 Time: 75:33 to 76:40 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video defines the Sharpe Ratio for a Portfolio which is basically just the z-score for that portfolio. The professor never used the word Z-Score, so a teaching strategy is to ask the students who have just watched the clip what statistic the Sharpe Ratio corresponds to. The students paying attention should be able to answer "Z-Score". The professor continues the explanation until 77:38 by saying exactly what statistics instructor say about the Z-Score, that the raw value is meaningless without knowing the mean and the standard deviation. Course Topic: Uniform Distribution Video Link: http://oyc.yale.edu/economics/econ-251/lecture-16 Time: 67:17 to 69:20 or 70:00 and then again to hear the solution from 72:49 to 73:16. Then at 73:52 to University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video poses the question if a man meets 1000 women who rate uniformly on [0,1]. After meeting each woman the man can either marry her or move on to the next and never go back. What is the man's optimal strategy? Although this does not give any math that is done in the statistics class that uses the uniform distribution, it is both funny and relevant to college students and "answers" the question why so many married men are seduced by the "other woman". Also the same theory works for how to decide which restaurant to stop at when driving down Main Street in a town you are unfamiliar with. The final part does the mathematics which is pretty simple and could be worth showing if there is time. Course Topic: The Normal Curve and the Central Limit Theorem Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-3 Time: 13:00 - 15:43 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video begins by explaining the Central Limit Theorem and the Normal Curve. Then it describes how it can be used to look at the rate of evolution of the Galapagos finches. This perfectly ties in the content from the statistics course to evolutionary biology and can be used as an effective introduction to the topic.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-2 Time: 59:02 to 67:25 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video explains that with a normal distribution, values far from the mean should occur very infrequently. He gives the example of the daily stock market change since 1928. The graph looks normal, but there are three tremendously large outliers in 1929 that correspond to the stock market crash and the one day rebound. If the distribution had been normal this basically couldn't have occurred. This is a long but interesting application of the normal distribution that can be used to emphasize that the normal distribution rarely occurs for populations that do not correspond to averages and these strong outliers kill normality.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-3 Time: 8:19 to 11:39 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the Central Limit Theorem and then goes on to explain its limitations for finance due to the fact that to apply the Central Limit Theorem the variance must be finite. In finance the variance may be infinite due to the large outliers that occur. This video can be used to show that the Central Limit Theorem is not just a math concept. Other fields such as finance and physics make great use of it. Since the video gives a full nonmathematical presentation of the theorem it can reemphasize what the elementary statistics instructor presents.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-5 Time: 9:09 to 10:37 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the formula for the standard deviation of a binomial experiment. The professor relates it to the insurance industry. He makes a mistake and his statement is not perfect, but even Nobel Laureates make mistakes. This short clip can be a nice motivator to explain why having many trials in a binomial experiment is preferred.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-7 Time: 7:05 to 10:01 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video shows how an investment firm took advantage of the normal distribution and basically sold off the right tail and beefed up its left tail to make more money most of the time. Unfortunately, when the left tail hits, the is catastrophic loss. This is a nice example of using the normal distribution in the real world before understanding the Central Limit Theorem.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-13 Time: 66:13 to 67:27 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video looks at the normal distribution and the professor notes at the end that "fat tails" makes the stock market not normal. He goes over the idea of the normal distribution especially that there are not many outliers. This is a quick application that shows the dangers of the normal distribution used in finance.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-24 Time: 70:13 to 71:00 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows that the graph of the daily returns of the S&P 500 is very close to normally distributed in 1972. This is a very clear example of a distribution following the normal distribution. One thing to note is that over a longer time, it is not normal since outliers are stronger now and the normal distribution is no longer a good model. This could be from the fact that we know that it should be normal so it isn't.
Course Topic: Confidence Interval Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-11 Time: 41:20 to 47:00 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: In this video, the professor first goes over what a 90% confidence interval, gives an example, and then asks the class to guess at a 90% confidence intervals for the world population (in 2011), the weight of the world in metric tons and how many languages there are in the world. This is a great way to explain what a confidence interval is and convince them that confidence intervals are used in other classes. If you just want to show this for the first question, you can stop at 43:43. Much later, he gives the answers, but an instructor can save time and just have the class do the exercise for the first question for the current date, show the correct answer via Google, then then see what percent got it right. Later at 50:10 he explains that people are over confident and fewer then 90% will get their question containing the population mean.
Course Topic: Hypothesis Testing Video Link: https://www.youtube.com/watch?v=CNr_7gPhYtY&list=PL2CD836B66D3CEBED&index=17 Time: 37:58 - 39:21 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: John Huelsenbeck Teaching Ideas: This video uses does a great job in explaining what hypothesis testing is all about in the context of seeing if a population follows the Hardy Weinberg Laws of population. It will make a wonderful introduction to hypothesis testing. It doesn't go over any specifics such as Type 1 and 2 errors, p-values, etc.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-30 Time: 37:30 - 38:33 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video uses the p-value and hypothesis testing to test the claim that an acid reflux medication is effective compared to the over the counter meds. The guest presenter explains that there were two studies done that showed statistical significance and one that showed no difference. This reinforces the idea that a low p-value is not a guarantee. This would be a good time to discuss type one and type two errors and their repercussions. Course Topic: Simplifying Linear Expressions Video Link: http://oyc.yale.edu/economics/econ-159/lecture-11 Time: 18:45 to 21:01 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video is looks at the genetics cooperation. The professor derives the payoffs of cooperation and defection. He uses the variable epsilon, so the instruction will need to remind elementary algebra students that epsilon is just like x. Although the professor solves the problem by simplifying both linear expressions, no steps are show. The students can be asked to fill in the details of the omitted steps. In a few more minutes the professor shows that the situation is the same if the population begins with mostly creatures that are defectors.. The algebra is very simple so can be given to any basic algebra class.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-7 Time: 22:30 - 25:59 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video looks at the equation that relates the difference in the pressure at different altitudes vs. the density, the acceleration of gravity and the change in elevation from top to bottom. This is basically the buoyant force. The algebra just involves dividing both sides by A and then subtracting the pressure at the top from both sides. There are several letters in the equation, so it will be tough for beginning algebra students to deal with, but they will all understand the science behind it as long as the instructor gives some explanation of the details.
Course Topic: Solving a Linear Equation Video Link: http://oyc.yale.edu/economics/econ-251/lecture-20 Time: 9:10 to 10:25 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows the professor solving a linear equation to find the probability that the Yankees will win the world series given gambling line of the game. This is a fun example of having to solve a linear equation. If you have enough time go another four minutes, you will see a way of taking advantage of the fact that some will be suckered into betting even odds and you can end up with no risk and a guaranteed win.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-9 Time: 45:13 to 46:58 (or until 51:55 to see both players' strategies) University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video looks at the optimal strategy for a tennis player hitting to the left a proportion of q times. There are two expressions and the solution is when the expressions are equal to each other. An instructor can pause the video just before the algebra is to be done and have the students work it out. Then the instructor can resume the video and the students can check if they got it correct. This is a very good class exercise for an elementary algebra class. Another possibility is for the students to watch the first example and then for them to work out the second example to arrive at the full Nash Equilibrium.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-10 Time: 48:20 to 54:39 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video is very similar to the tennis one just above, but the premise is the taxpayer community must choose a proportion to cheat on taxes and the IRS must choose a proportions of tax returns to audit. The math involves solving a linear equation. The professor skips most steps, but the students can be asked to attempt to solve the equation.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-12 Time: 46:03 to 47:09 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video looks at the Nash Equilibrium that models the evolutionary biology of aggressive vs. cooperative behavior. The video clip itself just gives the equations, is the instructor will need to explain to the students that it is modeling the behavior where there is a cost of confrontation between two aggressors, an aggressor gets all food when it meets a cooperative species and two cooperatives share the food when then encounter each other. The professor says "trust me" on the calculations. It would be effective to have the students not trust the professor and see if they can arrive at the solution by themselves. There are coefficients that are parameters in the equations, so group work might be needed to ensure that the whole class gets through the derivation. Later in the lecture, the professor notes that biologists have used this to find the ration of V to C. Course Topic: Slope of a Line Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-19 Time: 0:54- 2:48 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video shows the lines that relate income and clothing expense and income and food expense. The professor explains that the elasticity (slope) is greater for clothing. This is a very easy to understand example of how the slope of a line can be directly used to understand consumer habits. The professor does not give out any numbers or equations so this is just a conceptual example.
Course Topic: Equation of a Line Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-7 Time: 18:52 - 21:12 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video looks at the line that relates the concentration of ethoxide to the rate of the reaction that takes place. The professor particularly points out that the y-intercept is not 0. This is a great way to explain to the students the importance of the y-intercept of a line.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-4 Time: 32:15 to 36:47 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video plot the standard deviation (risk) vs return on investment. The professor's explanation is easy to understand and provides a meaningful example of understanding the slope and y-intercept of a line. The y-intercept of the line means the return on an investment with no risk. The slope is the increased return per risk on the investment. Just about every student will understand the importance of this example and why there is a linear model.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-23 Time: 57:13 to 58:37 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video defines the Sharpe Ratio as the slope of the line that determines the expected value of an investment as a function of the standard deviation. This will require some explanation of the economics behind the Sharpe Ratio, especially for elementary algebra students. He emphasizes that the slope of a line is the same for any two points.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-12 Time: 16:55 to 17:32 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the equation of the line that relates the positions of two people, the velocity of the one who is moving and the time that elapsed. This is a nice real world view of the equation of the line that does more than just write down x and y and an equation. The slope is the velocity and the y-intercept is the initial distance between the two people.
Course Topic: Regression Line Video Link: https://www.youtube.com/watch?v=2z0nrWrcHSA&list=PL2CD836B66D3CEBED Time: 7:23 - 9:59 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: Alan Shabel Teaching Ideas: This video shows two regression lines. The first is for number of cones produced by the tree vs. the width of the tree rings. The second is the fecundity vs. the probability of survival. This can be either used in a statistics class or a beginning algebra class. The content is basic enough that everybody should be able to understand it.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-16 Time: 12:40 - 14:08 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video looks at the relationship between the redshift and the distance between galaxies. The professor says that the data are strongly correlated to a line and gives the famous Hubble equation that results. This can either be shown in an elementary statistics class to talk about regression analysis or a beginning algebra class to talk about equations of lines that go through the origin.
Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-4 Time: 25:12 - 26:56 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video uses a regression line to understand the relationship between Muliken's Electronegativity, the energy required to pull an electron away from a charged atom, and Pauling's Electronegativity, a measure of the difference between cross atom bonds and the average of the same atom bonds. This may be too technical for an elementary statistics class, but in any class, the slope and y-intercept can be discussed.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-1 Time: 19:25 - 20:41 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows two scatter plots of trait vs. reproductive success. The first has a strong positive correlation and the second has a weak correlation. He then states that the first is the driver of evolution and the second is the driver of being able to use DNA sequences to infer history. This is a great way to show students the power of regression analysis.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-4 Time: 16:15 - 17:36 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows how to use a scatter plot and a regression line to predict the time at which two species shared a common ancestor. This is a straightforward use of regression analysis and clearly demonstrates how the regression line can be used to make predictions.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-11 Time: 13:52 - 14:58 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video displays the scatterplot and regression line to relate the observed age vs. predicted age when an animal will have its first offspring. The predicted age is based on a mathematical model that maximizes theoretical success based on Darwinism. The professor gives the correlation r = 0.93 and is careful to note that this does not imply causation. This can be used in the section of the statistics class where correlation is presented.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-12 Time: 33:50 - 35:15 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the social rank of the mother vs. the lifetime reproductive success for both male and female red deer on the same axes. There is a correlation for the sons but not for the daughter. This gives a nice comparison on what a correlated scatter plot looks like vs. one that is not correlated.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-16 Time: 28:55 - 33:55 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the scatterplot of adult life expectancy vs. male fidelity. The regression line is presented showing a strong correlation that indicates that fidelity is linearly associated with living longer. The correlation is not presented, but students may appreciate that at least for the species of Procellariforms, cheating on your mate does not tend to pay off.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-21 Time: 12:40 - 13:30 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the scatterplot of type 1 diabetes and worm infections. There is a clear negative correlation. The value of r is not shown, but the professor does refer to the negative correlation.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-25 Time: 11:20 - 12:34 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the scatterplot of temperature and rate of development of insects. The correlation is very strong and positive. The line is shown but r is not. This is an easy to understand example of the regression line used in biology.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-24 Time: 15:28 - 17:26 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video displays the connected scatter plot of the amount of global sea ice above the mean vs. year. There is a clear trend downward. The regression line is shown, but the correlation is not. This is a profound example of how regression analysis is used to understand trends.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-9 Time: 57:30 - 58:40 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a scatterplot of fertility drop vs. how Catholic people were in their voting patterns. There is a clear correlation. The scatterplot and regression line is shown, but the correlation is not discussed. The result is not surprising, but it is can serve as a reminder that one must collect data before making inferences.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-13 Time: 12:04- 13:56 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a scatterplot of the number of children women want vs. the number of children they have. The video shows both the regression line and R2. The professor compares this line with the line y = x which contains the points that corresponds to the what it would be if women had what they wanted. The difference is around 1. This is also the y-intercept. this video is a nice example of an application of regression analysis and shows an application of the y-intercept.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-14 Time: 12:10- 13:05 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a scatterplot of the use of contraception and the fertility per country. There is a clear negative correlation. The regression line is drawn, but if the correlation is shown it is too blurry to see. The relationship seems obvious, but until the data is shown it is just a guess.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-18 Time: 47:28- 49:00 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a scatterplot of the population growth rate and the growth of per capita GDP. The correlation is surprisingly zero. This is a good example of how something everyone expects to be true turns out not to be based on the data. This helped change US policy on family planning. Six minutes later the professor explains that there is a 20 year delay on the economic benefits due to the fact that the childrent cust first become working age.
Course Topic: FOIL Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-12 Time: 28:26 - 30:00 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video applies FOIL to multiply out a square of the sum of the orbital probabilities of two electron shells. He uses this to find the overlap density. The content will be pretty high level for elementary algebra students, but with some gentle explanation the students can appreciate that FOIL is used in chemistry.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 50:02 to50:53 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the standard parabolic form of a moving body with constant acceleration and the formula: t = (v - v0)/a. The professor says that he will not waste your time with the algebra and just shows the answer. It would be a good class exercise in college algebra or intermediate algebra for the students to verify the result. Since there are many letters, it would make a good group exercise.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-8 Time: 57:15 to 60:54 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the new velocity of a rocket after having expelled exhaust. There are many letters involved, but the algebra only involves FOIL and combining like terms. This demonstrates an interesting use of FOIL in the "real world" that beginning algebra students can understand.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 27:44 to 28:20 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video works out the problem of finding the square of the angular speed for an accelerating object. The professor does not show any steps. The steps will be a challenge for students since they involve FOIL and substitution, so it is recommended to have the students work in groups to come up with the steps.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-12 Time: 56:15 to 59:05 (or 60:16 if there is time) University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video multiplies two equations together using FOIL to work out Einstein's theory of relativity. The professor says "I want you to multiply the left hand side by the left hand side and the right hand side by the right hand side." The video can be paused at that point and the students will work it out using FOIL. Then continue to verify they got it correct. Then play the rest of the clip. The students will have no idea what the context is until the instructor tells them at the end that they have just done part of the derivation of Einstein's theory of relativity. These beginning algebra students will be impressed that they could do what Einstein did. Course Topic: Functions and the Vertical Line Test Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-2 Time: 44:17 - 45:14 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graphs of the temperature vs. distance from the surface of the earth. The graph is clearly not the graph of a function. This would be a great example of using the vertical line test to see if a graph is a graph of a function. The professor states that the temperature is a function of altitude. Discussion can be given why this is correct. The answer is that the graph shows the temperature as the horizontal axis and the altitude as the vertical axis, so x is a function of y. This is a nice critical thinking opportunity.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-31 Time: 9:08 - 10:22 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graph of the amount of ozone in the atmosphere vs. the altitude. The graph is clearly not the graph of a function. Similar to the example above, this would also be a great example of using the vertical line test to see if a graph is a graph of a function. What is interesting here is that there are two peaks: the beneficial ozone layer that protects us from the sun's UV rays and the harmful layer that is the smog we breath. Course Topic: Intersection of Two Lines Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-6 Time: 42:58 - 45:27 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video explains why the intersection of two lines can be used to determine the equilibrium state when there are two genes that are both advantageous at low frequencies. The professor explains why this explains gender balance, evolutionary stable processes, Nash Equilibrium, and the polymorphism of pathogen resistance genes. The algebra of solving the systems is not shown, but the graph is. This can be used to introduce solving two linear equations and two unknowns.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-3 Time: 58:30 to 59:53 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video shows the graphs that correspond to the payoff lines of a game where the two players have to choose from two selections and win according to a fixed rule set. This part of the video does not show the payoff values, but the graphs are displayed along with the equations. The professor asks the students how the intersections point is found. This would be a good time to stop the video and ask the class so see if they can use either substitution or the addition method to solve it. The equations are difficult to read, but there are two clear points on each line, so the students can get the equations by themselves.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-4 Time: 57:44 to 59:40 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses the definition of what is means to be a "best strategy" in game theory. The clip starts out with two equations and three unknowns, but with symmetry, it becomes two equations and two unknowns. The explanation of the premise comes well before, but it takes several minutes. Also calculus is used to arrive at these equations, so it would be a bad idea to show the prior minutes. It involves a synergistic profit sharing agreement. An instructor may want to just give a brief explanation of what is going on to the students including writing down the equations for the students. An instructor can give an example of the third variable "b", stop just after the professor writes down the equations and have the students find the values of the other two variables. The instructor can let the students know that the intersection point is called the Nash Equilibrium, the same Nash from the movie "A Beautiful Mind".
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-6 Time: 49:50 to 52:24 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses solves a system two linear equations to find the intersection point of two lines that correspond to best strategy of each of two players in a two player game. This gives the Nash Equilibrium and the Cournot Quantity for the optimal strategy using game theory. There are a lot of variables and parameters in the problem but the algebra is straightforward. Course Topic: Solving Two Linear Equations In Two Unknowns Video Link: http://oyc.yale.edu/physics/phys-200/lecture-3 Time: 54:43 to 58:02 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video produces and solves a linear system of two equations and two unknowns that represents the forces acting on two connected objects with two masses. The professor uses the addition method and shows every step in the process. This clip will be easy for the beginning algebra student to understand and can be used as a motivator for the addition method.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-3 Time: 62:12 to 63:07 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video is similar to the one above that produces and solves a linear system of two equations and two unknowns that represents the forces acting on two connected objects with two masses, but instead it looks at the tension of the rope that occurs when the rope is stretched by a force and there is a mass on the other side. The professor uses the addition method and shows every step in the process. The professor further explains how this will help you if you need to purchase a rope at the hardware store. This clip will also be easy for the beginning algebra student to understand and can be used as a motivator for the addition method.
Course Topic: Rational Expressions Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-5 Time: 22:29 - 25:03 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows looks at the mathematical model for genetic change in asexual haploids. The professor shows the results that involve rational expressions. It would be a great class exercise to derive the last expression. The second to the last is p/(1 - sq) where p = 1-q. The last is given by 1 - p/(1 - sq) and the result is q(1 - s)/(1 - sq). It is pretty challenging at the beginning algebra level, so it is recommended that the students work in pairs or groups to solve this.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-17 Time: 42:08 to 47:38 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the no risk price of an option in terms of the stock price, the fraction up, the fraction down, and the price of the call. The resulting expression is a rational expression in terms of these. The equation will be over many students heads but does show that rational expressions come up in finance.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-14 Time: 36:12 to 37:25 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video looks at the quantity that a first firm should produce if the first firm knows how much the competitor will produce in reaction to the first firm's decision. The professor adds the expressions for the two quantities to verify that the total will be greater than the total if the second firm did not know what the first firm would produce when the second firm made the decision for its quantity. There are a lot of letters in the expression, but the denominators only differ by a constant. The professor does not show any steps, so students can be asked to derive the solution themselves. The level would be appropriate for a beginning algebra class. Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-3 Time: 14:25 - 16:17 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video looks at the ideal gas law: PV = nRT and the air density formula. The professor says "If you follow the math ...". If an instructor writes the ideal gas low on the board, the students can be asked to "follow the math" and verify the formulas that the professor is presenting. The math is easy here, but there are several letters to deal with in the algebra.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-20 Time: 33:55 - 38:37 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video looks at the change in ocean salinity over time. The professor first looks at the change in salinity as a function of ratios of the mass of the salt and the mass of the water. Then he converts it to a function of the ratio of the masses of the water. Finally, he writes it as a ratio of expressions involving the initial and final depths of the ocean. This is a nice video that goes over each step in manipulating rational expressions and shows one of the consequences of global warming.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 17:22 to 19:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video subtracts the two rational expressions that come from the Lorenz transformations in Einstein's theory or relativity. The denominators are the same, so in principle the algebra is easy, but the equations are quite involved and students will need some explanation from their instructor. The denominator has a square root, but since the denominators are the same it can still be done in the section of rational expressions. This example shows that two people moving at different speeds will measure both length and time differently. Although the equations are complicated, with coaching students will get the gist of Einstein's theory.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 30:37 to 32:06 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video simplifies a rational expression where there is a substitution first. The example demonstrates the difference between Newtonian physics where speeds can exceed light speed and Einstein's relativity where the speed is never as fast as light. With some concentration and assistance from the instructor, students will be understand the algebra and maybe the physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23 Time: 53:09 to 55:18 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video goes over the efficiency of a Carnot engine. The last part of the derivation the professor uses the identity: (Q1 - Q2)/Q1 = 1- Q2/Q1. This is an excellent opportunity to remind the students that you can split numerators but not denominators.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 30:32 to 48:45 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the electric force produced by a dipole for a point along the line of the dipole. In the derivation, there is a difference of two rational expressions, The professor finds a common denominator and subtracts them. The subtraction involves two uses of FOIL, multiplying the minus sign through and combining like terms. This is an ideal example that will show most of the fine points of subtracting rational expressions. The FOIL part is not explicitly shown, so this would be a great exercise for the students to try and then see if their result is correct. The instructor just has to pause the video as soon as the video shows the difference.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8 Time: 10:32 to 13:27 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video first looks at a circuit with parallel resisters and then at a circuit with parallel capacitors. The physics will be far over the students heads, but the instructor can tell them that it helps to understand how anything with electronics works. The math is simple though. For each there a a single constant greatest common factor that is the same as the numerator (denominator for the second example) that cancels.
Course Topic: Complex Rational Expressions Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 32:18 to 33:18 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video starts with the complex rational expression that corresponds to the relativistic velocity equation and plugs in the speed of light for one of the velocities. The professor simplifies the expression and ends up with the speed of light again. It would be a good exercise for the students to try it themselves by pausing the video right after the professor plugs in c.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-19 Time: 30:36 to 34:05 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video defines the Doppler shift. The professor does a lot of algebra, most containing complex rational expressions. The pace is very fast, so beginning algebra students will most likely not be able to keep up with all the substitutions, but the topic is interesting and the instructor can add to it be explaining how the Doppler shift is used in astronomy.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-6 Time: 67:45 to 69:27 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video computes the capacitance between two charged spheres. The answer is a complex rational expression and the professor simplifies it without showing the work done. Students can be asked to complete the steps. The physics will be over the heads of most algebra students, but with guidance they can be made to understand at least the idea of capacitance.
Course Topic: Rational Equations Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-8 Time: 30:00 to 31:37 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the equation that relates the price, rate, and time for a discount bond. The equation is P = 100/(1 + r)T. The professor then solves for (1 + r)T to get (1 + r)T = 100/T. The professor does not show the steps involved. Student can be asked to fill in the steps to arrive at the professor's solution.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-3 Time: 41:20 to 42:13 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows the professor solving for a variable in a rational equation. The application is finding the price and quantity sold using marginal utility equations that Nash came up with. This is just a small piece of the derivation that involves more advanced math and economics. The problem is completed at 45:30. The details of the topic will be over the heads of the students, but with hand waiving they will understand that it has to do with how to price and stock goods. Video Link: http://oyc.yale.edu/economics/econ-251/lecture-3 Time: 55:40 to 60:42 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video uses the Cobb-Douglas equation to solve for the prices of two goods given that two people will spend 3/4 and 2/3 of their income on the goods. This has a nice historical framework and the examples involves fractions which will make students realize that math teachers are not just being mean when they give word problems with fractions. The problem is completely worked out.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-24 Time: 34:10 - 36:26 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video looks at Archimedes law to understand why we only see the tip of the iceberg. The professor starts with a regular algebraic equation and then turns it into a rational expression of proportions. Finally he arrives at the result that the tip of the iceberg is only 10% of the iceberg.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-7 Time: 40:20 to 43:00 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows that the circular orbit of a planet around the sun is possible. It uses Newton's laws and manipulates the resulting rational equation to solve for v2r, which tells us how fast the planet has to move given a fixed radius. The details will be over the heads of beginning algebra students, but the big picture will be clear to them. This will serve as a motivator as long as students know that they are not responsible for understanding it all. If you continue until 44:14 the professor uses another rational equation to investigate Kepler's third law.
Course Topic: Rational Inequalities Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-35 Time: 22:40 - 23:05 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video compares the strategy of being a guarder vs. being a sperm producer. There is an inequality given looks quite complex, but just multiplying both sides by the denominator simplifies it. The professor does not derive the final solution, so students can be asked to try to arrive what the professor arrived at.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-36 Time: 8:17 - 9:22 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video analyzes an inequality that measures the benefits of helping out a family member vs. not helping out. The inequality produced is: B/C > 1/r which means that the ratio of the benefit to the cost must be greater than the reciprocal of the genetic relationship. The professor just multiplies by Cr to get rid of the denominator. This is a simple example of a use of rational inequalities.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-22 Time: 16:55 - 18:22 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video discusses optimal strategy for the grim trigger strategy is in equilibrium. With the grim trigger strategy the player starts with cooperate and continue cooperating as long as the partner also cooperates, but as soon as the partner defects always defect forever. The inequality is given and the professor solves it with cross multiplication. He does not check to see if the inequality needs to switch signs when multiplied by 1-delta, but since delta is a depreciation, 1-delta has to be positive. This is a very simple inequality that does not break into cases.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-22 Time: 48:25 - 49:46 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video discusses another strategy for playing a two player game where each player can cheat or cooperate. This strategy involves cooperating if the last time either both cooperated or both cheated. This clip begins long after the explanation of the rules is given, so the instructor will need to fill the students in on what the rules are and what the equations mean. The algebra is involved, but nothing the students have not seen in elementary algebra when they are solving rational equations (since the denominator is always positive, the fractions can be cleared by just multiplying both sides by it. At around 67:00, the professor solves a similar problem that involves how much an employee's wage should be. Course Topic: Division of Exponents Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-4 Time: 17:43 - 19:05 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an easy application of dividing exponents using the subtraction rule to find how fast the earth is moving around the sun. The application comes because there is a quotient written in scientific notation. He avoids using a calculator by using basic approximations such as p/3 is about 1. For some students the answer of 30km/sec will be surprisingly fast. Course Topic: Power and Quotient Rules of Exponents Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-7 Time: 2:12 - 4:56 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an easy application of the power and quotient rules for exponents to show how much dimming there will be when a planet the size of earth passes in front of its star. The exponents arise due to scientific notation. The clip makes use of the area of a circle formula which squares the radius. The radius is given in scientific notation, so squaring requires multiplying the exponent by 2. The derivation demonstrates how much harder it is to see earth compared to a Jupiter sized planet at the same distance from its star. The professor explains that this is the reason why we cannot use the transit method to find an earth sized planet if all we have are earth based telescopes. Instead we must use telescopes in orbit.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-19 Time: 46:00 - 49:00 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an easy application of the power and quotient rules for exponents to give an estimate on the density of the universe. The exponents arise due to scientific notation. He uses 200 as an approximation of 63. In the end the finding is that the density is one-third the critical density. At the end the professor states that there is more to it, so it is important to let the students know that this is not the final result of the density since it does not take into account dark matter and energy.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23 Time: 34:59 to 37:36 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the power and sum rules of exponents to transform an equation that involves temperature and volume into on that involves pressure and volume. The professor asks his students if they know what he did. The instructor can pause the video and see if anyone from the class knows what happened.
Course Topic: Formulas With Square Roots Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-2 Time: 3:48 - 7:34 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the escape velocity formula in order to understand why our atmosphere stays attached to earth. The explanation is very clear and only uses gentle mathematics, so elementary algebra students will be able to understand this completely.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 32:27 to 35:06 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows a relationship between space-time in one frame of reference and that of another. The professor starts with two radical expressions and shows that taking the squared difference is invariant. This is a bit complicated, but demonstrates an interesting physical fact in the special theory of relativity.
Course Topic: Solving by Taking the Square Root of Both Sides Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-1 Time: 44:00 - 45:50 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an easy application for solving by taking the square root of both sides to find the orbital period of Jupiter given how far it is from the sun. The professor uses number sense by saying that the square root of 125 is close to 11. This is a nice application that avoids using a calculator and can be used to motivate solving problems by taking roots.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 64:05 to 66:08 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the equation E2 = p2 + m2 and explains that the plus or minus cannot be ignored. The professor explains that the negative energy solution corresponded with anti-particles and started a whole new field of physics. This is a simple example of algebra being used and will help students remember the plus or minus when taking the square root of both sides of an equation.
Course Topic: Cube Roots Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-4 Time: 15:47 - 17:05 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of the cube root to the size of a molecule. The idea is that when converting from volume to length, you take the cube root of the number of molecules per cubic cm to convert to the number of molecules per cm. This gives a real world application of cube roots and gives them a sense of perspective when talking about volume vs. length. Course Topic: Proportions Video Link: https://www.youtube.com/watch?v=9EXFqf6XgTA&list=PL2CD836B66D3CEBED Time: 22:20 - 26:10 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: Alan Shabel Teaching Ideas: This video shows an application of
proportions to estimating the number of individuals in a population. The
basic algebra is explicitly used. The application involves tagging a fixed
number of individuals such as birds, releasing them and then capturing a new
collection to find out the proportion that have the tag. Then the total
population is estimated using the formula:
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-2 Time: 9:15 - 11:30 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of proportions to related the binding energies of Coulombic, magnetic, strong binding, gravitational, and chemical bonds. Has the a "proportional to" or "varies as" symbol for the first, second and fourth of these. The student will have to be explicitly shown what the symbol means. An instructor can freeze the screen at the end and have the students write in words what each of these are. It is recommended to do this in pairs or groups since not all students will know that m stands for mass and r stands for the distance between the two. Furthermore, the instructor will most likely have to tell the students that q stands for charge.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-3 Time: 23:46 - 28:05 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of both "proportional to" and "inversely proportional to" to the idea of electrostatic force and how it relates to the distance from the charge. The professor goes over the difference between the 2D model and the 3D model and why in the 3D the force is inversely proportional to the square of the radius. Asking students to write down in words what that equations represent would be an effective lesson for the students. The professor states it is words, but the students may not be listening.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-6 Time: 11:59 - 14:30 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video first shows the relationship between the total power power emitted from an object goes like the fourth power of the temperature. Immediately after this he states that the wavelength that is emitted the most goes inversely as the temperature. This is very clearly stated in the video and students will easily see how the idea of direct variation and inverse variation are used. Then in the next few minutes the professor demonstrates this with a light bulb experiment.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-20 Time: 18:28 to 20:45 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains how to find the density of a fluid using a U-tube. The video begins with the formula for density and does some very basic algebra and comes up with a proportionality equation. An instructor can follow up with asking students to find the density of oil if the instructor sets up the problem in advance.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-20 Time: 20:49 to 26:31 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains how hydraulic lift works. The professor sets up the problem and writes down the equations to end up with a ratio problem. This example is something all students will understand but most did not know how the lift worked before the math was done. It is an easy but effective way of showing them an application of ratios.
Course Topic: Solving for a Variable Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-4 Time: 20:00 - 23:44 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of solving for a variable in order to find out how fast a star will move due to the force of an earth sized planet. In the example the subtraction rule for division of exponents is also used. The professor uses approximations such as 3x3 = 10 in order to avoid using a calculator. By starting at the suggested time, students will get a hint of the calculation for Jupiter and then will see the full derivation for the earth. The punch line is that we cannot detect earth sized planets with the wobble technique.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-18 Time: 32:30- 35:40 (or 39:24 to see the full unit conversion process) University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of solving for a variable in order to calculate the age of the universe. The professor begins with a simple example of finding when a person started driving given the speed and distance using a variation of the d=rt equation. Then he uses Hubble's law to calculate the age of the universe which is 1/H given the assumption that the expansion has been constant throughout time. He spend pretty much work in changing the units and ends up with the final answer of 17 billion years. Since the universe is 13.82 billion years old this demonstrates that the expansion of the universe has not been constant throughout time. This is a very simple example of solving for a variable.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-6 Time: 26:37 - 28:24 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video solves the equation that relates the albedo to the temperature. The professor solves this equation for temperature. The algebra involves dividing both sides by 4p and taking a 1/4 root of both sides. The explanation is very clear and demonstrates how math is used to understand how the temperature is related to the albedo. The albedo is basically how much of the sun's rays get reflected back into space. This demonstrates that when the polar ice melts, the temperature of the earth increases causing more ice to melt. This is a feedback loop that will exacerbate global warming. This is a very relevant example of the use of algebra to help us understand global warming.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-20 Time: 28:00 - 30:24 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video looks at the effect of wind to the temperature and heat in the ocean. He begins with an equation that relates heat to the mass, heat capacity and change in temperature and then he solves for the change in temperature. The algebra he does can be understood by any beginning algebra student, but there are many letters in the equation.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 49:35 to 50:25 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains that instead of using a clock, we can use a speedometer to measure time by knowing that the velocity of a falling body is linear. The professor begins with the linear equation: v(t) = v0 + at and solves for t. This can be used in beginning algebra sine there derivation is very simple, but the result is profound. Be sure to begin the video just after the mention of taking the derivative.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-3 Time: 42:24 to 43:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses Newton's law: F = ma, and the law of gravity F = -mg, sets them equal to each other, divides by m to get the fundamental law of gravity that a = -g. This tells us that the acceleration of gravity is a constant. This is a nice application of linear equations that looks different from the typical 10x = 30, but is actually the same. This will confuse beginning algebra students at first, but after some explanation they will have learned an important point of algebra: letters are numbers.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-4 Time: 63:30 to 64:22 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains the physics behind the loop the loop rollercoaster. The professor solves for the force that the track exerts in terms of the radius, the mass, the velocity and the acceleration of gravity. This is an example that will explain rollercoasters to the students while at the same time demonstrating basic algebra skills. The instructor can stop the video at 64:57 to see just the derivation and not the full interpretation. At this point the students can be asked to fill in the details of the steps that the professor skipped. It is also interesting to note that if the equality holds, then it is in orbit.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-10 Time: 38:41 to 40:12 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the velocity of a wheel at the bottom of a hill. The algebra is easy, but there are a lot of letters in the equation and the physics may not be something that the students know. Nevertheless, it can be an interesting example that students can relate to especially if the instructor explains that dropping a ball from a building results in a faster final velocity then rolling it down a hill of the dame height.
Course Topic: Multiplying Through Video Link: https://www.youtube.com/watch?v=CNr_7gPhYtY&list=PL2CD836B66D3CEBED&index=17 Time: 35:15 - 35:54 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: John Huelsenbeck Teaching Ideas: This video uses substitution and multiplying through to prove that the frequency of an allele of the next generation equals the frequency of the current generation just from random mating. This is a short and easy to follow clip of a nice application of beginning algebra. Course Topic: Squaring a Binomial Video Link: https://www.youtube.com/watch?v=CNr_7gPhYtY&list=PL2CD836B66D3CEBED&index=17 Time: 31:15 - 32:15 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: John Huelsenbeck Teaching Ideas: This video squares the binomial: (p + q)2 in order to compute the frequencies of the offspring when p and q are the proportions of two genes in a population. This demonstrates how algebra is used in conjunctions with the Hardy-Weinberg rules.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-2 Time: 38:46 - 41:00 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This is another video squares the binomial: (p + q)2 in order to compute the frequencies of the offspring when p and q are the proportions of two genes in a population. This demonstrates how algebra is used in conjunctions with the Hardy-Weinberg rules and emphasizes that after the second generation, the population does not change.
Course Topic: Solving by Squaring Both Sides Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-8 Time: 12:12 - 14:20 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application for solving by squaring both sides to find out how massive something must be to become a black hole. The professor goes through the derivation quickly without writing down the steps. After viewing this clip the students can be asked to fill in the details by explicitly showing all of the steps.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-18 Time: 45:15- 50:00 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of square root equations to decide whether our universe is a "Big Crunch" universe of a "Cold Dark" universe. The professor shows the equation that involves a square root and solves for a variable by squaring both sides. There are many letters in the equation, but the professor does a good job making it not so daunting.
Course Topic: Linear Inequalities Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-12 Time: 14:17 - 15:30 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video models the ability for a mutation to take hold based on a linear inequality. The example is general and does not have specific numbers, but can be used as an introduction to where linear inequalities are used. The variables in the example are Dm and Df which are the change in male and female fitness (spreading their DNA), so beginning algebra students will need to be reminded that these can be replaced with x and y. It also uses m and f as parameters, so an instructor can put example numbers and have have the students try to graph the resulting inequality.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 8:35 to 12:42 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video goes over the grandmother paradox in relativity. There is a pretty complex looking inequality that the professor works with and ends up with an inequality involving Δx and Δt and c. If the instructor shows the students that this can be looked at as the same as y > cx, then it demonstrates the area of space time that we can travel in. This is a fascinating example that will interest students and if done right can be shown to beginning algebra students.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 15:22 to 20:05 (or later or earlier depending on time) University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video goes over the same example as the one above, but in a graphical way. The professor draws the axes and talks about what points are allowed so that we cannot change the space-time past. If time permits, this example and the previous one can be shown. Otherwise either could work with the instructor's additional explanation.
Course Topic: Quadratic Formula Video Link: http://oyc.yale.edu/economics/econ-251/lecture-12 Time: 45:23 to 49:37 (or 45:50 if you just want to show the quadratic formula used) University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video uses the quadratic formula to solve an equation that becomes a quadratic. The premise is that you want to find the prices of apples and consumption of them per generation.. Only the math is done in this clip, so the instructor will have to watch the part before an after this clip so that the students can be shown what the equation refers to.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 61:10 to 64:04 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the standard question, "When will it hit the ground?" that is asked about position-velocity-acceleration for an object that is moving with constant acceleration. The professor uses the quadratic formula to solve it. He makes a big point about the fact that there are two answers one that is a positive time and one that is a negative time. He explains that the one that is positive is the answer to the questions, but the negative solution is also interesting.
Course Topic: Parabolas Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-11 Time: 25:30 - 26:59 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows that if you look at the number of eggs hatched in coordination with the probability of survival for that clutch size, then the number eggs laid vs. the number of successful eggs is a parabola. The vertex of the parabola is the expected number of eggs hatched. The professor uses calculus, but intermediate algebra or college algebra can used to find the vertex using the vertex formula. For a tongue and PowerPoint lesson inspired by this application click here.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 53:02 to55:44 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the standard parabolic form of a moving body with constant acceleration and shows a specific example y(t) = 15 + 10t - 5t2. The graph is shown in general. The professor makes a big point about the mathematical domain vs. the domain that comes from real world considerations. This is a very simple example of the parabola used in physics and can be shown when one first introduces the parabola.
Course Topic: The Cubic Function Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-34 Time: 11:00 - 13:12 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graph of the wind shear and wind power vs. altitude. the professor explains that the fact that the cubic nature of the graph is the reason that newer wind turbines are so tall. He also explains that if the wind gets too strong, the turbines can't handle the force so there is a best altitude.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-34 Time: 22:56 - 25:35 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video derives the formula for the total amount of energy that hits a wind turbine. He uses the formula for kinetic energy to calculate the wind power density. This is a middle level derivation that uses some physics, but the math is quite easy. Students who get excited about renewable energy will apreciate that there is a use for the cubic function. Course Topic: Understanding the Domain of the Square Root Function Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-9 Time: 40:22 - 42:50 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of finding the domain of a square root function to Einstein's theory of relativity. The professor shows a square root function that involves a quotient inside a root. Then finds out that when v approaches c (the velocity approaches the speed of light), the mass approaches infinity. Although the professor does not talk about it, it will be a learning opportunity to ask the students why the velocity cannot be faster than the speed of light. The answer would involve taking a square root of a negative number meaning that the mass is not even defined. Course Topic: Piecewise Defined Functions Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-11 Time: 29:15 to 31:54 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the "Weighting Function" which models psychological reactions to probability. In particular, people tend to round values close to 0 to 0, but when it reaches a threshold value, the overestimate the value. A similar phenomenon occurs for values close to 1. The professor explains the function and then sketches a representative graph. He does not write down the equation. It would be a good exercise for students to try to write down an equation for the Weighting Function. It is open ended, so it would be a good group activity to explore the concept of piecewise defined functions.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-17 Time: 24:55 to 27:10 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the call price of an option on the expiration date vs. the price of the stock. The professor explains the function and then sketches a representative graph. He does not write down the equation. The graph is on the x-axis until x is equal to the exercise price (out of the money) and moves up at 45 degrees after (in the money). Students can be asked to write down the corresponding expression for this piecewise function.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-7 Time: 7:50 to 8:48 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses looks at the demand function for a product when there is a single competitor. This function is a piecewise function that depends on whether the price is below, above or equal to the competitor's price giving a piecewise linear function. An instructor can provide the prices and have the students graph the function.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-4 Time: 34:38 to 35:36 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the equation that gives the electric field strength as a function of r for a constant density charge inside a sphere. The equation is a piecewise defined function with the two pieces being inside and outside the sphere. The professor explains that the same equations hold for gravity. This is a nice example of a piecewise defined function that most students will understand, especially for gravity.
Course Topic: Exponential Growth Video Link: https://www.youtube.com/watch?v=2z0nrWrcHSA&list=PL2CD836B66D3CEBED Time: 19:23 - 22:02 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: Alan Shabel Teaching Ideas: This video shows the first half of the derivation of the exponential growth model. It begins with birth and death rates and arrives at Delta N / Delta T = rN. It does not solve the differential equation but an instructor can easily show the class the final N = erT equation whether it is just to give it to them in an intermediate algebra class or pre-calculus class or whether the full derivation is shown in a calculus class.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-31 Time: 25:25 - 26:26 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video uses an exponential equation (K = 10-3/4DE) to approximate the equilibrium constant given the change in energy. He goes over the simple example when the change in energy is 4, dropping the equilibrium constant by a factor of 1000. This is a nice application that makes the exponential growth equation simple. It starts with a exponential with base e and then he approximates the base e equation with the base 10 equation to see that they are just off by a constant multiple of the exponent.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-36 Time: 43:45 - 46:25 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows uses several rules of exponents to derive the formula for the difference in entropy between gauch- and anti- butane. The derivation uses both the product rule and the rule that eln2 = 2. Since it would take a bit of explaining for the students to understand what entropy actually is, it is recommended to just hand waive through this part and not worry if the students really understand entropy. Instead focus on the use of the rules of exponents.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-8 Time: 37:27 to 38:46 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the equation that that gives continuous compound interest. The professor presents the equation Balance = ert, but does not give a numerical example. This can be used as a quick introduction to e and exponential growth. He uses 1 and the principle.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-10 Time: 36:37 to 37:59 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the equation for the mortgage balance in terms of the monthly payment, the rate and the time. The professor writes down the standard formula given that the payments are monthly and the interest is monthly. This is a standard example of an exponential equation.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-7 Time: 46:57 to 48:16 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video uses the annuity formula to solve the question, "At 6% interest what is a $12, 24 year annuity worth?" The professor uses the formula C/i [1 - 1/(1+i)T]. Although the formula looks difficult, the example is quite easy and can be given to an intermediated class. This is the same calculation as mortgages, except that we use a monthly interest rate and T is measured in months.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-3 Time: 46:40 - 51:34 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graphs and equations for pressure and density vs. altitude which both follow an exponential curve with negative growth constant. The graph that the professor shows has the dependent variable as the horizontal axis and the independent variable as the vertical axis, so that may need to be explained to the students. At the end, the professor uses the equations to show why airplanes must be pressurized. The atmosphere at the altitude of the typical flight level is only a quarter as dense as it is on the ground. This is a very simple example of exponential decay that has an application that most students can relate to.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-30 Time: 34:24 - 37:57 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video explains what exponential population growth is. The professor explains it in detail without doing any real mathematics. He arrives at the fact that if we stay at 1% per year growth rate then in 100 years, the population will increases by a factor of about 2.7 (e). This is a very basic explanation that could be used in an intermediate algebra class since it is too low level for anything more advanced. Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-7 Time: 64:56 - 66:14 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video shoes very rough sketches of he linear curve that Malthus believed that modeled the growth of agriculture and the exponential curve that models population growth. The professor explains that in any case, starvation will eventually result. This is a nice application of comparing linear vs. exponential growth models.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-7 Time: 57:52 to 61:27 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video sketches the graph of an exponential decay curve that is the graph of the charge as a function of time. The professor explains that this is why it is very dangerous to play with any electrical device even after it has been unplugged. He also goes over how a camera flash works. On a personal note, the author of this webpage had a colleague die while disconnecting a high power communication circuit at AT&T even though he had shut the power off before putting his screwdriver on the screw that was attached to the circuit.
Course Topic: Logs Video Link: https://www.youtube.com/watch?v=MdYzePZUSqk&list=PL2CD836B66D3CEBED&index=6 Time: 23:03 - 23:53 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: Alan Shabel Teaching Ideas: This video gives a log equation called the Shannon Diversity Index Equation. It is a nice example of logs used in biology and can be shown in an intermediate algebra, pre-calculus class, or calculus class when introducing logs.
Video Link: https://www.youtube.com/watch?v=-frfAZoaqDw&list=PL2Q_sOQgsm24ybtnVq75-TgZd86lMjm9m&index=8 Time: 58:00 - 1:00:46 (or 1:02:40 to see the sorting competition) University: Harvard Course: Computer Science 50 (Introduction to Computer Science) Professor Name: David Malan Teaching Ideas: This video looks at the run time for various sorting algorithms. It emphasizes that log2(n) < n for large n. The first two minutes derives the formula and the second two minutes shows it in action treating it like a competition.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-16 Time: 29:05- 34:55 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of logarithms to define the brightness scale of stars. This compares the magnitude of an object to the brightness of the object. After the minute or two of this definition, the professor goes over the rules of logs and then spends a couple of minutes preaching about the power of understanding logarithms. This is an outstanding video to show to an intermediate algebra class while to take away some of the skepticism from the students.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-16 Time: 35:00- 36:57 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of logarithms to find the apparent magnitude of the start Sirius. This takes off where the prior example left off. It does not use any of the properties of logarithms, so it can be shown when first introducing logarithms; however, it does use log x. The students will need to be told that log x means log10x. The professor uses the fact that 101/2 is approximately equal to 3. He humorously states that 3 is both p and the square root of 10.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-16 Time: 39:29- 43:01 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of logarithms to find the absolute magnitude of the star Sirius. This continues the discussion of the prior two videos. This application will serve as a reminder of negative exponents. It also reminds the students that log 1 = 0.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-16 Time: 45:09- 47:47 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of logarithms to use the standard candle method to find how far away a star is. This is a great example of solving a simple log equation that involves converting the log equation into an exponential to find the value of x that is originally inside of the logarithm.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-36 Time: 6:00 - 7:10 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video takes and exponential equation and takes ln of both sides to show that the minus the heat of formation of the carbon atom is the slope of the plot of the pressure of carbon atoms vs. 1/T, where T is the temperature. The graph of this line is shown at 12:50. The lecture is pretty high level and will be over the heads of intermediate algebra students, but may be of interest to college algebra students. They will have to be told what the instructor is doing chemically.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-36 Time: 30:00 - 30:34 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows Boltzman's key equation: S = k log w which relates the enthalpy S of an ideal gas to W which is the number of microstates corresponding to a given microstate. In other words, it relates the disorder of the collection of gas molecules to the number of configurations they can be in. If the students only see these 34 seconds, then they will see the equation in a historical framework with the equation on his tombstone. Their instructor can hand waive through the equation just giving students hints of what it is about.
Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-6 Time: 8:05 - 9:36 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video defines pH using the log function: pH = pKa - log [B]/[HB] The professor doesn't do much math here other then to talk about what happens if solution is half ionized then the ration will be 1 and log(1) = 0. This can work as a quick introduction to logarithms.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-17 Time: 51:20 to 52:41 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the Black-Scholes formula that gives the price of an option . The equation will be over all the students heads but does show that the ln x and exponential functions are used in advanced finance. The instructor should worn the student that they won't understand the equation, but can give a hand waiving explanation that the cost of an option is complicated.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-21 Time: 64:45 to 68:50 (can stop earlier) University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the probability of financial ruin for a dealer in playing a game repeatedly. This function is [(1 - p)/p]S where p is the probability of winning and S is the amount that the dealer starts with given that the bet is for $1. The professor notes that at p=1/2 the dealer will always go broke eventually. Students can be asked why. Then gives an example with p > 1/2. There are plenty of questions that students can be asked such as when will there be a 20% chance of going bust if the dealer begins with $5. Students must know how to take a root of both sides to answer this last question.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-1 Time: 12:50 to 14:01 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video presents the inflation adjusted DOW Jones over time on a log scale. The professor states "going up two of these is multiplying by 10". Students can be asked, "Why did the professor use a log scale?" This is an open ended question that can result in many good answers.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-12 Time: 9:50 to 11:13 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video writes down the utility function based on what people consume when they are young and when they are old. This is written as an expression that has logarithms in it. Much later, the professor shows how this is used to understand Social Security, but the instructor will need to explain how this works.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-19 Time: 0:54- 7:29 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video shows a log scale of number of children vs. per capita income over two time periods. Both log regression lines are provided. This is an interesting use of logs to explain an important fact about fertility.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-19 Time: 26:10 to 27:44 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video displays the formula for the intensity of sound which involves the common logarithm. The professor calculates the intensity for a simple example. This is a simple example that can be shown to students who are first introduced to the common logarithm. If you continue until 29:22, there is a nice explanation on how the decibel scale means that increasing by 1 increases the intensity by a factor of 10.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-24 Time: 55:50 to 56:48 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the entropy change that is experienced when a gas doubles in volume. The professor uses the power rule for logs. The students probably won't understand the physics that is being discussed so the instructor will have to provide some thoughts about entropy.
Course Topic: Solving Log Equations Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23 Time: 33:49 to 35:26 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the power, product, inverse and x-intercept properties of logarithms to solve an equation explaining how the temperature falls as the volume of a gas expands. This is a nice example to give when discussing log equations since it uses so many properties.
Course Topic: The Ellipse Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-12 Time: 44:00 - 45:50 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application of the ellipse to understanding the general theory of relativity. Before Einstein, the astronomers did not know why Mercury's orbit defied Kepler and Newton. This clip shows the orbit of Mercury and defines the perihelion distance and the precession of the perihelion. This can be used as a motivator of why the equation of the ellipse that includes the focus can be useful.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-7 Time: 12:48 to 14:27 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video presents Kepler's first law of planetary motion. The professor shows how to construct an ellipse using a string fixed to two points and moving the strong around the curve. He also shows the major and minor axes of the ellipse. This will serve as a nice introduction to the section on ellipses for an intermediate algebra or college algebra course.
Course Topic: The Hyperbola Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-4 Time: 42:57 to 47:35 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video plots the standard deviation (risk) vs. return on investment for various strategies on distributing a portfolio between bonds and stocks. The professor clearly shows a hyperbola, but it may not be clear why the hyperbola happens. Students will need to trust the speaker based on the fact that he won the Nobel prize. This is a very real application of the hyperbola unlike just about every other application that is found in textbooks. The next two minute of the video (47:35 to 49:30) gives a simple explanation of strategizing investments and why investing in only bonds is always a bad choice.
Course Topic: Sequences Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-18 Time: 15:00 - 16:01 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video sequences that arises from looking at chains of butadiene molecules. The resulting sequences are: N, 1/root(N), 1/N, N-1, and (N-1)/N. The professor notes that the last sequence representing the total overlap stabilization approaches 1 in the limit. This can be used as an introduction to sequences.
Course Topic: Series Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-2 Time: 19:33 to 24:19 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video gives the formal definition of variance and covariance. The professor gives it in the context of finance. No numerical examples are given, but he does give an example of return on investment over 10 years for the variance and the comparison of IBM and GM for the covariance. These examples are shown without providing the details. This can be used in an algebra class to introduce why sums are important.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-8 Time: 39:06 to 40:23 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the equation for the Present Discounted Value when payments are coming in annually and every six months. If the payments are the same each month then this is a geometric series. Otherwise they are not. This could be used to introduce infinite series in an algebra class. Course Topic: Geometric Series Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-11 Time: 35:19 - 37:00 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video compares the number of offspring an animal has if it has a possibly infinite lifespan vs. a finite lifespan where every day it has a 20% chance of death and each day it is alive it lays 10 eggs. The first scenario is an infinite geometric series and the second is a finite geometric series. The professor shows all the math for the infinite geometric series, but the math is not done for the finite geometric series. Students can be asked to fill in the details and derive the answer for the finite case. This would not be that difficult for them if they have the formula.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-12 Time: 62:00 to 62:30 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video looks at a formula that describes the price sequence over time which is p + p2 + p3 + ..., a geometric series. The point is that in solving for p, there is a quadratic formula used and the professor choses the smaller of the two root to ensure that p < 1. This is a good video to show to emphasize that the infinite geometric series diverges when |p| > 1.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-17 Time: 45:00 to 49:38 (or shorter) University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video looks at what a homeowner must pay in order to pay off a mortgage early. The geometric series is given, but the professor does not show the calculations. Students can be asked to verify the value that the professor gets. This would be a good exercise in using the finite geometric series formula for a real life application.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-21 Time: 65:55 to 67:01 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video presents the value of a 30 year 9% bond using a finite geometric series. He does not actual calculate it, but it would make a good exercise to have the students make this calculation.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-17 Time: 42:42 - 47:16 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video presents presents the optimal strategy for a 10 stage two player game where at each stage the players take turns offering to give $a to the other player where a is determined by the player making the offer. The games starts with $1 and at each stage the value of the money depreciates by a factor of delta. The solution turns out to be a geometric series which the professor evaluates. Students need to hear from their instructor the rules to the game which are given much earlier in the lecture. this is an interesting application that is not too difficult for intermediate algebra students or college algebra students.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-17 Time: 51:21 - 52:55 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video continues basically where the video above it left off, but taking the limit as the number of terms goes to infinity to get the infinite geometric series. The professor does a nice job explaining why the finite geometric series converges to the infinite geometric series. This in combination with the one above will make it so the instructor does not need to derive the two formulas.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-22 Time: 8:28 - 15:03 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video discusses optimal strategy on whether to cooperate or defect in a two player game. With the grim trigger strategy the player starts with cooperate and continue cooperating as long as the partner also cooperates, but as soon as the partner defects always defect forever. The value of cooperating forever is an infinite geometric series. The professor derives the infinite geometric series formula starting at 13:20. So save time the first minute of the video can be played and then jump to 13:20 and watch through to
Course Topic: Pascal's Triangle Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-23 Time: 21:50 - 26:20 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video Pascal's Triangle to look at the number of spin configurations of the protons of an organic compound. this is a clever use of Pascal's triangle, but the content is a bit difficult for the students to understand. An instructor either will have to spend a lot of time explaining or will have to just tell the students not to worry if they do not understand the chemistry. Course Topic: Mathematical Induction Video Link: http://oyc.yale.edu/physics/phys-200/lecture-24 Time: 63:20 to 65:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video first considers the number of arrangements of 100 coin tosses with 0, 1, and 2 heads. For 2 heads, the professor uses the formula for combinations. Then the professor explains that this relates to gas and entropy. The math is not difficult to follow, but the students may need some explanation of the context of Boltzmann's formula.
Course Topic: Mathematical Induction Video Link: http://oyc.yale.edu/economics/econ-159/lecture-15 Time: 1:32 - 4:42 for the statement and 12:23 - 31:25 for the full proof University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video presents Zermelo's Theorem. Which states that in a sequential game where there is a win, loss or tie then either player 1 can either force a win or a tie or player 2 can force a win. This works for checkers or chess. The first three minutes has the statement of the theorem and the second clip is very long but rigorously goes through the steps for mathematical induction. It would be a good idea to first show the students the first clip and then have them work in groups to see if they can provide a mathematical induction argument for the proof.
Course Topic: Radians Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 10:59 to 13:50 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video gives the definition of radians. The professor first finds the length of an arc at θ degrees and argues that the radian is the natural measure of an angle. It is a very simple explanation that all students will understand.
Course Topic: Using the Trigonometric Functions Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-13 Time: 28:30 - 31:53 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video goes over the Coriolis Force which is a formula that involves the sin of the latitude. This also refers to the force vector and gives the direction. Although no math is done with this formula, but this is an interesting use of the sin function and the Coriolis effect is incredibly important in meteorology but not that well understood by the general public. The professor does go over that since sin 0 = 0 the equator has no Coriolis force and the maximum Coriolis force occurs at the poles since the sin is a maximum there.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-17 Time: 40:32 - 42:26 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video explain why the seasons occur. The professor explains that the cos of the solar zenith angle determines what season it is. this is an east to understand use of the cos function to explain something so basic yet something that most do not know the mathematical reason.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-4 Time: 41:50 to 44:14 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video considers the physics problem of a block moving down an inclined plane where friction is not ignored. The question asks how steep an angle before it slides down. The professor's derivation involves dividing sin by cos to get tan. This is an effective video to play on the day that tan introduced in a trig class.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-4 Time: 57:34 to 60:06 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the physics application of a mass spinning around a circle when the mass is attached to a string that is held fixed at the top. The professor derives the acceleration and uses the definition of the tan function. This is an easy to follow example that student should be able to relate to.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-4 Time: 60:19 to 62:26 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at another physics application that asks what angle should a road bank to keep the car from slipping. This is a very relevant example and makes use of sin, cos, and tan. It can be shown when the students are first learning about the trig functions. He explains it further through 63:17.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-11 Time: 31:33 to 35:56 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the force that a wall exerts on a ladder that is leaning on a wall with a given angle. The derivation is lengthy, but it does use the cot function and complementary angles.
Course Topic: SOHCAHTOA Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-2 Time: 35:35 - 36:55 and 49:00 - 51:08 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video uses right triangle trigonometry to demonstrate how the sin x can be used to find the difference from a planet to its star. It is a very simple explanation. Afterwards he uses the approximation sin x is close to x. Then at 49:00, he uses it to show why "The idea of looking up and seeing a (exo) planet is going to fail".
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-4 Time: 31:35 to 38:24 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the classic physics problem of a mass sliding down an inclined plane. It makes extensive use of the trig functions sin and cos which come from looking at the diagram with the xy-plane positioned so that the x-axis corresponds with the incline. The derivation may be difficult for the students at the level of a trig class, but the professor does an excellent job explaining each part. This is a classic example of where math is needed to solve science problems. The clip is a bit long, but may be worth it due to its importance in physics and its heavy reliance on trig.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 56:00 to 57:47 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video calculates the work done to take a pendulum that begins vertical and bring it to an angle θ0. The professor accomplishes this by looking at the change in energy of the system and makes use of triangle trigonometry. This is a solid example of using the cos function in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-19 Time: 57:13 to 60:52 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the problem of where to set the boat in the water when there are two holes in the breakwater. It is a great application of triangle trigonometry and the double split experiment and most students will be able to follow.
Course Topic: Using Arc Trigonometric Functions to Solve for an Angle Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-4 Time: 18:32 - 23:42 (Skip 19:55 to 21:40 as the professor fumbles with the experiment) University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows how an inverse trigonometric function is used to measure lengths at the molecular scale. The professor describes how Newtons used "Newton's Rings" to measure something the size of 30 water molecules. This a a nice application of using sides of a right triangle to find the angle using the cos-1 function.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-14 Time: 3:30 - 6:37 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video uses a cos-1 function to compute the bond angle for hybridization structures. The professor pauses and has the students figure out the answers. This can be used in a similar way where the trigonometry class is prompted to come up with the bond angle for each. Course Topic: The Graph of the sin (or cos) Function Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-4 Time: 35:00 - 37:00 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows how understanding the period and amplitude of the sin function is used to detect and characterize exoplanets. The professor describes how the doppler effect is seen as a sin wave when the hot Jupiter orbits around its star causing the star to wobble in a sin wave. This can be shown when first going over the graph of y = sin x in a trig class.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-5 Time: 8:37 - 9:28 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video show the sense at which light is a wave. It presents both the graphs of both the time and the position vs. force on a charge that will make it accelerate. The professor uses a nice animation to show the second graph. This can be used when presenting the graphs of y = sin x and y = cos x for the first time to students.
Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-20 Time: 5:05 - 7:00 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video shows the way a the electron's time dependent wave function works when there is a 1s and 2p interaction. This motion is a sin wave. In particular the professor explains how the frequency determines the rate that the electron goes up and down.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-1 Time: 12:40 - 14:33 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video presents the graphs over time of the predicted tide due to the moon and the sun, the actual tide, and the difference. The predicted is clearly a sin wave. The professor states that there are two high tides each day. An instructor can ask the students what the frequency and period are based on this fact. Course Topic: Trigonometric Identities Video Link: http://oyc.yale.edu/physics/phys-200/lecture-19 Time: 41:10 to 46:45 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the sum to product formula to investigate constructive and destructive interference of waves. The professor explains the physics and manipulates the equation involving a sum of two cos functions. This can be used to motivate the sum to product rule in a trig class.
Course Topic: Parametric Equations Time: 33:15 - 40:36 University: MIT Course: The Early Universe Professor Name: Alan Guth Teaching Ideas: This video shows how a cycloid models the Big Crunch theory of the universe. The Big Crunch theory uses the forumal y = 1 - cos(θ) which shows that at the beginning the universe had no volume and after one period, we will be back to no volume (The Big Crunch). It turns out that the most recent evidence shows that the Big Crunch theory is incorrect and in fact we will have an accelerating universe finishing off with a cold dark lonely universe instead.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 59:11 to 63:11 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the problem of finding the angle one should throw a projectile in order to maximize the horizontal distance that it travels. The professor very clearly goes through the derivation and even uses the double angle formula for sin: sin(2x) = 2sin(x)cos(x). This clip can be used in a trigonometry class as it serves as motivation for parametric equations and using trig formulas.
Course Topic: Vectors Video Link: http://oyc.yale.edu/economics/econ-251/lecture-2 Time: 65:15 to 66:06 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video explains using addition of vectors that the sum of the endowments of a product x is equal to the sum of the endowments of a product y. This is a little difficult to follow, but with some explanation from the instructor the students can see why the vectors model the situation.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 8:05 to 10:43 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains what the definition of a vector is in the context of the position of a hiker and then in general two dimensional space. The explanation is very clear and can be used as the direct definition of a vector in math class for a vector described as a direction and a magnitude. The example is just a sketch without numbers shown.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 10:51 to 12:31 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the graphical definition of addition of two vectors in two dimensional space. The professor draws the vectors tip to tail to first define addition of vectors. Then he shows the commutative law by drawing the standard parallelogram. This is very "mathy" but it does show that math definitions are directly given in other courses. This video can be used to reinforce what is being done in a math class when introducing vectors.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 16:15 to 17:42 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video proves that the magnitude-direction definition of a vector can always be restated as x and y coordinates. The professor shows this just as a math instructor would, so the clip would help validate what is done in class as vectors are introduced for the first time.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 17:42 to 19:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video begins where the above one left off. It uses trigonometry to derive the formulas that relate the magnitude and angle of a two dimensional vector to its x and y coordinates. It is very easy to follow, but does not give an example. It can substitute for the proof that a math instructor will give in this section.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 55:50 to 58:38 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video generically solves the problem that asks if a car goes over a cliff, when will it hit the ground. It uses the standard vector valued function. No numbers are given so this is more of a derivation than an example. The professor does not complete the last steps, so it would be a good idea to as the students to fill in the details at the end.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 59:16 to 58:38 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video generically solves the problem that asks if a car goes over a cliff, when will it hit the ground. It uses the standard vector valued function. No numbers are given so this is more of a derivation than an example. The professor does not complete the last steps, so it would be a good idea to as the students to fill in the details at the end.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 20:12 to 21:13 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the definition of the magnitude of a vector to derive the kinetic energy formula in two dimensions. It is clearly stated and can be shown at the pre-calculus level without issue.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 24:43 to 27:26 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video introduces the concept of four dimensional vectors that correspond to points in the space time continuum. The professor states that numerical subscripts work better then changing letters since with many you will run out of letters. This is a nice way of convincing students that the n-vector has use. If you have extra time you can continue for another few minutes and the professor will mention the 10-vector in string theory.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 13:22 to 14:15 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at two charges in space and considers the displacement vector from the first charge to the second. The professor shows why this displacement vector is just the subtraction of the vector from the origin to the second charge and the vector from the origin to the first charge. The reason why this triangle depicts subtraction is very clearly stated and can be used to remind students how subtraction of vectors works geometrically. If you continue to 16:35, you will see this in use to find the total force between the two charges along with a detailed explanation of the magnitude of a vector and how to find a unit vector in the direction of a given vector.
Course Topic: Dot Product Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 24:13 to 27:14 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video begins by looking at the work done in two dimensions and notices that the form occurs often. This motivates the professor to define the general definition of the dot product in two dimensions. This combination of application and pure math will serve as a great start to the topic of the dot product either in a pre-calculus or a calculus class.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 28:02 to 30:43 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks demonstrates that the dot product of a vector with itself is the square of its magnitude. Then the professor uses the Law of Cosines to derive that A o B = |A| |B| cos θ. This is a very "mathy" clip, but shows that the math is important in physics.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-6 Time: 38:37 to 40:43 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the potential difference between two nearby points. The professor derives that it is a dot product. Then he uses the geometric definition to come up with the equation that equates the change in potential to a product of the electric field, the change in radius, and the cos of the angle between them.
Course Topic: The Cross Product Video Link: http://oyc.yale.edu/physics/phys-200/lecture-11 Time: 43:27 to 47:13 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video defines the cross product of two vectors in order to define the torque. The professor goes into a long explanation of how to find the angle using the right hand rule. This video can be shown as a replacement to the class explanation since it defines the cross product mathematically.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-11 Time: 49:27 to 51:01 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains three properties of the cross product: AxB = -BxA, AxA = 0 and ixj = k. Although this is a physics class it is straight math. This can be shown to reinforce these three essential properties of the cross product.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-11 Time: 52:46 to 55:36 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video defines the torque as the rate of change of angular momentum. The professor makes use of the product rule for cross products and also the fact that a vector crossed with a parallel vector is 0. This is an excellent example of the use of the product rule for cross products and would accompany the corresponding lecture in a vector calculus class.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 63:15 to 66:24 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the torque that results from a uniform electric field on a dipole. The professor derives the formula for this and then explains that it is just the cross product of the charge and the electric field. This is all done geometrically so no matrices are uses, just magnitudes and angles. This is a simple application of the cross product that students should be able to understand.
Course Topic: Limits Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-10 Time: 10:30 - 12:25 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is uses some basic limits to come up with three major results in relativity including the famous E = mc2. This can be used when first introducing limits to show why they are needed to understand profound ideas of our universe. The student needs to know very little about physics or calculus to understand this clip.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 62:52 to 64:37 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the momentum in the context of Einstein's special relativity and shows that the limit of momentum goes to infinity as the speed of the particle goes to the speed of light. The professor does not say or write down the word "limit" but he does talk about getting close to the speed of light resulting in large momentum. This can be used when talking about limits not existing but going to infinity.
Course Topic: Slope and the Derivative Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-2 Time: 11:49 - 16:20 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of the derivative to find how far apart bonded atoms need to get before they break their bonds. The professor shows that the force is the slope (derivative) of the energy. He has an animated diagram that shows the parabola and moving slope lines for Hooke's Law vs. a similar diagram for electrical charge forces. Then he shows what happens if there are two such forces which demonstrates that derivatives are additive. He next shows what a minimum looks like for the sum of the two energies. This is a helpful vides to show how the graphical version of calculus is used in chemistry. Course Topic: Derivatives Video Link: http://oyc.yale.edu/economics/econ-251/lecture-21 Time: 67:01 to 67:49 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video looks at the rate of change in the present value of a bond with respect to the interest rate. The present value formula is given as a geometric series and the professor calculates the derivate of each term. It is a simple example of the power rule with a negative exponent applied multiple times.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 33:28 to 36:30 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video justifies the definition of velocity as the derivative of the position function. The professor goes through the full derivation starting with the average velocity and then taking the limit as Dt goes to 0. He even discusses the tangent line and its slope. This will be an excellent reinforcement of the first day of the derivative.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 55:44 to58:04 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows the standard question, "How high does it go?" that is asked about position-velocity-acceleration for an object that is moving with constant acceleration.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 18:14 to 18:58 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video starts out with the work equation DK = Fd and divides both sides by Dt to get dK/dt = Fv which the professor describes as the Power. This is an example of taking the derivative of both sides of an equation and getting a new important physics property.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 25:25 to 28:47 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the relativistic formula that compares the velocity an object from two frames of reference. It shows that velocity is relative to the frames and depends on the speed of light. The professor uses the definition of the derivative (Δx/Δt) as both go to zero. This is a powerful example of the definition of the derivative in use and can be shown on the first day of derivatives.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-22 Time: 65:52 to 69:12 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the specific heat at constant volume. The professor takes a derivation of the very simple function 3/2 RT with respect to T. This can be shown at the very beginning of the lecture on derivatives.
Course Topic: Higher Order Derivatives Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 36:30 to 37:23 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains that once you know how to take a first derivative one can take the derivative any number of times. The professor emphasizes that the second derivate is the acceleration while after that the rest are not that useful. This is a very easy to understand commentary on higher order derivative that would be helpful to calculus students to see that the exact same concepts are discussed in physics classes.
Course Topic: Derivative of a Piecewise Linear Function Video Link: http://oyc.yale.edu/economics/econ-251/lecture-2 Time: 39:22 to 42:36 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video presents the marginal utility function based on a person who values one ticket more than two tickets and values three the least. The professor draws the graph of the piecewise linear function and then marginal utility function that is the derivative graph. This can be used to explain why the derivative of a continuous function may not be continuous. Course Topic: Tangent Lines Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-8 Time: 21:37 to 26:40 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video looks at the scenario where there are two people on an island who both grow grain. The first consumes a lot the first year and the second is a saver and consumes a lot the second year. The production and indifference curves are tangent to each other for each. They meet up and work out a loan to create a better economy for both. The professor uses tangent lines multiple times in order to solve this problem. It is shown completely graphically with no equations presented, but students will see a clear application of tangent lines. They will need to be told what a utility curve is in economics in order to understand and they may need to hear a little about how loans work. The background information is presented before this clip, so an instructor who is not familiar with economic theory will want to watch that part first so that the a brief explanation can be provided to the students. Course Topic: Product Rule Video Link: http://theoreticalminimum.com/courses/general-relativity/2012/fall/lecture-9 Time: 107:00 - 107:30 University: Stanford Course: General Relativity Professor Name: Leonard Susskind Teaching Ideas: This video uses the product rule to derive part of the Einstein Field Equations in general relativity. It relates the Einstein Tensor to the Ricci Tensor and the Curvature Scalar. This will be way over the heads of all of the students, but it is good to show them what will be coming in the future if they become physicists and want to understand that Einstein's general theory of relativity makes extensive use of calculus. Course Topic: Chain Rule Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-20 Time: 23:00 - 27:00 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video uses Hook's Law and Newton's F = ma to derive the fact that the frequency will be independent of the amplitude. At the end, the professor shows how this can be used to make a watch. The chain rule is used in taking the derivative of x = hsin(wt) with respect to t. This is an easy and elegant application of the chain rule.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-23 Time: 20:58 to 21:59 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video explains mathematically why there is less risk to diversify rather than put all your money into one stock. This is based on taking a derivative that uses the chain rule and noticing that the deriavtive is negative at zero hence the variance goes down by diversifying. The differentiation is not that difficult, but the chain rule is used.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 61:00 to 63:03 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video starts with simple harmonic motion and takes two derivatives using the chain rule each time. The professor states that the velocity's amplitude is multiplied by ω and the acceleration is multiplied by ω2. This emphasizes that the derivative of the inside is what makes velocity and acceleration differ from each other and from position.
Course Topic: Relative Extrema Video Link:
http://oyc.yale.edu/economics/ Time: 48:00 to 49:45 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video displays the futures curve for oil. There is a clear maximum and minimum. Of note is that in hindsight the futures market predicted the oil prices four years out almost perfectly. This can be used to show an application of relative maximum and minimum and increasing and decreasing functions. There is no explicit calculus used, but it is not difficult to infer how a model of the curve could be constructed and then analyzed. Course Topic: Second Derivative Test, Concavity and Inflection Points Video Link: http://oyc.yale.edu/economics/econ-159/lecture-6 Time: 30:45 to 33:17 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses the first and second derivative tests to find the optimal strategy using game theory. The calculus is explicitly shown and after a second derivative is taken there is a clear maximum. At 8:04 the professor describes the problem. A couple wants to meet up at the movies. There are three movies out: A guy movie, a chick flick, and Snow White. The couple forgot to tell each other their plan and they have no means of communicating with each other. They really want to meet up (first priority), but also would rather not see the opposite gender movie (lesser priority). The math solves the game theory strategy of what probabilities the man and woman should choose each of the three movies.
Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-20 Time: 38:00 - 40:08 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows an application of concavity to the future of the universe. The professor shows that the red shift observed demonstrates that the universe's expansion is accelerating which implies Dark Energy. The professor does not explicitly use the word "concave up" but if the students learn that F = ma = mx'', then they will see that the second derivative greater than 0 implies positive acceleration and positive concavity. The Dark Energy is often called "Einstein's Biggest Mistake" due to the fact that when Einstein first came up with the idea he thought is must be wrong and thus a mistake. Thus Einstein had it correct all along. The video does not state this, but this the the "+C" in the integration in Einstein's calculations.
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-2 Time: 19:00 - 20:25 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of the inflection point to finding when chemical bonds are broken. The professor shows a nice animation of the Morse Potential which measures the bond energy vs. the distance the atoms are apart from each other. He show that for three atoms, the potential is additive and at the inflection point, the point at which the curvature changes from "being this way to this way" as the professor shows with his hands. The students should at this point be asked what math words could have been used and the students should respond "concave up" and "concave down".
Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-7 Time: 47:43 - 49:17 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows both the graphs of the function and its second derivative on the same xy-plane to demonstrate the relationship between the Psi function from Schrodinger's equation and the potential energy function. The professor uses "curvature" instead of "concavity", so that will have to be explained to the students. The graph is busy but clear. The fives minutes of lecture before this explains how the second derivative graph is found geometrically.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-11 Time: 21:46 to 24:16 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the "Value Function" which models prospect theory which shows the way people value financial gains and losses. For a positive gain the function is concave down and for losses, the function is concave down. The professor draws a typical graph, but does not show the equations. It is not differentiable at the origin. This is because there is a big psychological difference between losing and gaining even if it is not a significant loss. For the next couple of minutes, the professor explains it and how businesses use it to exploit people. This is a nice application of concavity that will interest the students.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-11 Time: 14:20 to 15:55 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video shows the graph of the average wage in a career vs. the percent of the average wage that a person gets from Social Security at retirement. The professor specifically points out that the function is concave showing that Social Security is a better deal for low wage earners. This is a very practical use of the second derivative that just about everyone understands.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-4 Time: 38:16 to 43:46 (or 45:02 to see it finally solved) University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video uses the definition of what is means to be a "best strategy" in game theory from the prior clip with a specific example. The professor takes a derivative and then takes a second derivative noticing that the second derivative is negative so the point is a maximum. The explanation of the premise comes before, but it takes several minutes. It involves a synergistic profit sharing agreement. An instructor may want to just give a brief explanation of what is going on to the students including writing down the equations for the students. This is a very well explained used of the first and second derivative test. Course Topic: Differentials Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 56:54 to 59:43 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains the concept of differentials and gives the example F(x) = x2. This is almost exactly what is shown in the corresponding section in calculus class, so rather an an application it will show that the concept is so important it is explained again in physics. This is particularly necessary since most students consider this a minor topic that can be safely forgotten. If you watch until 62:50, you will see the example F(x) = (1 + x)n.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 63:23 to 65:01 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses differentials to approximate the relativistic mass that is typically done in a modern physics class. It makes use of differentials, but the instructor may have to fill in a couple of steps so that the students can see the connection.
Course Topic: Integrals and Sums Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-2 Time: 12:35 to 16:15 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video looks at the mean and expected value defined by sums and integrals. This is a nice example that can be shown to calculus students when they are first learning about integration to emphasize that integration is much more than just an area under a curve. The professor only gives and explains the definitions. No examples are provided here. An instructor can make up examples such as the uniform distribution to show students how this works. Course Topic: Fundamental Theorem of Calculus Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 23:32 to 24:39 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video presents the statement of the First Fundamental Theorem of Calculus in a clear manner. The proof is not given, but the statement is written down just as one would get in a calculus class. This would be a nice reinforcement to present right after the proof is completed in class. If you go until 26:22, the professor gives a simple example. Course Topic: Basic Integration Video Link: http://oyc.yale.edu/physics/phys-200/lecture-7 Time: 56:24 to 57:12 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the anti-derivative of 1/x2 in order to find the gravitational potential energy. The professor calls it "Mickey Mouse calculus" because it is so easy. The instructor can let students know that the power rule is used so often in calculus and physics that soon they will feel like it is "Mickey Mouse calculus".
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-1 Time: 58:30 to 62:59 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the force on a charge that is raised above a charged ring. Although there are quite a few messy constants, the actual integrand is just a constant. This can be shown to students who are first learning about integration.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 5:42 to 7:52 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the Force Energy Theorem using basic integration. The derivation also uses implicit differentiation backwards which will be a nice review of past material. This is an easy to follow video that employs simple calculus to arrive at an important physics theorem. If you watch the video until 10:05, the full conservation of energy law is derived.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-7 Time: 12:31 to 14:44 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video goes over the force and energy on a charge when there are two charges and an infinite plate between them. The integral is so basic, the professor doesn't even write down the integral sign. This is a quick and simple example of an application of electricity at requires integration.
Course Topic: Area Under a Curve Video Link: https://www.youtube.com/watch?v=fIB5AE4SRN4 Time: 49:22 to 50:29 University: Missouri University of Science and Technology Course: Engineering Geology and Geotechnics Professor Name: David Rogers Teaching Ideas: This video describes how the area under a curve is used to find the hydraulic radius of a stream. He shows the typical stream bed and the water above it and how the curve is not just a simple parabola, but must be determined by arduously collecting data. This could be used to discus why using area approximations is necessary. The application is very real life in that it looks at actual surveys that the professor has done in the field. Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-6 Time: 3:45 - 6:56 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graphs of the emitted radiation vs. the wavelength. He sketches what the graph looks like for a cool temperature, and intermediate temperature, and a height temperature. He describes the total radiation as the integral under the curve and the peak wavelength which is the wavelength that give the greatest radiation. The professor does not do any calculus and no equations are shown, but this example can show how calculus will be used.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-17 Time: 44:20 - 46:28 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the graphs of the amount of solar radiation throughout the year for four different latitudes. The professor notes that even though the poles have a greater maximum amount of solar radiation, their total radiation which is the area under the curve is much less, hence it is colder at the poles. There is no actual calculus done in this clip, but the students can easily imagine how the integral would be used to find the total annual radiation. Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-10 Time: 38:03 - 39:35 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays graphs of the number of people in more developed and less developed regions vs. the age groups. The professor notes that the area under the curve is the total population which is clearly larger for less developed regions. It is also interesting to note that the age distribution is skewed right for the less developed regions and relatively uniform for the more developed regions. The graphs are shown vertically which gives an instructor an excuse to remind students the difference between integrating with respect to x (dx) and y (dy).
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-12 Time: 24:00 - 24:40 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays the graph of the world population growth over time. Although the professor does not state it, the area under the curve represents the total population growth. The graph is on a grid, so it naturally leads itself to using rectangular approximations to the integral. Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 38:30 to 43:00 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at starting with a constant acceleration and finding the position function. The professor goes through the step by step logic to derive the solution. He does not use the integral sign, so it would be a good exercise for the students to fill in the details and write it as one would in calculus using integration twice. The professor's discussion is very easy to understand and would work well with the first day of learning about indefinite integration. Course Topic: Finding the Constants of Integration Video Link: http://oyc.yale.edu/physics/phys-200/lecture-1 Time: 44:35 to 47:16 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video begins with the solution of taking two integrals of a constant and then finding all three constants when the equation represents the position function with constant acceleration to arrive at the standard x(t) = g/2 t2 + v0t + s0. This is a simple example on where the +C comes up in physics. Course Topic: Substitution Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3 Time: 39:02 to 42:37 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video uses substitution to integrate a function that represents the electric field produced by a charged infinite plate. The answer surprisingly does not depend on the distance from the plate. The professor explains why and also explains why the integral is necessary to come up with such a conclusion. Every step in the substitution is explained in the clip. This is an excellent example of substitution.
Course Topic: Integration and Logs Video Link: http://oyc.yale.edu/physics/phys-200/lecture-8 Time: 60:42 to 62:24 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the new velocity of a rocket after having expelled exhaust. It begins after all the algebra has been done and shows just the calculus that is performed. The professor integrates -dM/M. He skips several steps which gives the students the opportunity to be asked to fill in all of the steps, an exercise that will be a challenge but not unreasonable.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-7 Time: 53:49 to 57:57 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the most basic RC circuit. Although students at this point in their education have not seen circuits, the instructor explains it well providing them with an excellent introduction to circuit analysis. The integration is simple and gives standard exponential growth for the solution. The professor does apply separation of variables, but students should be able to follow along even if they haven't seen that technique before.
Course Topic: Area Between Two Curves Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-26 Time: 32:18 - 33:23 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the graphs of both the birth and death rates vs. population density on the same set of axes and on a new set of axes shows the difference between the birth and death rates. The professor explains that the intersection of the two curve is the carrying capacity density. Although the professor does not show this, the area between the two curves can be interpreted the in total historical population if the rate of density growth is equal to the time. This could excite the students during the time in the calculus class when they are just looking at analytic geometry.
Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-12 Time: 25:48 - 30:19 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video explains why there is a heat exchange between the equator and the poles. The professor shows a diagram that shows a graphs of radiation from the sun and emitted radiation by the earth vs. latitude. The area enclosed by the two curves is the heat surplus and the area between on the tails is the heat deficit. There are no equations provided, but the picture is clear and provides strong motivation for needing to find the area between two curves.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-10 Time: 25:10 - 27:13 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays graphs of the birth numbers and death numbers over time for the country of Egypt. It is clear that there are many more births than deaths. The professor does not do any calculujs, but students can be asked to interpret the area under the curve. This is an easy to relate application of area.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-4 Time: 2:58 - 5:41 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a very rough diagram of human population over time showing three distinct stages in human history. The stages transition due first to farming and agriculture and second to the industrial revolution. Each segment follows a logistic growth curve. This is a nice example that shows that carrying capacity can change so the model must account for that as a piecewise function.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-10 Time: 45:17 - 47:12 (or 49:10 to see the UN projections) University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays graphs of the number of people in more developed and less developed regions vs. the age groups. The professor notes that the area under the curve is the total population. Moreover, he notes that after another generation even if the fertility rate magically becomes two, the shape will get from being approximately a triangle to approximately a rectangle and the population will still double. In actuality the rate is over 2 and there will be an additional contribution to the future population. An instructor can sketch some graphs that represent these various cases and analyze the future population using integration.
Course Topic: Derivative of the Inverse Function Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-30 Time: 13:42 - 15:36 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video starts with the equation that relates the earth's temperature to the flux (radiation from the sum per unit area of the earth). The professor takes the derivative using differentials and then flips it around to get the derivative of the inverse function. The derivation is easy to follow and provides an alternative approach to finding the derivative of the inverse. The application is within the section on global warming, but that is not evident from the video. The instructor will need to give the student the premise of the video clip and let them know that this does not include feedback and CO2. Course Topic: Inverse Hyperbolic Functions Time: 50:15 - 52:51 University: MIT Course: The Early Universe Professor Name: Alan Guth Teaching Ideas: This video integrates an integrand of the form 1/root(1+kx2) an obtains a hyperbolic sin (sinh x) function that gives the size of the universe as a function of time, showing the universe grows without bound. This disproves the Big Crunch theory and demonstrates the accelerating cold dark theory. This can also be used when doing inverse trig substitution to show that an alternative method of integration is with an arctan substitution. Course Topic: Separable Differential Equations and Exponential Growth Video Link: http://oyc.yale.edu/chemistry/chem-125b/lecture-1 Time: 38:20 - 38:50 University: Yale Course: Freshman Organic Chemistry II Professor Name: Michael McBride Teaching Ideas: This video shows the differential equations that are used to model zero, first and second order reactions. The differential equations are all separable and simple to solve. Students can be asked to solve each of them. The challenge for them will be to accept that it is ok to have [A], the concentration of A, be a variable.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-12 Time: 15:30 - 16:13 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays the graph of the world population over time. The professor notes that the rate of growth has increased over time so rather than it being an exponential growth, it is a hyper exponential growth. In a calculus class or differential equations class, this can be modeled by an equation such as dx/dt = kxt which can be solved by separation of variables. The professor does not go into this detail, but it would be a good exercise for students to play with different possible models.
Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-26 Time: 3:01 - 5:19 (or to 5:55 if there is time to show the doubling time derivation) University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video shows the mathematics behind exponential growth in the framework of population growth in a biology class. The professor goes through each calculus step just as a math professor would. This is a great reinforcement of what is done in the calculus class. After the derivation, if time permits the professor continues to derive the doubling time formula of 0.69/r.
Course Topic: Volume of Revolution Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-12 Time: 7:00 to 8:30 University: Yale Course: Freshman Organic Chemistry Professor Name: Michael McBride Teaching Ideas: This video displays the various orbitals of electrons. They are almost all solids of revolution. Although the video shows no mathematical formulas, we can stress to the students that finding these volumes is instrumental in answering questions that arise in chemistry such as density and what molecular configurations are possible. Course Topic: Work Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 12:27 to 13:31 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video gives a clear explanation that the work done by a force a distance d is Fd and that this is the distance in kinetic energy from beginning to end. This will give a reason behind the definition that is just presented as something to memorize in calculus.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 19:25 to 22:44 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the work done by stretching a spring. The professor begins with F = ks and then does the full derivation using rectangles just as a calculus instructor would do in a calculus class. Instead of integrating ks, the professor treats F as a generic function of position and presents the integral definition of work. This will reinforce what is done in that section of calculus.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 40:28 to 43:00 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the conservation of energy to show that the work expression that includes both gravity and pulling a spring in combination is a constant. An instructor may need to fill in the detail of the integration that the professor left out. This is a direct extension of the standard work calculations done in calculus.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 52:55 to 55:50 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video calculates the work done to take a pendulum that begins vertical and bring it to an angle θ0. The professor goes through all the steps of the process. This is a clear example of the use of calculus in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-22 Time: 56:32 to 59:20 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the work done by a gas along an isotherm. The derivation is easy to follow and is in line with what the calculus textbooks show. This can replace the corresponding piece of a calculus lecture on work done by gas.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23 Time: 39:00 to 42:21 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the work done in an adiabatic process. The professor integrates something with a power in the denominator using the power rule. This is a good time to stop and remind students that just because there is a denominator, it does not mean that the integral is a ln. Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 67:42 to 69:41 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the work done by an electric field on a dipole due to torque. The integration is very simple, but the application is that an electric field can cause a rotation such as spinning a tire in an electric car. The professor does not explicitly describe the application, but the instructor can ask the students to come up with some applications.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-7 Time: 32:31 to 33:38 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the work done to charge up a capacitor. The integration is very simple as is the full derivation. This would be a quick application to show students when talking about work. If you play the video to 35:30 more derivation occurs and relates the energy to the volume of the space between the capacitor plates.
Course Topic: Center of Mass Video Link: http://oyc.yale.edu/physics/phys-200/lecture-8 Time: 16:27 to 18:46 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at a one dimensional object of constant density and derives the obvious formula that the center of mass is at the center of the line segment. The professor goes through all the steps of breaking it apart and adding up all of the masses times the distances. Although this is a very simple example, it serves to explain the essence of the integral.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-8 Time: 23:42 to 28:33 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the center of mass of a triangle with constant density. The professor uses symmetry for the y-coordinate and uses integration for the x-coordinate. This is a very standard example that one would see in a calculus class. The professor give the full derivation and can show the students that the exact same math will be in their physics class. The professor uses similar triangles which serves as a nice reminder to the students that this technique occurs outside of math class. Course Topic: Moment of Inertia Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 63:22 to 66:02 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video calculates the moment of inertia of a disk about its center. The professor goes through each classic calculus step to solve this problem. His explanation is easy to understand so this clip can serve as an effective example of this concept.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 67:31 to 69:18 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video calculates the moment of inertia of rod about its endpoint. The calculus is easy and the explanation is clear. This can be an effective motivator for exploring this topic.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-9 Time: 70:07 to 71:53 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video is a continuation of the above video, but calculates the moment of inertia of rod about its center. The professor uses symmetry to assist in evaluating the integral explaining that the integral of an even function from -a to a is twice the integral from 0 to a. This is a classic use of symmetry in integration.
Course Topic: Inverse Trigonometric Substitution Time: 22:45 - 25:13 University: MIT Course: The Early Universe Professor Name: Alan Guth Teaching Ideas: This video goes through the steps of inverse trigonometric substitution in order to find equations involved in the Big Crunch theory of the universe. The professor integrates cos(θ) as cos(θ) by mistake, which makes for a learning opportunity. The physics is not shown, so we will have to explain that this is for the Big Crunch theory and that the answer y = 1 - cos(θ) shows that at the beginning the universe had no volume and after one period, we will be back to no volume (The Big Crunch). At 33:15 the professor shows the curve that is formed graphically and goes over the physical interpretation. It turns out that the most recent evidence shows that the Big Crunch theory is incorrect and in fact we will have an accelerating universe finishing off with a cold dark lonely universe instead.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3 Time: 22:57 to 28:20 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the integral of 1/(x2 + a2)3/2 using inverse trigonometric substitution. At the beginning, the professor asks his Yale students how to solve this integral and none of the students can do it. This is a good chance to let the students know that inverse trigonometric substitution should not be taken lightly. At the end the professor explains that this gives the electric field strength a distance a from a charged wire. This is an important application in physics and the professor goes through every step of the integration.
Course Topic: Logistics Growth Video Link: https://www.youtube.com/watch?v=mRtw4UOwyCM&index=5&list=PL2CD836B66D3CEBED Time: 2:22 - 7:00 University: UC Berkeley Course: Biology 1B (2nd Semester Biology) Professor Name: Alan Shabel Teaching Ideas: This video compares the exponential growth model and the logistics growth model. It exhibits the differential equations but does not solve them. It only shows the "S-Curve" graph. The last minute of two of the time range should be enough to get the main point of logistics growth. This can be used either in a calculus course or a differential equations course, particularly on autonomous differential equations.
Video Link: http://oyc.yale.edu/molecular-cellular-and-developmental-biology/mcdb-150/lecture-4 Time: 2:58 - 5:41 University: Yale Course: Global Problems of Population Growth Professor Name: Robert Wyman Teaching Ideas: This video displays a very rough diagram of human population over time showing three distinct stages in human history. The stages transition due first to farming and agriculture and second to the industrial revolution. Each segment follows a logistic growth curve. This is a nice example that shows that carrying capacity can change so the model must account for that as a piecewise function.
Course Topic: Improper Integrals Video Link: http://ocw.mit.edu/courses/physics/8-286-the-early-universe-fall-2013/video-lectures/lecture-23-inflation/ Time: 42:00 - 45:50 University: MIT Course: The Early Universe Professor Name: Alan Guth Teaching Ideas: This video uses an improper integral to calculate the event horizon of the universe. This is the distance such that an object must be from us so that light from it will never reach us due to the acceleration of the universe and the speed of light. This is a fascinating application of improper integrals that will encourage student interest.
Video Link: http://oyc.yale.edu/economics/econ-252-11/lecture-8 Time: 40:24 to 40:43 University: Yale Course: Financial Markets Professor Name: Robert J Shiller (Nobel Laureate) Teaching Ideas: This video presents the equation for the Present Discounted Value when payments are coming in continuously. If time permits, the minute of lecture before this clip shows the infinite series for annual and biannual payments which is an infinite series. This could be used to introduce how an infinite series becomes an improper integral.
Course Topic: Taylor Polynomials and Series Video Link: http://oyc.yale.edu/economics/econ-251/lecture-7 Time: 25:50 to 29:09 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video proves the rule of 72 which says that the time to double your money is about 72/(100i) where i is the interest rate as long as i is not that far from 7%. In the derivation, the professor uses the Taylor expansion from ln(1+i) finding the first three terms. This is a pretty simple example of Taylor polynomials being used and can be show to the students why the Taylor polynomial his helpful for basic estimations.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-22 Time: 6:25 to 8:58 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video goes over Bernoulli's St. Petersburg Paradox which looks at an infinite expected value where people are not willing to pay much for that bet. Instead there is a utility function that is logarithmic. He goes over this infinite series and realizes it as a logarithm. He does not give the details of the power series that gives this log, but it is something that can be given to the students as an exercise.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-5 Time: 66:33 to 68:02 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video gives the full statement of the Maclaurin Series formula just as a calculus instructor would give. No applications are provided, but the explanation is easy to follow. If you watch until 69:38 you will see the example ex. This provides reinforcement to what is done in a calculus class.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 0:59 to 8:05 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video gives an argument for the McLaren Series. The professor starts with the constant approximation. then moves on to the linear approximation and continues with the quadratic approximation. Finally he writes down the full series. The professor state that he is doing it in the way physicists do it, but it is no different from the way it is done in calculus class. This can replace the instructor's introduction of McLaren Series.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 8:58 to 12:00 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the McLaren series for the function 1/(1-x). The professor does not refer to any physics, but the mathematics he does is the same as what students see in a calculus class. Next the professor uses it to show what is happening with 1/(1 - 0.1). This is a nice example of showing numerically what is happening with the geometric series. If time permits, the continuation through 14:20 discusses the convergence of this series.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 17:30 to 18:37 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the McLaren series for ex. He does this quickly and clearly. Although this is a physics class, he just does the math here and does not go into how it can be used in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 20:15 to 22:09 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the McLaren series for cos x. Like the clip above, the professor does this quickly and clearly. Also as above, he just does the math here and does not go into how it can be used in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-16 Time: 25:21 to 27:13 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses McLaren series to derive Euler's formula. Like the clips above, the professor does this quickly and clearly. Also as above, he just does the math here and does not go into how it can be used in physics.
Course Topic: Series Expansion of a Binomial Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-10 Time: 4:30 - 7:19 and 8:30 - 10:22 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video shows is an application using the Taylor polynomial to realize where Newtonian and Post Newtonian physics differ. The professor does not actually show the calculus, but it would not be difficult for the students to fill in the details. This could also be used in a first quarter calculus course as an application of the tangent line approximation to a curve. Then at 8:30, the derived formula is applied to gamma the relativistic factor.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-7 Time: 64:24 to 66:43 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the second degree Taylor expansion of the binomial to derive the classic formula from physics that the potential energy of a object subject to gravity is GMm/R2. Many students know the inverse square relationship, but this clip shows them why and convinces them that Taylor Series can be useful.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-14 Time: 64:40 to 67:58 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video works out the energy of a particle by looking at the power series expansion of its energy formula. The second term of the Taylor series is 1/2 mv2, the kinetic energy. The first term is mc2 which is the rest energy and where Einstein came up with his famous equation. This is a pretty easy derivation and may be the first time that students understand the famous equation. Students will have a great sense of why power series are important.
Course Topic: Quadric Surfaces Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-6 Time: 17:00 - 18:53 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video shows the dynamics of a plume of air that is released and shot out horizontally by a wind if there is a steady state output of pollutants. After a fixed number of minutes, the plume will be in the shape of a paraboloid with vertical cross sections as circles and horizontal cross sections as parabolas. No equations are given, but the hand drawn pictures are clear and the professor nicely demonstrates that drawing cross sections is a good graphing technique. Course Topic: Contour Diagrams Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-3 Time: 19:00 - 16:26 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video shows an application of a contour diagram to understanding energy states and activation energies. The professor clearly demonstrates how the gradient curve (although he does not use these words) demonstrates what must occur in order for a reaction to take place. This can be used either when introducing functions of several variables or when discussing gradients or extrema of multivariate functions. Course Topic: Spherical Coordinates Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-9 Time: 24:40 - 27:57 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video uses spherical coordinates to simplify the Schrodinger equation for the one electron hydrogen atom. The professor explains how the use of spherical coordinates allows us to write the Schrodinger equation as a product of three functions, each a function of a single variable. He uses r instead of r and switches the roles of theta and phi, so the students will have to be told that in applications, the variable names are not standarized. Course Topic: Partial Derivatives Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-11 Time: 38:07 - 39:00 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video looks at the partial derivative partial derivative of fitness (number of offspring) of a male with respect to further survival. It just shows the graph based on age but not the equations. The graph is given and shows that "after the age of 46 evolution doesn't care if you are there anymore." This gives a very meaningful lesson based on partial derivatives.
Time: 9:00 - 10:59 University: MIT Course: Statistical Mechanics Professor Name: Mehran Kardar Teaching Ideas: This video uses the fact that the mixed partial derivatives are independent of order to prove one of Maxwell's results about thermodynamics relating energy, temperature, force, position, momentum and enthalpy. The physics will be way over the heads of the students, but it is helpful for them to see what graduate level physics looks like.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-2 Time: 48:50 to 52:38 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video presents the diminishing marginal utility which looks at the partial derivatives with respect to x and y and notices they are negative so diminishing. This is a very simple example of an application of partial derivatives.
Video Link: http://oyc.yale.edu/economics/econ-251/lecture-3 Time: 31:25 to 32:02 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video looks at the utility functions and explains that the marginal utility of x (partial derivative with respect to x) divided by the price of x equals the same quotient in y. The lecture is somewhat scattered, but it is a common example of using partial derivatives. Students will need to be told what the professor is talking about.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 2:30 to 4:28 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video defines the partial derivative with respect to x. Although this is a physics class, this clip could have just as well been taken from a calculus class. This could be shown as a reinforcement to the motivation behind partial derivatives or it can just replace the in class lecture on the subject.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 5:15 to 8:58 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows an example of taking both the first and second partial derivatives of a polynomial function in two variables. The professor clearly shows the notation and the solutions and makes a point at the end to explain, but not prove, that the mixed partial derivatives are equal. This could come right out of a math class, but since it is a physics class students will realize how important this concept is. The proof is shown over the next 7 minutes until 16:30. Course Topic: The Gradient Vector Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 61:26 to 63:14 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video shows that the gradient of the electric potential gives back the electric field. The professor computes the partial derivatives and packages them as a vector field after taking the gradient. The professor never states the word "gradient" but the instructor can tell the student that that this is what happened.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-6 Time: 40:24 to 42:42 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video explains that the electric field points in the greatest change in potential. Then the professor mentions that going in the direction in the gradient vector will be most efficient. Finally, he shows the physics formula for the directional derivative. The notation looks different and the words "directional derivative" are not used, but it would be a good question of the students to see if they can answer the question of what he has just defined.
Course Topic: Chain Rule for Partial Derivatives Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 19:34 to 24:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the derivative of the kinetic energy with respect to time in the context of finding the work done. The professor begins with the standard kinetic energy theorem in two variables and then takes a partial derivative with respect to t on both sides. In the derivation he makes use of the chain rule with the composition function R1 -> R2 -> R1. The explanation if clearly stated and can be used to introduce the chain rule or to give an example of this type of chain. The clip can be ended at several points along the way if time is a big issue.
Course Topic: Finding the Extrema for Multivariate Functions Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-9 Time: 46:30 - 47:55 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video applies the technique of taking the partials and setting them equal to 0 to find the extrema for the 2p orbitals of a hydrogen atom.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-4 Time: 24:10 to 28:18 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video gives the definition of what is means to be a "best strategy" in game theory. The definition just is the definition of the maximum value of a two variable function. The professor does not do any computation, but the definition can help students see where extrema of multi-variable functions can be useful. The application that the professor goes over earlier is in soccer: kick left, middle, or right. Course Topic: Constrained Optimization Video Link: http://oyc.yale.edu/economics/econ-251/lecture-3 Time: 3:55 to 5:11 University: Yale Course: Financial Theory Professor Name: John Geanakoplos Teaching Ideas: This video writes down the utility function for two goods subject to two constraints that are based on the amount of product available. The professor does not solve it here, but it is a great class exercise to ask them to solve it using Lagrange Multipliers. The solution is not very difficult, but looks tough due to having six variables. The professor solves it in the next five minutes, but spend a long time with a not that clear or simple solution. Course Topic: LaGrange Multipliers Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-32 Time: 5:30 - 8:51 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video demonstrates that the optimized line through the origin for deciding to stop searching for food in a patch of land and start searching in another one is found by rotating it until it is tangent to the payoff vs. time curve. This is similar to the search of the maximum value given a constraint that the method of LaGrange multipliers finds. This can be used to introduce constrained optimization problems.
Video Link: http://oyc.yale.edu/economics/econ-159/lecture-14 Time: 28:15 to 33:44 University: Yale Course: Game Theory Professor Name: Ben Polak Teaching Ideas: This video looks at the quantity that a first firm should produce if that first firm knows how much the competitor will produce in reaction to the first firm's decision. An equation is presented and the professor indicates that one could substitute the constraint equation into the max/min equation and then take a derivative, but another method is the method of LaGrange multiplier. This is a nice reminder to calculus students that there are two ways to solve constrained optimization problems. The professor decides to solve using substitution rather than LaGrange multipliers. Students can be asked to do the problem, but it is a bit of alphabet soup. This can also be used as an application of first quarter calculus. Course Topic: Vector Valued Functions Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 58:27 to 62:51 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at two parallel oppositely charged plates and considers the motion of a particle that starts in the middle with a given initial velocity. The professor explains that the path of the particle is described by a vector valued function. The professor derives the equation for this path and then explains that this is how TV screens work. This is a relevant application of vector valued functions that students may not have seen before and could get them excited to learn this topic.
Course Topic: The Derivative of a Vector Valued Function Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 38:18 to 41:32 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula for the derivative of a two dimensional vector valued function. The professor gives a very clear mathematical derivation, but this clip has no physics applications contained in it. It would be good to show that what happens in a math class occurs in the same way in a physics class.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-2 Time: 49:05 to 51:26 (or to 52:29 to hear a physical application of this formula.) University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the derivative of the vector valued function that describes a particle moving at a constant velocity around a circle. Then the professor takes a second derivative to show that the acceleration (a(t) = r''(t)) is just a constant times the original vector valued function r(t). This is an important and simply derived application of derivatives of vector valued functions. He then reveals in a funny way that a(t) = v2/r. Course Topic: Double Integrals in Polar Coordinates Video Link: http://oyc.yale.edu/physics/phys-201/lecture-4 Time: 72:50 to 75:03 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video shows how to perform a double integral in polar coordinates to find the area of a triangle with vertices (0,0), (1,0), and (1,1). Although the professor just does the math without explaining how it applies to physics, the students get to see how a physicist approaches a double integral using slightly different notation.
Course Topic: Triple Integrals Video Link: http://oyc.yale.edu/chemistry/chem-125a/lecture-12 Time: 28:26 - 35:47 University: Yale Course: Freshman Organic Chemistry I Professor Name: Michael McBride Teaching Ideas: This video infers the application of triple integrals to finding the overlap density of two atoms' electron shells. The professors only talks about the integral and never states that it is a triple integral due to the fact that this is a freshman class; however, more mathematically sophisticated students will be able to see that the integral written down is a triple integral. Course Topic: Vector Fields Video Link: http://oyc.yale.edu/physics/phys-201/lecture-2 Time: 30:32 to 32:15 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video sketches the vector field that shows the electric field induced by a single charge. The professor is very clear about focusing on the magnitude and direction for this vector field that obeys the inverse square law. This is a great way to introduce vector fields to students.
Course Topic: Line Integrals Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 34:15 to 36:13 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the line integral in the context of comparing kinetic energy and work. The professor talks about cutting up a curve into tiny pieces and adding up all the work components. This will be an nice reinforcement and application when first introducing line integrals.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 47:50 to 51:52 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows how to evaluate a line integral to find work by going along a path y = x2. It makes use of the fact that y is a function of x and does not talk about parameterized curves. The first few seconds summarizes another path, so this video emphasizes that the choice is path can be important. This video nicely goes over two concepts in a short time and can serve as a tie between what is done in calculus and physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-23 Time: 20:48 to 23:22 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video finds the work done by a gas in going through a cycle that begins along an isotherm continues along a constant pressure line and then up the constant volume line to get where one started. The professor shows that this is a line integral and stresses the fact that going around the other direction results in the opposite work done. This is a nice clip when talking about orientation of a curve.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 14:56 to 17:24 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the formula that the change in kinetic energy is equal to the total work done, and thus the conservation of energy theorem in three dimensions. The derivation uses derivatives of vector valued functions and line integrals. It is easy to understand and can serve as a powerful example to illustrate the use of the line integral.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 21:14 to 24:19 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video gives an example of a force that is not independent of path. The professor goes through the line integral calculations along both paths. It is a simple example, but it demonstrates the process of finding line integrals well.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8 Time: 4:24 to 6:17 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video explains what the EMF is for an electric circuit. The professor derives it using a line integral, but it is intuitive enough that he does not have to parameterized the curve and integrate. This is more of a motivating example of why line integrals are important than an example of how to work out a line integral.
Course Topic: Fundamental Theorem of Line Integrals Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 61:51 to 63:53 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video explains how the fundamental theorem of line integrals in two dimensions works. The professor does not name the theorem specifically, but he does check that the appropriate partials are equal. This is a clear example of the FTLI in physics.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 63:55 to 64:51 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows that gravity is a conservative vector field and thus satisfies the conservation of energy law. This is an example where although the language of math and physics may differ, the fundamentals are the same. The example is very easy since the vector field is a constant, but the importance in physics is clear. To see the concept applied to a rollercoaster go until 67:57.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-6 Time: 69:47 to 70:26 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video presents a challenge for the students where the instructor posits that if the force is parallel to the position then it is conservative. Since the professor just asks the question without presenting the solution, an instructor can ask the students to solve the problem in class. Having students do an application right out of a physics class will show them that solving math problems is more than just an exercise.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 25:06 to 26:44 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video shows that independence of path for line integrals is equivalent to the statement that the line integral over any closed curve equals zero. This is identical to how it is introduced in a vector calculus class, but it can be motivating to see it happen in a physics class.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 27:16 to 31:15 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video presents the statement of the fundamental theorem of line integrals. The statement is the same as it is in a vector calculus class even though the professor is presenting it in the context of physics. This will serve as a reinforcement of this important theorem.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 45:05 to 47:05 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video demonstrates the fundamental theorem of line integrals applies to gravity. This is a very basic example, but shows that the FTLI applies to something as important as gravity.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-5 Time: 55:30 to 59:12 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video uses the FTLI by selecting a convenient path to find the potential energy difference from one point to another through an electric field. This serves as a solid application of the FTLI and goes through the process of integrating well. The professor does make an error and fixes it, but that will not confuse the students.
Course Topic: Flux Video Link: http://oyc.yale.edu/geology-and-geophysics/gg-140/lecture-12 Time: 35:06 - 37:59 University: Yale Course: The Atmosphere, the Ocean, and Environmental Change Professor Name: Ronald Smith Teaching Ideas: This video defines the mass flux and the heat flux through a cylindrical conduit. In the second part, the professor explains a slide that shows that there is heat flux while there is no mass flux. It only defines it for constant vector fields and is not very high level, but it can be used as a motivator when teaching flux in a multivariate calculus course. Course Topic: Surface Integrals Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3 Time: 57:46 to 60:14 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video explains Gauss' Law that states that the flux through any closed surface containing a charge q is equal to the charge over ε0. This is a nice introduction to flux integrals. The professor describes it geometrically which explains the idea of what a surface integral is.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-4 Time: 33:27 to 34:36 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video looks at the flux through a sphere with a charge inside. The professor explains that since the direction is outward, the cos is1 and the field strength is a constant along the sphere. Thus the integrand is just a constant giving the constant times the area of the sphere. This is a nice example that shows the idea of the flux integral.
Course Topic: The Divergence Theorem Video Link: http://oyc.yale.edu/physics/phys-201/lecture-3 Time: 62:26 to 64:23 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This video relates Gauss' Law to a triple integral. It will not be obvious from the video, but this basically shows that Gauss' Law is a result of the divergence theorem. Students may need some more reference for this and an article that explains it is at: http://www2.ph.ed.ac.uk/~rhorsley/SII08-09_mp2a/lec20_2.pdf (See the part on Source and Sinks).
Course Topic: Payoff Matrices and Game Theory Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-33 Time: 11:19 - 16:12 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video explains basic game theory and payoff matrices using having a hawk vs. dove attitude. The professor goes into detail on how it works and defines what it means to be a stable strategy. At the end he asks the students to work out what strategy is best and why. In class students at this point can also discuss with each other the best strategy. Course Topic: Coordinate Transformations and Matrices Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 9:12 to 10:10 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video writes down the picture and the equations of the coordinate transformation that corresponds to rotation of the plane by an angle θ. The professor does not write down the corresponding matrix, but linear algebra students can easily be asked to write down this matrix. The instructor will need to tell the students that the professor is providing an analogy to the Lorenz transformations that occur in Einstein's theory of relativity.
Course Topic: Linear Transformations Video Link: http://oyc.yale.edu/physics/phys-200/lecture-13 Time: 13:28 to 13:07 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video points out that the Lorenz transformations that occur in Einstein's theory of relativity are linear transformations. The instructor can tell the students that in order to understand Einstein's theories, one must first have a firm grounding in linear transformations and linear algebra in general.
Course Topic: The Magnitude of the Displacement Vector is Invariant Under the Rotation Matrix Video Link: http://oyc.yale.edu/astronomy/astr-160/lecture-12 Time: 4:30 - 6:40 University: Yale Course: Frontiers and Controversies in Astrophysics Professor Name: Charles Bailyn Teaching Ideas: This video looks at the invariant effect of the rotation matrix on the displacement vector. For the next few minutes he discusses how this also works for the space-time metric tau which is also invariant under 4x4 matrix rotations. This can be looked at when discussing the generalized inner product of a space and how the main application is to the metric of space-time. Course Topic: Leslie Matrices, Eigenvalues and Eigenvectors Video Link: http://oyc.yale.edu/ecology-and-evolutionary-biology/eeb-122/lecture-26 Time: 19:00 - 20:34 University: Yale Course: The Nature of Evolution: Selection, Inheritance and History Professor Name: Stephen C. Stearns Teaching Ideas: This video introduces the idea of a Leslie matrix to come up with the population in the next generation given the current generation's adult and child populations and their rates. The professor just refers to the fact that a matrix can be constructed and linear algebra can be used, but does not show the matrix. It is recommended that after presenting this clip, the class is show another website that actually gives the Leslie matrix. One such is here: http://en.wikipedia.org/wiki/Leslie_matrix. This is a nice application that clearly shows the use of matrices outside of mathematics. A class can further use eigenvalues and eigenvectors to show that the dominant eigenvalue is the long term growth rate and the eigenvector is the stable age distribution. Course Topic: UC Functions Time: 21:11 - 23:45 University: MIT Course: Circuits Professor Name: Anant Agarwal Teaching Ideas: This video uses the UC function "A" to solve an LRC circuit. He goes through the process in 4 steps. The first 2 1/2 minutes involve describing the steps. Then if there is time, the next five minutes solve the problem. This nicely mirrors what is done in a differential equations class, but uses the LRC circuit as the example.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17 Time: 60:36 to 66:53 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the physics problem of an object subject to a spring like force, friction, and a forcing function that is a cos function. the professor uses a trick involving a combination of complex numbers and UC functions to solve it without using any trig functions. The professor explains how to obtain the real part in the next seven minutes. This shows an alternate method of solving such differential equations.
Course Topic: First Order Linear Differential Equations Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8 Time: 18:48 to 21:03 University: Yale Course: Fundamentals of Physics II Professor Name: Ramamurti Shankar Teaching Ideas: This videoStarts with the equation that corresponds with an RC circuit. This equation is a first order linear separable differential equation. The professor splits up the problem by writing the solution as the limiting solution plus the rest. Although this is not the traditional mathematical way that this type of equation is solved, it demonstrates that there are many techniques to simplify differential equations that make them easier to solve.
Course Topic: Second Order Differential Equations Homogeneous With Constant Coefficients Video Link: http://oyc.yale.edu/physics/phys-200/lecture-3 Time: 47:30 to 48:25 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video derives the differential equation -kx = mx'' which gives the position of a mass on a spring. The professor states, "then you go to the math department and say please tell me what's the answer to this equation." At this point the instructor can tell the class that they are the math department and they must tell the professor the answer.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17 Time: 18:19 to 24:19 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the second order differential equation x'' + ω02x = 0 using the technique of substituting y = Aeαt into the differential equation in order to come up with simple harmonic motion.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17 Time: 25:16 - 29:00 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video shows that the harmonic oscillator is a linear differential equation, thus any linear combination of solutions gives a solution. This is easy to follow, but the physics is not apparent in the explanation.
Video Link: http://oyc.yale.edu/physics/phys-200/lecture-17 Time: 39:00 43:44 University: Yale Course: Fundamentals of Physics I Professor Name: Ramamurti Shankar Teaching Ideas: This video solves the differential equation that corresponds to a harmonic oscillator with friction. The professor uses the standard technique that is used in a differential equations class, so this can replace the derivation that occurs in the differential equations lecture on solving second order homogeneous differential equations with constant coefficients. The instructor should note that the professor made a mistake in calling the solution -α where it should have been α. The professor later notices the mistake, but spends an unreasonable time correcting it. Instead of showing this error fixing clip that comes next, the instructor can just continue it and demonstrate the cases. This can easily replace the lecture on this material starting with the professor's clip and finishing with the instructor's comments about damped, critically damped, and over damped systems. |