About
Statistics
Algebra
Trigonometry and Vectors
Calculus (1st Year)
Advanced Math
Algebra and Pre-Calculus
- Writing Down an Algebraic Equation from Words
- Definition of a Linear Equation
- Simplifying Linear Expressions
- Solving a Linear Equation
- Slope of a Line
- Graph of a Line
- y-Intercept of a Line
- Equation of a Line
- FOIL
- Functions
- Intersection of Two Lines
- Solving Two Linear Equations In Two Unknowns
- Rational Expressions
- Complex Rational Expressions
- Rational Equations
- Rational Inequalities
- Geometry
- Exponents
- Division of Exponents
- Power and Quotient Rules for Exponents
- Formulas With Square Roots
- Solving by Taking the Square Root of Both Sides
- Cube Roots
- Proportions
- Solving for a Variable
- Multiplying Through
- Squaring a Binomial
- Solving by Squaring Both Sides
- Linear Inequalities
- Quadratic Formula
- Parabolas
- A Survey of Graphs
- The Cubic Function
- Understanding the Domain of the Square Root Functions
- Graphing With Roots and Denominators
- Solving Nonlinear Systems of Equations
- Determining a Rational Equation from its Graph (College Algebra)
- Piecewise Defined Functions
- Exponential Functions
- Logs
- Solving Log Equations
- The Circle
- The Ellipse
- The Hyperbola
- Sequences
- Series
- Geometric Series
- Pascal's Triangle
- Mix of Many Algebra Topics
Course Topic: Writing Down an Algebraic Equation from Words
Video Link: https://www.youtube.com/watch?v=dHakIrJSz-4&index=12&list=PL0A0E275BC354C934
Time: 41:14 to 45:30
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video first describes in words the relationship between the hydraulic gradient and the recharge, discharge and height for a water table and the relationship between hydraulic head and the recharge and discharge points. The first part shows it in words and the second part shows the equations. An instructor can pause the clip at the first part and see if the beginning algebra students can write down the equations after having seen the words. Then the instructor can resume the video and the students can see how they compare with what the professor did. The professor's lecture is very entertaining and has a special way of lightening up any topic.
Course Topic: Definition of a Linear Equation
Time: 7:10 - 9:29
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video shows a circuit and its corresponding equation and explains that it is linear. The physics will be difficult for the elementary algebra student, but the explanation of what it means for an equation to be linear in a variable is very simply stated.
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.
Time: 36:24 to 38:07
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video shows how to use the rate law to find the rate constant. The method involves plotting the line: ln[A]t = -kt + ln[A]0. Although this equation involves the natural logarithm function, students do not need to know anything about logs to understand this. The professor tells the students that ln[A]t is y, t is x, k is m and ln[A]0 is b. The professor's explanation clearly shows that the rate constant m is the slope. Her explanation is easy to follow even for beginning algebra students.
Video Link: https://www.youtube.com/watch?v=6f59FQGDOa8&index=21&list=PL48DE756A5800ED5F
Time: 25:26 to 27:05
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video shows the line that depicts the body size vs. age at first breeding and also the body size vs. frequency of breeding. The slope of body size vs. age at first breeding is positive. The slope of body size vs. frequency of breeding is different depending on which animal: mouse, dik dik, deer and rhino. The slopes are all given but the different graphs are not. It would be a good exercise for the students to sketch each of the graphs to practice what the slope is. No particular points are given, so this is just to focus on the slope.
Course Topic: Graph of a Line
Time: 18:19 -21:49
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video goes over the voltage output for a mosfet amplifier. The goal is to search for the point at which it stops behaving like an amplifier. The professor writes down the equation of the line with slope 1 and y-int 0. Then he shifts the graph to the right and writes down the new equation. The axes are labeled VI and VS, so the students may need to be told that these are like x and y.
Video Link: https://www.youtube.com/watch?v=aovCX4vqPNI&list=PL-XXv-cvA_iARKmuLqeDrJ4vWHck36c2Z&index=9
Time: 44:14 -46:14
University: UC Berkeley
Course: Developmental Psychopathology
Professor Name: Stephen Hinshaw
Teaching Ideas: This video compares different treatments for ADHD and shows a graph of the effectiveness of each over time. The equations are not given, so it would be an effective exercise to have students find the equations of each of the four lines given to connect the beginning and ending points.
Course Topic: y-Intercept of a Line
Time: 31:57 to 33:49
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video looks at the second order rate law for a reaction. The original equation looks nothing like a line, but when the professor shows what the variables are and displays y = mx + b on the PowerPoint, the equation is clear. This is a nice example of lines coming up even when an equation does not look linear at all. She explain that kinetics is all about getting data to plot the line. She finishes with a clicker question asking what the intercept of the line is. Beginning algebra students should be able to get it after having been taught that b represents the y-intercept.
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.
Time: 9:05 to 10:15
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video shows the tree diagram for a heap structure for computer storage. There are four linear equations given, but without some explanation from the instructor basic algebra students will not see that they are equations since they do not use x and y. This is a good chance to show students that in the real world linear equations come in many flavors. This will also introduce them to the important logic tool of the tree diagram. The students can be asked, for example what value corresponds to the left child for a parent at i = 4.
Time: 23:22 to 26:48
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video shows the equation that relates the change in Gibbs free energy, heat, temperature and enthalpy. If you keep the temperature and heat fixed then this is the equation of a line with slope -T (temperature) and y-intercept ΔH (change in heat). This will look very different from what algebra students are used to, but will show that they the symbols used in math and science differ greatly. The instructor can have them graph an example line with say ΔH = 4 at constant temperature T = 3 degrees Celsius.
Time: 10:01 to 15:50
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video uses the same equation as the one above, but really shows that this is the equation of the line. This time the temperature is the independent variable. The professor's PowerPoint shows y = mx + b and she explicitly finds the slope and y-intercept. She spends time explaining how to interpret the y-intercept that it tells us when the reaction becomes spontaneous. This is a great example for students that goes through the main aspects of lines and how they are used. It is definitely worth the five or six minutes of class time.
Video Link: https://www.youtube.com/watch?v=wV8Zs-s8uzw&index=4&list=PL2Q_sOQgsm24ybtnVq75-TgZd86lMjm9m
Time: 46:24 to 52:24
University: Harvard
Course: Introduction to Computer Science
Professor Name: D. J. Malan
Teaching Ideas: This video explains how to write a C program to convert Fahrenheit to Celsius. The professor shows the equation: C = 5/9 x (F - 32) and purposely makes the mistake of omitting the parentheses in the code. He asks his students what the mistake was. An instructor can pause the video and ask the students too. This will serve as a great reminder of order of operations and being very careful to pay attention to detail in math formulas.
Video Link: https://www.youtube.com/watch?v=c8OHZZ1AX5I&index=12&list=PL48DE756A5800ED5F
Time: 12:45 to 14:29
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video goes over the relationship between group size of a species and percentage of time that ostriches vigilantly watch for predators. The professor does this for both the total amount of watching and watching per individual. He plots the points that make up the lines but does not show the equations of the lines. This would make an excellent activity for students who are learning how to find the equation of a line given two points.
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-18
Time: 17:02 to 19:02
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video notes that the intensity of light is the square of the wave function. The professor uses FOIL to show that although wave functions are additive, intensities are not. The FOIL is the square of a sum and the point is to remind students that there is a middle term.
Time: 36:43 - 43:02
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video goes through the mathematics to show that a presented electric circuit is a linear amplifier. One step of the derivation involves FOIL and the result is a linear equation through the origin. The derivations involves alphabet soup, so passive observance might be the only way beginning algebra students will persevere through the clip. The larger A is, the greater the amplification. Amplifiers are not just used to make music louder. Their main application is for digital signals so that a receiver can better distinguish between 0s and 1s. This is how computers get information through the Internet.
Time: 15:28 to 17:23
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video goes over the idea that the probability for an electron is the square of the wave function. The example given is (1sa + 1sb)2 = (1sa)2 + (1sb)2 + 2(1sa)(1sb). The professor highlights the third term and notes that it corresponds to destructive interference. The students will not understand the at a deep level, but they can feel good about being able to do the math of quantum mechanics.
Time: 35:29 to 39:02
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video is similar to the above clip. It also goes over the idea that the probability for an electron is the square of the wave function. The example given is (2pxa + 2pxb)2 = (2pxa)2 + (2pxb)2 + 2(2pxa)(2pxb). The professor again highlights the third term and this time notes that it corresponds to constructive interference. The students will not understand the at a deep level, but they can feel good about being able to do the math of quantum mechanics. They will need to be told that the 2's do not get squared because they are part of the variable name and not actually numerical values.
Course Topic: Functions
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 graphs of the ozone concentration 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, but would be if the axes were switched. This could lead to a discussion on choice of the independent variable and inverses.
Time: 33:08 to 37:19
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video goes over Newton's method and then shows that it has quadratic convergence so that the number of iterations needed for n digit precision is logarithmic. This is above and beyond what is learned in the standard calculus class, but it is an important point in any algorithm.
Video Link: https://www.youtube.com/watch?v=nfkX5W8-oBU&index=9&list=PL8A25592E6D32C753
Time: 45:29 to 49:06
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video goes over constraint propagation in graph coloring. The professor looks at the domain x < y - 1 and explains how this can improve efficiency by getting rid of possibilities. This will be difficult for beginning students to follow, but might be intriguing for students enjoy artificial intelligence.
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.
Time: 4:47 - 8:00
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video uses the principle of superposition to find the voltage when there are two resisters and two batteries. The equation's right hand side is a rational expression and when the resisters are put together, we get a sum or rational expressions. The equations look quite messy, but they are actually quite easy to solve. The professor skips the algebra steps, so it would be a good exercise to have the students work it out in class.
Time: 14:32 - 17:13
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video shows a very complicated circuit. The professor works out the corresponding math using standard techniques in simplifying rational expressions to simplify the voltage expression. The math is not that tough for the students to follow, but the circuit will be way over the students' heads. Based on the mathematical conclusion, the circuit is a subtractor.
Time: 26:37 - 28:56
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video goes over the particle wave duality of light. The professor humorously states that de Broglie got his Nobel prize for just doing basic algebra. The equations are shown on the PowerPoint and an instructor can ask the students to see if they can perform this "Nobel Prize" algebra too which is very simple even for an elementary algebra student.
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.
Time: 46:25 - 48:36
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video works out the math in order to solve for the voltage in a circuit in series. The diagram will be too complex for the students to understand and the algebra will be a challenge, but students may enjoy watching some advanced work. The instructor can highlight the one piece where the professor multiplies both the numerator and denominator by the same expression.
Time: 34:56 - 37:57
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video, similar to the above video, works out the math in order to solve for a voltage. The diagram will be too complex for the students to understand and the algebra will be a challenge, but students may enjoy watching some advanced work. This is actually an equation example, but there are complex rational expressions within it.
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-16
Time: 16:43 to 19:09
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video shoes the equation that relates the focal length to the object's distance to the parabolic mirror and the image's distance to the parabolic mirror. The equation is a rational equation. The professor doesn't so anything with it, so this is just a motivational clip for the student so see a rational equation applied to imaging.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-17
Time: 39:48 to 43:03
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video finds the vertical location on a spherical lens a light ray must go through in order for it to travel to the same location as though it want horizontally only. A lot of algebra is used including the parabolic approximation to the square root function. Some will be over the student's heads, but some will be clear.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-17
Time: 66:50 to 70:14
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at the equations that relate to a magnifying glass. All students should be able to relate to the application and the math shown can be followed with some work. The first step is to divide two simple rational expressions.
Time: 33:49 to 35:34
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video derives the formula for finding the second order half life. The equation is a rational expression and the algebra that the professor does is standard for rational equations. This is a great example of why knowing how to solve rational equations is so important.
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: Geometry
Video Link: https://www.youtube.com/watch?v=CgQ6U_ub5Uo&list=PL8A25592E6D32C753&index=15
Time: 6:12 to 12:52
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Anupam Basu
Teaching Ideas: This video explains how to prove geometry theorems using propositional logic. Students will get a sense of why understanding the formalities of geometry is important for understanding how computers come up with conclusions when given a language based question. Because of the symbols, this is a difficult concept but uses the same basic logic principles that are used in geometry.
Course Topic: Exponents
Time: 30:01 - 32:42
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video considers the wave function of the 1s orbital. There is an equation shown where an exponent is simplified, but the professor does not show the work. The instructor can pause the video at this point and see if the students can figure out how the work was done. It involves breaking up a 3/2 power to a 1/2 power cubed and combining two like powers.
Video Link: https://www.youtube.com/watch?v=pjFYiBEp1Ug&list=PLE73AA240E8655D16&index=8
Time: 27:48 - 32:33
University: Oxford
Course: Quantum Mechanics
Professor Name: James Binney
Teaching Ideas: This video explains the radioactive decay from U238 to Th234. The professor calculates the number of times that particle attempts to cross the barrier up to an order of magnitude. He uses the quotient rule for exponents along with negative exponents to come up with 1040 attempts as the average number of tries before it makes it. This is a great example that both uses the rules of exponents and provides a sense of the enormity of numbers and can lead to discussion about large numbers in general.
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 for 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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-14
Time: 54:59 to 56:48
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video uses a very complicated formula to show that the electrons in a circuit travel at the speed of light. The math is just working with scientific notation and exponent rules, but the physics is quite complicated. This is a major result in physics and students at the algebra level will at least understand a part of the derivation.
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.
Time: 36:32 - 37:32
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video shows how to find the radius of the nucleus of an atom using probabilistic techniques and backscattering. The math involves isolating r by taking the square root of both sides. The professor uses the 1/2 power instead of the square root symbol. There is a physical demo of this, but you have to go until 43:25 to see it through.
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:
x/n = m/N, N = mn/x
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.
Time: 11:00 - 14:33
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video shows how the electron was first discovered by Thompson. It uses proportions and some basic algebra including canceling and absolute values. The math is pretty gentle and the concept is something every student should already know. No numbers are given but the clip uses algebra at the level of a beginning algebra student.
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-9
Time: 67:56 to 69:02
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video derives the formula for the magnetic field that is created by a solid cable. The physics may be new to the students, but the solving for B is just division by its coefficient.
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.
Time: 30:34 to 33:38
University: MIT
Course: Principles of Chemical Science
Professor Name: Sylvia Ceyer
Teaching Ideas: This video shows the wave function for the 3S electron shell. The professor explains why understanding the wave function is important and then sets it equal to 0. The professor gives the two answers without showing any work, but with some help from their instructor students can work it out using the quadratic formula. The instructor will have to tell them to divide out the exponential and substitute x = r/a0.
Time: 40:08 to 42:46
University: MIT
Course: Principles of Chemical Science
Professor Name: Christopher Cummins
Teaching Ideas: This video sets up the equation that gives the change in the reactant when solved. The professor states that it is a cubic equation, but his example shows a quadratic equation after multiplying by the denominator. Since he does not give the solution, the students can be asked to find the solution using the quadratic formula. This will be a challenge for them since one of the constants is written in scientific notation.
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-16
Time: 16:43 to 17:32
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video explains that with a parabolic mirror, light shined in parallel rays will reflect onto the focus. No equations are given, but the picture is sketched. This is a relevant application of parabolas that can serve as a hook.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-16
Time: 40:39 to 46:20
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video proves that the geometric formula, directrix and focus, for the parabola gives the quadratic equation y2 = 4xf. This is an easy to follow derivation that can replace the instructor's derivation.
Time: 44:04 -48:20
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video looks at the graph of a mosfet amplifier. The professor shows how a region on the x-axis gets mapped to a region on the y-axis. Since it is a parabola, the y-axis range is much larger than the x-axis range clearly demonstrating how a parabola is an "amplifier" of the x-axis. This is a clear and memorable way to show what the graph of a function is all about. Students often just follow rules instead of get an intuitive feel for mathematical ideas. This will give them that intuitive feel. At the end, the professor shows a real amplifier and graphs both the x and y horizontally, but in different colors.
Video Link: https://www.youtube.com/watch?v=ySTr4TO3IiU&list=PL8A25592E6D32C753&index=32
Time: 37:10 to 39:14
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video talks about learning. The example given is a table of numbers 1 through 4 and their squares. The professor explains that we can only hypothesize the solution for being the squaring function. This is a unique example of the parabola in that it is used as an example of how we learn.
Course Topic: A Survey of Graphs
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-22
Time: 38:53 to 40:14
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video looks at three possible potential wells that are piecewise, parabolic and y = 1/x graphs. The professor doesn't do much with them, but does sketch the graphs. This is a vey complex application of a survey of graphs that students will not understand at all but sometimes it is effective to show them what advanced physics looks like.
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 appreciate 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: Graphing With Roots and Denominators
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-13
Time: 17:37 to 21:36
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video graphs the function of current vs. driving force for an LRC circuit. There are no asymptotes, but there is a removable singularity and a maximum to investigate that corresponds to the "heard frequency". This is a nice application of graphing based on an equation's features that shows students how radio, cell phones and other devices that uses frequency detection work. If there is time for another few minutes, there is further explanation on changing the station.
Course Topic: Solving Nonlinear Systems of Equations
Time: 23:04 -27:35
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video solves the nonlinear system of equations (line and parabola) algebraically to find the point at which an amplifier changes. The computation is messy and involves the quadratic formula with parameters as the coefficients, but it demonstrates the algebra well.
Course Topic: Determining a Rational Equation from its Graph (College Algebra)
Time: 30:48 - 34:06
University: MIT
Course: Principles of Chemical Science
Professor Name: Elizabeth Vogel Taylor
Teaching Ideas: This video shows the energy as a function of distance between two hydrogen atoms. There is a vertical and horizontal asymptote, an x-intercept, and a minimum. The professor doesn't do this, but a good exercise for college algebra students would be to try to come up with a rational equation that has this graph. The x-axis units are not given, so this is an open ended problem. This will be challenging, so group work is recommended.
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.
Time: 37:00 -41:17
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video explains how an amplifier works. The engineering principles will be way over the heads of the students, but the piecewise function is well explained. The pieces are horizontal and quadratic. This is an application that everyone can relate to even if the engineering is tough. If there is time for another five minutes of play, there is a T-table drawn that clearly explains why it is an amplifier (quadratics stretch out the y-axis). Then he does an physical experiment.
Video Link: https://www.youtube.com/watch?v=5K1to94YQtU&list=PL8A25592E6D32C753&index=28
Time: 54:20 to 55:15
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Anupam Basu
Teaching Ideas: This video goes over a fuzzy logic formula called the S Curve. It is a nice example of a piecewise defined function that comes up in AI. There are four pieces in the definition and the graph and equations are given.
Course Topic: Exponential Functions
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-8
Time: 22:38 to 23:25
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video sketches the graph of an exponential curve that represents the charge on a capacitor after starting it up as a function of time. This is a variation on the standard exponential graph that is reflected and shifted. This is a nice example of how to transform the standard exponential.
Time: 6:02 to 10:48
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video finds the computing time to compute the nth Fibonacci number. The time turns out to be exponential in n. This is a fun example that mixes math and computer science.
Time: 10:25 -12:42
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video investigates a fictitious exponential circuit. The professor shows the equation and then graphs it. He points out the y-intercept and shape. This is a very simple example of an exponential.
Time: 45:51 to 47:02
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video sketches the graph of an exponential function that represents the voltage over time of a circuit with a capacitor. This particular exponential has both a vertical and horizontal shift and the professor explains how to interpret these in terms of how the capacitor works. This is a great integration of math and circuit analysis that can be used both to present how to solve use shifting to graph a general exponential and why one might need to sketch such a graph. If you watch it for two more minute, you get an actual demonstration of the circuit being read by a meter.
Time: 27:33 - 32:22
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video looks at a very complicated circuit whose voltage equation simplifies to a exponential. The professor goes over both when the function is exponential decay and exponential growth. This is an excellent way to show students the difference between the shapes of the graphs for growth and decay.
Video Link: http://www.extension.harvard.edu/open-learning-initiative/bits Then go to Moore's Law: Exponential Growth
Time: 0:00 to 3:29
University: Harvard
Course: Bits
Professor Name: Gordon McKay
Teaching Ideas: This video explains exponential growth and Moore's Law. It shows Kurzweil's graph of Moore's Law that includes species on the y-axis. This is the most powerful clip of all clips in this entire collection since it explains why the biggest event in the future of mankind will happen in the mid 2020's. I have created a PowerPoint to go along with it that can be found here: https://docs.google.com/presentation/d/1YqHJcex6a5rHZO1gM5OTmuJ5cwLL9GvCyYszBypjqYU/edit?usp=sharing
Video Link: http://www.extension.harvard.edu/open-learning-initiative/bits Then go to Moore's Law: Opening a Bike Lock
Time: 0:00 to 3:41
University: Harvard
Course: Bits
Professor Name: Harry Lewis
Teaching Ideas: This video first explains how exponentials work. The professor explains the growth of numbers as you repeatedly multiply them. This leads to the definition of bit strings and the fact the an 8 bit number can represent any number up to 256. There are no visuals other than seeing the professor speak, but it is a simple example of exponents being used.
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.
Time: 19:58 to 22:24
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video explains that a sorting algorithm with a binary search takes O(nlgn) time where lgn is the log base 2 of the number of items to sort, n. This is a quick clip that will need some additional explanation by the instructor in order for the students to have an idea of what is going on. In particular, above this example on the board is a pairwise swap search that takes O(n2) time. Students can be asked why the binary search is faster than the compare and swap search.
Time: 34:08 to 38:24
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video explains how to arrive at the time for the above binary search sorting algorithm. The professor creates a tree to represent the times at each stage. This will take some explanation in order for the students to understand what the professor is doing, but can be helpful to show students that logarithms are needed in computer programming.
Time: 2:16 to 4:17
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video compares the height of a balanced binary tree to a completely unbalanced tree. The balanced tree is of order lg(n) or log2(n) and the completely unbalanced tree is of order n. This clearly demonstrates the difference between the logarithm function and the linear function and works well to remind students why the logarithm function is important.
Time: 49:50 to 52:01
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video finds the time for the Radix Sort algorithm which is kind of like alphabetizing in a dictionary. The proof uses the property logn(nk) = k. The specific steps are not shown, so an instructor can ask the students to fill in the details. The instructor will have to give background information about what the professor is doing and what Radix Sort is all about.
Time: 28:05 - 30:14
University: MIT
Course: Circuits
Professor Name: Anant Agarwal
Teaching Ideas: This video starts with an exponential equation that relates the time it take to reach a voltage in a circuit to the voltage. The professor solves the equation for time to get the inverse. The steps are not shown, so it would be a good exercise to have the students fill in the steps.
Time: 27:56 - 31:45
University: MIT
Course: Circuits
Professor Name: Anant Agarwal's Substitute: Name not given
Teaching Ideas: This video explains that the relation between time and voltage is an exponential. The professor then finds the time when the capacitor will lose enough charge to produce a bad signal. The math involves solving a simple exponential. The professor does not show the work, so students can be asked to fill them in.
Time: 18:46 to 19:54
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video displays the equation that relates the change in Gibbs free energy to the equilibrium constant and the reaction quotient in a chemical equation. It makes use of the quotient rule for logarithms with the natural log, ln. This is a short clip that will give students an idea of where logs and log properties are used in chemistry.
Time: 27:50 to 30:55
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video subtracts two equation that have logs and uses the quotient rule for logs to find a new equation, The Van't Hoff Equation, that relates the ratio of the equilibrium constants to the corresponding temperatures during the reaction. The professor goes over the math quickly, so an instructor will want to pause it and have the students fill in the steps that are not written down. At the end of the clip, she uses the equation to compare equilibriums for an exothermic reaction. This is a useful way to introduce the quotient rule for logarithms.
Time: 15:51 to 17:06
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video calculates the equilibrium constant for pure water. The professor shows that lnK = -32.24 and then states that K = 1.0 x 10-14. At this point the instructor can ask the students how she arrive at this value for K. This a a great example in turning an exponential equation into a log equation. The students will need their calculators to verify the value of K.
Time: 19:09 to 21:09
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video calculates the pK for water, pKw. The professor uses both the product rule for logs and explains that log 1014 = 14. This is a great example of applying the rules of logarithms in chemistry.
Time: 12:05 to 15:04
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video finds the the relationship between the pKa and pKb (pK for the acid and base in a reaction) by comparing it to the pKw (pK for water). The computation involves taking the log of both sides and using the product rule for logarithms. It is a pretty simple example, but the students will need some explanation from their instructor about what the chemistry is.
Time: 20:44 to 22:48
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video derives the Henderson Hasselbalch Equation that gives the pH of a solution. The derivation uses the product rule for logarithms. The presentation is easy to follow and can show students where logs are used in chemistry.
Time: 38:12 to 41:04
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video shows how to calculate half-life. The professor talks about its importance in determining how long one must store nuclear waste. She goes through the ln equation and simplifies, although there is an error in her parentheses. She finally gets the equation t1/2 = 0.6931/k. This is a relevant example that uses the natural log function.
Time: 2:13 to 2:58
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video explains the historical work of Arrhenius that shows if you plot the reciprocal of the temperature vs. the ln of the rate constant, you get a straight line. This is a creative and prolific use of the ln function. The explanation is clear and the historical references are interesting.
Time: 13:48 to 17:00
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video derives the equation that give the ln of the quotient of rate equations to digest sucrose vs. the energy and temperature for a reaction. The professor uses the quotient rule for logs and other algebra, but does not show the steps. Students can be asked to fill in the steps. At the end the professor explains in a humorous way why this shows that it is essential for the body to regulate temperature.
Time: 23:10 to 23:59
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video finds the ratio of the concentrations of two biochemical compounds but using the pH equation. The math involves isolating a log and then converting it to an exponential. The science is technical but may be interesting to students who like biology.
Time: 15:32 to 19:52
University: MIT
Course: Principles of Chemical Science
Professor Name: Christopher Cummins
Teaching Ideas: This video looks at an equation that relates the electrical work to the Gibbs free energy, temperature and the equilibrium constant. The equation involves a logarithm and the professor solves for the K which is inside the log. He does not show the work, so students can be asked to fill in the details by turning it into an exponential equation.
Video Link: https://www.youtube.com/watch?v=vuuJ8cGcl30
Time: 11:40 to 13:07
University: Harvard
Course: Introduction to Computer Science
Professor Name: D. J. Malan
Teaching Ideas: This video is a summary of a binary search algorithm. The professor explains that the time to complete is log2n where n is the number of thinks to search. This will not work unless the instructor has the class try it out before watching the video clip. Here's what needs to be done. The instructor randomly hands each student a unique number from 1 to n where n is the number of people in the class. The goal is to find the largest number in as little time as possible. First the instructor has the students all stand up. They recite their numbers one at a time and sit down if theirs is not the largest yet recited. This takes n recitations. Next all students stand up and in the first step compare with another student. The largest of the pair stays standing. Then the ones still standing pair up and compare with the larger of the pair staying standing and so on. This takes log2n steps. The video clip shows the graphs of n, n/2 and log2n on the same axes and demonstrates why the binary search algorithm is so much faster. This is a great activity for learning about logarithms.
Video Link: https://www.youtube.com/watch?v=-frfAZoaqDw&list=PL2Q_sOQgsm24ybtnVq75-TgZd86lMjm9m&index=8
Time: 14:03 to 17:54
University: Harvard
Course: Introduction to Computer Science
Professor Name: D. J. Malan
Teaching Ideas: This video explains the binary search algorithm using an activity involving hidden numbers. The professor has a list of hidden sorted numbers and asks a student volunteer to find the number 7. He coaches her to use the binary search algorithm. Then after it is found explains that it is found in log2n time. The professor gives an elegant explanation of why this is a logarithm. An instructor can easily do this activity in the class and have the students explore the logarithm function.
Video Link: https://www.youtube.com/watch?v=Khe8X5tn3i8&list=PL0A0E275BC354C934&index=4
Time: 1:40:42 to 17:54
University: Missouri University of Science and Technology
Course: Engineering Geology and Geotechnics
Professor Name: David Rogers
Teaching Ideas: This video presents the graph of the degree of consolidation as a function of the log of the time. The professor does not do any algebra but this is a practical use of logs that relates to road construction.
Time: 24:50 to 27:51
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video defines the disorder of a set in terms using log2. The professor writes down the function and looks at a special case which when plugged in uses the identities -log21/2 = log22 = 1. The instructor will need to explain that the professor is using this in order to decide what question to ask first in order to get at an answer as quickly as possible when there are many possible questions to ask. As a side note, this formula is also used to measure diversity in biology. This is a relevant application of logarithms.
Video Link: https://www.youtube.com/watch?v=J1jlwaMIYD0&list=PL48DE756A5800ED5F&index=3
Time: 13:21 to 14:32
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video looks at the exponential growth model of a rabbit. The professor first shows the regular model and then looks at the same function using a log scale. He notes that the log scale gives a linear pattern for the data. This is a nice use of logarithms and it at an very elementary level.
Video Link: https://www.youtube.com/watch?v=mN3xVvjE4ms&index=13&list=PL48DE756A5800ED5F
Time: 7:40 to 10:00
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video presents the Species-Area Effect which relates the number of species expected within a habitat of a given area. The equation is S = cAz, where S is the number of species and A is the area. The professor takes the log of both sides to get: ln(S) = ln(c) + z*ln(A). The professor does not show the log properties work, so this would make an excellent exercise to ask the students to fill in the details. The professor emphasizes that the reason to do this is to make this a linear equation with y = ln(S) and x = ln(A). Then you get y = zx + ln(c) which is the slope intercept form of a line.
Video Link: https://www.youtube.com/watch?v=KdUR8Av4yW4&list=PLFA75A0DDB89ACCD7&index=8
Time: 9:33 to 13:45
University: University of Chicago
Course: Global Warming
Professor Name: David Archer
Teaching Ideas: This video goes over the relationship between the CO2 concentration and the energy increase. He shows that this is logarithmic and the doubling time is of interest. It is very simple to understand and will work great when introducing the logarithm. The professor emphasizes that
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.
Time: 20:49 to 25:38
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video finds a formula for the number of nodes of a almost balanced binary try given the height. The solution involves so many mathematical gems that it may be worth showing even if the application is above most algebra student's heads. It involves the Fibonacci sequence, the Golden Ratio, the change of base theorem, and changing a log equation to an exponential equation.
Time: 25:48 to 28:10
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video shows an alternative estimate for the number of nodes in the almost balanced binary tree. It tackles exactly the same problem as the clip above, but does an approximation to simplify the derivation. At the end the professor takes an exponential equation and turns it into a log equation. He talks through the work but does not write down the steps on the board. One step is the power rule for logs, so this clip would be appropriate to show during a lecture on properties of logarithms. It is not as flashy an exciting as the previous clip, but will be easier for the students to grasp. Both clips can also be shown if there is time.
Time: 32:44 to 35:13
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video shows the log equation and the corresponding exponential equation that relates Gibbs free energy to the equilibrium constant for a reaction. An instructor can ask the student to derive this relationship by solving the log equation or the exponential equation for the other variable. Next the professor shows a clicker question that asks the students to decide what the Gibbs energy will look like if the equilibrium constant is large. This is a wonderful activity that the instructor's students can work on to see if they can answer it. It quizzes their ability to understand the shape of log and exponentials.
Course Topic: The Circle
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-16
Time: 48:28 to 49:30
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video works out where the focal point is for a spherical mirror which is drawn as a circle. The professor recalls what the equation of a circle is and FOILs it out to make it look like the equation of a parabola. The professor uses the word "sphere" buy the equations are for circles. The clip ends by having the professor asking to compare the circle to the parabola. The instructor can ask the student to compare and note that f = R/2. After the clip is over, the professor does go over it, but it is along explanation and may not be worth the time to show it.
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.
Video Link: http://oyc.yale.edu/physics/phys-201/lecture-16
Time: 34:28 to 37:10
University: Yale
Course: Fundamentals of Physics II
Professor Name: Ramamurti Shankar
Teaching Ideas: This video poses the question that asks if you are in an steel elliptical room and you and your rival are standing at the foci, what direction should you shoot your gun so that you will hit your enemy? The explanation is tongue and cheek but the professor shows hints of the geometrical argument. This is a fun video to lighten up the subject of ellipses. If you run it for another couple of minutes, you get the argument that involves the sum of the distances to the foci is a constant.
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.
Video Link: http://oyc.yale.edu/psychology/psyc-123/lecture-9
Time: 00:50 to 1:50
University: Yale
Course: The Psychology, Biology and Politics of Food
Professor Name: Kelly Brownell
Teaching Ideas: This video presents the original work by Malthas in the 17th century that states that the growth in food production is arithmetic while the growth in population is geometric which will eventually result in mass starvation. This is a great example of how arithmetic and geometric sequences are used.
Video Link: https://www.youtube.com/watch?v=6f59FQGDOa8&index=21&list=PL48DE756A5800ED5F
Time: 33:14 to 34:37
University: UC Berkeley
Course: Environmental Science
Professor Name: (Not Provided in Video)
Teaching Ideas: This video looks at the population each year when there is constant annual population growth. The professor derives the equation that gives the nth term of a geometric sequence. The presentation looks a bit different than a sequence of numbers with commas, so the instructor will need to show at the end an example and explain why this is a sequence.
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.
Time: 10:57 to 14:28
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video finds the time it takes if one grows a table by one each time it needs to expand vs. doubling it each time it needs to expand. The first is the arithmetic series: 1 + 2 + 3 + ... + n and the second is the geometric series: 1 + 2 + 4 + ... + n. The professor shows that the first is of order n2 and the second is of order n. It would be a good exercise for the students with some hints from the instructor to fill in the details.
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
Time: 50:00 to 50:54
University: MIT
Course: Introduction to Algorithms
Professor Name: Srini Devadas
Teaching Ideas: This video creates a tree diagram for a recursive algorithm corresponding to a search. The professor computes that total computer work that is needed. The last term of the computation dominates the sum of all the previous ones. The professor does not actually sum the geometric series, but the students can be asked to sum them up and verify that the sum is less than the last term. This is a difficult but important application that the students can work on to experience what computer scientists do.
Time: 35:15 to 40:27
University: MIT
Course: Artificial Intelligence
Professor Name: Patrick Winston
Teaching Ideas: This video goes over the calculations involved to determine the amount if work it takes to conduct progressive deepening in the Mini Max game using artificial intelligence. This is the process that computers go through when they play chess or any other two player game. The derivation uses a finite geometric series and the professor actually derives the formula that is derived in an algebra class. This is an application that will interest many students since most play computer games.
Video Link: https://www.youtube.com/watch?v=ayP0KCeBK_U&list=PL8A25592E6D32C753&index=4
Time: 41:30 to 44:48
University: India Institute of Technology
Course: Artificial Intelligence
Professor Name: Sudeshna Sarkar
Teaching Ideas: This video shows how long a breadth first search will take in the worst case scenario. The formula involves a finite geometric series which the professor states but does not derive. At the end she gives the example of depth 12 with 11 children at each node and comes up with 35 years to process using 111 terabytes of memory. This is a short and simple example of geometric series being used.
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: Mix of Many College Algebra Topics
Time: 23:27 to 28:33
University: MIT
Course: Introduction to Algorithms
Professor Name: Erik Demaine
Teaching Ideas: This video proves the fact that a particular sorting algorithm is on the order of nlg(n) time where lg(n) = log2n. The argument involves permutations, factorials, the product to sum formula for logs, the quotient to difference formula for logs, the graph of the log function, series notation, and even hints at an integral and Sterling's formula although he does not use calculus. This would be a great capstone video to show during the last week of a college algebra course.
Time: 11:32 to 15:49
University: MIT
Course: Principles of Chemical Science
Professor Name: Catherine Drennan
Teaching Ideas: This video find the pH of a buffer solution. The work involves rational expressions, approximation, solving linear equations and quadratic equations, and calculating a logarithm. This can actually be given in an intermediate algebra class with some assistance from the instructor. In class, you will want to ask the students to show the work in solving both the linear equation and the quadratic equation. At the end the students can explain why the approximation technique that the professor uses is reasonable.