Students learn how simple machines, including wedges, were used in building both …
Students learn how simple machines, including wedges, were used in building both ancient pyramids and present-day skyscrapers. In a hands-on activity, students test a variety of wedges on different materials (wax, soap, clay, foam). Students gain an understanding of how simple machines are used in engineering applications to make our lives and work easier.
This video lesson presents a real world problem that can be solved …
This video lesson presents a real world problem that can be solved by using the Pythagorean theorem. The problem faces a juice seller daily. He has equilateral barrels with equal heights and he always tries to empty the juice of two barrels into a third barrel that has a volume equal to the sum of the volumes of the two barrels. This juice seller wants to find a simple way to help him select the right barrel without wasting time, and without any calculations - since he is ignorant of Mathematics. The prerequisite for this lesson includes knowledge of the following: the Pythagorean theorem; calculation of a triangles area knowing the angle between its two sides; cosine rule; calculation of a circle's area; and calculation of the areas and volumes of solids with regular bases.
This lesson teaches students about the history of the Pythagorean theorem, along …
This lesson teaches students about the history of the Pythagorean theorem, along with proofs and applications. It is geared toward high school Geometry students that have completed a year of Algebra.
Students analyze a cartoon of a Rube Goldberg machine and a Python …
Students analyze a cartoon of a Rube Goldberg machine and a Python programming language script to practice engineering analysis. In both cases, they study the examples to determine how the different systems operate and the function of each component. This exercise in juxtaposition enables students to see the parallels between a more traditional mechanical engineering design and computer programming. Students also gain practice in analyzing two very different systems to fully understand how they work, similar to how engineers analyze systems and determine how they function and how changes to the system might affect the system.
Working in small groups, students complete and run functioning Python codes. They …
Working in small groups, students complete and run functioning Python codes. They begin by determining the missing commands in a sample piece of Python code that doubles all the elements of a given input and sums the resulting values. Then students modify more advanced Python code, which numerically computes the slope of a tangent line by finding the slopes of progressively closer secant lines; to this code they add explanatory comments to describe the function of each line of code. This requires students to understand the logic employed in the Python code. Finally, students make modifications to the code in order to find the slopes of tangents to a variety of functions.
This lesson aims to help students with quadratic functions y = ax2 …
This lesson aims to help students with quadratic functions y = ax2 + bx + c. This is the next step after linear functions bx + c. The lesson begins with three quadratics and their graphs (three parabolas): y = x2 - 2x + (0 or 1 or 2). The prerequisite or co-requisite is some working experience with algebra, like factoring x2 -2x into x(x-2). The objective is to connect four things: the formula for y, the graph of y (a parabola), the roots of y and the minimum or maximum of y. The particular example y = x2 – 2x could be repeated by the teacher, for emphasis. The lesson will take more than one class period (and this is deserved!). The breaks allow time to consider parabolas starting with -x2 and opening downward. A physical path would be one (dangerous?) activity.
Video lecture on quadratic equations and their graphs. The video connects the …
Video lecture on quadratic equations and their graphs. The video connects the equation, the graph, the roots, and the minimum or maximum of the quadratic function.
This question provides students with an opportunity to see expressions as constructed …
This question provides students with an opportunity to see expressions as constructed out of a sequence of operations: first taking the square root of n, then dividing the result of that operation into s. Students studying statistics encounter the expression in this question as the standard deviation of a sampling distribution with samples of size n when the distribution from which the sample is taken has standard deviation s.
This textbook is intended to support courses that bridge the divide between …
This textbook is intended to support courses that bridge the divide between mathematics typically encountered in U.S. high school curricula and the practical problems that natural resource students might engage with in their disciplinary coursework and professional internships.
This course develops logical, empirically based arguments using statistical techniques and analytic …
This course develops logical, empirically based arguments using statistical techniques and analytic methods. Elementary statistics, probability, and other types of quantitative reasoning useful for description, estimation, comparison, and explanation are covered. Emphasis is on the use and limitations of analytical techniques in planning practice.
A two-semester subject on quantum theory, stressing principles: uncertainty relation, observables, eigenstates, …
A two-semester subject on quantum theory, stressing principles: uncertainty relation, observables, eigenstates, eigenvalues, probabilities of the results of measurement, transformation theory, equations of motion, and constants of motion. Symmetry in quantum mechanics, representations of symmetry groups. Variational and perturbation approximations. Systems of identical particles and applications. Time-dependent perturbation theory. Scattering theory: phase shifts, Born approximation. The quantum theory of radiation. Second quantization and many-body theory. Relativistic quantum mechanics of one electron. This is the second semester of a two-semester subject on quantum theory, stressing principles. Topics covered include: time-dependent perturbation theory and applications to radiation, quantization of EM radiation field, adiabatic theorem and Berry's phase, symmetries in QM, many-particle systems, scattering theory, relativistic quantum mechanics, and Dirac equation.
This task asks students to find the amount of two ingredients in …
This task asks students to find the amount of two ingredients in a pasta blend. The task provides all the information necessary to solve the problem by setting up two linear equations in two unknowns.
This task has some aspects of a mathematical modeling problem (SMP 4) …
This task has some aspects of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are given all the relevant information on the nutritional labels, but they have to figure out how to use this information. They have to come up with the idea that they can set up two equations in two unknowns to solve the problem.
This tasks is an example of a mathematical modeling problem (SMP 4) …
This tasks is an example of a mathematical modeling problem (SMP 4) and it also illustrates SMP 1 (Making sense of a problem). Students are only told that there are two ingredients in the pasta and they have a picture of the box. It might even be better to just show the picture of the box, or to bring in the box and ask the students to pose the question themselves.
The purpose of the task is to show students a situation where …
The purpose of the task is to show students a situation where squaring both sides of an equation can result in an equation with more solutions than the original one. The reason for this is that it is possible to have two unequal numbers whose squares are equal.
In this number tracing activity students create a rainbow number line. This …
In this number tracing activity students create a rainbow number line. This can be a colorful tool with a personal connection to the student that may be used as a reference. It can serve as a visual and motor reminder when reading and writing numbers because the student went through the tracing motion.
In this task students are asked to analyze a function and its …
In this task students are asked to analyze a function and its inverse when the function is given as a table of values. In addition to finding values of the inverse function from the table, they also have to explain why the given function is invertible.
This task requires interpreting a function in a non-standard context. While the …
This task requires interpreting a function in a non-standard context. While the domain and range of this function are both numbers, the way in which the function is determined is not via a formula but by a (pre-determined) sequence of coin flips. In addition, the task provides an opportunity to compute some probabilities in a discrete situation.
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