The purpose of this task is to provide an opportunity for students …
The purpose of this task is to provide an opportunity for students to reason about equivalence of equations. The instruction to give reasons that do not depend on solving the equation is intended to focus attention on the transformation of equations as a deductive step.
Student teams are challenged to evaluate the design of several liquid soaps …
Student teams are challenged to evaluate the design of several liquid soaps to answer the question, “Which soap is the best?” Through two simple teacher class demonstrations and the activity investigation, students learn about surface tension and how it is measured, the properties of surfactants (soaps), and how surfactants change the surface properties of liquids. As they evaluate the engineering design of real-world products (different liquid dish washing soap brands), students see the range of design constraints such as cost, reliability, effectiveness and environmental impact. By investigating the critical micelle concentration of various soaps, students determine which requires less volume to be an effective cleaning agent, factors related to both the cost and environmental impact of the surfactant. By investigating the minimum surface tension of the soap, students determine which dissolves dirt and oil most effectively and thus cleans with the least effort. Students evaluate these competing criteria and make their own determination as to which of five liquid soaps make the “best” soap, giving their own evidence and scientific reasoning. They make the connection between gathered data and the real-world experience in using these liquid soaps.
The purpose of this task is to identify the structure in the …
The purpose of this task is to identify the structure in the two algebraic expressions by interpreting them in terms of a geometric context. Students will have likely seen this type of process before, so the principal source of challenge in this task is to encourage a multitude and variety of approaches, both in terms of the geometric argument and in terms of the algebraic manipulation.
In this undergraduate level seminar series topics vary from year to year. …
In this undergraduate level seminar series topics vary from year to year. Students present and discuss the subject matter, and are provided with instruction and practice in written and oral communication. Some experience with proofs required. The topic for fall 2008: Computational algebra and algebraic geometry.
Seminar for mathematics majors. Students present and discuss the subject matter and …
Seminar for mathematics majors. Students present and discuss the subject matter and write up exercises. Topic for Fall 2002: Classical geometry, beginning with Euclid's Elements and continuing to applications of Galois theory that solve the geometry problems of antiquity. No prior knowledge of Galois theory required. Instruction and practice in oral communication provided.
This word problem is based estimating the height of a person over …
This word problem is based estimating the height of a person over time. Note that there is a significant amount of rounding in the final answer. This is because people almost never report their heights more precisely than the closest half-inch. If we assume that the heights reported in the task stem are rounded to the nearest half-inch, then we should report the heights given in the solution at the same level of precision.
Build coin expressions, then exchange them for variable expressions. Simplify and evaluate …
Build coin expressions, then exchange them for variable expressions. Simplify and evaluate expressions until you are ready to test your understanding of equivalent expressions in the game!
The typical system of equations or inequalities problem gives the system and …
The typical system of equations or inequalities problem gives the system and asks for the graph of the solution. This task turns the problem around. It gives a solution set and asks for the system that corresponds to it. The purpose of this task is to give students a chance to go beyond the typical problem and make the connections between points in the coordinate plane and solutions to inequalities and equations. Students have to focus on what the graph is showing.
Represent inequalities on a number line. Represent inequalities using interval notation. Use …
Represent inequalities on a number line. Represent inequalities using interval notation. Use the addition and multiplication properties to solve algebraic inequalities and express their solutions graphically and with interval notation. Solve inequalities that contain absolute values. Combine properties of inequalities to isolate variables, solve algebraic inequalities, and express their solutions graphically. Simplify and solve algebraic inequalities using the distributive property to clear parentheses and fractions.
Sal solves a linear system with 3 variables by representing it with …
Sal solves a linear system with 3 variables by representing it with an augmented matrix and bringing the matrix to reduced row-echelon form. Gaussian Elimination is the method, reduced row echelon is just the final result.
Introduction to a selection of mathematical topics that are not covered in …
Introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory and optimization techniques. Emphasis on basic ideas and on applications in mechanical engineering. Selection will change every year. This course forms an introduction to a selection of mathematical topics that are not covered in traditional mechanical engineering curricula, such as differential geometry, integral geometry, discrete computational geometry, graph theory, optimization techniques, calculus of variations and linear algebra. The topics covered in any particular year depend on the interest of the students and instructor. Emphasis is on basic ideas and on applications in mechanical engineering. This year, the subject focuses on selected topics from linear algebra and the calculus of variations. It is aimed mainly (but not exclusively) at students aiming to study mechanics (solid mechanics, fluid mechanics, energy methods etc.), and the course introduces some of the mathematical tools used in these subjects. Applications are related primarily (but not exclusively) to the microstructures of crystalline solids.
This problem provides students with an opportunity to discover algebraic structure in …
This problem provides students with an opportunity to discover algebraic structure in a geometric context. More specifically, the student will need to divide up the given polygons into triangles and then use the fact that the sum of the angles in each triangle is 180_.
Parts (d) and (e) of this task constitute a very advanced application …
Parts (d) and (e) of this task constitute a very advanced application of the skill of making use of structure: in (d) students are being asked to use the defining property of even and odd functions to manipulate expressions involving function notation. In (e) they are asked to see the structure in the system of two equations involving functions.
Playing the role of engineers in collaborations with the marketing and production …
Playing the role of engineers in collaborations with the marketing and production teams in a chocolate factory, students design a container for a jumbo chocolate bar. The projects constraints mean the container has to be a regular trapezoidal prism. The design has to optimize the material used to construct the container; that is, students have to find the dimensions of the container with the maximum volume possible. After students come up with their design, teams present a final version of the product that includes creative branding and presentation. The problem-solving portion of this project requires students to find a mathematical process to express the multiple variables in the prism’s volume formula as a single variable cubic polynomial function. Students then use technology to determine the value for which this function has a maximum and, with this value, find the prism’s optimal dimensions.
Roller coasters projects are frequently used in middle and high school physics …
Roller coasters projects are frequently used in middle and high school physics classes to illustrate the principle of conservation of mechanical energy. Potential energy transforms to kinetic energy and vice versa, with gravity being the driving force during the entire process. Even though friction force is mentioned, it is rarely considered in the velocity calculations along the coasters’ paths. In this high school lesson, the friction force is considered in the process. Using basic calculus and the work-energy theorem for non-conservative forces, the friction along a curved path is quantified, and the cart’s velocity along this path is predicted. This activity and its associated lesson are designed for AP Calculus. Practice problems/answers, a PowerPoint® presentation and student notes are provided.
This 10-minute video lesson shows that three points uniquely define a circle …
This 10-minute video lesson shows that three points uniquely define a circle and that the center of a circle is the circumcenter for any triangle that the circle is circumscribed about.
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