This module explores whether it’s time to buy an electric car by examining its cost to own and convenience to operate. The module has two stand-alone parts that can be used together or separately. The first component examines the cost of owning anelectric car relative to a traditional hybrid or a gasoline-powered car. Students learn the basics of building a cost model, making assumptions, and confronting uncertainty through scenario analysis with a spreadsheet. The second component looks at convenience by asking the question, “Can I get there from here?” In so doing, it introduces the correspondence between maps and graphs, relates graph connectivity to vehicle range, and explores locating charging stations to ensure that feasible routes exist. This module is appropriate for mathematics classes or other types of classes that address technology in society, business, or life skills.
When a group of people must decide on whom to select out of a group of eligible candidates, how do they decide? Who should have input into the decision?
Given all the people eligible for a heart transplant how do you decide who gets one?
Given all the people eligible to be CEO for a company, how do you decide?
How does a professional sports team choose a player in the draft and how does a player maximize his position? How does the NFL promote fairness among teams?
The students develop ranking procedures for both heart transplantation and the NFL draft. They examine the similarities and the differences, and they look at difficulties unique to each context. They consider how to the measure “success” of their rankings, and compare various alternatives based on spreadsheets and solutions from other members of their class. This module is appropriate for mathematics, computer science, and social studies classes in grades 9-12.
3) Network Capacity Expansion and UtilizationThis module uses networks familiar to students (text messaging, cell phone, Internet) as a motivating point to model capacity and demand in simple networks, such as school hallways, and then builds on this to consider more complicated networks. Students develop the skills and perspectives used in modeling real-world situations using networks as an example. They simulate the likelihood of congestion delay using rolls of dice (and later Excel spreadsheets) to consider issues of capacity when constructing networks and/or routes. Students come to understand the difference between average and peak demand and the costs associated with adding new capacity. As students run simulations many times, and as they look at more complex simulations, they see the essential role that computers play in this approach. This module is appropriate for mathematics and computer science classes grades 11-12.
4) Internet Privacy ModuleHow can people use and enjoy technology such as social networks (like Facebook and Twitter) while still having some kind of privacy? Through a series of case studies students explore uses of data that are intrinsic to the value that users gain from a social network site, as well as uses of data that are valuable for business resource (such as use for advertising). Students evaluate the case studies to determine the effects on the overall usefulness of the site to its users and to its owners based on a variety of privacy policies and privacy policy settings. A simple example of computing without revealing information is another case study (e.g., the pricing of crops without revealing the quantities of crops grown). The module incorporates critical analysis of data collection and sharing strategies, management of interfaces, and a focus on the interplay among technology, society, and policy. This module is appropriate for social studies, media studies, and computer science classes in grades 9-12.
5) Tomography: A Geometric and Computational Approach Module
Tomography is the science of examining internal structure with external measurements. Most people think of tomography
in the context of medical testing, such as CT scans, but tomography is also used in food safety analysis, ocean acoustics, oil pipelines, optics, etc. any time it is impossible to directly look inside something. Students tackle activities in which they are challenged to determine what is inside some object. They study how CT scan images of an object are created using 3-D reconstruction of 2-D slices of the object using shadows, pin prints, graphs, and more. The main questions of the module are: How can 3-D images be created from 2-D images (i.e. slices) of it?; How much computational power and skill are required to do these reconstructions; and what do they depend on? This module is appropriate for high school classes in computer science, mathematics, biology, environmental science, and physics.
A brief history of the use of codes in fighting wars launches the module. This includes the role of Native Americans as codetalkers during both World Wars and the use of secure communication to both inform and misinform. The module then moves to current uses of codes, such as for protecting credit card information in online purchasing. Students are encouraged to think about how one produces a foolproof code for transmission of information and to think of and compare possible encoding devices and/or schemes. The module discusses challenges faced in developing a foolproof encryption that can still be decrypted. The role of codes in data compression is also discussed and placed in the context of genetic codes. Specific codes such as binary codes, picture codes, codebooks, distortion codes, RSA codes, and others are introduced. Ciphers are distinguished from codes, and their computational issues discussed. This module is appropriate for high school classes in computer science and mathematics. The one-page teasers could be used in social studies, history, or biology classes as well.
In 1998, the National Resident Match Program changed how they match medical students with hospitals for their residencies. Motivating this change was a concern about fairness. This module looks at a variety of different types of matching problems and discusses the properties that are desirable in each case. Students learn about the classic Gale Shapley algorithm as a means of understanding the notion of stability and fairness as two such desiderata. They discuss how to measure fairness in problem instances that admit multiple stable matchings. Through several prompts, students are motivated to define and compute several different measures of fairness and to compare and contrast their various definitions of fairness. The module concludes by presenting recent results from the research literature that show a surprising convergence of a local measure of fairness, that considers each individual’s median level of satisfaction, with a global measure of fairness based on the median distance measured within a partially ordered set defined on the set of stable matchings. This module is appropriate for high school classes in computer science and mathematics and the associated mini-module is appropriate for social studies classes.
2) Polynomiography and Art
This mathematically inspired computer medium is based on algorithmic visualizations of one of the most basic and fundamental tasks in sciences and mathematics: solving a polynomial equation. Polynomiography has numerous applications in education, computer science, mathematics, fine arts, and design. It helps student to think about why (since antiquity) solving for the unknown in a polynomial equation has remained such a difficult task, why we need algorithmic methods to do it, and whether they can be approximate or must be exact. With the increasing role of visual tools and technologies, the search for these solutions introduces a striking appreciation of the connections between creativity in art and the intrinsic beauty of science and mathematics. Students are able to visually discover the Fundamental Theorem of Algebra, symmetry in shapes, the concept of iteration in science and nature, the notion of convergence, sensitivity of solutions as data parameters change. Students are encouraged to structure their own investigations with the module and to share their images and discoveries through social networking. This module is appropriate for high school classes in art, design, science, computer science, and mathematics.
Connect Four is a popular sequential game played on a 6x7 grid. Players take turns dropping red or black checkers into the tops of columns, where they fall to the lowest unoccupied space in the column. The first player to get four of their colored checkers to line up vertically, horizontally, or diagonally wins. A draw occurs when all 42 spaces are filled without a winner. It is a game with simple rules and can be demonstrated easily in a classroom. With multiple sets of Connect Four and groups of four, students can experiment with winning strategies. For instance, is possible to find an algorithm to produce perfect play from any configuration (even if mistakes have already been made)? This is always possible with a powerful enough computer, by checking all the positions. However, students are encouraged to find efficient algorithms that will work on computers readily available. Students consider what an algorithm is, what it means to be efficient, and they develop strategies to solve connect four from different vantage points. This module is excellent for helping early high school students appreciate what it means to solve a problem when brute force exists, but is too time consuming.
4) Competition or CollusionRock, paper, or scissors? Fastball or curveball? Bluff or fold? Work together or backstab the competition? Game theory offers a computational approach to optimal decision-making in conflicts (or competitions) between "players" who are each choosing from a variety of possible strategies. While recreational games and sports are natural examples, game theory is also effective in modeling business decisions, political campaigns, biological evolution, and decisions in many other real-life contexts. Students will play a series of simple games and discuss their decision-making process in choosing strategies. Students will then learn to model real-life situations using the concepts of game theory and develop computational tools for optimal decision-making under a variety of conditions. This module is appropriate for high schools classes in mathematics, computer science, and economics.
5) Streaming InformationThis module introduces students to the issues, methods, and challenges in successfully transmitting information. Topics include error detection, error correction, data authentication, data compression, and efficient transmission.
6) RecursionThis module encourages students to think recursively by investigating and inventing their own recursive definitions, such as the definition of n!. They consider functions defined recursively, such as parsers in computer science, and recursion in algorithms like those for dynamic programming.
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