
The Value of Computational Thinking across Grade Levels 912 (VCTAL)
VCTAL has developed a set of instructional modules, for use in high school classrooms. The modules are intended to cultivate a facility with computational thinking in students across different grade levels and subject areas. The project was administered by DIMACS at Rutgers University in collaboration with the Consortium for Mathematics and its Applications (COMAP).
What is Computational Thinking?
Computational thinking is a high level thought process that considers the world in computational terms. It begins with learning to see opportunities to compute something, and it develops to include such considerations as computational efficiency, selecting appropriate ways to represent data, and making approximations. A person skilled in computational thinking is able to harness the power of computing to gain insights. Computational thinking is not programming. It is a thought process that can be brought to bear not only in programming but also in a wide array of other contexts. It relates to mathematical thinking in its use of abstraction, decomposition, measurement and modeling, but is more directly cognizant of the need to compute and the potential benefits of doing so.
Modules
Click here to view a list of VCTAL authors.
Student Material
Teacher Material

It's an Electrifying Idea!
This module can be used in a variety of ways to suit classroom needs. It can be taught in
its entirety using 1.5 to 2 weeks of class time. Alternatively, it may be viewed as two
standalone "minimodules," each of which takes about 5 days. Units 1 and 2 combine to
form a minimodule that explores whether an electric vehicle is cost effective. The
remaining units (3 through 5) combine to form a minimodule that looks at the
convenience of driving an electric car given its limited driving range between charges.
These two minimodules form Parts 1 and 2 of this module. For a very short introduction,
Unit 1 can be viewed as a oneday "teaser" on the topic. Likewise, Unit 3 can be viewed
as a oneday activity, introducing ideas in modeling with graphs.
The time it takes to cover the entire module will vary, depending on the abilities of the
students and their access to computers. The module requires the use of spreadsheets and
specialpurpose applets. If there are computers in the classroom and the students also
have access to computers outside the classroom, the module can be completed in 6 to 8
days of class time. In most cases, it will likely take longer. A good option may be to do
either Part 1 or Part 2 in its entirety and then do the first unit of the other part to give
students exposure to both parts.
The module includes a large number of exercises associated with both inclass activities
and homework. It is not necessary for students to do all of the exercises. In fact, doing so
may feel redundant to many students. We expect that most teachers will select a
representative sample.
Types of Classes in Which This Module Can Be Used
This module could be used in a number of different classes, many of which may have the
flexibility in curriculum to incorporate nonstandard materials. Examples of potential
target classrooms include those that discuss technology in society, business decisions, life
skills and personal financial decisionmaking, or planning in society. These could include
technology, business or business math, and social studies classes. Alternatively, Part 1
could be used in environmental science or algebra classes and Part 2 for a class in
discrete mathematics.
Support Files
ExploreCostUnit1.xls
ExploreCostUnit2.xls
ExploreCostUnit6.xls
Applets
ElectricCar
ElectricCar_Simple 
Student Material
Teacher Material

Heart Transplants and the NFL Draft
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 912.
Types of Classes in Which This Module Can Be Used
This module could be used in a number of different classes in grades 912, many of
which may have the flexibility in curriculum to incorporate nonstandard materials.
Potential target classrooms are those that discuss technology in society, decision
making, or planning in society, including computer science, business, social studies,
mathematics, or health classes. 
Student Material
Teacher Material

Network Capacity
This module develops the skills and perspectives used in modeling realworld situations as
networks. Using the demand on networks familiar to students as a motivating point (text
messaging, cell phone, Internet), the module attempts to model some simple networks (school
hallways) and build on that to more complicated networks. It concentrates more on allowing
students to experiment with and think about networks and network capacity, rather than
describing the algorithms used to determine optimal network settings or behavior.
As the simulations increase in complexity, the idea that computers are essential for solving the
more complex problems is introduced.
Key Ideas
Network, capacity, traffic, overload, repeated simulation 
Student Material
Teacher Material

Your Data and Your Privacy:
Do you know what "they" can tell about you?
This module introduces students to privacy issues that are created, worsened, or
solved by computer technology and the collection of data. The module is driven by a
series of case studies drawn from various wellknown websites. It also examines a
surprising way in which a computational strategy can protect privacy. The
concluding activity has students create a proposal for the design of a ComputeaDate dance website; in their proposal, the students are expected to apply what
they've learned about privacy issues through the case studies in this module.
The focus of the module is on how our ideas about privacy and personal conduct
need to change in response to evolving technologies and the proliferation of data
collection tactics. Often there are no clear answers to questions about privacy, as
evidenced by numerous court cases. When answers are not clear, the focus should
be on becoming more informed and holding careful and informed discussions.
Prerequisites
No computerprogramming experience is required or involved. Some familiarity
with Internet sites such as Google, Facebook, and Netflix would be helpful. Basic
algebra skills are required for one activity.
Materials and Resources
Copies of handouts and printed materials are provided for activities. Simple
calculators for Unit 3 are optional but could be helpful for some students. Internet
access during class is optional but could be used to provide direct experience with
some of the sites used in the case studies or to show online videos that enhance the
module.
Suggested Uses
Although this module can be used in any subject and in grades 9 through 12, it is
perhaps best suited to 11thgrade socialstudies courses on ethics or law that
examine issues of rights or laws protecting rights or to a technology class. 
Student Material
Teacher Material

Tomography: A Geometric and Computational Approach
The purpose of this module is to provide a general background on computed
tomography (CT) and study how CTscan images are created, using
threedimensional (3D) reconstruction of objects using twodimensional (2D)
pieces (slices) of the object. Currently, CT scans are used in medical imaging,
archeology, food safety, structural integrity, virtual autopsies, and many other
applications.
The main questions of the module are:
How can 3D images be created from 2D images (pieces) of the object?
How much computational power and what computational thinking skills are
required to do these reconstructions, and what does this reconstruction depend
on? 
Student Material
Teacher Material

Cryptography: Lqwurgxfwlrq wr Fubswrjudskb
(Cryptography: Introduction to Cryptography)
The purpose of this module is to introduce students and teachers to the theory and practice of
cryptography. As this is a vast subject with numerous applications, any introduction necessarily has to be
brief and must concentrate on a few jewels of the subject; we have chosen to emphasize RSA because of
the richness of the mathematics and the importance of its applications. More material and exercises are
included than can be covered in one week in order to allow teachers flexibility in terms of topics and depths. In particular, the homework problems range from straightforward calculations to some very
challenging exercises (included for advanced students).
Types of Classes in Which This Module Can Be Used
This module could be used in a number of different classes in grades 912, many of
which may have the flexibility in curriculum to incorporate nonstandard materials.
Potential target classrooms are those that discuss technology in society, decision
making, or planning in society, including computer science, business, social studies,
mathematics, or health classes. 
Student Material
Teacher Material

Fair and Stable Matching
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.
Types of Classes in Which This Module Can Be Used
This module is appropriate for high school classes in computer science and mathematics and the associated minimodule is appropriate for social studies classes. 
Student Material
Teacher Material

Polynomiography: Visual Displays of Solutions to Polynomial Equations
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.
Types of Classes in Which This Module Can Be Used
This module is appropriate for high school classes in art, design, science, computer science, and mathematics. 
Student Material
Teacher Material

The Analysis of Games
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.
Types of Classes in Which This Module Can Be Used
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. 
Student Material
Teacher Material

Competition or Collusion?
Game Theory in Sports, Business, and Life
This module introduces students to gametheory concepts and methods, starting
with zerosum games and then moving on to nonzerosum games. Students learn
techniques for classifying games, for computing optimal solutions where known,
and for analyzing various strategies for games in which no optimal solution exists.
Finally, students have the opportunity to transfer what they've learned to new
gametheoretic situations.
Prerequisites
Students should be able to use the skills learned in Algebra 1, including the ability to
graph linear equations, find points of intersection, and algebraically solve systems of
two linear equations in two unknowns. Knowledge of basic probability (material that should have been learned by 9th grade) is also required; experience with
computing expected value would be helpful, but it can be taught as part of the
module. No computer programming experience is required or involved, although
students with some programming knowledge may be able to adapt their knowledge
to optional projects..
Types of Classes in Which This Module Can Be Used
The module can be used with students in grades 1012 in almost any class, but it is
best suited to students in mathematics, economics, political science, or computer
science courses. 
Student Material
Teacher Material

Gently Down the Stream:
The Mathematics of Streaming Information
This 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.
Types of Classes in Which This Module Can Be Used
This module could be used in a number of different classes in grades 912, many of
which may have the flexibility in curriculum to incorporate nonstandard materials.
Potential target classrooms are those that discuss technology in society, decision
making, or planning in society, including computer science, business, social studies,
mathematics, or health classes. 
Student Material
Teacher Material

Recursion  Problem Solving and Efficiency:
How to Define an Infinite Set Finitely
Recursion can be used in many areas, from modeling population growth or the spread
of disease to determining how much money you will have in an account at retirement.
The purpose of this module is to provide a general background on the process of
recursion, a method for solving problems where the solution to a problem depends on
solutions to smaller instances of the same problem. A recursive process is one in which
objects (a whole system) are defined in terms of other objects (stages of the system) of
the same type. The whole system can be built knowing only a few initial values or
stages of the system and applying a small set of rules. Computers routinely use
recursion in performing many standard operations or processes.
Types of Classes in Which This Module Can Be Used
This module could be used in a number of different classes in grades 912, many of
which may have the flexibility in curriculum to incorporate nonstandard materials.
Potential target classrooms are those that discuss technology in society, decision
making, or planning in society, including computer science, business, social studies,
mathematics, or health classes.


This material is based in part upon work supported by the National Science Foundation under Grant Number DRL 1020201. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation. 



