In earlier grades, students define, evaluate, and compare functions and use them to model relationships between quantities. In this module, students extend their study of functions to include function notation and the concepts of domain and range. They explore many examples of functions and their graphs, focusing on the contrast between linear and exponential functions. They interpret functions given graphically, numerically, symbolically, and verbally; translate between representations; and understand the limitations of various representations.

**NOTE: The New York State Education Department shut down the EngageNY website in 2022. In order to maintain educators' access, nearly all resources have been uploaded to archive.org and the resource links above have been updated to reflect their new locations.**

In earlier modules, students analyze the process of solving equations and developing fluency in writing, interpreting, and translating between various forms of linear equations (Module 1) and linear and exponential functions (Module 3). These experiences combined with modeling with data (Module 2), set the stage for Module 4. Here students continue to interpret expressions, create equations, rewrite equations and functions in different but equivalent forms, and graph and interpret functions, but this time using polynomial functions, and more specifically quadratic functions, as well as square root and cube root functions.

**NOTE: The New York State Education Department shut down the EngageNY website in 2022. In order to maintain educators' access, nearly all resources have been uploaded to archive.org and the resource links above have been updated to reflect their new locations.**

This site teaches Reasoning with Equations and Inequalities to High Schoolers through a series of 5909 questions and interactive activities aligned to 36 Common Core mathematics skills.

This site teaches Structure in Algebraic Expressions to High Schoolers through a series of 3482 questions and interactive activities aligned to 26 Common Core mathematics skills.

Algebra and Trigonometry provides a comprehensive exploration of algebraic principles and meets scope and sequence requirements for a typical introductory algebra and trigonometry course. The modular approach and the richness of content ensures that the book meets the needs of a variety of courses. Algebra and Trigonometry offers a wealth of examples with detailed, conceptual explanations, building a strong foundation in the material before asking students to apply what they’ve learned.

This course provides an introduction to the language of schemes, properties of morphisms, and sheaf cohomology. Together with 18.725 Algebraic Geometry, students gain an understanding of the basic notions and techniques of modern algebraic geometry.

This is a textbook for first year Computer Science. Algorithms and Data Structures With Applications to Graphics and Geometry.

This textbook is an introductory coverage of algorithms and data structures with application to graphics and geometry.

This web page contains a free electronic version of my self-published textbook Algorithms, along with other lecture notes I have written for various theoretical computer science classes at the University of Illinois, Urbana-Champaign

In-depth study of an active research topic in computer graphics. Topics change each term. Readings from the literature, student presentations, short assignments, and a programming project. Animation is a compelling and effective form of expression; it engages viewers and makes difficult concepts easier to grasp. Today's animation industry creates films, special effects, and games with stunning visual detail and quality. This graduate class will investigate the algorithms that make these animations possible: keyframing, inverse kinematics, physical simulation, optimization, optimal control, motion capture, and data-driven methods. Our study will also reveal the shortcomings of these sophisticated tools. The students will propose improvements and explore new methods for computer animation in semester-long research projects. The course should appeal to both students with general interest in computer graphics and students interested in new applications of machine learning, robotics, biomechanics, physics, applied mathematics and scientific computing.

This course treats various methods to design and analyze datastructures and algorithms for a wide range of problems. The most important new datastructure treated is the graph, and the general methods introduced are: greedy algorithms, divide and conquer, dynamic programming and network flow algorithms. These general methods are explained by a number of concrete examples, such as simple scheduling algorithms, Dijkstra, Ford-Fulkerson, minimum spanning tree, closest-pair-of-points, knapsack, and Bellman-Ford. Throughout this course there is significant attention to proving the correctness of the discussed algorithms. All material for this course is in English. The recorded lectures, however, are in Dutch.

In this game activity students practice comparing shapes and naming something that is alike or different about them.

Students learn about the differences between types of water (surface and ground), as well as the differences between streams, rivers and lakes. Then, they learn about dissolved organic matter (DOM), and the role it plays in identifying drinking water sources. Finally, students are introduced to conventional drinking water treatment processes.

Commercial fishing nets often trap "unprofitable" animals in the process of catching target species. In this activity, students experience the difficulty that fishermen experience while trying to isolate a target species when a variety of sea animals are found in the area of interest. Then the class discusses the large magnitude of this problem. Students practice data acquisition and analysis skills by collecting data and processing it to deduce trends on target species distribution. They conclude by discussing how bycatch impacts their lives and whether or not it is an important environmental issue that needs attention. As an extension, students use their creativity and innovative skills to design nets or other methods, theoretically and/or through hands-on prototyping, that fisherman could use to help avoid bycatch.

Bycatch, the unintended capture of animals in commercial fishing gear, is a hot topic in marine conservation today. The surprisingly high level of bycatch about 25% of the entire global catch is responsible for the decline of hundreds of thousands of dolphins, whales, porpoises, seabirds and sea turtles each year. Through this curricular unit, students analyze the significance of bycatch in the global ecosystem and propose solutions to help reduce bycatch. They become familiar with current attempts to reduce the fishing mortality of these animals. Through the associated activities, the challenges faced today are reinforced and students are stimulated to brainstorm about possible engineering designs or policy changes that could reduce the magnitude of bycatch.

Students learn that fats found in the foods we eat are not all the same; they discover that physical properties of materials are related to their chemical structures. Provided with several samples of commonly used fats with different chemical properties (olive oil, vegetable oil, shortening, animal fat and butter), student groups build and use simple LEGO MINDSTORMS(TM) NXT robots with temperature and light sensors to determine the melting points of the fat samples. Because of their different chemical structures, these fats exhibit different physical properties, such as melting point and color. This activity uses the fact that fats are opaque when solid and translucent when liquid to determine the melting point of each sample upon being heated. Students heat the samples, and use the robot to determine when samples are melted. They analyze plots of their collected data to compare melting points of the oil samples to look for trends. Discrepancies are correlated to differences in the chemical structure and composition of the fats.

This set of a teacher and student guides provides instruction on a 2-3 day series of activities about Le Chatelier’s principle, which shows the effect of changes to conditions in an equilibrium reaction. Students work in pairs or groups to develop their concepts of equilibrium and the effects of changing the amount of reactants or products on an equilibrium system. The concepts are presented and analyzed using graphical representations, qualitative lab data, and modelling. The first part addresses the misconception that equal amounts are required for equilibrium through using a mini-activity that involves the transfer of water between beakers. The second part is a lab activity where students will see how an equilibrium system reacts to a change in concentration. The third part uses manipulatives to understand how an equilibrium operates using the mathematical equilibrium constant (Ksp) at the particulate view.

Students learn about linear programming (also called linear optimization) to solve engineering design problems. As they work through a word problem as a class, they learn about the ideas of constraints, feasibility and optimization related to graphing linear equalities. Then they apply this information to solve two practice engineering design problems related to optimizing materials and cost by graphing inequalities, determining coordinates and equations from their graphs, and solving their equations. It is suggested that students conduct the associated activity, Optimizing Pencils in a Tray, before this lesson, although either order is acceptable.

In this textbook the focus will be on the bacteria and archaea (traditionally known as the “prokaryotes,”) and the viruses and other acellular agents.

Students define and classify alloys as mixtures, while comparing and contrasting the properties of alloys to those of pure substances. Students learn that engineers investigate the structures and properties of alloys for biomedical and transportation applications. Pre- and post-assessment handouts are provided.