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Patterns, Functions, and Algebra
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Overview of the Course

"Patterns, Functions, and Algebra" explores the "big ideas" in algebraic thinking, such as finding, describing, and using patterns; using functions to make predictions; understanding linearity and proportional reasoning; understanding nonlinear functions; and understanding and exploring algebraic structure. The course consists of 10 two-and-a-half hour sessions that each include video programming and activities, provided online and in a print guide. The 10th session explores ways to apply the algebraic concepts you've learned in K-8 classrooms. You should complete the sessions sequentially.

Session 1, Algebraic Thinking
Discover what it means to think algebraically, and learn to use algebraic thinking skills to make sense of different situations. Topics covered include describing situations through pictures, charts, graphs, and words; interpreting and drawing conclusions from graphs; and creating graphs to match written descriptions of real-life situations.

Session 2, Patterns in Context
Explore the processes of finding, describing, explaining, and predicting using patterns. Topics covered include how to determine if patterns in tables are uniquely described and how to distinguish between closed and recursive descriptions. This session also introduces the idea that there are many different conceptions of algebra.

Session 3, Functions and Algorithms
Investigate algorithms and functions. Topics covered include the importance of doing and undoing in mathematics, determining when a process can or cannot be undone, using function machines to picture and undo algorithms, and recognizing that functions produce unique outputs.

Session 4, Proportional Reasoning
Look at direct variation and proportional reasoning. This investigation will help differentiate between relative and absolute meanings of "more" and compare ratios without using common denominator algorithms. Topics include differentiating between additive and multiplicative processes and their effects on scale and proportionality and interpreting graphs that represent proportional relationships or direct variation.

Session 5, Linear Functions and Slope
Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships and the relationship of rates to slopes and equations.

Session 6, Solving Equations
Look at different strategies for solving equations. Topics include the different meanings attributed to the equal sign and the strengths and limitations of different models for solving equations. Explore the connection between equality and balance, and practice solving equations by balancing, working backwards, and inverting operations.

Session 7, Nonlinear Functions
Continue exploring functions and relationships with two types of nonlinear functions: exponential and quadratic functions. This session reveals that exponential functions are expressed in constant ratios between successive outputs and that quadratic functions have constant second differences. Work with graphs of exponential and quadratic functions, and explore exponential and quadratic functions in real-life situations.

Session 8, More Nonlinear Functions
Investigate more nonlinear functions, focusing on cyclic and reciprocal functions. Become familiar with inverse proportions and cyclic functions, develop an understanding of cyclic functions as repeating outputs, work with graphs, and explore contexts in which inverse proportions and cyclic functions arise. Explore situations in which more than one function may fit a particular set of data.

Session 9, Algebraic Structure
Take a closer look at "algebraic structure" by examining the properties and processes of functions. Explore important concepts in the study of algebraic structure, discover new algebraic structures, and solve equations in these new structures.

Session 10, Classroom Case Studies
Explore how the concepts developed in "Patterns, Functions, and Algebra" can be applied at different grade levels. Using video case studies, observe what teachers do to develop students' algebraic thinking and investigate ways to incorporate algebra into K-8 mathematics curricula. This session is divided into three grade bands: K-2, 3-5, and 6-8.


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