Module 0: Practicing Symbolic Reasoning
Evaluating, Simplifying and Solving: A Conceptual and Practice Oriented Review. We review methods and rationale for evaluating and simplifying expressions and solving equations. Practice problems and detailed solutions provide all students opportunity to gain fluency with procedural aspects of algebra.
Module 1: The Meaning of Quantities, Variables, Expressions and Formulas
This module begins by investigating what is involved in identifying and relating quantities in the context of learning to solve simple word problems. The idea of variable is introduced as a way of representing the values that quantities can assume. Students are supported in developing precise use of language when defining variables so that the varying quantity and units of measurement are conceptualized in the mind of the student. Students receive practice in constructing terms, expressions and formulas that are meaningful to students. They develop the vocabulary to describe what each part of an expression represents in the context of an applied problem. They are also supported in learning to conceptualize quantities as a foundation for writing meaningful formulas to relate two varying quantities in word problems. Arithmetic of signed numbers is introduced and students are supported in both understanding and practicing procedures for simplifying expressions and evaluating formulas. The laws of exponents are introduced and practiced. This module concludes by exploring the meaning and methods for solving equations. Students will have many opportunities in future modules to revisit and practice developing meaningful formulas, simplifying expressions, and solving equations.
Module 2: The Function Concept: Representing and Modeling Function Relationships
This module introduces the idea of function as a more formal and useful way to describe and represent relationships between two varying quantities. Function notation is introduced and students receive practice using and interpreting function notation. Students develop an understanding of the definition of a function as a mapping of each input value in a function’s domain to a unique output value in the function’s range, while developing powerful connections between various representations of a function. The module ends by introducing students to the idea of a sequence, sequence notation, and arithmetic sequences and how they are related to linear functions.
Module 3: Constant Rate of Change and Linearity
Students learn how to represent changes in the value of a quantity and explore what it means for two quantities to change together at a constant rate of change. They develop a conceptually powerful understanding of constant rate of change and learn to use this meaning to define, graph and interpret linear functions. Students learn about constant rate of change as a proportional relationship between the changes in two quantities, and thus understand how the change in one quantity can be used to find the corresponding change in another quantity. When students apply this reasoning to find new function values they are able to develop meaningful formulas that represent linear relationships. Piecewise functions are introduced and used to define function relationships that have different rules for different intervals of a function’s domain.
Module 4: Understanding and Using Linear Equations and Inequalities
Students see linear equations as emerging from situations where the output of a function relationship is known and we want to find the value of the input variable that makes the equation true. Students receive more practice solving linear equations and are introduced to inequalities and how they are useful for representing constraints. The methods of solving inequalities algebraically and graphically are introduced and practiced in various applied contexts.
Module 5: Systems of Equations
Students explore contexts involving simultaneous equations by generating tables of values to describe how pairs of values are related and change together. Tables and formulas are used to develop students’ understanding of what it means for a system to have a common point. Students are supported in understanding and applying graphical and algebraic methods for solving linear systems. They then investigate contexts in which entire solution regions satisfy the indicated constraints, leading to a study of linear inequalities and systems of linear inequalities.
Module 6: Exponential Functions: Understanding and Representing Multiplicative Growth
Students explore contexts involving multiplicative growth and use their reasoning to represent multiplicative growth with exponents. Students explore contexts that require them to contrast and describe linear and exponential growth, and explain that linear functions grow by equal differences over equal intervals and exponential functions grow by equal factors over equal intervals. Students will review properties of exponents and apply these properties to represent partial growth factors (for example, 1 month growth factors when the 12 month growth factor is known). Throughout the module students will build tables, formulas and graphs from applied contexts and be prompted to make interpretations that demonstrate understanding and connections.
Module 7: Quadratic Functions and Equations
Students explore simple contexts that involve quadratic growth between two quantities and practice defining and evaluating quadratic functions. The idea of average rate of change is introduced and applied in the context of describing quadratic growth in applied contexts. Students will learn methods for factoring quadratic expressions and apply these methods to determine zeros of quadratic functions. They will be supported in seeing structure in quadratic expressions and learn the method of completing the square as a way to write an equivalent form of a quadratic function. Throughout the module students will practice order of operations, combining like terms, and algebraic methods to represent the same expression in a different form. These methods will be applied when graphing quadratic functions.
Module 8: Polynomial Functions and Equations: Exploring Non-Linear Growth Patterns
Students will be introduced to the general form and behavior of polynomial functions. They will be supported in using polynomial functions to model quantitative relationships using formulas, tables and graphs. They will examine how two quantities change on an interval of the function’s domain and be supported in using this reasoning to explain how the average rate of change of the output variable with respect to the input variable changes over an interval of the function’s domain. They will apply reasoning that they have learned throughout the course (considering how two quantities values change together) to describe a polynomial function’s behavior. Methods for adding, subtracting and multiplying polynomial functions will be introduced and applied.
Module 9: Statistical Models
- Determine Scale and Units for Representing Quantitative Data
- Representing Data with Dot Plots, Histograms and Box Plots
- Representing and Comparing the Center of a Data Distribution
- Understanding and Using Ideas of Mean and Median to Represent the Center of a Date Distribution
- Understanding and Using Ideas of Interquartile Range and Standard Deviation to Represent the Center of a Data Distribution
- Examining Data Sets: Interpreting Differences in Shape, Center and Spread
- Exploring and Summarizing Categorical Data for Two Categories
- Represent Data on Two Quantitative Variables
- Determining and Understanding the Correlation Coefficient of a Linear Fit