Chapter 1: Fitting Functions to Tables

1A: Tables

This investigation begins the Algebra 2 thread of functions and fitting. Students investigate how to use constant differences and other cues to fit linear and quadratic rules to input-output tables. They use a functional modeling language on their calculator to model and experiment with functions.

1B: Fitting the Data

Students begin to develop a statistical perspective, in the sense of thinking of data in terms of trends, rather than as individual points. Students investigate whether or not a data set can be reasonably approximated by a linear function, and they study alternatives to linear trends. Students investigate relationships among mean, median, variance and standard deviation.

Students encountered recursively-defined functions in Investigation 1A. Here, they investigate recursion in greater depth by analyzing a recursive function that determines the monthly payment on a loan. Students also investigate the factorial function — a recursively-defined function with no simple closed form.

Chapter 2: Functions and Polynomials

Students develop a "functional perspective:" what is a function? Students learn to decide whether a given pairing is a function from a table, a graph, or an equation. Students also revisit notation and develop precise definitions of domain and range. Students investigate the arithmetic of functions, including composition. Inverse functions are introduced.

2C: Factors, Roots, and Zeros

The Factor Theorem and the Remainder Theorem are introduced and studied. Students learn the relationship between roots and factors of polynomials. Students also learn to divide polynomials by monic linear polynomials.

Students study various polynomial forms and methods for factoring them: chunking, scaling, and grouping. Rational expressions are also introduced.

Chapter 3: Complex Numbers

3A: Introduction to Complex Numbers

Students encounter problems whose solutions require calculations with square roots of negative numbers, and they then extend the real numbers to include these square roots. They investigate arithmetic of complex numbers.

3C: Complex Plane, Graphing, Complex Numbers

This investigation offers an exploration of some more advanced topics in complex numbers, including in-depth investigations of magnitude and direction, roots of polynomials, and roots of unity.

3B: The Complex Plane

Students learn how to graph complex numbers, and how to interpret arithmetic geometrically. Magnitude and direction are introduced.

Chapter 4: Linear Algebra

4A: Gaussian Elimination

Students solve systems of linear equations. Matrices and Gaussian elimination are introduced as tools for solving linear systems.

4B: Matrix Algebra

Students learn to solve matrix equations. Dot product is introduced. Students also learn to multiply matrices and find inverses of matrices, by hand and with their calculators.

4C: Applications of Matrix Multiplication

Students learn to use matrices to represent sequences of geometric transformations, model the evolution of a system over time, and analyze sequences of repeated probabilities.

Chapter 5: Exponential and Logarithmic Functions

5A: Working with Exponents

This investigation begins with a review of laws of exponents, including a review of zero and negative exponents. Students investigate arithmetic and geometric sequences, and they use these to extend the laws of exponents to include rational exponents.

5B: Exponential Functions

Students explore exponential functions via tables and graphs.

5C: Logarithmic Functions

Students learn what logarithms are, how to work with them, and how to graph logarithmic functions. Logarithmic scales are introduced.

Chapter 6: Graphs and Transformations

6A: Transforming Basic Graphs

Students enlarge their toolkit of basic graphs to include circles and graphs of simple cubic equations. They investigate how graphs are translated or stretched when the variables are changed with simple transformations.

6B: Affine Tansformations

Students investigate the algebra of functions of the form x → ax+b. This algebra is applied to completing the square for quadratic equations and to reducing cubic equations to one of three simple forms.

6C: Graphing Using Affine Transformations

Students study an alternative way to understand the effect of translations and dilations on graphs—instead of transforming the graph they transform the axes.

Chapter 7: Sequences and Series

7A: The Need to Sum

Students investigate how to sum integers.

7B: Sum Identities

Students investigate definite and indefinite sums, and they develop the formula for the sum of the first n integers and sums of powers of a base.

7C: Arithmetic and Geometric Sequences and Series

Students see identities for sums of finite arithmetic and geometric series and for convergent infinite geometric series. The emphasis here is on the "linearity of summation."

7D: Pascal's Triangle and the Binomial Theorem

Students spend time investigating both Pascal’s Triangle and the Binomial Theorem.