Chapter 1: Fitting Functions to 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.
1C: More About Recursive Models
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
2A: About Functions
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.
2B: Making it Fit
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.
2D: Advanced Factoring
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.