Algebra 1 |
Geometry |
Algebra 2 |
Precalculus |
Chapter 1: Fitting Functions to Tables
Investigation 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.
Investigation 1B: Fitting and 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.
Investigation 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.
Project: More on Monthly Payments
Chapter 2: Functions and Polynomials
Lesson 2.0: Polynomial Basics—Optional Review
Investigation 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.
Investigation 2B: Making It Fit
This investigation introduces Lagrange Interpolation — a method to find the least-degree polynomial function that passes through a given set of points. This investigation is algebraically technical, and as a result, students develop fluency in arithmetic with polynomials and the use of a computer algebra system.
Investigation 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.
Investigation 2D: Advanced Factoring
Students study various polynomial forms and methods for factoring them: chunking, scaling, and grouping. Rational expressions are also introduced.
Project: Heron’s Formula
Chapter 3: Complex Numbers
Investigation 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.
Investigation 3B: The Complex Plane
Students learn how to graph complex numbers, and how to interpret arithmetic geometrically. Magnitude and direction are introduced.
Investigation 3C: Complex Numbers, Geometry, and Algebra
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.
Project: Factoring a Sequence of Polynomials
Chapter 4: Linear Algebra
Investigation 4A: Gaussian Elimination
Students solve systems of linear equations. Matrices and Gaussian elimination are introduced as tools for solving linear systems.
Investigation 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.
Investigation 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.
Project: More Matrix Operations
Chapter 5: Exponential and Logarithmic Functions
Investigation 6A: 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.
Investigation 6B: Exponential Functions
Students explore exponential functions via tables and graphs.
Investigation 6C: Logarithmic Functions
Students learn what logarithms are, how to work with them, and how to graph logarithmic functions. Logarithmic scales are introduced.
Project: Functional Equations
Chapter 6: Graphs and Transformations
Investigation 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.
Investigation 6B: Affine Transformations
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.
Investigation 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.
Project: A Group of Functions
Chapter 7: Sequences and Series
Investigation 7A: The Need to Sum
Students investigate how to sum integers.
Investigation 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.
Investigation 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."
Investigation 7D: Pascal's Triangle and the Binomial Theorem
Students spend time investigating both Pascal’s Triangle and the Binomial Theorem.
Project: The Line of Best Fit Contains the Centroid
Chapter 8: Introduction to Trigonometry
Lesson 8.0: Right Triangle Trigonometry—Optional Review
Investigation 8A: Trigonometric Functions
Students learn the definitions of sine, cosine, and tangent as they apply to angles between 0 and 360 degrees, and learn to solve simple equations involving trigonometric functions.
Investigation 8B: Graphs of Trigonometric Functions
Students learn to graph sine, cosine and tangent. Some basic trigonometric identities are introduced, including the Pythagorean Identity.
Investigation 8C: Applications to Triangles
Students investigate the area of a triangle from a trigonometric perspective, and look at a geometric form of angle addition formulas. Law of Sines and Law of Cosines are introduced.
Project: Brahmagupta’s Formula