Restructuring Algebra 2

Algebra 2 has been a hard course for me to teach for several reasons:

• My students’ understanding of linear and exponential functions has faded during Geometry, but these topics are a crucial foundation for Algebra 2.
• My students have not seen quadratics before, and the polynomials part of my unit becomes bloated.
• I’m supposed to cover some Statistics, but it feels like a deviation from the themes of the rest of the course.
• I need to balance accessibility and rigor. Every student takes this course, and their paths diverge afterward.

Last year, my course was broken into units according to the different types of functions, with an additional unit where we looked at transformations and regressions. As I was looking for ways to chop polynomials into some manageable chunks, I noticed that we used three different non-algebraic representations in that unit alone. I wondered: what if my units were designed around non-Algebraic representations instead of function types?

What would that change? I’m hoping that this will (a) spread polynomials throughout the course, (b) meaningfully integrate both review and statistics, and (c) keep us focused on access and structural thinking. I decided to go for it. Below is a sketch of how I plan to organize the course this year.

Unit 1: Delta Columns

• Make tables to analyze data and visual patterns
• Analyzing tables with a delta column, recursive rules, and a hockey stick to write closed form rules (see CME Algebra 2 Chapter 1)
• Reviewing linear and exponential functions, while previewing quadratic (and cubic) polynomials

Unit 2: Sign-Analysis Graphs

• Making graphs without really paying attention to the y-axis
• Finding x-intercepts of polynomials using the zero-product property
• Finding end behavior

Unit 3: Area (Rectangles & Squares)

• Factoring (aka making a rectangle) and using that to solve/sketch
• Polynomial division (i.e. given area and one side, find the other side)
• Completing the square (i.e. making a square with some extra stuff) and finding the vertex

Unit 4: Intersections

• Solving equations using graphical methods
• Connect to linear systems unit, but expand to any function types
• Making connections/comparisons between Algebraic methods and the graphical method

Unit 5: Double Number Lines

• One “addition” line and one “multiplication” line (aka a log scale)
• Review exponent rules in a new context, including negative and rational exponents
• Evaluate logs, solve exponent/log equations, make sense of log properties

Unit 6: Complex Plane

• Introducing i as a rotation
• Playing with conjugates
• Exploring the consequences of complex numbers in polynomials functions/equations

Unit 7: Normal Curve / Pascal’s Triangle

• Binomial Theorem and some applications
• Binomial distributions end up approximately normal
• An intro to working with the normal curve

Unit 8: Scatterplots / Regressions

• Comparing clean and messy data
• Using context and R^2 to determine the appropriateness of the function and fit
• Collecting data, interpolating, extrapolating, and having goofy fun at the end of the year

This is very much a work in progress, and many of these units will be new or totally different than before, so any feedback, ideas, or questions would be greatly appreciated.