**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.**

I wonder about regression much earlier, especially the way Desmos rewards knowing algebraic structures. Then can use that in many of the units.

I like starting with the tables and difference quotients; wondering if graphing should follow that or the regression following that. Differences help classify functions -> regression for forms -> graphing properties. ?? Hope you write about this as it goes.

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Long term reflections after trying this? I’m fascinated by this idea but I can’t see the practice of it

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I’m hoping that this year allows more time to write, but briefly: 1) Some units more naturally fit with this approach than others, 2) I need to emphasize the theme of non-algebraic representations more with students, 3) I should shift the sequence around a bit.

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