Our year always started with an introduction to nonlinear functions and how their graphs compare to linear functions.

When is a graph increasing? When is it decreasing? We always started the year with an overview of towards and away motion graphs. As Lila and her friends raced to school, they got tired so they slowed down. The distance between the friends and home decreased as they ran from school to home, then increased as they raced back to school.

Teachers at my school also used motion sensors and a water lab to introduce motion graphs. There is also a virtual Desmos water lab.

**Domain and range**

We always spent extra time on domain and range since it comes up throughout the school year.

Students often had some trouble finding the domain and range of graphs. Dragging a pencil across the grid, then asking students to say, "stop!" when the pencil hits the graph, then again when the pencil left the graph, helped them identify the intervals' start and end points. Here are the practice cards we used and here is the reference sheet.

Warm-ups get pencils in hands and students working as soon as they enter class. We used this algebra 2 template throughout the year to graph functions.

**Function vs. not a function**

Identifying when a relation is a function always got a quick review.

We looked at relations represented in tables, graphs, mappings, equations and sets of coordinates to determine which were functions.

We can remain the same height year after year, but can we be two different heights at the same time? One is a function, one is not a function in that it doesn't make sense in real life. I drew these two graphs on the board and asked students which made sense. This brought us into the vertical line test for functions.

**Evaluating functions**

Leaving non-functions behind, we'd then get into evaluating functions given graphs, equations, word problems and tables.

We'd also learn to compose functions given different function forms.

**Parent functions intro - absolute value, quadratics, radicals**

Before teaching students how to graph functions in vertex form, I gave them a See, Think, Wonder sheet and put an absolute value graph (part of our algebra 2 word wall) and equation on the board. I asked students to take a minute to write down what they see, then what they think, then what they wonder.

My 11th grade algebra 2 students weren't totally familiar with this format, but I wanted to give them a few minutes to make the connection between the equation and vertex of a function in vertex form before we jumped into learning about them.

**Graphing absolute value functions - vertex form**

At this point in the year, my students started to need visual supports and reminders, which I provided through our algebra 2 word wall (with some algebra 1 references added in) and math cheat sheets.

We graphed most functions throughout the year by completing tables. This provided extra practice evaluating functions, and the structure of always using a table seemed to work well.

This absolute value gallery walk was a more successful algebra 2 activity than I had expected. Students worked in groups and found all of the characteristics of the first absolute value graph they visited. When groups started getting distracted, I took it as a cue that they were done and it was time to rotate to the next graph. Each group checked the previous group's work and made any necessary edits.

**Graphing quadratics - vertex form**

We completed another See, Think, Winder comparing the absolute value graph and equation from earlier with a quadratic graph and equation in vertex form.

For a quadratic with a value = 1, the pattern from the vertex -- up 1, over 1; up 3, over 1; up 5, over 1, up 7 over 1 -- helps find additional coordinates once the vertex is found. I don't point this out immediately, but it's a nice way to check if our tables are accurate.

We graphed quadratics in vertex form by completing a table then plotting coordinates (video).

**Graphing radicals - vertex form**

We analyzed the relationship between the graphs of quadratic functions and radical functions to set the stage for later in the year when we learned about inverse functions.

**Function operations**

We then switched gears into function operations-- adding, subtracting and multiplying. We revisit this later in the year during our polynomials unit.

In small group, we focused on adding, subtracting and multiplying functions. We also reviewed evaluating functions.

**Factoring**

Next came factoring. One year I did decide to give students a multiplication chart, which helped those who did not know their multiplication tables. Students were generally thankful for the table, which I hadn't expected. I thought giving them a multiplication table to use during factoring would be an insult, but it didn't turn out that way.

Before jumping into trinomial factoring, we played a game of figuring out two numbers that multiply to a certain number and add to a certain number.

A lot of factoring was a review from algebra 1. It seems to be one of those topics that can be reviewed every year. We completed a lot of factoring activities. Finding GCFs of binomials and trinomials was more difficult for students than I had expected. The multiplication table helped as a support.

**Quadratic formula**

The Quadratic Formula was another algebra 1 review.

Even so, we practiced with the formula a lot.

**Complex numbers**

We did a quick jump to learning about complex numbers, then went back to the Quadratic Formula so see how they show up there.

When students asked why there are imaginary numbers, I flicked the lights on and off and mentioned that we need the square root of -1 to explain the way electricity flows.

**Graphing quadratics in all forms**

We compare the axis of symmetry formula to finding the mean between 2 numbers. We also use the up 1, over 1; up 3, over 1... pattern when the quadratic's a value = 1.

I like point out the pattern that emerges when we graph quadratics. For a quadratic with an a value of 1, the pattern from the vertex -- up 1, over 1; up 3, over 1; up 5, over 1, up 7 over 1 -- helps find additional coordinates once the vertex is found.

**Quadratic word problems**

This is one of my favorite units in algebra 2 because of how much students learn.

We had a quadratic keywords poster as part of our math word wall so that students could get a quick vocabulary refresher if needed.

Students needed a lot of practice and reminders for how to identify left bound/right bound cursor placement on the graphing calculator.

**Solving radical equations**

We analyzed the "invisible" part of a radical graph that would be there if the radical was 1:1 with its quadratic inverse, and related it to the radical equation's extraneous solution.

**Polynomials**

Sketching polynomials was another favorite unit because of all the learning that happened. At first, students think the equations are far too complicated-looking to ever be able to graph. But within a few days they are able to do it. We started with naming polynomials then moved on to sketching.

The polynomials section of our algebra 2 word wall helped students identify zeros, bounces, crosses, zeros and end behavior.

We used another template and cheat sheet for sketching polynomials. With all of the warm-up templates, students grabbed one on the way to their seat and completed it based on the information on the board.

**Inverse functions**

Usually the last unit we'd get to in small group was our inverse functions unit.

A hole punch can be used to investigate inverse functions in the coordinate plane (post and video).

By this time of the year, my juniors were the oldest students in the building because the seniors had already left, and were very excited for summer.

**In algebra 2 inclusion:**

To help support students in our inclusion algebra 2 class, I made a series of cheat sheets for student notebooks:

**Polynomial long division**

**Synthetic Division**

**Exponential functions**

**Logarithms**

**Graphing rational functions**

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