Year Long Revised Plans for Pre-AP Algebra 2 and My Rational

Last year, I agreed to teach Pre-AP Algebra 2 as an overload about two weeks before the school year began. Although I had taught Algebra 2 before, it had been over 20 years ago. I definitely felt like a first year teacher as I was teaching AP Environmental Science for the first time and was switching from the Precalculus curriculum that I had been teaching to the AP Precalculus course. The order and focus of AP Precalculus was different enough from my class before, that I pretty much needed to scrap my old curriculum and start over...fortunately Math Medic had put a a whole set of lessons together. All of this to say, I was strapped for planning time. As a result, I followed the Pre-AP Algebra 2 lessons as presented by College Board, and just found or created lessons only for the topics that the curriculum said were not addressed.

Here were some of my frustrations: 

1. Although the lessons were described in detail in the teacher materials, not all of the materials needed were compiled for the student facing documents, specifically, the formative assessments. And not all of the parts of the lessons were in a presentation. Some of the questions could be presented verbally, but for many of the lessons, I needed to put together a Google Slides presentation. This was different from Pre-AP Biology, where all parts of the student lessons were provided as well as Google Slides presentations for lessons as needed.

2. The lessons were great at getting the students thinking, but I felt like they were missing consolidation. Students could work through the lesson, but in the end didn't know/realize what they were doing. The Pre-AP units will tell you what percent of the topics are covered and I'd be surprised that the course map would say 40% of instructional time was covered by model lessons in Unit 1 when in the unit outline, there was only one learning objective out of 9 that was not addressed by the model lessons. Maybe the percentage was lower because the lessons were missing consolidation or maybe because students needed more time to practice? I think this is the case, but I'm not sure of what College Board's reason was for the difference in % vs. learning objectives covered.

3. Besides needing consolidation, there was an assumption that students remembered everything they learned in Algebra 1, like completing the square and different forms for the equation of a parabola. My students definitely needed a refresher. I learned this as we went through the year. I needed to remind myself that this was not a complete curriculum, and by unit 3, I was adding lots of refreshers, lessons and practice. 

By Unit 4 (Trigonometry) I decided to take an entirely different approach. I went to Math Medic for most of the lessons. I also figured this would be a good experience for my students as they would be in my AP Precalculus class the next year, where the class is primarily taught through Math Medic lessons. I then tucked the Pre-AP model lessons where I thought they worked best in the progression. Occasionally, the Pre-AP lessons would be used as an introduction, sometimes as additional practice. I was much more purposeful about consolidation. 

This summer, I decided to give Pre-AP Algebra 2 a complete makeover, including refresher lessons, for topics that students needed to recall, just in time for the lesson that would take that topic deeper. I added in Math Medic lessons for each of the learning objectives, and where I decided that using just a Pre-AP model lesson was sufficient, I added in consolidation. My consolidation included a page with a box for quick notes, Math Medic style. Usually, I would have boxes for quick notes for two lessons on a page so as to not waste paper. I kept all of the topics in each of their set Pre-AP units, and mainly in order, since I do love Pre-AP's assessments and wanted to be able to use all of them. These made-over plans also include homework for extra practice. Sometimes I just used the formative assessments from Pre-AP for homework practice, and sometimes from our textbook, which is Big Ideas Algebra 2 by Larson and Boswell. 

Besides Pre-AP's learning checkpoints and performance tasks, I also give my own quizzes and tests. Problem-attic.com is my go to test-building site. Besides using Algebra 2 Regents questions from the site to build assessments (I am in New York), I would often go to Illustrative Math questions. Those questions are great thinking questions and often matched well with the thinking that the Pre-AP Algebra 2 lessons were geared toward. 

Here's my Algebra 2 Plans for the year. I did remove the time for each lesson column from the Pre-AP template. Each of the Tuesday/Thursday classes are 80 minutes and every other Friday when we meet is 60 minutes.

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Reflections on the EFFL Workshop by Math Medic

I've just completed 106 hours of professional development this summer and decided that if I'm truly going to apply what I've learned, I need to reflect on the takeaways before I forget them. The summer began with the AP Biology Read in Kansas City, MO which accounted for 70 hours. Last week, I participated in an online APSI for AP Environmental Science for another 30 hours. Today was the last day of a 6 hour, online workshop by Math Medic on EFFL (Experience First, Formalize Later). Since this workshop is the freshest in my mind, I'll start with this one. 

I love Math Medic. First of all, they write a whole set of lessons that correlate to the AP Precalculus course. And secondly, they teach math the way I have always wanted to teach math...students learn by doing some activity that they can relate to (Experience First), but points them to the main ideas of the lesson. Anyway, by the end of the year last year I was also using Math Medic lessons for Pre-AP Algebra 2, especially for the way they consolidated the lessons (Formalize Later).

So, here are my takeaways: 

1. Project a blank "Experience" page that I direct students to write in certain work that their group has done. Last year, I just left them blank as I gave margin notes and pointed to student work up on the boards (my classes use whiteboards or windows to write their group work), but the work was hard to see.

2. Have students write the margin notes as I do, and they should write those notes in a different colored writing utensil just like I do, to help the notes stand out. 

3. Only focus on the challenging questions from the Experience page and chose different students to explain their thinking before connecting their explanation to the margin notes that are pointing to the main idea.

4. Know the main ideas before the lesson, so that I can better ask focusing questions that point students to the main idea versus just pointing then to how to get the right answer. Connect their thinking to the main idea(s).  Here's a Math Medic blog post about focusing versus funneling questions.

5. Taking this workshop also gave me the idea to share the essential knowledge statement(s) that go with the lesson and work with the students to determine what would be the most helpful information to write in the quick notes section. There are essential knowledge statements for me to use for both AP Precalculus and for Pre-AP Algebra 2.

I'm excited to see how my math classes will go in the coming school year while using these practices. 

Completing the Square with Blocks

Several years ago, when my Precalculus class made it to our Conic Sections unit, I realized that most of my students did not remember how to complete the square from when they learned it in Algebra 1 (and maybe 2). I was reading through the 5 (Math) Practices in Practice and Building Thinking Classrooms at the time and developed this activity to help walk students through developing their understanding of completing the square.

I divided the activity into four sections. The first samples start with just an expression with "x^2" and an even numbers of "x"s (b). They are asked to build a square, and are allowed to fill in any missing parts with single unit blocks (c). Eventually, we move into an odd number of "x"s, and then expressions with single units as well. 

By the fourth section of exercises, they are thinking through negative b values and the whole quadratic equation to complete the square for. For many students, being able to see the squares being built helps them to remember what to do when they see questions on tests that require completing the square to solve. 

Although I've traditionally done this activity in Precalculus at the start of conic sections, I'm going to add it to my Algebra 2 class next year as well when we are looking at quadratics. 

For this activity, I use some Algebra blocks I have as well as Cuisenaire rods paired with "x^2"s that I made out of card stock. I still didn't have quite enough, so I also cut some sets out of scrapbook paper I had on hand. I actually like the paper sets best when we need to only use half an "x" piece when b is odd. Then we can just tuck half of the rod under the square. 

Here are the documents that I made to go with this activity. This document includes the directions for each section. This document has each of the expressions or equations to work with in each section.  The examples are formatted so that either each group of students can have a quarter sheet of the examples for each section or a large sheet that I post somewhere in the classroom for them to refer to.

Two Labs to Introduce Rational Functions in Algebra 2

 

All the possible sizes for a "Cube-ie"

As our class was working through the Unit 3 topics for Pre-AP Algebra 2, we arrived at rational functions. Rational functions in the Pre-AP curriculum have two key learning objectives with several corresponding Essential Knowledge statements. Rational functions are one of the topics that need to be teacher developed. As I was thinking through LO 3.2.5 Construct a representation of a rational function, I decided that I wanted my students to discover rational functions in a hands-on way. I put together two lab exercises for students to do.

The first exploration involved measuring the length of time it takes a "cubie" to somersault 50 cm along a meter stick. A "cubie" is a segmented worm with anywhere from 1 cube body segment to 10 cube body segments (domain restriction). All we needed were a meter stick, several linking cubes, and a timer. 

One of the group's "cubie" somersaulting its way to 50 cm.

I borrowed the second exploration from an AP Environmental Lab on solar insolation. Students held a flash light above graph paper at different angles, counted how many squares were illuminated, and then divided the pre-measured brightness of the flashlight by the number of squares illuminated. This gave a measure of brightness per square.  We determined the brightness of the flashlight using the free Arduino SJ app on my school iPad. Phones should be able to do this too.

One thing I will do differently next year is to use actual graph paper instead of my big cling whiteboard graph paper because the squares were too big to see much of a difference as the angle of the flashlight changed. I had hoped that using the bigger squares wouldn't leave them counting tons of squares, but it turns out that regular graph paper does work better for this. 

Here's the lab I gave to students.

A Regression Lab for AP Precalculus

Starburst Grab

 I'm teaching AP Precalculus this year. Although I am loving the course developed by the College Board and the thinking it is requiring of my students, I'm a little overwhelmed with all of the prep. It doesn't help that I am also teaching AP Environmental Science for the first time. 

Speedy Squares

Anyway, I spend plenty of time looking through posts from the AP Precalculus teachers Facebook group. One of the posts talked about regression labs that a teacher was doing when they got to topic 2.6. None of these regression labs are my creations, but I put them all together into a form that I could use with my students that guided their thinking, hopefully into deeper understanding. The original activities can be found on Math Equals Love and Stats Medic.

Measured Race Station

I modified Starburst grab so that it would be a cubic regression, although it wasn't a fabulous fit. Then we had two linear and one quadratic regression. 

The labs were a big hit with students. One change I'm planning for next year is to have students answer the questions about each station at the same time that they are making their graph and finding the regression equation and residuals. 

Here's the document that I put together that includes both the handouts that I gave to each student as well as the directions that I placed at each station.

Filling in Unit 2 Lessons for Pre-AP Algebra 2

We've just started Unit 2 in Pre-AP Algebra 2. We've completed the lessons on the composition of functions and have moved onto looking at transformations as composition of functions. There weren't lessons for Concept 2.2 on Transforming Functions. 

The first two learning objectives focus on how different compositions of functions cause different transformations. I decided to make a Desmos activity that would address those LOs. Here's the link for Transformations as Composition of Functions. 

Then we moved our focus to representing a sequence of transformations graphically and paying attention to the order that these transformations needed to be done. We did this with another Desmos activity that I edited from someone else. This activity was Transforming Functions-Shifting and Reflecting. We mainly did this together since there were a lot of questions and students were still building their understanding. Next year, I plan to do Transforming Functions first, and then will move onto the Shifting and Reflecting activity.

After doing the above Desmos activities, I assigned "Talking Transformations and a New Origin" by Julie Reulbach. The worksheet can be found on her i speak math blog here. I also sent them home with notes on the order that transformations need to be performed from MathBitsNotebook Algebra 2.

In our next class I'll challenge students with the Transformations Word Search by Sarah Carter that you can find on her blog Math Equals Love. One word of warning, the pdf file has a transformation of -4f(x), that doesn't actually spell a word with the 4 letters. All of the rest of the transformations do spell common words.

If we have time, I'll also have students work on a matching activity that asks them to match a graph with a function and a list of transformations. The document with the card matching game can be found on this blog post: Partially Derivative.

At the end of Unit 2, the LO for 2.3.3 about restricting a domain of a function so that it is invertible. I found a Khan Academy lesson that covers this, so I plan to use that lesson with my students. It's from the Precalculus curriculum and is titled: Restricting Domains of Functions to Make them Invertible.

Update: I made a set of notes for students to practice restricting domains of functions to make them invertible. The link to that document is here. Since I included a couple of absolute value graphs, students were reminded of how to convert absolute value functions into a piecewise function to be able to find the inverse function.

Fillable Lab Sheets for AP Environmental Science Labs

Another one of my new classes for this year is AP Environmental Science.  Since we are a one to one device school, my goal is for students to complete as many lab analyses by Google Docs as possible. This saves lots of photocopies. A surprising bonus is that I am finding the labs faster to grade this way. Labs that are turned in through Google Classroom as google docs can be commented on. I can quickly type in a few comments to students' lab and count comments to determine the grade. 

One other I like about making fillable labs in Google docs is using 1-cell tables for the answers. Otherwise I find myself searching for answers that blend in with questions.

The labs themselves with teacher editions and student editions with procedures can be found in AP Classroom for Environmental Science if you are an approved teacher. 

I'll add links to the three labs that I've made fillable answer sheets for. If I could find a fillable lab sheet that someone else had done, I used them and didn't reinvent the wheel.

AP Lab #1: Primary Productivity

AP Lab #2: Species Diversity

AP Lab #7: Physical and Chemical Properties of Soil