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

Finishing Up Unit 1 of Pre-AP Algebra 2

One afternoon toward the end of summer, I got a phone call while at the orthodontist with my youngest daughter. I was a little surprised to see the call was from the head of the math department at my school.  The call was to ask me if I would consider taking on one of the Pre-AP Algebra 2 classes. I was actually delighted to take on another math class, although it meant an overload in an already busy year for me. My class load already included AP Environmental Science, which is brand new for me, and AP Precalculus which is a brand new curriculum. The only prepped and familiar class for this year is AP Biology. 

One of the benefits of Pre-AP Algebra 2 is that the curriculum is primarily put together, although I'm finding that it still takes some time to prep the lessons. Fortunately, unit 1 had lessons for each of the topics except for the last one. Key concept 1.3: Graphing Absolute Value as Piecewise Defined Functions didn't have a lesson. Before the year started I decided to teach the concept through a Desmos activity. I'm still about a week away from using this lesson, but wanted to share it in case it may be helpful to someone else. Feel free to let me know if you catch any errors. I'll also make adjustments as my students try it and help me find any issues.

Here's the link to Absolute Value as a Piecewise Function.

Hexagonal Thinking in AP Biology

I happened to be scrolling through Facebook a couple of weeks ago and saw a post about reviewing for an AP Biology unit test with hexagonal thinking. I was immediately intrigued. I used to give students some time in class to start working on their progress checks in AP Classroom before the test, but I had several students who were not engaged and others who preferred a quieter environment to complete these questions. Although I love a good Kahoot!, I don't always find them incredibly helpful for reviewing for an AP unit test. 

The hexagonal thinking exercise engages students and makes them think about the concepts we were studying and how all of these concepts were related. I decided that this is what we needed to strengthen our review sessions. I went to Amazon to start looking for potential hexagons to use. In the end, I settled on these cards because there were a lot of them, they were a comfortable size for writing on and fitting several on a table, and they were fairly inexpensive. Because they are just matte card stock, I laminated them. It was a little time consuming to laminate and cut out, but now I have a set of 120 cards that can be used over and over. 

I listened to this Cult of Pedagogy podcast about hexagonal thinking to get a feel for how it might work in my classroom. The podcast talked about students not just building connections with their hexagons, but also to choose several key connecting points to describe why there was a connection there. There are several digital versions that are free to use to do the hexagonal thinking exercise virtually, but since we're in class, I enjoy my students being able to get their hands on. 

When we were ready to do this in class, I made a list of key words and phrases for the unit as well as sets of 5 connection arrows and a shared document for students to share their connections and a picture of their hexagonal web. 

So far we have made hexagonal webs for our Ecology unit and our Metabolism unit. Here is the copy of the document that I made for unit 8 and also the document students used for unit 3.