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.