This post provides a handout that guides students through the various steps for completing the square to transform an equation into vertex form. Students are guided through each step in isolation.
Students are presented each step in a separate chunk of the lesson. Then the steps are chained together, with scaffolding that is faded. This is a different approach than presented in a previous post. The chunks, examples, and scaffolding help make students more independent in completing the work. This frees up the teacher to provide more 1 on 1 support.
Chunks of the Lesson
The initiation addresses prerequisite skills: factoring, perfect squares, fractions, and doubles. In lieu of having students divide by 2, I focus on identifying fractions that add to the linear coefficient as you will see in the second page.
The students identify the constant that results in a perfect square. They do so by identifying doubles that result in the linear coefficient (e.g., 6 = 3 + 3). The examples help guide them through this process. This section could be presented after a hands on activity on
Students are then tasked with factoring perfect squares in isolation, including those with fractions. The doubles are modeled for whole numbers first, generalized to fractions.
At this point, the students have identified the constant to complete the square and then factored expressions. The next sections have students complete the square and then factor in equations. Note that the equations are structured as a step after the students would have subtracted the original constant, leaving the quadratic and linear terms on the right.
The last section chains all the steps together, first with scaffolding then without. Additional practice would be generated with other handouts that have problems in isolation.
Dr. Po-Shen Loh shared a possible new method for factoring a quadratic. This post provides a layman’s attempt to share the steps that teachers may find intriguing and possibly useful, especially for complex roots.
The premise of Loh’s method is that in lieu of considering two factors separately, you can focus on the following:
Average of the two factors which is the coefficient of x (linear coefficient) divided by 2. In the case below, that is 8/2 = 4.
The common distance of each factor from the average, d. In the case below, the factors are converted into the expressions 4-d and 4+d because both are d units away from 4.
This results in a single unknown, the distance d.
The aforementioned expressions with d replace the factors. Now we have an easy quadratic equation to solve using square root. Once d is determined, the factors of 12 are now known and we are on our way.
This method works for complex| factors. This makes Loh’s method less time intensive than using the Quadratic Formula, and there is no formula to memorize.
I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).
Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.
I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.
The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.
Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.
The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.
I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.
A reader asked about an algebra 2 problem and shared (below) his effort to cut up the math into bite-sized pieces. I greatly appreciate his effort because he is trying to meet student needs. While this post is very “mathy” I want to make a couple of points to the readers. First, I wrote out a detailed response (2nd photo below). Second, in both of our efforts we attempted unpack as much as possible. This is what our students need. Also, the reader is developing his ability to do this unpacking and if he continues he will become increasingly more adept at this skill (growth mindset). That means his future students will benefit!