We are shifting gears today and working with something that feels very different than the other topics in this chapter - and we are unfortunately abandoning the Purrrrrfect Cat Toy Company for the day. However, we will be using systems of equations to help us. The purpose of today's lesson is to take rational expressions and write them as the sum of partial fractions.
To start, I give students the problem on slide 2 of the PowerPoint and have them perform a simple addition problem with two rational expressions. Then I give them a sum and see if they can work backwards to find the two fractions that were added together to get the sum (slide 3).
I find that my students will understand that the denominators must be (x + 2) and (x – 3), but many will have difficulty finding the numerators. Some will get the answer by using guess and check. Some may even use variables - I encourage them to follow this track. Even if a student can’t get the answer, it can be beneficial for them to establish if the numerators should be constants or variables. I will give students a few minutes to work on this problem.
The bulk of today’s lesson will consist of a whole class discussion with breaks in between for students to work on the problems. After a student finds the correct fractions that will add together to give the fraction on slide 3 of the PowerPoint, I explain that we found the partial fraction decomposition and explain what partial fractions are. One of the main purposes of teaching students this concept is that it will help them out when they take calculus and must take the integral of something like (3x + 11)/(x^2 – x - 6). Telling your students that it will be helpful next year might alleviate the stress of learning a topic that does not seem accessible to them.
On slide 6 I will go through the process of finding the partial fraction decomposition by setting up the two fractions with the linear factors in the denominator and variables in the numerators. Then we can perform the multiplication and set up a system of equations to find the value of the variables. To drive home the point that the expressions are equal, I will graph both on Desmos to show that the graphs are identical. This is also a good refresher on rational functions and their asymptotes.
For the next slide, we have an example that is much more difficult. The rational expression is to the third power and it has a repeated factor. I walk through my teaching strategies for this one in the video below.
For the last example, the denominator is already factored into a linear and quadratic factor. Students will try to factor the quadratic denominator but will quickly find that it is not factorable. Thus, for this decomposition there will only be two factors. In the video below I highlight my teaching strategy for this example.
After going through these three examples, the last slide of the PowerPoint asks students to write down some of the "rules" of partial fraction decomposition that we discussed today. Here are the ones that I will be looking for as we discuss.