I thought I'd write about how we structure our rational functions unit. I'm curious what others do with it.
(Background ... we are Texas - a non common core state; we have 90 minute classes every other day)
We split the unit into two parts.
Day 1: Data Collection: How is the resulting graph like others we have studied? How is it different? (We start every function unit with a data collection day). For rational functions we used the spaghetti cantilever activity. Find it here. Students work in teams, collect data, answer questions about independent, dependent variables, discrete vs continuous functions, and domain and range. Then they try to find a function that fits their data.
Day 2: Transformations: Transformations are a huge part of our curriculum and by this time of the year, students are very familiar with how "a," "h," and "k" affect functions. So we start with the basic rational function in transformation form. Students use Desmos to track transformations and describe them. With this day we talk about asymptotes and how they are a major attribute of this function. (Asymptotes were introduced in our exponential/logarithm unit). Last we express the domain and range of functions!
Day 3 & 4: We start this day with a review of factoring and how to use factoring to simplify algebraic fractions. Then we being the work of dissecting the attributes of the rational functions based on the factored forms of functions. We start with discontinuities - the vertical asymptotes and holes. Then we discuss the horizontal asymptote; next we find the intercepts; and last we write the domain and range.
Day 5: We review ... presenting the content in various ways. Here is a sample.
Day 6: We test ... this test is short and could be combined with a introductory activity to the second half of the unit. I chose to give my students time to work on their Desmos Creative Art unit.
(This is where we are now)
Day 1: Students use their understanding of simplifying algebraic fractions to multiply and divide fractions. This is a skill practice day. We also solve simple rational equations - the factors cancel leaving simple linear equations to finalize. Students play Tic Tac Toe after doing some routine work.
Day 2: Students use rational functions in introductory problem solving. This year I am using Illuminations, "Light It UP" unit. Students will work in groups ... I may help them get started - use leading questions - but then I want them to grapple with the problems themselves as much as possible. I will set up the experiment ... maybe in 2 - 3 stations so that students can do the experiment as well.
Day 3: I have to get pumped up for this day ... we will add and subtract algebraic functions and simplify complex fractions. My students are so squeamish about fractions. I would really like to talk with their previous teachers about how fractions were handled in middle school, algebra 1, and geometry. This will be a skills practice day! I'm thinking I will give every student a rational function on an index card. They will partner up with another student - find the sum and difference. Then I'll play a few seconds of music. When the music stops, they partner up again - find the sum and difference again. We may do this 4 or 5 times.
Day 4: We solve rational equations. I haven't thought this far yet ... but I'd like to create some form of partner practice that is self-checking.
Day 5: We will tackle the more traditional word problems that involve rational functions like rate of work problems.
Day 6: I will use this as a catch up day. I know 90 minutes sounds like a long time but it's every other day ... and so often skills that I think we master in our 90 minutes need reviewing.
Day 7: This will be our official test review day.
Day 8: We will test!
This about 5 school weeks! The only other unit that we spend this much time on is the quadratic unit.
So ... how do you address rational functions? Do you have any activities that help students with adding and subtracting? What are your favorite application problems for rationals?