I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative specifically explains this lesson’s Warm Up- Inverse Functions which asks students to solve an equation for both x and y and compare the results.
I also use this time to correct and record the previous day's Homework.
Why do we have functions?
This question is not only meaningful to this lesson but also to this unit and all of Algebra. I love to tell my students that mathematicians are always looking for shortcuts (Math Practice 8), not because they don’t want to do the mathematics but because they want to do it in the most efficient way. A formula or function provides a MASSIVE amount of information in a very small amount of space which makes it a great "shortcut".
For example, two numbers add to 1000. Even if I limited myself to whole numbers, this would take me a very long time to list. If I included all real numbers, I could NEVER finish. However, if I write x + y = 1000, this represents ALL possibilities.
Inverses in Real Life
I introduce inverse functions through a real life scenario. I found that you have to be extremely careful in the problems that your pick to explore inverse functions. Both variables have the possibility of being the independent variable. This excludes any function with time as one of the variables.
The Jamisons got lunch at the Nampa Farmer’s Market. They bought drinks for $1 and hotdogs for $2 and spent $10.
The students begin by finding an equation to model this situation(Math Practice 4). It should look something like d + 2h = 10. We then solve this equation for both d and h, look at the equation as both a function of hot dogs and drinks, and then compare our findings (Math Practice 7). This will lead us to the concept of inverse functions.
I use my class pairings extensively during a guided investigation style lesson like this. I have the students work through whatever step and then check with their partner. We then discuss it as a class. I vary my methods for choosing a person to share their work. Sometimes I will call on someone randomly. Sometimes I will ask for volunteers. If the problem has a limited number of solutions like yes/no or true/false, I will ask for a thumbs up/down from everyone.
Once we have finished this problem, we explore some of the cases of inverses situations in everyday life. This provides an opportunity for students to develop a deeper conceptual understanding of inverses of functions.
Inverses in Algebra
Students must also be able to find the inverse of a function not connected to a real life scenario. I am showing the traditional method of switching the x and y values. It is important to make it clear that it becomes a DIFFERENT function. There is no way that y = 3x - 5 is the same as x = 3y - 5. This may seem obvious but not always to the students. The notation f(x) and f-1(x) represent two different functions. The fact that they both use x is confusing. I point them back to the hotdog/drink problem and remind them that the x stood for both drinks and hotdogs depending on which function we looked at.
This Guided Practice has students find the inverse of linear as well as a quadratic functions and includes a real life scenario. To focus on having the students verbalize the connections between the equation and the scenario (Math Practice 3), a useful method is to have one student explain their thinking and then the other paraphrase. This also ensures that they are all participating and building understanding.
Another strategy that I use is to walk around the room listening for particularly insightful ideas or explanations. I then ask students to share. This is especially helpful when I can pick a student who doesn't usually share to help build their mathematical confidence.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
Today's Exit Ticket asks students to find the inverse of the linear function f(x) = 4x - 7.
This Homework begins with a variety of linear and power functions. The students are asked to find the inverse for each one. The goal of this portion is to reinforce the skill learned in the lesson. The final two problems give the students a real life scenario involving two unknown quantities that are related. The students are asked to write a function for each variable and then explain how the functions relate to each other and to the graph.