The Overtime Problem
Lesson 1 of 12
Objective: SWBAT write inequalities in one variable and to graph inequalities using a number line. SWBAT describe the relationship between the number of hours worked and a person's total earnings when they are being paid for overtime hours.
This warm-up exposes students to the big idea of this learning unit: Absolute Value and Inequalities. It also provides an opportunity to help students talk through any misconceptions or partial understandings they have about absolute value and inequalities. I spend a lot of time referring to number lines while students work on the first two problems, so it can be helpful to have a large number line displayed throughout this lesson and this unit.
Students often say that they forget what the inequality symbols mean. It helps to have a simple reference poster that shows the meaning of these symbols and to keep this poster in the same place in the room throughout the unit. You can also demonstrate the use of open and closed circles on the number line, so that students who have forgotten this or never fully understood it have a place to look.
One of the inequalities in the second problem has no possible solutions (-1 < x < -2), but represents a common misconception. Students often write this compound inequality without realizing that it makes no sense. Asking them to find numbers that satisfy this inequality gives them a chance to realize on their own that it makes no sense (MP1, MP2).
The third problem is challenging for students to solve, but not too difficult for them to think about. I like to make many copies of the Graph Organizer (front and back) and pass them out to students once they begin this problem. I purposely don’t provide them right away, because I want students to think about the problem in a more open-ended context and then give them some structure
The Graph of Problem 3 was created using Desmos (Note: it does not show the open and closed circles on the endpoints.) In class, I have this already set up on my computer so that I can display it on the projector if students want me to. I always ask them, “Do you want me to show you my answer, or do you want to think about it more?” I try to set up a classroom culture in which “getting the answer” is not the goal. I tell students that I can tell them, but I also say, “As soon as I tell you the answer, I have permanently taken away your chance to figure this out yourself.” I also tell them, “I only want to show you the answer once you have really thought about this.” In this case, I might display the answer briefly (without showing the functions), and ask them: “How do you think I created this graph? What type of equations do you think I used?”
At this stage of the unit, it is important for students to start thinking about these kinds of relationships. The second page of the warm-up provides opportunities for advanced students to explore problems with absolute value and inequalities more deeply. Allowing students to attempt page 2 gives you more time to discuss the third problem with students who need more support. I tell students, “You don’t need to fully understand the third problem today—just figure out as much as you can.”
The Overtime Problem
Before distributing this problem to students, I tell them that this new unit will involve combining their prior knowledge about linear functions with a new idea: inequalities. Then we briefly discussed the idea of overtime pay. I did NOT explain what "time and half" means; we just talk about how you get paid more for those hours beyond the first 40. Then I ask them to answer the two questions on the page and to figure out how to create a data table, a graph and possibly an equation for this situation.
I give students about 20 minutes to work on this investigation. I tell them that I do not expect them to fully understand the multiple representations, but just to get started thinking them. I circulate and ask students to explain their thinking about the data table, or the graph or even the equation. I don't evaluate their thinking formally; rather I ask them questions about why they thought whatever they thought.
I like to use the closing to remind myself (and hopefully the students!) about the key ideas of the lesson. I present the two questions to students and I explain to them:
"This function is a great example of the key idea of this entire unit. It is called a piecewise function--why do think it is called this? What do you think that means?"
I am not expecting very articulate answers to these questions, I really just want to students to think about the main ideas of the lesson and to talk about them a little bit as the lesson closes.
In this lesson, I ask students to write and think for a minute or two, then I ask some students to share their ideas. Usually a few students are able to share some helpful thoughts (and it doesn't tend to be the same students over and over again, refreshingly). I highlight some thoughts that may be useful and I share any of my own ideas that I want to present to them.
This is a very quick activity. Then, I ask students to write their "Check-outs." Checkouts are a few sentences or bullet points written on a piece of scratch paper. I stand at the door and look briefly at each student's checkout to have a quick check-in with them about whatever they wrote.