In this lesson, students are asked to take a systems of inequalities word problem from start to finish. At this point in the unit, students have learned all of the different steps for solving a system of inequalities as well as three different ways to solve a system of equations. This task allows them to pull all of that work together. I use this task as an opportunity to address misconceptions about solving systems of inequalities, as I find there are a number of details that often confuse students.
I begin today's class by having students read today's task, Food for Fido and Fluffy, out loud. I let them know that after the work we've done so far in the unit, they are ready to take a problem from start to finish on their own. If I am working with a group of students who really struggle with this content, I might take some time at the start of class to generate a list of steps they will follow in order to graph a system of inequalities. Otherwise, I let students get right to work and address the steps in the Discussion section of the lesson.
Next, I let students get to work, either in small groups, pairs, or individually. The Food for Fido and Fluffy task outlines the steps students should follow, but I may let my students skip the graphing of each inequality individually, and allow them to move right toward a combined graph. It all depends on the particular class or groups of students.
One of the reasons that I like this task is that I find it gives me a good opportunity to circulate around the room, work with students 1-on-1, and address some questions or common misconceptions.
The issues I am most likely to address are:
The discussion for this lesson can go in many different directions depending on how the students work through the task. I might share some student made graphs or generate a class graph together. If I've noticed that several students have the same misconceptions, I might spend a lot of time addressing those, or have other students explain how they handle those situations.
I sometimes ask for volunteers to help generate a list of steps we can use when we solve a system of inequalities.
I also like to take the time here to point out how systems of equations can come into play. Once we have a feasible region, I like to ask students, "What if we want to know the exact point where two lines cross?" In this graph, the point where the fat and protein line crosses at the point that might not be exactly clear. This point would be the place where the pet got just enough protein (met the exact minimum) and exactly 18 grams of fat (met the maximum exactly). I find that the word exactly sometimes helps student distinguish between a systems of equations problem and a systems of inequalities problem. I ask students how we can find out, using algebra, what that point is. This is a good opportunity to link the usefulness of systems of equations in the context of a systems of inequalities.
I usually end class with a reflection question. Today, I will do an Exit Ticket with students and ask them to respond the following prompt:
What step of solving a system of inequalities is easiest for you? Why? What step is the most challenging? Why?
I like to type up and share their responses (anonymously) at the start of the next class so students can see where there are certain trends (where everyone struggles) or see that different students have different strengths and weaknesses.
This is a good time in the unit to give students some practice working with a system of inequalities on their own. I might use this activity as a homework assignment or an assessment to check in and see which students need additional targeted instruction.
Food for Fido and Fluffy is licensed by © 2012 Mathematics Vision Project | MVP In partnership with the Utah State Office of Education Licensed under the Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported license.