Finding the Feasible Region: From Start to Finish

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:

• One of the most common misconceptions I see with inequalities is the use of the phrase "at least" within a word problem.  Students often automatically translate "at least" to a < sign. In this problem it comes up with the protein constraint, where it says "Each meal should consist of at least 12 grams of protein."  I like to ask students a question like, "Well, if your pet got 14 grams of protein, would that be ok?"  For me, always connecting back to what is possible with sample numbers seems to help students choose the appropriate sign.

• Students often struggle when asked to write an inequality where the variables do not have coefficients.  So in this task, I often find that students are able to write the fat, carbohydrate, and protein inequalities fairly easy, but struggle to write the last one, which is the total ounces of food.  I find this to be a recurring issue in a number of word problems, both for systems of inequalities and systems of equations.  I try to address this issue by asking students combinations of numbers that would fit the problem.  So I might so, "ok, so they can't have more than 10 ounces of food, what's a possible combination of Tidbits and Flakes that would work for them?"  I might keep track of those combinations for students and then ask them something like, "ok, so what do you think about an inequality? What math operation are you doing to check and see if your combination works?" Ironically, this inequality is simpler than the others, so sometimes students get confused by it.

• I also find students are often confused by the differences of units of measure because the problem references both grams and ounces.  By this point, we've worked with enough systems to try to draw a comparison to another problem. Many systems of equations problems work with money, so I try to show that when a problem has money in it, the other constraint does not need to also be restricted by money.  It might have to do with something entirely different.  Because the grams of protein, fat, and carbs, are contained within the ounces of food, we can again look at combinations that work or don't work and see how the issue plays out.

• Another misconception I see is that when students are scaling their graphs, they often think the graph has to go as high as what the inequality has to be greater than or less than.  For example, if the inequality is 4T + 2F < 18, students will think the axes have to go as high as 18.  In order to address this misconception, I try to work with students to graph inequalities using the x and y intercepts so they can see the maximum or minimum values of a real world quantity that will work. So for this problem, I might say, well if you only gave your pet Tidbits, how many ounces would you have to give him so he got exactly 18 grams of fat? Students should then be able to see that the highest number they will have to go to on the Tidbits access will be 18 divided by 4.  

• Lastly, less so in this problem, but in many inequalities, students are confused about what side of the line to shade.  I always encourage them to choose a point that they think will work and plug it back into the inequality to see if the inequality remains true.

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.