Writing and Graphing Inequalities to Represent Constraints

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SWBAT write inequalities in two variables based on a real world problem. Students will be able to graph inequalities in two variables and show the solution.

Big Idea

Application practice! Students solidify their understanding of writing and graphing inequalities with a new set of constraints.


5 minutes

The purpose of today's lesson is to allow students to practice what they have learned about writing and graphing inequalities thus far in the context of a new situation.  I don't take much time to introduce the Pampering and Feeding Time task as it should be fairly straightforward for students.  I begin class by reading through the problem together and then I let students get to work. Homogeneous grouping can work nicely for this task as I can work directly with small groups of students who need more assistance and can provide an extension activity for students who are ready to move faster.


40 minutes

I give students a good chunk of time to work on Pampering and Feeding Time. As I circulate around the room, I double check to make sure students have incorporated and kept track of some of the extraneous math in this problem (changing hours to minutes, and multiplying by 2 for the pampering time that happens twice per day).  

If a student is struggling, I try to have him/her look back at the previous lesson's work to see which method (Carlos' or Clarita's) they prefer.  In my experience, most students prefer to write the inequality Clarita's way, by thinking through the math they need to do to take care of pampering time for both cats and dogs.  Once they have the inequality written, they my need some prompting to remember Clarita's method of graphing involves the x-intercepts. That is, they will need to think about how many cats they could pamper if they didn't take any dogs and vice versa. I like how this task gives them two opportunities to practice the graphing piece.

If students are interested in (or prefer) Carlos' method, can you help them set up a table and remind them the first value will need to be 0 dogs.  They can then look at how many less cats they can pamper if they start to increase the value of dogs.  From there, they will have the y-intercept and should be able to reason about the rate of change of the line.

If some students finish before others, I give them more word problem practice.  I might use a problem from the IMP curriculum. They have a good word problem with two inequalities on page 131 of their Year 2 curriculum. Any real world problems that have students write and graph inequalities in two variables will work.  I emphasize graphing them on separate axes, as in the next lesson will put it all together to create a feasible region.

Discussion + Closing

15 minutes

Once everyone is finished with both graphs, I gather the class back together to discuss.  It's great if at least two students have chosen different methods for writing and graphing the inequalities and we can highlight them both.  What's important is that all students have a method they can use to graph inequalities in two variables without plotting individual points that work or do not work.  

I also remind students again about using the phrase "half plane" and we can talk about the points extending in both directions infinitely.  This is also a good place to discuss the "constraints" of the real world problem and we can talk about negative numbers of cats and dogs not making sense for our problem, but still fitting the actual inequality.

For my students who often struggle with math, having some steps written out can be helpful when we spiral back to this content at the end of the unit.  I like to take time at the end of class for them to reflect on their method to write and graph inequalities.  As today's reflection activity, I ask them to create a series of steps that another student could follow to write and graph an inequality in two variables.


  1. Pampering and Feeding Time 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.