Introduction to Inequalities: Working with Constraints

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SWBAT explore the idea of constraints in the context of a real world problem. SWBAT to examine combinations that work and don't work for a business model.

Big Idea

Before learning about systems of inequalities and equations, students jump in to a real world business problem and explore what it's like to work within certain constraints.


10 minutes

Because this is the start of a brand new unit for students, I like to start class with a brainstorm about being a business owner.  I begin class by asking students a question like, " What are some conditions that business owners have to think about when they start a new business?"  I let students share out their answers as I write them on the board.  We look over the list together and I point out to students that some of these conditions can be thought of as limitations or restrictions.  

Then I tell students that this unit will focus on them helping two young people start a business as pet sitters.  We will learn about some math concepts that will help them advise the business owners about some important decisions.  The business owners will have to work within certain limits, and of course, will be interested to know how much profit they can make.  

Next, I hand out pet sitters.pdf, and we read it out loud together.


35 minutes

Next, I let students get to work in small groups or pairs on the task.  If students have trouble getting started, you can encourage them to check out what happens with the dad's suggestion of 12 cats and 12 dogs.  Students should realize that although the dad's combination would use up all of the space available, it would use up more start up money than they have.  Students may then be ready to adjust the number of cats and dogs from there.  If I need to drum up more interest among students (which is usually not the case with this problem), I might encourage them to see who can find the combination of cats and dogs that yields the most profit.

Other issues I watch for:

  • I watch for students who come up with some understanding of what happens when you take away one cat or dog and what effect that has on the other variable.  A student who has an understanding of this concept might be able to relate it to the slope of the inequality later.
  • I also watch for students who think about using all cats or all dogs.  This can lead to a good discussion on graphing an inequality using intercepts in a future class.
  • I look for different ways that students organize the information. I may have students share out how they kept track of the info in the next section of the class.


I have found the idea of profit and start up costs to be confusing for students in this part of the work.  I encourage them to think about the start up costs as money they do not have to pay back so they can think of the profit as something entirely separate. This can be a difficult concept for students as some of them want to think about how much they have to spend on one type of animal versus how much profit they bring in.  For the purposes of this problem and keeping the inequalities and profit equation separate, I ask students to put that kind of thinking on hold for now.

Discussion + Closing

15 minutes

Next I lead students in a whole group discussion to explore some of the combinations that they found.  I always try to show at least one combination that doesn't work, along with many that do, so students can see that a certain combination may fit one set of constraints but not another.  This idea will connect well to the graphs of inequalities in a later class.

I have students share out some of their best combinations.  I like to have students write out the math that they did to check to see if the combination worked.  This makes a nice connection to Common Core Math Practice 8: Look for and express regularity in repeated reasoning. I want students to recognize, for example, that in the space constraint, they are multiplying the number of cats by 6 and the number of dogs by 24 and adding those two numbers to make sure they are less than or equal to 360 over and over again. If students can see this work written out, I find it's easier for them to generate the inequality that matches this situation.

After we have a bunch of combinations (and the math behind them) written out, I ask students if they can tell me, in precise language, what math they have to do to check one of the constraints (I start with the space constraint).  We write out a paragraph like "First, I take the number of cats and multiply by 6. Then I take the number of dogs and multiply by 24.  Then I add those two numbers together and check and make sure the sum is less than or equal to 360."  Next, I ask students if they can translate this paragraph into algebra.  I then ask students to work on the other inequality on their own by following the same steps.

Lastly, I like to distinguish between the inequalities of space and start up costs and the profit. I ask students if there are any limitations on the profit and if it is a constraint.  I then ask them if they can generate an algebraic expression for the profit.  I like to discuss the difference here between an inequality and an equation.

I often end my classes with an exit ticket.  For today's class, I ask students to write a response to the following prompt: "What is your understanding of the word constraint?" Students write their answers on index cards and I collect them at the door.  





Pet Sitters 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.