Out of Bounds

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SWBAT represent constraints by equations or inequalities.

Big Idea

Find your limits, or at least find the limits to the equations and inequalities in an assortment of problems.

Set the Stage

15 minutes

You will want copies of the Toy Store Challenge before this lesson.  I begin by telling my students about a friend of mine who owns a toy company that makes wooden doll houses.   I say that she wants to expand her business to include rocking horses, but isn't sure how it will affect her profits.  I ask my students to help me figure this out for my friend.  I tell them to work with their right-shoulder partner to determine what else they need to know to set up this problem.  While they're working I walk around offering encouragement and redirection as needed.  For example, if a team is struggling I might ask questions like "What are the lowest values possible for each of your variables?" or "What information should to be included in calculating costs and profits?" (MP1, MP2, MP4)  When everyone is done I ask for volunteers to tell the class what they discussed.  While my students are sharing, I act as scribe and record their ideas on the board, summarizing as we go.  After all teams have shared I ask the class to review the board and see if I've missed anything.  I never know for sure what my students will come up with for this part of the lesson, but I've found that taking time for this now gives my students the confidence and experience to make things go more smoothly later in this lesson and in the unit.  Generally their list includes: how many of each toy can be made each hour/day/week, how much it costs to make each toy, and the retail price of each toy.  Once we've summarized this piece, I give them the Toy Store Challenge with the information I have and tell them we need to create equations or inequalities to represent each of these components.  (MP1, MP2, MP4)  I walk around again while my students are working encouraging and redirecting them, but also looking for teams that have recognized some of the additional constraints (like not having negative dollhouses or rocking horses!).  When all the teams are done I ask those teams I've noted during my observations to share what they've done with the class.  I emphasize the constraints to this problem even though they haven't been clearly stated so that my students begin to look for these kinds of limits.  I'm not asking them to solve for a maximum or minimum, I'm just looking for the constraints. 




Put it into Action

35 minutes

I tell my students that they get to work with their front-partner to find the constraints for today's Limits Challenge problems.  I remind them that they each need to write their own responses but that they can definitely collaborate to figure out how to handle each challenge.  I also tell them to be ready to present at least one problem of my choosing to the class.  I pass out the Limits Challenge problem sheet and ask if there are any questions, then tell them they have about 20 minutes to complete the problems. (MP1, MP2, MP4) While my students are working I walk around offering encouragement and assistance as necessary.  For example, if a team is having trouble coming up with constraints, I might ask leading questions like "What is the lowest value that will work for x?" or "How are x and y related?"  As they finish up the challenge I have each team draw from a cup of numbered popsicle sticks to determine which problem they'll present.  I ask each team to present in order starting with problem #1 and have the rest of the class critique their method and solution.  The presenting team has an opportunity to respond, then the next team presents and so on until we've gone through all the problems. (MP3)

Wrap it Up

5 minutes

To reinforce the idea of constraints on equations and inequalities I give my students each a notecard and ask them to summarize what constraints they worked with for today's problems.  I explain that I don't want the details of each problem but instead am looking for a more general statement like, "most of the problems had values that couldn't be negative".  I explain the value of these notecards in my video.