The Great Inequality Debate

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Objective

SWBAT understand and apply the properties of inequalities. SWBAT decide which expressions are greater, equal, or cannot be determined.

Big Idea

Which expression greater? How do you know? Students think deeply about inequality relationships and the properties of inequalities.

Introduction

10 minutes

This task provides a great opportunity for students to work with inequalities and simultaneously practice the standard for mathematical practice #3: Construct viable arguments and critique the reasoning of others.  A lot of today's work will be spent discussing the problems and letting students provide examples and counter examples to build their arguments. 

We begin today's class by reading through the instructions of Greather Than. I draw students attention to the fact that they have three things to do with each problem: 

  1. Decide which expression is greater.
  2. Show an example or counter-example.
  3. Explain their reasoning.

I let them know that our discussion today will have a particular focus on explaining our reasoning.

We start with the first problem together. I give students a minute or two to work on the problem and then ask someone to share out their thinking. I am hoping for students to actually make a mistake here and say that x-y will always be greater. Then I ask for another student to try to prove them wrong. My students often ignore negative numbers as possible values for variables so I will be looking for someone who took negative numbers into account.  Students may be surprised to see that in Question #1 a greater than relationship cannot be determined.

Investigation

20 minutes

Next, I let students get to work in small groups or pairs to start working through the problems.  I remind them that they have to give examples and reasoning for each problem.  I encourage them to move around on the page and come back to a problem they might get stuck on. I'll be circulating and looking for students who are able to come up with strong counter examples. 

Students often have trouble with the number line questions on #7 and #8. Here, I encourage them to choose numbers for the dots to represent and work from there. Students may also struggle with the exponential expression in Question #10.

 

 

Debate + Discussion

25 minutes

The debate is the main part of today's lesson. Depending on how much time we have, I might choose the problems that are particularly rich for students to debate.  Some Questions that I like in particular are: #5, 9, 11 - 15.  I put the problems up around the room and have different groups write up their answers.  Then we debate and look for counter examples to see if the greater expression really can be determined.  Because we are debating each other's work and ideas here, it is important to return to our classroom norms about discussion. I want to make sure students use appropriate language to disagree with another student.  See my previous lesson on Generating Classroom Discourse for resources about discussion norms and sentence stems.

 

 

Closing

5 minutes

A lot of students are surprised to learn that many greater than relationships cannot be determined. I want to hear about some of their ah-ha moments in today's reflections. I ask them to write a short reflection in response to the following prompt:

  • What surprised you about some of the inequality relationships? Why?

Citations

Greater Than? 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.

http://www.mathematicsvisionproject.org/secondary-1-mathematics.html