Teaching Inequalities

Date: 04/02/2009 at 08:08:33
From: Aileen
Subject: teaching inequalities using greater than or equal to 

Hi.  We're looking for the best way to help students understand the
idea of greater than or equal to, including some real-life examples of
how it might be used.  Can you help us?

Date: 04/02/2009 at 09:42:23
From: Doctor Peterson
Subject: Re: teaching inequalities using greater than or equal to

Hi, Aileen.

I think the usual difficulty is what "or" means here, and why the idea
is needed.  When you first give a simple example, it can seem almost
silly:

  7 >= 6   and   6 >= 6

(I'm using ">=" to mean "greater than or equal to")

To say "7 is greater than or equal to 6" seems redundant; why bother 
saying "or equal"??  And to say "6 is greater than or equal to 6" 
seems even sillier.  So why bother with the idea?

It becomes meaningful when you aren't talking about specific numbers, 
but about a variable.  Then it starts making sense to have options 
connected by "or".  Let's take an ordinary English statement that 
incorporates the same idea:

  I am 6 feet tall or more.  (Or, I am at least 6 feet tall)

I know my own height, and so in describing myself I would not need to 
be so vague; I could just say "I am 6 feet tall."  A situation that 
might call for this statement would be

  Only people who are 6 feet tall or more can go on this ride.

This is a general description of ALL people who are allowed to do 
something, and does not just cover one person with a specific 
height.  It says that someone 6 feet tall may go, and also someone 6 
feet 1 inch, and so on.  Anyone whose height is EITHER 6 feet OR 
greater fits the description.

In the same way, using a variable, if I say

  x >= 6

I am saying that x can be any number from 6 up: 6, 6.0001, 7, 1023, 
or whatever.  As long as EITHER x > 6 OR x = 6, the statement 
(inequality) is true.

At this point you could present examples like this, asking for 
examples of numbers for which an inequality is true and numbers for 
which it is false, and then presenting specific numbers and asking 
whether it is true for them.  This might be done in the context of 
examples like mine of height.  You could also translate some English 
statements like "The president must be at least 40 years old" into 
symbols.  Graphing inequalities can also help to clarify their 
meaning, since this emphasizes the idea that many numbers satisfy 
the inequality, and that the "or equals" part causes the boundary 
point (6 in my example) to be part of the solution set.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/02/2009 at 14:25:32
From: Aileen
Subject: Thank you (teaching inequalities using greater than or equal to )

Thank you.  This is very helpful.