Roots and Radical Expressions – Why so Negative?

Students can really struggle with this unit on Roots and Radical Expressions if not taught correctly right away. This is usually the first time that the concept of a math problem having 2 answers ever arises. If you aren’t careful with the way you present this you will lose your students that aren’t big fans of math because they will feel like it’s too much on top of how stressed about math the already are.

When I present this lesson I start with showing them very clearly any number multiplied by itself is ALWAYS positive. I give them plenty of examples. Once I have them 100% confident that they believe any number multiplied by itself is positive I then go into the lesson. You can’t teach them roots and radical expressions without a clear understanding of the basics. It is better to use 5 minutes of class to re-cover this concept.

Introducing Square Roots

Start with the definition in terms they can understand. For example, if the inverse or (opposite) of squaring a number is finding the square root then when we take the square root we should get the number we started with when we squared it. Here is an example…

(-4) squared is 16 right? They will all agree…

So then the Square Root of 16 should be (-4) right?

ALWAYS START WITH AN EXAMPLE INVOLVING NEGATIVE NUMBERS AND DO NOT USE (-2)!

This is like ripping the band-aide off quickly. It shocks them for a second but the pain of negative numbers goes away fast.

Almost 100% of the time a student will ask, “why isn’t the square root of 16 positive 4?”

This is where the magical ah ha moment happens…

You simply respond, IT IS!

Now you have their full attention and can explain to them this crazy new concept of a math problem having two answers.

Give them examples and counter examples, test their brains. Once you have them understanding the math behind the concept, have them each make a list of things where the negative answer would not be needed. For example, if you were talking about distance we never say we walk negative 5 miles. Only the positive answer would be necessary. Then you can give them examples of times that the negative answer is just not needed.

Side Note:

It is very important on the first homework assignment you make them give both answers to the roots. You do not want to let them get into the habit of only giving you the positive answer just because it is most commonly used. Make them give you both until they clearly understand there are two!