How to Teach Square Roots Conceptually

If you have taught for any length of time, you’ll surely have seen one of these two things below.

$\sqrt{24}=6\sqrt{2}\text{ or }\sqrt{4}=\sqrt{2}$ 

Sure, this can be corrected procedurally.  But, over time, they’ll forget the procedure and revert back to following whatever misconception they possess that has them make these mistakes in the first place. 

I’d like to share with you a few approaches that can help.   Keep in mind, there is no way to have students seamlessly integrate new information with their existing body of knowledge.  There will always be confusion and misunderstanding.  By focusing in on the very nature of the issues here, and that is lack of conceptual understanding and lack of mathematical literacy, we can make things smoother, quicker, and improve retention.

Step one is to teach students to properly read square roots.  Sure, a square root can be an operation, but it is also the best way to write a lot of irrational numbers.  Make sure you students understand these two ways of reading a square root number.

 

$\begin{array}{l}1.\sqrt{2}\text{}\text{ asks, "What squared is 2?"}\\ \text{2}\text{.}\text{ If you square the number, }\sqrt{2},\text{ the product is 2}\text{.}\end{array}$

Students are quick studies when it comes to getting out of responsibility and side-stepping expectations.  Very quickly, when asked “What does the square root of 11 ask?” students will say, “What squared is the radicand?” 

When pressed on the radicand, they may or may not understand it is 11.  But, they’ll be unlikely to have really considered the question for what it asks.  Do not be satisfied with students that are just repeating what they’ve heard.  Make them demonstrate what they know.  A good way to do so is by asking a question like the one below.

$\text{How is }\sqrt{9}\text{ like }{x}^2=9.$ 

Another way to test their knowledge is to ask them to evaluate the following:

$\sqrt{2}\times \sqrt{2}.$ 

We do not want students saying it is the square root of four at this point.  To do so means they have not made sense of the second fact listed about the number.  An alternative to using a Natural Number as the radicand is to use an unknown.  For example:

$\sqrt{m}\times \sqrt{m}.$

Step two requires them to understand why the square root of nine, for example, is three.  The reason why it is true has nothing to do with steps.  Instead, the square root of nine asks, “What squared is 9?”  The answer is three.  There is no other reason.

Once again, students make excellent pull-toy dolls, saying random things when prompted.  Once in a while they recite the correct phrase, even though they don’t understand it, and we get fooled.  It is imperative to be creative and access their knowledge in a new way.

Before I show you how that can be done with something like the square root of a square number, let’s consider the objections of students here.  Students will complain that we’re making it complicated, or that we are confusing them.

First, we’re not making the math complicated.  Anything being learned for the first time is complicated.  Things only become simple with the development of expertise.  How complicated is it to teach a small child to tie their shoes?  But once the skill is mastered, it is done without thought.

The second point is that we are not confusing them, they are already confused.  They just don’t know it yet.   They will not move from being ignorant to knowledgeable without first working through the confusion.  If we want them to understand so they can develop related, more advanced skills, and we want them to retain what they’re learning, they have to understand.  They must grasp the concept.

So how can we really determine if they know why the square root of twenty-five is really five?  We do so by asking the same question in a new way. 

$\text{Given that the number }{k}^2=m,\text{ what is }\sqrt{m}?$ 

Another way to get at the knowledge is by asking why the square root of 25 is not 6.  Students will say, “Because it’s five.”  While they’re right, that does not explain why the square root of 25 is not six.  Only when they demonstrate that 6² = 36, not 25, will they have shown their correct thinking.  But, as is the case with the other questions, students will soon learn to mimic this response while not possessing the knowledge.  So, you have to be clever and on your toes.  This point is worth laboring!

Step three involves verifying square root simplification of non-perfect squares.  This uncovers a slew of misconceptions, which will address. Before we get into that, here is exactly what I mean.

$\text{Is this true: }\sqrt{24}=2\sqrt{6}\text{ ?}$ 

Have students explain what is true about the square root of twenty-four.  There are two ways they should be able to think of this number (and one of them is not as an operation, yet). 

1.      What squared is 24?

2.      This number squared is 24.

The statement is true if “two times the square root of six, squared, is twenty-four.”  Just like the square root of 9 is three only because 3² = 9. 

The first hurdle here is that students do not really understand irrational numbers like the square root of six.  They’ve learned how to approximate and do calculation with the approximations. Here is how they see it.

$\sqrt{2}=1.4$ 

$3+\sqrt{2}=4.4$ 

$3\times \sqrt{2}=4.2$ 

  What this means is that students believe:

1.      Addition of a rational number and an irrational number is rational.

2.      The product of a rational and irrational number is also rational.

a.       This can be true if the rational number is zero.

This misunderstanding, which naturally occurs as a byproduct of learning to approximate without understanding what approximation means, is a major hurdle for students.  It must be addressed at this time.

To do so, students need to be made to understand that irrational numbers cannot be written with our decimal or fraction system.  We use special symbols in the place of the number itself, because we quite literally have no other way to write the number.

A good place to start is with π.  This number is the ratio of a circle’s diameter and its circumference.  The number cannot be written as a decimal.  It is not 3.14, 22/7, or anything we can write with a decimal or as a fraction.  The square root of two is similar.  The picture below shows probably over 1,000 decimal places, but it is not complete.  This is only close, but not it.

 

Students will know the Pythagorean Theorem.  It is a good idea to show them how an isosceles right triangle, with side lengths of one, will have a hypotenuse of the square root of two.  So while we cannot write the number, we can draw it!

The other piece of new information here is how square roots can be irrational.  If the radicand is not a perfect square, the number is irrational.  At this point, we cannot pursue this too far because we’ll lose sight of our goal, which is to get them to understand irrational and rational arithmetic.

This point, and all others, will be novel concepts.  You will need to circle back and revisit each of them periodically.  Students only will latch on to correct understanding when they fully realize that their previously held believes are incorrect.  What typically happens is they pervert new information to fit what they already believed, creating new misconceptions.  So be patient, light-hearted and consistent.

Once students see that the square root of two is irrational, they can see how they cannot carry out and write with our number system, either of these two arithmetic operations:

$3+\sqrt{2}\text{ or 3}\times \sqrt{2}.$ 

This will likely be the first time they will understand one of the standards for the Number Unit in High School level mathematics. 

Students must demonstrate that the product of a non-zero rational and irrational number is irrational.

 

Students must demonstrate the sum of a rational and an irrational number is irrational.

Keep in mind, this may seem like a huge investment of time at this point, and they don’t even know how to simplify a square root number yet.  However, we have uncovered many misconceptions and taught them what the math really means!  This will pay off as we move forward.  It will also help establish an expectation and introduce a new way to learn.  Math, eventually, will not be thought of as steps, but instead consequences of ideas and facts.

Back to our question:

$\text{Is this true: }\sqrt{24}=2\sqrt{6}\text{ ?}$

Just like the square root of nine being three because 3² = 9, this is true if:

${\left(2\sqrt{6}\right)}^2=24.$ 

Make sure students understand that there is an unwritten operation at play between the two and the irrational number.  We don’t write the multiplication, which is confusing because 26 is just considered differently.  It isn’t 12 at all (2 times 6)! 

Once that is established, because of the commutative property of multiplication,

$2\sqrt{6}\times 2\sqrt{6}=2\times 2\times \sqrt{6}\times \sqrt{6}.$

There should be no talk of cancelling.  The property of the square root of six is that if you square it, you get six.  That’s the first thing they learned about square root numbers. 

$2\times 2\times \sqrt{6}\times \sqrt{6}=4\times 6.$

As mentioned before, students are quick studies.  They learn to mimic and get right answers without developing understanding. This may seem like a superficial and easy task, but do not allow them to trick themselves or you regarding their understanding.

A good type of question to ask is:

$\text{Show that }mn\sqrt{m}=\sqrt{m^3{n}^2}.$ 

To do this, we students to square the expression on the left of the equal sign to verify it equals the radicand.  This addresses the very meaning of square root numbers.

Last step is to teach them what the word simplify means in the context of square roots.  It means to rewrite the number so that the radicand does not contain a perfect square.

The way to coach students to do this is to factor the radicand to find the largest square number.  This is aligned with the meaning of square roots because square roots ask about square numbers.  When they find the LARGEST perfect square that is a factor of the radicand, the rewrite the expression as a product and then simply answer the question asked by the square roots.  Here’s what it looks like.

$\text{Simplify }\sqrt{48}.$ 

$48=2\times 24,3\times 16,4\times 12,6\times 8.$

$\sqrt{48}=\sqrt{16}\times \sqrt{3}$ 
Write the square root of the perfect square first so that you do not end up with $\sqrt{3}4,$ which looks like $\sqrt{34}.$ 

$\sqrt{48}=4\times \sqrt{3}$.

At this point, students should be ready to simplify square roots.  However, be warned about a common misconception developed at this point.  They’ll easily run the two procedures into one.  They often write things like:

$\text{Simplify }\sqrt{18}.$ 

$\sqrt{18}=\sqrt{9}\times \sqrt{2}$

$\sqrt{18}=3\sqrt{2}$

${\left(3\sqrt{2}\right)}^2=9\times 2$

$9\times 2=18$

$\sqrt{18}=18.$

The moral of the story here is that to teach students conceptually means that you must be devoted, diligent and consistent with reverting back to the foundational facts, #1 and #2 at the beginning of this discussion.

This approach in no way promises to prevent silly mistakes or misconceptions.  But what it does do is create a common understanding that can be used to easily explain why $\sqrt{12}$ is not $3\sqrt{2}.$  It is not “three root two,” because ${\left(3\sqrt{2}\right)}^2=18,\text{}\text{ not 12}\text{.}$ 

This referring to the conceptual facts and understanding is powerful for students. Over time they will start referring to what they know to be true for validation instead of examination of steps.  There is not a step in getting $\sqrt{12}=3\sqrt{2},$ that is wrong.  What is wrong is that their work is not mathematically consistent and their answer does not answer the question, what squared is twelve?

If a student really understands square roots, how to multiply them with other roots, and how arithmetic works irrational and rational numbers, the topics that follow go much more quickly.  After this will be square root arithmetic, like $5\sqrt{2}-3\sqrt{8},$ and then cube roots and the like.  Each topic that you can use to dig deep into the mathematical meaning will, over time, quicken the pace of the class.

In summary:

1.      Square roots have a meaning.  The meaning can be considered a question or a statement, and both need to be understood by students.

a.       This meaning is why the square root of 16 is 4.

2.      Square roots of non-square numbers are irrational.  Arithmetic with rational and irrational numbers is irrational (except with zero).

3.      To simplify a square root is to rewrite any factor of the radicand that is a perfect square.

a.       When rewriting, place the square root of the square number first.

4.      The simplification of a square root number is only right if that number squared is the radicand.

I hope you find this informative, thought-provoking, and are encouraged to take up the challenge of teaching conceptually!  It is well worth the initial struggles.

For lessons, assignments, and further exploration with this topic, please visit: https://onteachingmath.com/squareroots/