On Teaching Math Podcast
Teaching Square Roots Conceptually
E3

Introduction:

Welcome to the On Teaching Math Podcast.  This is the podcast where we discuss all things related to the development of mathematical literacy in your students.  I’m your host, Philip Brown. 

 

Sponsor:

Today’s podcast is brought to us by the numbers 60, 72, 84, 90 and 96.  Take a minute, think of what they have in common … and don’t be ashamed if all you come up with seems like an easy grab, sometimes those are the most difficult observations to make.

It turns out that 60, 72, 84, 90, and 96 are the numbers, less than 100, with the most factors.

Having students explore this might be a good task. 

A great question that this might lead to is, “What number, less than 100, has the most unique prime factors?”

The benefit students will realize is not going to be found in what they find, but in devising a plan, and then monitoring and adjust the plan along the way.

In math we spend too much energy telling students what the plan is, and then they have to execute it.  It leads to all sorts of frustration on both parts.  They often don’t understand a process because they don’t know how each step fits in with the others and they don’t understand the goal.

But, we can’t just expect them to be master problem-solvers overnight, they need practice.

 

Story Time:

 Today’s topic is one that is near and dear to my heart.  I believe that teaching square roots, followed by exponents, conceptually, is a great way to help students, and yourself, break through to a new way of teaching and learning.  And I also believe that the new way of teaching and learning is far more effective, it is faster, students understand more deeply, and develop skills that will transcend your classroom.

But, before we get into that, I wanted to share a story that happened between myself and a student this week.

This freshman girl walked up to me after class.  I fully expected her to lament the academic struggles she’s experiencing.  She is taking a very difficult college-preparatory course called Cambridge IGCSE with me.  All freshman have a notoriously difficult time adjusting to the course expectations, their responsibility and so on. 

Instead of hearing her complain she said, “Mr. Brown, we’re birthday twins.”

I knew what she meant, but didn’t quite trust her correctness.  I asked what she meant.

She said, “We have the same birthday.”

Students don’t know my birthday typically.  You’re kidding…what’s my birthday then?

“April 16^th,” she said.

We exchanged a high-five and some chuckles.  I explained that this birthday might be difficult for me because my grandmother died over the summer.  I am the oldest grandchild and she and I shared a birthday.  Especially as adults, we were very close, though we always had a common sense of humor.  One year, when I lived in Germany, we sent each other the same exact birthday card.  When I got mine in the mail I scrutinized it over and again to see if white-out had been used, and then written over!

The girl gave a wry smile and said, “Well, that’s why you have me now.”

It was a cute moment.  We often get so focused on the frustrating parts of teaching that it is easy to lose sight of this kind of exchange.  So I thought I’d share. 

 

Today’s Topic:

If you listened to last week’s episode you heard the story about my math professor unpacking a probability course from a handful of facts.  It was a powerful experience for me as a student, one that I strive to provide for my students.

Today’s topic will be about how to approach square roots in a conceptual fashion.  Perhaps you’ll even gain a different way of thinking about square roots yourself!  Regardless, by getting kids to read and understand the mathematics, they’ll be developing a skill that transcends whatever particular topic in mathematics they’re learning.  Square roots is a great topic for beginning the process of getting students to change how they engage with math, how they think about it, and to develop mathematical literacy.

What I’ll do for you here is outline the progression of ideas I have used with students to help them learn square roots.

The first idea is that square roots are number.  The square root of four is just a number.  Yes, it can be rewritten without changing its value, but it is just a number.  Like all square roots, it has a unique property.  If you square it, you get four.

So here’s the take-away, the thing you have to get students to understand initially.

Square root numbers are just numbers.   They have a property.  If you square them, the product is the radicand. 

Now of course you’ll have to teach them what the radical symbol is as well as what a radicand is.  But that’s the idea. 

The square root of two is irrational.  We cannot write it more accurately another way.  But we do know that if you take that number and square it, the product is two.

${\left(\sqrt{k}\right)}^2=k$

Square root numbers can sometimes be written with a simple notation, but sometimes not.  The square root of nine, for example, is the same as three.  This is true because if you square 3 you get nine, and if you square the square root of nine, you also get nine.

But with irrational numbers that are square roots, we do not currently possess a more effective method of writing those numbers.

The second idea is a consequence of the first idea.  A square root number can be thought of as asking a question.  For example, the square root of four is asking, what squared is four.  To simplify a square root, we are just answering that question.

$\sqrt{5}=\text{what squared is five?}$ 

${\text{What}}^2=5,x^2=5,\sqrt{5}\mathrm{...}\text{ all the same meaning}\text{.}$ 

The third idea is that a square root is simplified if that simplified number squared is the radicand of the square root number.

The reason the square root of nine is three is because three $–$ squared is nine.  It is not true, or false, for any reason related to steps, process, or anything else.  

The reason $\sqrt{24}=2\sqrt{6},$ is because ${\left(2\sqrt{6}\right)}^2=24.$  Before students can really understand this, they have to understand arithmetic between rational and irrational numbers.

The fourth idea is a square root number whose radicand is not a square number is irrational.  Irrational numbers are numbers that cannot be written as a ratio of integers.  Irrational numbers can be written in a fraction form, $\frac{\sqrt{2}}{2},$ but that is not a ratio of integers.

Students have seen, but likely misunderstand, irrational numbers.  They think that the number pi is about 3.14.  They don’t realize that it is a ratio of a circle’s circumference and diameter.  We only use the symbol for brevity and clarity.  It is with this desire for concise communication in mathematics that we use $\sqrt{2}$ to mean, the number squared that is two.  We cannot write the number more accurately, and the description is cumbersome.

The fifth and final idea, before we actually start getting into how to simplify square roots, is how the arithmetic between rational and irrational numbers works.  This is really an issue of the order of operations, but it is confusing to students.

In the past they’ve learned to approximate square root numbers, which changes irrational square roots into rational numbers.  So they students are unlikely to have any experience dealing with arithmetic dealing with irrational numbers.

Let’s consider multiplication only at this point.  Students will probably understand that something like 2x times 3x is 6x².  That is, the coefficients get multiplied and the unknowns of same variable get multiplied. 

They’ll be unlikely to be able to do the same thing with the number $\pi .$ $2\pi \times 3\pi$ will probably be six-pi, according to freshmen students, not 6 pi-squared. 

However, they’ll be likely to multiply 2x with 3y, and get 6xy. 

Pulling those ideas together, students can begin to understand that $2\sqrt{5}\times 2\sqrt{5}=4\times {\left(\sqrt{5}\right)}^2.$ Going back to our first fundamental fact about square roots, the square root of five squared is just five.    So, we multiply the rational portions together and we multiply the irrational portions together.  We leave those two parts written as multiplication.

 

In summary, students need to understand the following five facts about square roots:

1.     Square roots are numbers.

a.    The property of square root numbers is that if you square one of these numbers, the product is the radicand.

2.    Square root numbers can be read as a question.

a.    What squared is the radicand?

3.    A square root is simplified correctly if the simplified number squared is the radicand of the square root number.

4.    Many square root numbers are irrational.

a.    They cannot be written more accurately.

5.    When multiplying irrational numbers, the rational components are multiplied together, and the irrational components are multiplied together.

 

It is often said that we teach to mastery.  I believe this is a platitude at best.  We cannot teach to mastery.  We can teach to proficiency.  Mastery comes with practice, and requires full buy-in from students.   Students will not fully understand without misconception, these five foundations of square roots.  However, in developing all that follows, if you frequently revert back to these five foundations, and encourage students to do so on their own, their conceptual understanding will become stronger.  Also, their practices and procedures will, over time, become more closely aligned with what it is they know, opposed to what they’ve been shown to do.

Now, how you get students to know these five facts is a bit tricky.  Simply providing them with the information alone will not do it.  In fact, it is likely the students won’t even believe they’re learning math yet because they’ve not seen any real steps. 

It will be important to ask questions that force students to cite these facts for justification of their answers, and to do so continually.

Prerequisite:  There are two pieces of prerequisite knowledge students must have.  They must know how to factor a number and they must know what square numbers are.  The square numbers from 1 to 400 should be readily accessible to students.  These skills should not be taught in isolation, but in context as they naturally occur with the lessons you deliver.

A good way to begin the next portion of instruction is with a question like,

Why is the square root of 16 equal to four?

The acceptable answer is that the square root of sixteen squared is sixteen.  The number four, squared, is also sixteen.  So they’re the same number.  This is fact one in action!

Why is the square root of 16 not equal to five?

The answer to the question here is because 5² does not equal 16. 

Why would it make sense to simplify a square root number by factoring the radicand to find the largest square number?

We are looking for students to connect the first two facts about square roots.  The property they possess and how they can be read as a question.  The square root of 24 is asking, what squared is twenty-four?

It is here that I offer some practical procedural advice.  I suggest that students factor the radicand to find the largest square number, then rewrite the square root number as the product of two square root numbers (which they should understand from fact #5).

So, in simplifying the square root of 48, students should find that 16 is the largest square factor of 48.  So they write $\sqrt{48}=\sqrt{16}\times \sqrt{3}.$ 

 

Since the square root of sixteen is four, the square root of forty-eight is the product of four and the square root of three.

How do we know if this is correct?

We need students to explain that because of the third idea, we can check if the simplification is correct.

The Trickiest Part

Students will be unaccustomed to factoring to find square numbers.  Having them write a table of values and factoring completely can help with this.  Once all factors are written, then it is easy to find which factor is the largest square.  If the largest square factor is one, the square root cannot be simplified!

 

Square Root Arithmetic:  Students should be able to articulate that an operation between the rational and irrational portions of the number $5\sqrt{2},$ exists, and is multiplication. 

Why can’t we multiply five and the square root of two?

The students will need to understand that the square root of two cannot be written more accurately.  That is why we leave the number “five-root-two,” written as it is.

Once this understanding is established it is a good time to introduce the product of a rational number and a square root that can be simplified, but is still irrational.  For example:

$4\sqrt{48}.$ 

I picked this number in particular because the answer will be sixteen times the square root of three.  But, students will often want to write 16 as 4.  If that happens, the student is NOT seeing square root numbers as numbers, but as an operation.  They’re performing steps without conceptual understanding.  This is natural and normal, and should be exposed before the students take quizzes or tests.

Once students understand how to handle such numbers, it is time to step it up for them.  Students will need to be able to add, subtract and multiply things like:

$5\sqrt{2}-3\sqrt{8}$ 

$\sqrt{18}-2\sqrt{27}+3\sqrt{72}-5\sqrt{3}$

${\left(5\sqrt{2}-3\sqrt{8}\right)}^2$

All of this might be difficult to understand in a Podcast, so we won’t explore this much further.  However, you can find more information about this on https://onteachingmath.com/podcast/E3

The thing we can discuss clearly in this format is how the order of operations plays a key role here.  Students will not yet understand why square roots and exponents hold the same priority in the order of operations, but should know they do.

Because of the order of operations, when dealing with something like, “five-root-two minus three-root-eight,” the subtraction cannot be performed until the square roots are dealt with.

Now it will turn out that those numbers are like-terms.  Students may or may not know why you can add or subtract like terms, even though multiplication is clearly an operation written in 6-root-two. 

This is a great time to dive into what’s happening.  Have you ever wondered why you can subtract something like 5-root-2 and 6-root-2, when subtraction comes AFTER multiplication?

The reason is simple, yet, elusive.  Multiplication is repeated addition.  It is not repeated addition like 3 + 5 + 7 + 8, but repeated identical addition.  Because 5-root-2 and 6-root-2 are the same numbers being repeated added, we can combine them before carrying out the multiplication.

It is perhaps clearer to see with something like 5 times two plus 6 times two being 11 times two.  You can add the coefficients before carrying out the multiplication.

At this point, there is only one thing left … rationalizing the denominator.  This skill has fallen out of favor in Algebra classes, which is unfortunate.  Algebra 1 is a foundational course and things learned should prepare students for other classes.  Understanding basic trigonometric function values without understanding how rationalizing the denominator works is nearly impossible.

Beyond that, there are fundamental misconceptions about division and reducing that students possess that make this idea difficult.  Those misconceptions will not go away, they’ll only come out later, with more complicated math (like rational functions), and cause future heartache and frustration.

To help students to understand rationalizing the denominator it is a good idea to lay out a few things they know are true.  However, how this is done is key.  Like is almost always the case, just telling them won’t do much good.  It will be enough for some, but not most.

Students need to get the fact that a number divided by itself is one (except for zero of course).

 

Since the square root of two is a number (fact #1), if we divide this number by itself, the result is one.

 

The second idea at play here is that multiplying by one does not change value.  Since the square root of two divided by the square root of two does not look like one, but it is, multiplying by it will not change the value.  It will be similar to getting change for a $100 bill.  It might look like more money, but it is the same value.

The last idea, before getting into how and why to rationalize the denominator, is that a square root number times itself is the radicand.  The radicand is rational.

 

Let’s consider the classic number $\frac{1}{\sqrt{2}}.$  This number has a denominator that is an irrational number.  The number itself is irrational, which is okay.  But, fractions that have rational denominators are much easier to deal with.  So we want to rewrite this number with a rational denominator.

Students should be able to lead you through the entire process, by referring to the three key ideas we introduced here. 

 

This is a good chance to shore up some misconceptions.  Have them rationalize the denominators with things like:

$\frac{\sqrt{8}}{\sqrt{2}},\frac{\sqrt{8}}{2},\frac{8}{\sqrt{2}}.$ 

The idea here is to make sure students are NOT losing sight of the fundamental facts about square roots as they get into more complicated procedures.

Summary:  So let’s review how this works.  To help students learn square roots conceptually significant time and frequent (even continual) revisiting, in developing student conceptual understanding must be invested. 

That means we are not teaching how to simplify square roots.  We are teaching what square roots are, what it means to simplify square roots.  We of course can, and should, offer procedural advice.  We don’t want students having to reinvent the wheel, but we do want them to connect the procedures and guidance we provide with those concepts.

Square roots are numbers with a specific property.  That property allows us to substitute other numbers with the same property in their place, if possible.  The square root of four is a number, that if squared, is four.  The number two is a number, that if squared, is four. 

Therefore, the numbers the square root of four and two are the same.

 

If you’d like more information, specific lessons, assignments, even assessments, that are aligned with the philosophies and practices outlined in this episode, please visit my website:  https://onteachingmath.com/squareroots.  On that page you will find a series of lessons, assignments, student resources, and lesson guides that will help you approach this topic in a new way!

 

If you’ve made it this far, thank you for spending your valuable time listening to what I have to offer.  If you’ve found it valuable and informative, please leave me a four-star review on whatever platform you use for your podcasts.  If you feel I deserve anything less than a four-star review, please give me a chance to fix it.  Email me at: [email protected]  Feel free to email me with any comments, suggestions, concerns or needs!  I would love to hear from you.

I hope you have a wonderful week teaching.  Until next Thursday … this is Philip Brown, signing off.