Square Numbers?

beginner square roots, square root manipulatives

Building square numbers teaches a lot of math. Very young children start out simply learning their shapes. Here you add a little math to this. They will discover the difference between building squares and building rectangles. A square is a specific type of rectangle, one whose sides are all the same.

To start, ascertain whether the child knows what a square is. If not explain by showing a one block and a hundred square...draw a square, get them to draw a square. Then start building a two square, using two 2's...and let them build the next one with 3's and then 4's on up...these are square numbers.

square numbers, base ten manipulatives, Mortensen Math

There are all kinds of discoveries to be made here. Counting the sides and making them the same takes a little work for young students. Older students will knock these out quickly. Ask questions along the way...how many in a three square?

How many in a four square...?

If we make 16 into a square, what is one side?

If all the sides are same if we know one side then we know all the sides...there is a symbol for this: square root sign
That symbol says we take ANY number make it into a square and then count one side. That little handle thingy or check mark part tells us we want to know about just one side. It's like a division symbol which just means form a rectangle and count the sides, but it's not any rectangle: it's a square. Some numbers are easier to make into squares than others. To start we are going to focus on the easy ones. Even if the student is in high school.
The symbol means form a square and count one side.

I recall vividly being in seventh or eighth grade thinking, "how did nine get to be a square number? It's all bulbous and round!" Many of my classmates agreed.


We knew only symbols not what the symbols stood for or represented and this can cause problems and make math seem mysterious or magical. IT HAS NOTHING TO DO WITH THE SYMBOL! The symbol 9 represents nine and nine can be formed into a certain geometric shape: a square. One side of that square is 3. The square root is visually obvious. Suddenly square roots are child's play.

In order to make it super easy, they need to know what the symbols MEAN. Square numbers are the ones that can be shaped into a square using the blocks...any number can be shaped into a square using imagination and math but only certain numbers are naturally square.

square numbers, kindergarten math actvities, Mortensen Math

Making them into a pyramid is not only fun it gives them a chance to see geometric progression first hand. Children love to build things. Students are always quite pleased with themselves when they build this all by themselves. It is, of course, fun to knock down too.

Here is a screencast off my blog, which is why I talk about other pictures and things that aren't on this page (I thought it would be good to include it here):

If you want to see the page where I'm talking about the towers and skip counting go to my blog...and look for the post called Make Math Fun. (Or just click the link.)

We can compare a six square and a three square and see that even though six is twice three, a six square is a lot more than twice as big as a three square.

square numbers, Mortensen Math, base ten blocks, math manipulatives

One student who was having a little trouble making squares figured out that if you put the numbers in the tray "sideways" you would always get a square.

A parent called this "cheating." How so? They used a tool to make sure they got square numbers every time and discovered that all they had to do was count and make the sides the same...I don't care how they do it; my instruction was "make squares." Then let them. Direct their discoveries by making observations and asking questions.

1, 4, 9, 16, 25, 36, 49, 64, 81, 100 all have something in common. They are all square. This is an SAT question:

What do the numbers in the set 1, 9, 25, and 100 have in common?

a) They are even.
b) They are odd.
c) They are square.
d) None of the above.

Super easy, IF you have had some experience. If you can "see" what they mean. Otherwise you have to remember they are all square or be able to do some "complex" math...and under test stress the student might not get it.

Young students might have to get out tens and measure how many 5 fives are, the bigger square numbers are more fun to measure...a wealth of knowledge is to be had just doing that because they see that (when discovering how many are in a nine square) 2 nines is 18 and three is 27 and so on...

[Note, my grammar Nazi friends, my grammar is correct here because it is short for "is the same as" instead of are. And BTW it's a math site.]

They learn their multiplication facts while learning about square numbers, they don't have to know their times tables in order to start square roots. Certainly it helps make things go faster if they know their multiplication facts but it shouldn't stop you from teaching them lessons about squares.

Later we can use the blocks to estimate square roots of numbers that don't make squares easily, and start using our imagination with concepts like the square root of two and three...but this is where you start. Even for older students because they need to see the shapes and understand the basic concepts.

This page is only 1 thru nine, what about bigger square numbers? Build and draw 11 squares on up thru 20. Plenty of fun and discovery to be had...

Here is a quick 10 minute explanation for TEACHERS, older students and parents:

After you have watched the videos up there this will make more sense, and after just 15 or 20 minutes on this page, you ought to have a handle on square numbers.

I did not mention that with directed discovery and some practice they will be able to tell you "the rules" on how to express square roots with radicals...and should they come across a text book that gives them the rules they will make sense.

And now that we have some experience with square numbers lets learn a little about completing the square with a square number while playing with polynomials.

Completing the square makes a lot more sense when you realize we are in fact talking about squares. This page goes from basic like this to more "advanced" algebra.

Go from Square Numbers back home.

Go here for a bit of adult humor and poetry...from Harold and Kumar, the square root of 3 poem. Uses the "F" word in the clip. So not suitable for all audiences. As you will find as we go along this site is not always "G" rated. Eventually I will partition off the adult material from the children's...here is the poem for those who don't want to be assaulted by an English word.

The Square Root Of 3

I fear that I will always be
A lonely number like root three
A three is all that's good and right,
Why must my three keep out of sight
Beneath a vicious square root sign,
I wish instead I were a nine
For nine could thwart this evil trick,
with just some quick arithmetic
I know I'll never see the sun, as 1.7321
Such is my reality, a sad irrationality
When hark! What is this I see,
Another square root of a three
Has quietly come waltzing by,
Together now we multiply
To form a number we prefer,
Rejoicing as an integer
We break free from our mortal bonds
And with a wave of magic wands
Our square root signs become unglued
And love for me has been renewed.

Speaking of meaning what does square mean? Are we square? A question asked after an exchange of money for example...or after a deal between two or more people...square: everybody is same, or even or equal. Fair.

Someone who is square could mean someone who doesn't smoke or drink. Why? Sometimes it means honest, a square dealer...

Use the student's language to explore and discover more meaning. Then tell them one of the reasons we love the Mathematics is it's always the same. Square root means make a square, count the sides every time: today, yesterday and tomorrow. The square numbers don't change...and they are the same in every language.

Eventually this is where we are going with square numbers:

“Adam and Eve are like imaginary number, like the square root of minus one... If you include it in your equation, you can calculate all manners of things, which cannot be imagined without it.” ~Philip Pullman

For most people that quote means NOTHING.

But before we go there we have to visit Pythagoras, and learn to square root any number, as well as under stand what it means to count the side of an eight square and get two square roots of two...

So meantime perhaps this has a little more meaning:

“If I can't get people to commit themselves on whether or not there is a square root of two, then I won't touch on God or anything here” ~Tom Lehrer

Contemplate the diagonal on a hundred square, pretending that the sides are ones instead of tens...but I am getting ahead of myself.