Home School Life Journal From Preschool to High School

Home School Life Journal ........... Ceramics by Katie Bergenholtz
"Let us strive to make each moment beautiful."
Saint Francis DeSales

It's Hip to be Square!

"You might think I'm crazy,
but I don't even care
I can tell what's going on
It's hip to be square"
Squares can be so much fun to play around with.
First, I made a huge pile of multi-colored and multi-patterned squares out of scrapbooking paper so that the boys would enjoy playing with them. I let them just have some freeplay fun with them. When it seemed that they were ready to move on, I asked them "How many squares does it take to make the smallest square?"
This was such a simple question that the boys were confused at first and then someone piped up, "One square is a square, isn't it?"
"Yes! Put one square on your work space -any color and pattern you like. And what is the next smallest number that makes a square?”
The kids looked around, played a bit with the papers, and then set four of them together.
“Four!" They shouted almost at once.
"Okay. Wrap enough of another type square around your first square, so that they all equal a four square-square." They did so.
"What is next?" This time the answer came a bit quicker even though it was more squares to put together because they were getting the pattern.
"Nine!" was practically shouted.
“Numbers that can form a square when you put them together in a grid are called Square Numbers. See how many square numbers you can form.”

Once they had been at this a while, I got out a small white board and asked them, "Let's look at your squares in a little bit of a different way. Okay, we started with one, and then we added how many more squares to get the next square?" Again, a pause while the took in what I was asking them. "You mean how many of the next color squares?"
"Three?" Still a little hesitancy.
"Yes. Okay. 1+3=4 because four is the total amount of color A and color B, right? Okay, what is next then? We will start with four since that is how many we have totally now. What did we add to that?"
"We added five of the next color."
"Yes and 4+5=9, right? So, let's look at the pattern in the numbers that is emerging..." There was some silence as they pondered what I meant.
"You mean that you take the answer from the first problem as the starting point for the next problem?"
"Yes, that is true. And what about the number you are adding to it?"
They thought a moment, a little frustration showing as they struggled to understand what I meant?"
I circled the numbers to make the pattern more clear.
"They are always just two more."
"Yes, and are they odd numbers or even?"
"Can you predict the next number then?"
Of course they could.
"Now get out that number of the next color/pattern. Then add them to the square."
"So, what kind of number do you get from adding two odd numbers together, odd or even?"
Easily answered.

"Like I said, these are square numbers."
I showed them how these with the superscript.
"You just count the number of squares that make up each side of the square. So, two on each side make four totally, or two squared is four."

"Okay, now we can look at it from the other angle. This is called a Square Root symbol. It tells us that like the roots of a tree the Square Root is the root of the number. Look at the square with four squares in it. How many squares are on the side?"
"That’s the square root, or what make up the square."
We looked at the square roots of all the squares we had before us.
Then we got out calculators and played with the square root button, first confiming the ones we had already looked at, and then going on to play with other numbers."
Then we made mosaics with our lovely square patterns.
And their reward for all their hard work?
We found some rewarding squares!

I found a similar lesson at A Child's Play, where she uses mosaic tiles.
And another at A Pilgrim's Heart.
And another at Educating Risa with counters.
If you have a similar lesson, or know of one, let me know and I will add it to the list.


  1. I love the way you make your kids think! great job!! We just dealt with square numbers the other day... and saw the pattern. I love how you did it with the paper squares. WAY more fun! I will keep it in mind for Javen. For the other boy in particular (the one working on square numbers), he would have been annoyed if I'd taken the time to show him the whole process!

    I think I would like to use the squares to make cool patterns or imagine making pretty quilts without all the sewing! ;)

    amy in peru

  2. A very clear lesson - but now I have that song stuck in my head! :)

  3. I'm thinking that last square is the most rewarding.

    And that song is stuck in my head now too, all 1 line of it that I know.

  4. Amy,
    I have found it very effective to do lessons like this with a mixed age group. It teaches some patience. It introduces concepts that you might not think to introduce to the younger ones. I am constantly suprised how much they absorb. It gives them things to think about...to ruminate on. I know this post makes it seem like I am running the whole show, but really there is a lot of interplay between their own discoveries and my making suggestions that is hard to convey. It is a little like Nature Study or Picture Study, in that you don't just give them all the answers. You make them look, make them search for it and once they have it, it is their own to keep.

  5. This is great! A bit over my Monkey's head yet, but really cool. What a fun way to teach these concepts!

    This would make a great addition to the Classical Homeschooling Carnival.

  6. I have a lot of scrapbooking paper I don't know what to do with. I might have to cut a lot of squares out for JC to play with and see what she discovers. She is kind of young for square roots but she is not beyond seeing patterns. Great lesson! I'm thinking you can use the squares, cut some in half and use it to demonstrate the pythagorean theorum!

  7. That is the fun thing about lessons like this. You can stop at any point and then pick it up again at a later time, when more mature.

  8. This is a great lesson. You know, I have studied so much math, but I didn't know about the next odd number. It's fascinating how math works.

  9. Hm. This is really interesting. My oldest who claims to hates math and stubbornly refuses to learn her addition/subtraction/multiplication/division loves things like the Fibonacci Sequence and Exponential Growth when told as a proof against vampires (think the chess and rice story). I think this would be a fun thing to do with her.

  10. We tried it with magna tiles and it worked great!


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