Homeschool Math: Perfect Squares Hiding inside Area Models

Syd was simplifying radicals the other day, and not having a fun time of it. She was struggling to link the concept of factoring the radicand to simplifying, and I was trying, as usual, to think of hands-on manipulatives that might clarify the process. 

I did NOT find a way to model simplifying radicals using manipulatives, alas, but while I was playing around with the decanomial square I DID find a hands-on enrichment that kids who are first learning the concept of perfect squares might enjoy.

I like this little activity because it connects the mathematical definition of the perfect square with the Montessori-style sensorial skill of eyeballing it, or even measuring it by feel. Although you're technically not allowed to eyeball stuff as mathematical proof, pattern recognition via the senses is very important. That's how kids learn to read, for one thing, and it's how IQ tests are built, for another. 

Use this activity with a kid who's first learning, or reviewing, the concept of the perfect square. You can do it with paper area models that a kid can draw and color on, or you can do it, as I've done here, with the decanomial square model, which is extra fun because it has pieces you can manipulate. Kids could try to find the largest perfect square(s) that would fit inside the area model, or just find any perfect squares that would--whatever they find fun and you find helpful! Here are some models that show examples:

These first two are when I was still thinking I might figure out a way to model simplifying radicals. I LOVE combining manipulatives with a dry-erase board to help kids connect the model to the algorithm it represents.

For all these examples, I've pulled an area model from our decanomial square, and we're arranging the perfect squares on top of it, leaving, of course, a remainder since the area models aren't themselves perfect squares.

You can write algebraic equations with these, showing how to use the Order of Operations and/or solve for x. For example:

5 + 5 x 5 = 30

or

8^2 + 2^2 + y = 80

You just can't, you know, use them to model how to simplify radicals...

The search continues!

P.S. Here are the resources that I used to help both kids master radicals.

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A DIY Binomial Square/Trinomial Square Manipulative

This trinomial square manipulative is an extension of the binomial square, and if you own a decanomial square manipulative you don't have to make this, because you've already got more than enough to model binomial and trinomial squares. I only made this a separate manipulative because I wanted its faces to match the DIY trinomial cube that I also built.

Because synergy!

I was a little bummed that I couldn't find enough cubes keyed to a centimeter standard to make my trinomial manipulatives in centimeter measurements. Instead, the smallest square in my DIY trinomial square is 1"^2, and the smallest cube in my DIY trinomial cube is 1"^3. So if you're trying to build a real Montessori-style trinomial cube, this is not the project for you. Keep searching for cubes measured in centimeters, or buy a zillion literal centimeter cubes and get to gluing! But because I started with inches, I was able to save myself some work when I made the trinomial cube by buying 1", 1.5", and 2" blocks, and gluing .5" blocks to them to make the prisms.

But that's a totally different project, which I made AFTER this. Here's how to make this project!

To make a trinomial square whose smallest square is 1", you will need the following materials:

• 81 blocks, each measuring .5"^3. I am profoundly devoted to Casey's Wood Products, and so I bought these .5" wooden blocks from them. 
• acrylic paint in the primary and secondary colors. Sooo... red, yellow, blue, purple, orange, and green.
• glue. You can use wood glue, but it's not my favorite. I prefer E6000!
• paint brushes.

You are going to glue together the following rectangles. Remember that these are area models, not volume models, so don't be stacking any blocks on top of each other. Everything is just one block tall!

• 2x2 (you need one of these)
• 2x3 (you need two of these)
• 2x4 (you need two of these)
• 3x3 (you need one of these)
• 3x4 (you need two of these)
• 4x4 (you need one of these)

Here's what it should look like when it's finished!

If you did an exceptionally bad job gluing, you can pause and sand each rectangle smooth, but don't feel like you need to get caught in the weeds with this project--a few bumps and drips are fine. Nobody needs their trinomial square to look like it came from IKEA!

You are going to paint the faces that represent the areas of the trinomial square, and either paint the .5" tall faces black or leave them unpainted (I left them unpainted--no weeds for me!). If you want to keep your trinomial square at least Montessori-adjacent, then make your 1"^2 faces yellow, your 1.5" faces blue, and your 2" faces red.

Here's another big veer away from Montessori-style: I painted the area models that are adjacent to the squares the secondary color represented by combining the primary colors of those two squares. I think it makes logical sense, and it's pretty!

As another optional step, you can seal these, but if you used acrylic paint and your kids aren't going to play roughly with them, you don't have to.

The main purpose of this manipulative is to illustrate (a+b+c)^2. You can go through a billion machinations to expand this trinomial square via calculations, but just by looking at this physical model and copying what you see, you can clearly see that it's a^2 + b^2 + c^2 + 2ab + 2bc +2ac.

How much sense does that make, and how easy is that to remember?

Here's the entire trinomial square lesson that I do with my kiddos. We tend to spiral in our math projects, so ages ago the kids built binomial squares to practice pattern-building and to see what equations with variables look like. We delved back into it when the older kid's algebra curriculum started factoring. We're back again because now it's the younger kid studying algebra and the older kid studying geometry, and this makes a lovely intersection. To add interest and rigor, I introduced trinomials, and next time we find our way back to it, I imagine that we'll find something else new to explore!

Speaking of something else new to explore: here's another fun bit of spatial reasoning play that you can do with a trinomial square: it's a puzzle! We know how to make a perfect square one way, but how many other ways can you find?





These perfect squares should look familiar, because they're binomial square models!

If you enjoy this type of puzzle, you should really check out pentominoes. I am low-key obsessed with them--honestly, I can't imagine anyone who's a visual learner or enjoys spatial reasoning who wouldn't go mad for them!

P.S. If you need an anchor chart or a poster for display, there's a good graphic of the trinomial square and its measurements here.

P.P.S. Want to see more handmade homeschool stuff, and the adventures that we have with them? Check out my Craft Knife Facebook page!

How to Square Binomials and Trinomials using Area Models

Let's say that you have two lengths: a and b. You would like to know what area would be covered by a square whose sides are each of these lengths combined.

The equation for that is (a+b)^2.

But how do you actually multiply that?

The algebraic way is to use the FOIL method: First, Outside, Inside, Last. This gives you (a+b)(a+b)=a^2 + ab + ab + b^2, which you simplify to (a+b)(a+b) = a^2 + 2ab + b^2.

That's fine algebraically, and you should totally memorize it, but here's what you should VISUALIZE when you do this, because here's what makes sense:

Visualize sitting on the rug in your family room. It's a Friday afternoon, soooooooo close to the end of your school week, and you'd very much rather be done with school and go walk your dog or listen to your music, but your mother wants to do one final project together before she sets you free. She hands you and your sister the decanomial square manipulatives and asks you to set them up:

Stacking the area models is NOT part of setting them up, but is also nearly irresistable.

Fun fact: a decanomial square is the same concept as a binomial square or a trinomial square. Whereas a binomial square is (a+b)^2 and a trinomial square is (a+b+c)^2, a decanomial square is the representation of (a+b+c+d+e+f+g+h+i+j)^2. How would you like to factor THAT puppy without being able to visualize an area model to make sense of it?

Your mother asks you to choose two squares from the decanomial. You'll label one square as a^2, with sides measuring length a, and the other square as b^2, with sides measuring length b. The challenge is build an area model of a square whose sides are each length a+b; to complete this challenge you may use a^2, b^2, and two other rectangles of your choice.



Visually, it's not hard to find the rectangles that match to complete the square, but it might take a little longer to notice that these rectangles each have two sides that match the length of one of the squares.

The solution to the puzzle, then, is (a+b)^2 = a^2 + ab + ab + b^2. Since you have two rectangles labeled ab, you can simplify your equation to (a+b)^2 = a^2 + 2ab + b^2.

This is exactly what you get when you the FOIL method, but doesn't it make a little more sense to see it?

Unfortunately, your cruel mother now tells you that you have to add a THIRD square to your puzzle. She. Is. So. MEAN!

You add a third square and label it c^2, with sides measuring length c. The puzzle is now a trinomial square, (a+b+c)^2, and your challenge is, once again, to complete the square. Your mother does not tell you how many pieces you have to use to complete this trinomial square, so you make yourself a shortcut:

  Actually, this works algebraically!

The equation you've created is (a+b+c)^2 = a^2 + 2ab +b^2 + 2((a+b)c) + c^2. As long as you can complete the puzzle with area models that have sides that relate to lengths a, b, or c, you can translate the solution algebraically... but this isn't the simplest solution algebraically.

THIS is the puzzle solution that's also the simplest algebraically:

(a+b+c)^2 = a^2 + 2ab + b^2 + 2bc + c^2 + 2ac.

You can make infinitely bigger squares with an infinite number of terms. I mean, think of how many terms that decanomial square has, and yet it still follows the pattern you can see in the binomial and trinomial square.

And as soon as you memorize the binomial square and trinomial square equations that you created, you can go listen to music out on the back deck with your cat!

Montessori Pink Tower and Cuisenaire Rod Extensions for a Sixth-Grader

When Syd's sixth-grade Math Mammoth curriculum covered exponents, she and I (and her sister, on occasion...) did a lot of hands-on sensorial work with exponents. It's easy to forget that even bigger kids benefit from hands-on math, but when you set something down in front of them and watch them become totally immersed in it, you're unlikely to forget again.

Soon after seeing how invested Syd was in working all the Montessori pink tower extensions, and how quickly she moved through them, I asked her if she wanted to join me in creating some extensions for the pink tower and Cuisenaire rods. Both sets of manipulatives are keyed to the centimeter, and so we found that they worked quite well together!

Here are some of the combinations that Syd and I found:

For some reason, Syd really enjoyed making a pattern with the pink tower, and then repeating it with the Cuisenaire rods. It's none of my business why or what she's getting out of a particular experience--the fact that she's happily engaged and invested in her work is proof enough that there's something of value in it for her.

We made a log cabin quilt block!

You can play a lot with perspective when you explore these two materials together. Each Cuisenaire rod is only one centimeter wide, so many of the patterns are best seen looking straight down from above.

I thought that this diagonal patter that Syd made was extremely clever. You can see that she doesn't have it quite worked out in this photo, but I can tell that she's noticed that two pink tower blocks can share a Cuisenaire rod. 

I think she might be exploring along the same lines here, as she's omitted the centimeter cube that she was originally using to cap all the corners of her creations.

This was just a "play" day for us, but you could make this activity more academically rigorous, and in some cases cross-curricular, by adding more investigations to it:

• Children could be the ones in charge of photographing their designs.
• Children could diagram their designs on graph paper. To continue extending it, they could add photographs of the completed designs, write a description or instructions, hand-paste or use a graphic design program to make a book, and then bind that book themselves.
• Children could use clip art versions of pink tower blocks and Cuisenaire rods in a graphic design program, designing patterns that are impossible to create in real life.
• Children can design and perform STEM challenges, such as creating the tallest free-standing tower or the longest possible bridge with supports.
• Combine these materials with the decanomial square to explore cubes, or add more pattern possibilities. Bonus points if you use foam core and/or foam sheets to make your decanomial square pieces one centimeter thick!

Most outside resources for these materials focus on extensions best suited for young children, but here are a couple that I've found that are sophisticated enough to intrigue an older child:

• Build a tower, an inverted tower, and connecting towers from Cuisenaire rods. You can challenge your fine motor skills with this one! This takes some serious problem-solving, and you'll be well-suited for architecture or engineering afterwards.
• Long division with Cuisenaire rods. This is how I modeled long division when the kids were first learning it, but it's a useful--and mind-blowing!--model to use anytime to make the connection that MATH MAKES SENSE.

Montessori Pink Tower Extensions for a Sixth Grader

As we have been playing a lot with exponents lately, I finally hit the big red button and purchased a Montessori pink tower.

Well, actually I specifically purchased an unpainted one, so I guess it's a "pink" tower. FYI: I've bought a few Montessori materials over the years, and I've always found the best prices at Alison's Montessori. That stuff is still spendy, though, so let me know if you ever find a cheaper place!

The kids first used the pink tower with the tower of squares that they'd previously made:

To make that tower of squares, you need a looooong roll of butcher paper and several sheets of cm-gridded paper. The kids are to make a tower by cutting squares from the cm-gridded paper, going from 1cm^2 to 20cm^2. They're to arrange it nicely on the paper to make a tower (it's dealer's choice if the tower is centered or aligned at one edge), and then they are to annotate each square with its exponent (2^2), its exponent in long form (2x2), and its total units (4). Keep it forever, as you'll be pulling it out for extension work forever, as you can see above!

In the activity above, the kids matched each cube to its square footprint (it became immediately clear that our cm grids weren't perfect centimeters, so there was a bit of averaging). There were cubes for the first ten squares on their chart. Then they put the same information--exponent, long form, and total units--on index cards, and matched them to the cubes. It was a quite informative visualization!

Even though it was still valuable for Will to engage in the work, and have her hands on those exponents, this activity was really more at Syd's sixth grade level, which became clear when as soon as the project was complete Will abandoned it to go do something else, and Syd continued to fool around with the tower. I was amused to see that she built it several times as perfectly as possible, just like a good Montessori schoolgirl, but she did quickly move on to exploring extension ideas:

After I saw that, I researched pink tower extensions, and printed out this set of pink tower extension cards for Syd to explore.

I think she liked them!

I've watched the kids as Montessori preschoolers, so it was especially interesting for me to see this new work presented to them. Both kids were interested and engaged in the exponents work, but Will had no interest in sensorial exploration with the blocks beyond that. Syd had a great interest in further sensorial exploration, and concentrated on the blocks quite deeply for a while. Just as a preschooler would, she started by building the tower, but whereas a preschooler would possibly do this dozens upon dozens of times, Syd got all she needed from doing it just a few times, and then seamlessly moved into exploring other patterns. She was deeply engaged for a while in making these patterns, and then she and I invented some patterns that also used Cuisenaire rods (I'll show those to you another time), and then, just like that, she was done. The tower is still sitting in a pile in the playroom, untouched for a week now, so this weekend I'll have her put it away.

But think of that process--Syd was just as engaged as a preschooler would be in this sensory material, and her experience was no less valuable just because she moved through the entire process in a week rather than three years, and no less valuable just because she's eleven, and not four. It clearly fed something in her, and I don't need to key it to state academic standards to know that, and I don't even need to know what, exactly, she took from the exploration--she took something, was engaged and happy and productive, and therefore it was a great school day.

Homeschool Math: Use the Decanomial Square to Explore Binomial Squares

If you took algebra in middle school, you probably remember that squaring a binomial makes no sense.

Let's say you've got to square the expression (a+b). You square the first term, take twice the product of the two terms, and square the second term:

a squared + 2ab + b squared.

And you're all, "Umm... okay?" And your teacher is all, "Just memorize it. It'll be on the test."

And you do. And you earn an A on the test. And then you forget what you've just memorized, on account of it makes no sense and you have nowhere to put that information, nor anything to use it on other than more algebra.

Well, there's a deep, dark secret at the heart of Algebra 1, and that deep, dark secret is that the binomial square?

It is REAL:


Will hasn't been asked to square binomials yet, and Syd certainly hasn't, either, so this is something that we're exploring well in advance, so that when each child is then asked to square her first binomial in her math curriculum, I can say, "Oh, that! Remember that we do that with the decanomial square. You remember what that looks like." Takes the fear out of math before the kid even knows there's something there to fear.

To start, I asked the kids to build the decanomial square and then to place the labels that let us use algebraic notation to name each piece of the square. Then, I asked each of them to pick two random squares from the decanomial square, and to place them kitty-corner to each other (kitty-corner, you probably know, is the official, scientific term). Then I asked them to figure out how to complete the puzzle to make a square:


I did have to redirect Syd, as her first instinct was to piece together the sides with many small pieces. I told her that the perfect solution would be just two pieces, one for each side.

And you can see as well as I can what both kids discovered, that it's two identical pieces that complete this square:

If they don't quite notice that these two pieces have the same lengths as square a and square b, they do notice that as soon as I ask them to write down the expression that represents what they have:

You can see in the image above (you can also see that I need to recondition our very old dry erase board) that Will has written the expression correctly: c squared + cb + bc + b squared.

She's written "cb" and "bc" because that's how they're labeled on the square, one in row c and column b, and one in row b and column c. But she can see that they're perfectly identical, so that and her knowledge of the commutative property of multiplication (a*b=b*a) allows her to simplify cb+bc into 2cb.

The simplified expression, then is this:


You can also look at the square that you've made and pick another way to write the expression, one that's even simpler: each side is made of length a + length e, so to represent the square as a whole, you can simply write (a+e) squared.

(a+e) squared = a squared + 2ae + e squared

Congratulations! You've just squared your first binomial!

See, that wasn't so hard, was it?

Homeschool Math: The Decanomial Square and Its Extensions

Just yesterday, I told a friend that if I'd had preschool to do all over again, I'd have skipped Montessori and saved the money.

When I said that, I was thinking of 1) how expensive Montessori was, and 2) how I'd primarily thought of myself as using it for childcare and social experiences, and so I'd perhaps have been better served with a a cheaper alternative.

It didn't take much reflecting on my statement, however, to realize that I was wrong. I'm glad that we didn't continue with Montessori, of course, because homeschooling has been vastly superior to any available alternatives, but Montessori, I think, has deeply affected how my children view learning, and how I mentor and mediate their learning.

Montessori is where I developed my firm philosophy of--other than screentime limits--never disturbing a busy child. Children's focused concentration is precious and not something to be lightly interrupted. That focused concentration, whether directed at "work" or play or nothing at all, as far as you can tell, is children thinking deeply and hard and they'll later be able to apply that focused concentration to their other work and studies.

Montessori is where I first saw work plans, and now I couldn't imagine our homeschool without them. For the kids, a work plan is expectation management, empowerment in how they're going to conduct their day, and a good model of clarity and organization that leads to a solid work ethic.

Montessori is where I became firmly convinced of the vital importance of hands-on manipulatives, especially for math. We think of math as cerebral, but it's also visceral and and intuitive and sensorial; you may know how to do long division on paper, for instance, but when you physically do long division, you understand how it works for real.

Much of our own math manipulatives, then, are Montessori-style, even if I create them myself and alter them from the specified Montessori format. The decanomial square, for instance, is a physical model of the multiplication table. In Montessori, kids compile a slightly abstract paper model (Montessori relies on the reinforcement of the color coding of numerical values, which kids have long internalized by this time, so that they can get away with having a less concrete representation) that they then will recreate with the bead manipulatives that they use. Kids can do some really sophisticated extension work with this material, including bringing out some of their old preschool manipulatives (the pink tower and the brown stair, in particular) to make clear the relationship between all of these different ways of representing numbers.

We use Cuisenaire rods instead of the Montessori bead materials, and we sure as heck don't have a school's worth of them to build ourselves a decanomial square with, so I didn't want to use a decanomial square that relied on color-coding to impart much of its crucial information. Instead, I wanted to make it gridded with centimeters throughout, and make the relationship of each piece to the number that it represents (as well as its area and perimeter and the length of each side) clear that way.

Thank goodness that my husband is a graphic designer!

Matt designed the decanomial square that we used (we need to figure out how to best format it for a home printer, and then my goal is to make it available for sale if there are any other Montessori-obsessed homeschoolers out there), and although he color-coded it to our Cuisenaire rods, I printed it onto cardstock in shades of grey, on account of I'm too lazy to replace the color ink cartridges on my printer. Syd then helped me assemble the pieces into the complete square and then cut them out.

In a Montessori setting, you would model the assembly of the decanomial square in an organized way, but I'm mean, so I gave it to the kids as a puzzle, with no other clue than that the finished piece would be a perfect square. I made them figure it out completely without my assistance, and wouldn't you know it, but they eventually (after some griping, and then some settling down to get to work) did present me with a perfectly assembled decanomial square!

That process was one complete math enrichment work for one school week, but last week we played some more with the decanomial square, extending our understanding of what it can offer. First, I had the children build the square again, which they did this time with minimal fuss (other than losing the 2x2 square, sigh, which we actually really needed for this lesson. Oh, well... I do intend to replace this particular model with the colorful one as soon as I can get Matt to replace the ink cartridges for my lazy ass, so then this entire greyscale one will be just spare parts).

Then, I demonstrated one of my favorite equations in all of math: the Pythagorean theorem. Will has encountered this before, and Syd has worked with squares before, so it was a good lesson for them both. I set up the 3x3 square and the 4x4 square at right angles to each other, and told them that these were two sides of a right triangle, sides a and b. They needed to find the square whose side made the perfect hypotenuse, or side c.

And they did!


A squared plus b squared equals c squared! If you worked hand-in-hand with a good graphic design program, it wouldn't be hard to print out physical models on the spot of the hypotenuse of various other right triangles. You could print out a square with sides exactly 5.3851640713 cm long, for instance, to go with your 2x2 square and 5x5 square.

Hmmm, maybe that's another set of models that Matt should design for me?

Most of our work, however, involved using the pieces of the decanomial square to build equations (pre-algebra for the win!). We set out one piece of the square, then covered it completely by puzzling together other pieces--

  

--then wrote an equation to represent that model:

For instance, one equation might read:  5 squared (ugh, I wish I could find the superscript hotkey without looking it up! Soooo lazy!) = 4 squared + 4 + 5. (I didn't mention it, but do you notice that this is also the Pythagorean theorem? So cool!) Another might read: 3 squared = 3 + (2x3).

That was the extent of this particular extension lesson, but there are so many more things to do with the decanomial square, and we'll be revisiting it often throughout algebra and geometry and possibly into trigonometry. I'm currently on the lookout, for instance, for a cheap version of the Montessori pink tower (you *can* DIY it, but with a to-do list as long as mine...) so that we can have physical models of the cubes represented in the decanomial square.

When I've finally gotten my hands on one, I'll be VERY curious to see if the kids remember it from their own Montessori preschool days. Between the two of them, I wouldn't be surprised if they'd built it a hundred times over the years that they were there.