A DIY Binomial Cube/Trinomial Cube Manipulative

If you've figured out that you can make a manipulative to literally represent (a + b + c)^2, then it's not a big stretch to figure out that you can also make a manipulative to literally represent (a + b + c)^3.

One is a literal square. The other is a literal cube!

I made this DIY trinomial cube to match faces with my DIY trinomial square. This means that it's also set to a 1" standard, NOT a 1 cm standard as are most of my other DIY math manipulatives. But since it was soooooo much easier to find 1", 1.5", and 2" blocks than it was to find 1 cm, 1.5 cm, and 2 cm blocks, it also means that I could take some major shortcuts on this build and save myself a lot of time gluing teeny-tiny wooden blocks together.

I'll take the shortcut every time!

The math isn't exactly tricky on this, but there's a lot of it, and a lot of moving parts, so bear with me.

The trinomial cube has three layers. Layer A is a trinomial square in which every prism has Height a. For my project, a = 1".

Layer B is a trinomial square in which every prism has Height b. For my project, b = 1.5".

Layer C is a trinomial square in which every prism has Height c. For my project, c = 2".

The base block for each layer is a wooden cube that I purchased from Casey's Wood Products:

From left to right, that's a 2" cube, a 1.5" cube, and a 1" cube.

Let's go back to Layer A. We already know that every prism in Layer A will have a height of 1". The cube in Layer A is a^3, with a measurement of 1" x 1" x 1". This is nothing but a 1" wooden block, with no extra pieces added to it. As part of this layer you're also going to have a piece that's b^2, and a piece that's c^2, each with a height of 1", and the prisms that combine the measurements of ab, ac, and bc, each with a height of 1"

For every prism in this layer, you can start with a 1" wooden block, and glue on either .5" wooden blocks or 1" wooden blocks to make the correct sizes.

Here are all the prisms for Layer A:

• a^3 (1" x 1" x 1"). This is a 1" wooden block.
• ab^2 (1.5" x 1.5" x 1"). This is a 1" wooden block plus 10 .5" wooden blocks.

You need one ab^2 for Layer A and two ab^2 for Layer B.

• ac^2 (2" x 2" x 1"). This is four 1" wooden blocks.

You need one ac^2 for Layer A and two ac^2 for Layer C.

• two copies of  ba^2 (1.5" x 1" x 1"). This is a 1" wooden block plus four .5" wooden blocks.
• two copies of ca^2 (2" x 1" x 1"). This is two 1" wooden blocks.
• two copies of abc (1" x 1.5" x 2"). This is two 1" wooden blocks plus eight .5" wooden blocks.

You need two copies of abc for every layer, so six in total.

Every prism in Layer B will have a height of 1.5". The cube in Layer B is b^3, with a measurement of 1.5" x 1.5" x 1.5". This is nothing but a 1.5" wooden block, with no extra pieces added to it. As part of this layer you're also going to have a piece that's a^2, and a piece that's c^2, each with a height of 1.5", and the prisms that combine the measurements of ab, ac, and bc, each with a height of 1.5".

Here are all the prisms for Layer B:

• ba^2 (1.5" x 1" x 1"). This is a 1" wooden block plus four .5" wooden blocks.

You need two ba^2 for Layer A and one ba^2 for Layer B.

• b^3 (1.5" x 1.5" x 1.5"). This is a 1.5" wooden block.
• bc^2 (1.5" x 2" x 2"). This is a 1.5" wooden block plus twenty-one .5" wooden blocks.

You need one bc^2 in Layer B and two bc^2 in Layer C.

• two copies of ab^2 (1.5" x 1.5" x 1"). This is a 1" wooden block plus 10 .5" wooden blocks.
• two copies of abc (1" x 1.5" x 2"). This is two 1" wooden blocks plus eight .5" wooden blocks.
• two copies of cb^2 (2" x 1.5" x 1.5"). This is a 1.5" wooden block plus nine .5" wooden blocks.

You need two cb^2 for Layer B and one cb^2 for Layer C.

Every prism in Layer C will have a height of 2". The cube in Layer C is c^3, with a measurement of 2" x 2" x 2". This is nothing but a 2" wooden block, with no extra pieces added to it. As part of this layer you're also going to have a piece that's a^2, and a piece that's b^2, each with a height of 2", and the prisms that combine the measurements of ab, ac, and bc, each with a height of 2".

Here are all the prisms for Layer C:

• ca^2 (2" x 1" x 1"). This is two 1" wooden blocks.

You need two ca^2 for Layer A and one ca^2 for Layer B.

• cb^2 (2" x 1.5" x 1.5"). This is a 1.5" wooden block plus nine .5" wooden blocks.
• c^3 (2" x 2" x 2"). This is a 2" wooden block.
• two copies of abc (1" x 1.5" x 2"). This is two 1" wooden blocks plus eight .5" wooden blocks.
• two copies of ac^2 (1" x 2" x 2"). This is four 1" wooden blocks.
• two copies of bc^2 (1.5" x 2" x 2"). This is a 1.5" wooden block plus twenty-one .5" wooden blocks.

I painted these prisms the same way I painted the trinomial square prisms, except for the cube, every face will be painted. All the 1"^2 faces are yellow, the 1"x1.5" faces are green--



--the 1.5"^2 faces are blue--



--the 1"x 2" faces are orange, the 1.5" x 2" faces are purple, and the 2"^2 faces are red:



Some prisms in different layers are identical, so here are the total number of prisms by type that I needed:

• one a^3 (1" x 1" x 1"). This is a 1" wooden block.
• one b^3 (1.5" x 1.5" x 1.5"). This is a 1.5" wooden block.
• one c^3 (2" x 2" x 2"). This is a 2" wooden block.

Here's a^3, b^3, and c^3.

• three ba^2 (1.5" x 1" x 1"). This is a 1" wooden block plus four .5" wooden blocks.

• three ca^2 (2" x 1" x 1"). This is two 1" wooden blocks.

• three ab^2 (1.5" x 1.5" x 1"). This is a 1" wooden block plus 10 .5" wooden blocks.

You need one ab^2 for Layer A and two ab^2 for Layer B.

• six abc (1" x 1.5" x 2"). This is two 1" wooden blocks plus eight .5" wooden blocks.

you need two abc prisms for each layer.

There are a lot of abc blocks, so thankfully Spots was willing to help me photograph them!

• three cb^2 (2" x 1.5" x 1.5"). This is a 1.5" wooden block plus nine .5" wooden blocks.

three cb^2

• three ac^2 (1" x 2" x 2"). This is four 1" wooden blocks.

three ac^2

• three bc^2 (1.5" x 2" x 2"). This is a 1.5" wooden block plus twenty-one .5" wooden blocks.

three bc^2

 Spots is sorry that she knocked over my blocks, so here's a blep to make up for it:

And also a boop for good measure:

 And here's what it looks like when it's all finished!

LAYER A

LAYER B

LAYER C

Spots is still not helping...

The kids wandered over as I was finishing photographing the cube (meaning that now I had two teenagers AND a cat helping me...), so I told them this was a puzzle that I'd just finished building and invited them to see if they could put it together:

They didn't notice that it was a pattern, at first, and so made some unworkable choices:

Eventually, though, a light bulb came on...

And behold! It's laid out unusually, but it's our trinomial cube!

Here's the Montessori setup for the binomial and trinomial cubes. They're normally given to small children to work as a puzzle, which is a great way to make that higher level math familiar and not scary at all when they come to it later.

And here are a couple more references, if you want to plan a lesson around the binomial or trinomial cube:

• Using the binomial cube in algebra.
• Video walk-through of the trinomial cube.
• Printable binomial cube nets. 

Want to see what other mischief we (and the cats) manage to get up to with our brand-new DIY trinomial cube? Check out my Craft Knife Facebook page for updates!

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!