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!

Day Trip under Duress: Spring Mill State Park

Do you remember what life used to be like, when you could roll out of bed on a Sunday, look at your two grumpy teenagers staring at tiny little screens, decide that enough was enough, and drag them, furious and loudly complaining, out to the car and take them on a spontaneous day trip to a public space?

What a luxury that was!

Honestly, as stir-crazy as this pandemic sheltering in place is making us, the kids would probably still have to be dragged, furious and loudly complaining, if I tried to take them on another day trip to Spring Mill State Park--the older kid, because dogs aren't allowed in the pioneer village or caves and so I wouldn't let Luna come on this particular trip that we took last summer, and the younger kid, because other than in-person ballet classes and visits her friends, both of which she is devastated to be without, she is the biggest homebody in the world, is NOT stir-crazy, and even 39 days into sheltering in place has no desire to go anywhere where ballet or her friends are not.

Maybe when it's safe to stop sheltering in place Matt and I will leave the kids home and just go back to Spring Mill State Park by ourselves!

On this particular enforced day trip, we mostly hung out in the pioneer village, dodging the historic reenactors by sneaking in to explore the buildings whenever they popped out of them:

Here I am, pretty excited to be on an adventure out in the sun. Notice that there are no grumpy teenagers in the frame!

Fortunately, even grumpy teenagers can't outlast an entire day of fresh air, sunshine, and interesting places to explore. Eventually, the younger kid found that checkers was calling our names--

--and by the time she'd beaten me thoroughly, the older kid and Matt had wandered off to their own adventures and the younger kid and I wandered away, too, discovering...

...A COTTAGE GARDEN!!!!!!!

I am OBSESSED with historical gardens. They make me burn with envy. I cannot get enough of pretending that I can copy them.

Because of that, when Matt and the older kid came back from their wanderings they figured out exactly where to find us!
 

I... don't know why this particular teenager is wallowing in the historic cottage garden, actually. She doesn't look grumpy, though!

Here's an old stagecoach route that the older kid was absolutely revved up about following uphill for so long that I'm pretty sure it was her revenge for me forcing her into this day trip:

It doesn't look it, but it was REALLY STEEP! Poor horses! Poor ME! 

 Here's an interesting rock that the younger kid spotted in the creek:

And here's just one tombstone from the historic cemetery that I also dragged the children through, mwa-ha-ha:

When we're finally allowed out in public again, I'll drag my furious and loudly objecting children back here, because even though we spent the whole day, we still didn't see the Gus Grissom Memorial or tour the caves!

The Kid Designed a Mentos Launch Tube, or, How to Make a PVC Pipe Guy Explosively Vomit

Y'all are not going to believe what Matt found when he cleaned out my cluttered, disorganized, impossible-to-find-anything-in-even-though-I-really-needed-the-linoleum-carver-right-that-minute homeschool closet.

One two-liter of diet soda, leftover from WAAAAY back when the kids were obsessed with Mentos and soda nucleation and so we did it every week!

It turns out that I could really use that empty two-liter bottle, actually, for the decomposition models that I'd like the kids to make this week. But even I am not going to advocate the consumption of an eight-year-old Diet Coke--that's just one thrifty rung up the frugality ladder too far.

Fortunately, Syd is working to earn the retired Junior/Cadette Science Sleuth badge, which is basically a miscellaneous hands-on science study that encourages a girl to stretch herself into a variety of scientific sub-fields through DIYs, model-making, and experimentation.

Would Syd like to explore engineering by building a custom Mentos launch tube that would propel the soda explosion according to her own design?

She would!

Syd built the launch tube using this PVC pipe Mentos launch tube tutorial--Matt had already planned a trip to our local big-box hardware store to get supplies for several DIY projects before our state's shelter-in-place order began, so I could tack the PVC pipe bits that we needed to his shopping list. If you don't have PVC pipe bits on hand, though, I think you could make due with cardboard and duct tape.

Syd followed the tutorial to build the launch chamber, but engineered her own paper clip trigger system. The big fun, though, was in cutting the end cap to create her own soda spray pattern!

Would she drill lots of holes in it to make a sunburst?

Cut a single line for a wide spray?

Try out the narrowest hole possible to see if it makes the soda more explosive?

Um, no. It turns out that what the world actually needed was a Mentos launch tube whose exit was designed to mimic a person explosively vomiting:

Also, soda is going to spew out of their eyes:

Little buddy looks so happy right now, all decked out with Mentos and a paperclip. It has no idea what's waiting for it out on the driveway...

The paperclip snagged a couple of the Mentos and kept them from falling until after a delay, but otherwise I think the young engineer was quite satisfied with her design. Also, this thing happened that I think is gross:



Can You Lick the Science, indeed?

Awww, our little guy feels much better now!

And now we have the two-liter bottles that we need to decompose banana peels and eggshells in our family room!

For Syd, this was an engineering project, not a chemistry one, so we didn't dwell much on the process of nucleation, itself. But if you want to do more with nucleation, it's also represented in this cloud in a jar model and in the formation of crystals.

P.S. The homeschool closet is spanglingly clean right now, and yet there is still no linoleum carver! I very much fear that we will have to clean out my cluttered, disorganized, impossible-to-find-anything-even-though-I-really-need-the-linoleum-carver studio closet next.

A Magical Day at the Children's Museum: Anne Frank, Mo Willems, and the Stories that We Tell

One week before our community's pandemic closures began, back when we were still happily anticipating a spring full of field trips, fashion shows, and fun adventures with friends, the kids and I had one more magical day at one of our favorite places in the world, the Children's Museum of Indianapolis.

As usual, it was service learning that led us there:

This was an especially fun activity, and both kids, who worked at a table separate from me, declared that it was one of the favorite activities they've ever led here! At both of our tables, we had lots of die-cuts of living things--people in real colors and people in fun colors, dragons, frogs, bears, etc., along with markers, foam stickers, and sticky-backed googly eyes that were incredibly difficult to separate from their backings.

Small children would wander up, with their adults or with a school group, and I would invite them to "make a character," and tell them that when they were done, I wanted them to tell me all about the character they'd made. The kids would choose a die-cut, settle in with markers and stickers, and one-by-one I'd ask each kid if they wanted googly eyes for their character. If they said yes, I'd ask how many they wanted, and then painstakingly unpeel the backings of that many eyes for them. Normally, it's really important for kids to do their own work, but those backings were practically IMPOSSIBLE to peel. It was bonkers how difficult they were!

When each kid had finished, they were excited to tell me about the character they'd created. I'd ask them to tell me what their character looked like on the outside, and as they did so we'd talk about how that was a physical trait. Then I'd ask them to tell me what their character felt like on the inside, and as they did so we'd talk about how that was a personality trait. Then, if we had time, they could tell me a whole story about their character, and if we didn't have time, I'd remind them that they could tell a story about their character when they were home.

OMG the kids were SO INTO THIS ACTIVITY! I don't know if it was the open-ended nature of the activity, the unusual materials they could access, or the agency they felt in story-telling, but they universally loved the snot out of this activity! And mentoring an activity is always much more fun when the kids are into it, so it turned out to be a terrific way to spend our morning.

We generally volunteer in the mornings, so afterward our tradition is to eat our packed lunch in the museum's big cafeteria. We do get an employee discount, but even with that the food is too expensive to justify three entire lunches every time we go, so we bring our lunch, but we always bring something that requires a bounty of condiments, because one notable fact about the Children's Museum is that its condiments bar hosts EVERY CONDIMENT. I'm talking ranch dressing. I'm talking honey mustard. I'm talking barbecue sauce. I'm talking hot sauce!

Seriously, it's, like, our favorite thing. You haven't lived until you've gotten up early to bake frozen chicken strips, put them in a Children's Museum-branded lunch bag, and eaten them cold with fourteen different dipping sauces.

Ugh, I'm craving it right now!

After lunch, we've generally got a few items on our museum to-do list before we make the drive back home. On this day, there was a brand-new exhibit on Mo Willems to explore!

Here is my favorite Mo Willems book:



Here is my second-favorite Mo Willems book:



I'm not as into the Elephant and Piggy series, but the kids definitely blew through them all when they were each learning to read. And yep, we sat in the museum gallery's reading area and blew through them all again.

And then learned to draw them for ourselves!

Will is smiling like a brat in this photo because in the video Mo Willems has just said, "Write your name on your drawing," and she has done so:

Most of this gallery is geared to the very young, but one of the many terrific things about the Children's Museum is that the galleries always include awesome stuff to educate and engage big people, too. Check out this exhibit of Mo Willems' original sketches for his books!

Syd was VERY interested to learn that he uses charcoal pencil for these sketches. He must be a very tidy artist, and we want to know how he avoids smudging charcoal all over his paper!

Fun side fact: Mo Willems is currently a Kennedy Center artist-in-residence, and during the pandemic he's putting out a daily series of videos. There's some cool how-to-draw stuff, but also really interesting inside info about his creative process and how he makes his art:



After Mo Willems, we of COURSE had to visit the dinosaurs--

--and then we made another visit to Anne Frank. The kids are currently working on a short study of her for this monthly patch program through Girl Scouts of Central Illinois, and of course I've used the patch as an excuse to also review the Holocaust through the lens of personal accounts of child victims, and to incorporate a diary-writing practice. Old or young, in circumstances ordinary or extraordinary, we own our own stories and we have the power to tell them. 

The kids and I have had a lot of conversations about what makes people like Anne Frank or Eva Kor ordinary, and what makes them extraordinary (and as I write this, it's just now occurred to me to make the connection between this and the character trait activity that we led on this day!), and I'm always interested to see how visiting the same exhibit we've been visiting for the kids' whole lives, but with a new focus in our minds, leads us to notice different things. This, for instance, is possibly the first time I've noticed this particular photo, in which a young Anne Frank is attending a Montessori school much like, if you count the beads and bead cabinet you can clearly see--

--the Montessori school my own kids attended for a time. 

It's important not to do any tale-telling about what I see and hear when the kids and I are on duty, but here we were just guests, and so I feel free to tell you that while we were all sitting on benches in Anne's exhibit, watching a short documentary on her life, a child sitting in front of us turned to her adult during a scary part of the film and asked, "Does Anne die?"

Friends, that adult said, literally, and I quote, "No, Anne doesn't die."

I gasped in horror! That's... I mean... that's so not right! I would have said something right then and there except that I've seen this little documentary a dozen or more times, and so I was watching the kid more than the film when it came to the part where the narrator explains how Anne and Margot die. The kid made a noise when the narrator said that, and shot her head around to give a betrayed look to her adult, but her adult was across the room on her phone and so didn't see it.

The second-to-last thing that we always do at the Children's Museum is ride the carousel:

And the last thing that we do is wander the gift shop. I don't normally have a lot of patience for gift shops, but this one is legitimately cool--they deliberately vary their stock so they're always adding new things, like this authentic made-in-Greece Greek dress that Syd talked me into buying for her by telling me that she'd wear it all the time AND use it as part of her Halloween costume this year:

In this photo I'm making her hold my ouzo because it's also Greek...

I never buy myself anything (but if you want to buy me a museum-branded hoodie and a messenger bag, feel free!), but I do take pictures of books that I'm going to request from the public library as soon as it opens again:

I love these magical days at the museum. I love connecting in ever-new ways with a place that we've been visiting since my girls were very small. I love working with children, and the challenge of mentoring a brand-new-to-all-of-us activity for a revolving cast of kiddos. I love our traditions of dipping sauces and carousel horses, and keeping track of how many people we tell the location of the nearest bathroom to.

I'm really, really, really looking forward to getting back there.