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!

Ink Blot Prints that Demonstrate Bilateral Symmetry

In math, the younger kid gets frustrated with a lot of computation, which is fine (for now--the older kid also gets frustrated with a lot of computation, and yet I still require it of her, because apparently I'm meaner to older kids), so since she's long finished up any kindergarten math requirements that I had of her, we've been luxuriating in those kinds of hands-on, in-depth, sensorial math activities that internalize concepts, and that the kid absolutely loves.

For instance, it wasn't necessary to spend weeks on bilateral symmetry. And yet... bilateral symmetry is so fun! Here are some of our activities:

• cutting out and folding shapes to discover and test their lines of symmetry
• using graph paper to draw shapes that have bilateral symmetry, then cutting them out and folding them to test them
• putting ANYTHING up against a mirror to see it in symmetry
• taking a nature walk to collect leaves, then sorting them into groups of symmetrical and non-symmetrical, then folding them to test those theories, then drawing in the lines of symmetry using Sharpies
• painting on one side of a paper, then folding the paper and pressing it down, then unfolding it to look at your magical Rorschach-style print

By the way, these BioColor paints are the BEST at that last one!

This is just the kind of activity that my little kid likes. She made print after print after print, then extended it to finger painting (discovering for herself that printing doesn't work if the paint has had time to dry), then moved on to some very colorful handprinting, then added more and more and more paint and found that she loved the feeling and the look of painting through all the layers...

...and made a GIANT mess!

And yes, to her infinite credit, she cleaned it up completely independently, including washing off the paint bottles, scrubbing the table, and giving herself a bath. That makes the activity even MORE satisfying, don't you think?

For kids whose current special interest is bilateral symmetry, here are a few more fun activities for enrichment and exploration:

• cut an object or picture in half, glue to a piece of drawing paper, then recreate the other half of the drawing
• make Rorschach ink blots, then have fun creatively interpreting them
• set up these super simple popsicle stick symmetry puzzles for the youngest learners
• print free worksheets for kids who like that sort of thing!

And our course it wouldn't be a homeschool project without lots of books!

P.S. Want to follow along with my craft projects, books I'm reading, dog-walking mishaps, road trips, and other various adventures on the daily? Find me on my Craft Knife Facebook page!