Unraveling the Arithmetic and Algebraic Roots of Binomial Multiplication using Multiple Representations

Okay, so this is long overdue.  My first official blog entry.  I’ll admit – I have long been the lazy type, satisfied with the reflection that occurs internally (in my head) after teaching a great math lesson.  But since meeting and hanging out with cats like Graham Fletcher, Mike Wiernicki, and Jenise Sexton, I’ve realized that my private musings are nothing more than pure selfishness.

After exploiting great tasks over the last few years from the likes of Graham, Mike, and Jenise, as well as Joe Schwartz, Robert Kaplinsky, and Nathan Kraft, I’ve been inspired to create and share some of my own via social media (I can only hope that they are great – that will be your charge as the reader – do as I have done: take these ideas, put them into practice in your classrooms, and let me know how it goes and most importantly, how to make them better).

So, to the task…

I first must thank Mike Wiernicki for introducing me to this engaging and just flat out cool concept.  As a writer of professional learning for a newly formed Algebra course for Georgia students, Mike penned this fantastic inquiry-based activity that will get students engaged in each of the 8 mathematical practice standards.

And before I forget…  I hope you discover something about multiplying binomials that you didn’t realize when you were an algebra student in 8th grade or high school.  I know I did, and that has been the most gratifying part about being a math teacher and crusader for teaching mathematics for understanding over the last 15 years.

So let’s get to it.  Since this involves what some might call a number “trick”, I thought it might be fun to set the stage with Arthur Benjamin’s famous Ted Talk.  So, I did.  After playing about 3 minutes of this clip for a group of about 30 fifth graders, I posed the following question to them: How do you think he does that?  While some students were still picking their jaws up from the floor, others started arguing – their observations ranged from “It’s magic!” and “He must have a really good memory!” to “He sees groups of numbers and patterns in way that most people don’t!”

I then explained to the class that we are going to discover a number trick (and then dissect it with multiple representations and intense curiosity because as John Van de Walle once said, “If you can’t explain the method [or trick], then you shouldn’t be using it!”)  In the spirit of JVW and SMP#1, we were not going to settle for just another procedure without connections to meaning task.

So here is the slide I projected.

1st screen

Followed by this one.

2nd screen

Students independently looked at the number line representations as they made sense of them.  I then asked them to turn and share their observations with a peer.  Although some had difficulty articulating it in precise mathematical language (which is typical when students are exploring new concepts), they quickly realized the relationship between the square of a number (x) and the product of the one less than the number (x – 1) and one more than the number (x + 1).  Without all the fancy algebraic notation, of course (as that was not the point).

But the best part was what followed – I simply asked, “What do you wonder?”

“I wonder if this will always work,” one student said.  See SMP#7.

“And will it work with bigger numbers?” another student added.  See SMP#8.

3rd screen

“I wonder why it works and who figured this out!” someone wondered.

They continued to ask questions, each one seemingly emerging in sequence like a chain reaction. “Will it work with decimals?”

“I wonder if it will work with negative numbers too!”

And then one young lady summed up what everyone else was getting at: “I wonder if we can find a case where it doesn’t work!”

Brilliant!  The best part about asking “What do you wonder?” is that you don’t have to impose your plan or agenda on the students.  Their natural curiosity takes over!  In a way, that curiosity takes the reins and charts the path of the lesson.

I had not anticipated these questions (even though I probably should have!).  It mattered not – these kids were brimming with curiosity.  I knew I had to deviate from my plan.

“Alright, go ahead and play with your calculators,” I said.  “Take all of these conjectures, and try them out.”

For the next 10 minutes, they feverishly punched numbers in the calculator to find out if this always works.

“6.2 x 6.2 is 38.44 and 5.2 x 7.2 is 37.44!  It works with decimals!!!” one boy shouted from the back of the room.

One curious girl even tried using pi.

Reluctantly, I stopped them and went back to the young lady who asked why.  “Let’s investigate why this works,” I told the class, as we pulled out the two-color counters to model this relationship.  See SMP#4.

Students then constructed a 6 x 6 array and I charged them with the task of transforming the array into a 5 x 7 in one single move.  After thinking for a bit, several groups of students did this:

5 by 7 minus 1a  5 by 7 minus 1b

“Oh!  There’s the minus one!” (pointing to the the extra dangling counter) an excited student shrieked.

We didn’t stop there.  I posed this slide next and had them conjecture, “What would be the difference in this case?”


Some students yelled, “It’d be 2 less.”  Others argued, “No, it’d be 4 less!”  So we investigated, essentially repeating all of the previous steps, but this time with a factor two less and a factor two more than the root.  Here are the images/animations I used to summarize their discoveries:

4th screen

5th screen

Then, because our focus was algebraic reasoning, we attempted to generalize without using manipulatives, but a tabular representation instead:

7th screen

From the students came 2 overarching conjectures:
1) Either we will always subtract 3 more each time we increase the number of “steps out”
2) The amount to subtract is always equal to the square of the number of “steps out”

Just as students were about to dive into debating these two theories, the fire drill siren began to blare.  It was one of those good old fashioned day-to-day necessary interruptions of instructional time.

Fire Drill

But, in the spirit of TKES standard 8.1 (Academically Challenging Environment)…


we didn’t allow learning to stop.  The students asked if they could take their calculators outside so they could keep trying to find different examples (and some wanted to continue their pursuit of the elusive counterexample!).  Wish I would have taken a photo of 26 fired up young mathematicians standing in line outside feverishly punching numbers and operations into their calculators!  It was a sight to remember!

When we came back in, we hit SMP#2 by expressing these relationships abstractly with algebraic symbols.  And finally, the moment I had been waiting for since May of 2013 (my last appearance as an official classroom teacher) — it was time to assign homework!  (but not the regular, boring and painful kind…)

Here was the assignment – go home and try this out on mom or dad.  And, just maybe, it could go something like this:

6th screen

And when Mom or Dad asks, “How did you figure that out?” I hope they all say something like this: “Well, how would you like me to show you?  Number line?  Array?  Area Model?  Equation?  Or better yet, how about all 4?!”


One thought on “Unraveling the Arithmetic and Algebraic Roots of Binomial Multiplication using Multiple Representations

  1. Hooray! I’m so glad you are blogging, and so glad you are part of the math nerd club. Can’t wait for another glimpse inside the Bri Brain.

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