Learning Algebra through the Montessori Magic- Square of a Binomial

Did you start learning algebra with (a + b)2and till today, don’t really know what it means than just mere memorization.

I had always read and heard about the montessori curriculum being connected. Fortunately I got to experience it first hand as I continued to climb the Montessori ladder by teaching in a Primary (3-6 years) classroom for a few years and then moving on in my journey by joining the Lower Elementary (6-9 years) classroom.

Last week I invited my third graders for a lesson on the ‘Square of a Binomial,’ that is an introduction to square root. As they saw me holding the binomial cube, I got to hear,

“We know this material already.”

“I’ve done this so many times.”

“This is so easy, I did it in Kindergarten.”

As much as their comments were fascinating and made me smile, I still continued with the planned lesson. I revised with them, the binomial cube with a ‘silent lesson.’ The concept of a ‘silent lesson’ is common in a Primary classroom but not really in a Lower El classroom. Surprising my third years were quite engrossed as they saw me remove the prisms from the box and build them on the lid of the binomial cube box.

As I completed building it, I placed it on the side and took out a ‘One hundred Square’ (golden beads) and asked the boys what it was and how we could write it on paper.

10 x 10 = 100

We took two rubber bands and started by playing a game of making combinations. We began to count from the top left side and went across till bead 7, and placed our rubber band vertically. The same count of 7 continued, but this time downwards and placed the rubber band horizontally.

Connecting the shapes together as we compare

Together we noticed that we had two squares and two rectangles and made the connection of the shapes with the binomial cube. The first square was on the top left side was a square of 7, and second one, on the bottom left was of square of 3. We took a bead square of 7 and placed it on top of our square and did the same with the bead square of 3, placing it on top of the second square. The two rectangles were combinations of 7 x3 and 3 x 7 respectively, and used the bead bars for comparison. As the boys got excited with these observations, I had to remind them that we were not done yet.

“Let’s learn to write this as an equation.”

Beginning with the square of 7, we recorded

(7 x 7) + (3 x 3) + (7 x 3) + (3 x 7)

49 + 9 + (7 x 3 ) + (3 x 7 )

Following this step we talked about rewriting the above equation with a special sign, to indicate the square of a number. A 2 high off the line is written.

7 2 + 3 2+ (3 x 7) + (7 x 3)

The raised two is called an ‘exponent.’ We also discussed that when the same factors are multiplied, there is a simpler way to record it.

(3 x 7) + (7 x 3) can also be written as

2(3 x 7)

With this information we were able to record the equation:

7 2+ 2(3 x 7) + 3 2 = 100

Working on combinations of 7 + 3 and 5 +5

In this lesson, basically we connected their Primary sensorial material with Elementary algebra (math) material. It was a complete ‘ah-ha’ moment for them!! Montessori math materials absolutely amaze me, or i should say sensorial materials do it, maybe language and cultural… well, each and every material does its own trick. Even after 5 years into this, it puts me in awe!!!

Maria Montessori, you sure were a genius!!!

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