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December 01, 2013

The how of the two digit squaring method (1)

The sixth graders I work with are (have become) a curious lot.

Following the exercise of p^2-1 (prime number square is divisible by 24) there were many questions of how things worked. My goal had been for them to look at the beauty of numbers and I had asked them which their burning question was why p^2-1 is divisible by 24 or how the technique of squaring works.

I had found the property of primes more interesting and as if to remind me that I am an adult the children overwhelmingly selected the squaring method deserved their attention.

There were three aspects that puzzled the kids about the method:
1) Why write the square of the units place is written as a two digit number even if the result is a single digit i.e. 1^2=01, 2^2=04 and 3^2=09.
2) Why the square of the tens place didn't need the same.
3) Why after multiplying the two numbers we needed to double it.
Of course, there was the addition of all these numbers, but apparently that was no puzzle :).

Distributive property of multiplication over addition
I needed to explain the distributive property of multiplication over addition to proceed. I realized that
the algebraic proof that came naturally to me was irrelevant as they had not encountered algebra meaningfully. I used that place value to show the same graphically.

A place value kit has blocks of ones, tens, hundreds (and thousands that I have now shown).

I had used the blocks to explain multiplication as the area of a rectangle i.e. 6x13 is the same as calculating the area of a shape having 6 rows and 13 columns (or 6 columns and 13 rows depending on how you look at it):

In that case, 6x(10+3) implies a split in the figure above of 13 into 10 + 3 i.e.

Which is the same as: 6x10 + 6x3

As a note, many aspects of algebra can be beautifully shown with the place value kit. I have been able to touch upon the following multiplication, squaring, decimals, fractions, area, volume with the same.

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