Sunday, April 25, 2010

Multiplying on the Grid

Learning to multiply whole numbers using a grid makes far more sense than the traditional method which has been utilized since well before any currently living person was born!

Furthermore, multiplying on a grid can help students with mental math, but more importantly, make it easier to understand multiplying and factoring polynomials later in algebra.

As seen in the first two images below, the method is remarkably simple. First sketch a grid to accommodate the number of digits in each factor. For instance, if multiplying a one digit factor by a two digit factor, you'll need a 1 by 2 grid. Use a 2 by 2 grid if the factors are each two digit numbers, and so forth.

Place the digits of the factors on the outside of the grid, each digit having been multiplied by its place value. For example, if one of the factors is 64, put 60 (6x10) and 4 outside the grid.

Next, just as in a common "times table", fill the grid spaces with the products of the numbers on the outside. Finally, add the numbers in the grid to get the product.

Image 1
one by two digits

Image 2
two digits by two digits

The following images illustrate how grid multiplication can facilitate polynomial multiplication and factoring as students progress through mathematics.

The size of the grid is determined by the number of terms in each polynomial factor. Multiplying a binomial by a trinomial would require a grid 2 by 3. Then each term of the factors is placed outside the grid just as the digits of the whole number factors were placed. The spaces of the grid are filled with the products of the monomial terms outside the grid. Finally, again just as with the whole numbers, the product of the two polynomials is the sum of the terms inside the grid.

Image 3
monomial by binomial

Image 4
binomial by binomial

Image 5
binomial by trinomial

Image 6 shows how the grid concept can be used to help factor a binomial. Place the first and last terms of the trinomial in the first and last grid spaces. Then find numbers for the sun ☀ and the cloud ☁ whose product is the constant term (20) and whose sum is the coefficient of the linear term (9), 4 and 5. Therefore the factors are (x+4) and (x+5).

Image 6
factoring binomials

I have long thought that just making this simple change in the way we teach a basic skill might have a subtle but significant impact on how students learn and understand higher level skills later. Why do we continue to teach the traditional method of whole number multiplication? Because that's the way we've always done it. That's what we know. That doesn't mean it's the best way.

Add-ons:. Try applying the grid method to mutiplying mixed numbers.
You can put a tech spin on this method by having students implement it in a spreadsheet.


Christiansen said...

This makes even more sense from an application standpoint if you consider multiplication as finding the area of a rectangle.

Thomas Boito said...

You make an excellent point

Mrs. Tenkely said...

Why, why wasn't I taught this way? Mental math is infinitely easier with this model! Now to teach students with it. Thanks!

Emily Starr, CEO said...

This is a great method for teaching students to multiply that I think many teachers aren't even aware of! (In the US at least) On StarrMatica, we share 3 different ways of approaching multi-digit mutliplication problems--one of the ways is the grid method.

The BBC has a nice online activity that practices solving problems with this strategy:
