Tuesday, February 4, 2014

Alternative Algorithms for Multiplication

Common Core standard 4.NBT.5 reads as follows:

“Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.”

Many of us are quick to jump right into the traditional algorithm that we learned as 4th grade students. It is an efficient strategy. The problem, however, is that our students end up memorizing efficient steps instead of understanding the “whys” behind “what” they’re doing.  Essentially, we’ve been teaching multiplication backwards.

Students need time with other methods of multiplication before we introduce the “shortcut” algorithm we know and use. They need lots and lots of hands on time! Research says that students could need as much as two years of conceptual practice with multiplication before we introduce the algorithm. This might be why the words “standard algorithm” for multiplication isn’t mentioned in the CCSS until 5th grade (5.NBT.5 Fluently multiply multi-digit whole numbers using the standard

So how do teachers teach multiplication if we aren’t using the algorithm?

Let’s start the way any good lesson might begin- with the use of manipulatives. Most students have used the base ten blocks in the early grades, so it’s a nice way explore multiplication.  Let’s take the example of 24 x 6.

To use base ten blocks, we’ll multiply 20 x 6 and 4 x 6. The partial products will be listed as base ten blocks.

 From here, students should add the partial products together to find the total product, 144. I like this method because I can visually see and touch the product.

This method transitions well into my next suggestion. Once we get the idea of how to use an area model of multiplication, we can replace the base ten blocks with numbers.

I like this method because it’s a nice reference to the area of a rectangle too. When we add up the area of the smaller rectangle and add it together, we find the area of the larger rectangle.

I’d also like to suggest that you try teaching multiplication based on partial products and place value. Place value is again one of those topics that the early grades really emphasize, so it seems natural to use their understanding of place value to multiply.

Let’s use 245 x 6.

     x  6
       30     6 x 5
+   240    6 x 40
   1200    6 x 200

The great thing about this method is that students really begin to see that they’re multiplying 6 times each place value- the hundreds, tens, and ones. As a matter of fact, they can multiply ANY place value first and get the same product!

Students need a lot of time to explore multiplication through various representations before they’ll understand the standard algorithm. Yes, they might be able to memorize the steps, but wouldn’t you want their understanding of multiplication to go a bit farther?  You don’t have to teach every alternative algorithm out there. Students don’t need to master every alternative algorithm either. You know your students and their learning styles, so focus on strategies that best fit your students. Allow them to find a few methods that they enjoy and go from there.

Looking for a few resources on the area model of multiplication? Check out my Multiplication Round-Ups!

These cowboy themed seat scoots focus on multiplying a 2 digit number by a 1 digit number and 3 digit by 1 digit using the area model method. Students will find partial products and add them together to find the product of various multiplication problems. But watch out! Some cards have missing factors as well!

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