# A New Look at Partial Products

### Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.
##### Grades 3-5 | Number Sense | Operations

I’m working on a professional development session for tomorrow (don’t judge me…), and I’ll be introducing mental math computation strategies for multiplication to my 3rd-5th grade teachers using the book Number Talks.  One great feature of this book is the DVD that’s included.  It contains video clips from classrooms and, as the saying goes, a picture is worth a thousand words. It’s just very cool to see and hear kiddos talking about their strategies for mental computation.

One of the goals of number talks is to increase students’ understanding of properties.  We probably all remember memorizing the commutative, associative, and distributive properties in school, but did they really mean anything to you?  Many teachers are familiar with using the  partial products method for introducing multi-digit multiplication.  Partial products is really based on the distributive property, because you are breaking the numbers into parts and then using the distributive property to combine them.  Traditionally, it looks like this (notice the distributive property notation beneath it):

But partial products is also a strategy for single-digit multiplication.  Consider 7 x 8. A common strategy that kiddos use to solve 7 x 8 is to think of 7 x 7 and add one more group of 7.  Hmmm, wouldn’t that be the distributive property?

The thing I love about teaching math this way is that it stresses number sense and true understanding of place value.  You may worry that the kiddos won’t have their own strategies.  Well, you never know until you ask them. 🙂
One additional note–YOU likely did not learn math this way.  There’s a good chance it might not make sense to you initially.  You might even be tempted to delete this post from your inbox and pretend it never happened.  But to quote a very old commercial, “Try it, you’ll like it!”  🙂

1. Thanks for spotlighting this! Our math program (ThinkMath) teaches one digit multiplication this way, using “split arrays” to support but also linking the equations, and new teachers are often confused. It is worth grappling with this! It is also a strong way to build multiplicative thinking, rather than just thinking about “adding on another group of 7.” Love your blog. 🙂

• Thanks so much, Jenna! You’re right about the fact that we, as teachers, need to grapple with this! A little scary, but a lot of fun! And a big payoff when the kiddos develop that deep understanding. 🙂

2. I absolutely LOVE partial product multiplication!! Just last week I did a lesson with my 4th grade students that involved decomposing numbers from 13-19 to help them multiply them by single digit factors! We used grid paper to make area models. It was so much fun to see their faces light up when they realized all the different ways we could decompose to make simpler problems to solve. 🙂

• What I love about this book is that it emphasizes that it’s not always splitting the numbers into tens and ones. Different kiddos will split the numbers different ways.

• I teach that they should start with what they know. Usually the 5s. So 8×7 would be 8×5, 8×2.

3. I was really disappointed that the authors of the common core used the word “traditional algorithm” for operations. There are HUGE advantages to both partial products and partial quotients in computing. Double-like this!

• I agree with you

4. I was really disappointed that the authors of the common core used the word “traditional algorithm” for operations. There are HUGE advantages to both partial products and partial quotients in computing. Double-like this!

• Our newly TEKS in Texas specify the standard algorithm as just one of multiple strategies children should know.

5. Just discussing this very thing in PLC meeting today. Trying to embed a video into a document I can post on a teacher web page for parents to understand this great strategy they might see their kids using. Thanks for these marvelous posts.

• Oh, right, it’s a tough sell for some parents. You wouldn’t be able to embed the videos from the DVD on a web page, but you could certainly show them at a parent math night.

6. I gave a presentation Monday on my time at Confratute this summer. One of the sessions I attended was “Math For Those Who Think They Can’t Do Math” by Rachel McAnallen. Fabulous session, and using partial sums/products for operations was one of the things that really stuck with me.
Carol
Still Teaching After All These Years

• Sounds like a great session! A lot of people really DO think they can’t do math.

7. This is such a Great book!

• Yes! It’s a great resource!

8. I loved this post and I just recently started trying to do this with my own math that I am studying! Math talks was a big topic in the Stanford class I just finished with PhD Jo Boaler. I too am trying to re-train myself with this and other methods.