How it Works: Integer Multiplication

This past week, Josh Zagorski forwarded a tweet from John Fritzky, a middle school principal looking for an explanation of how integer rules work for multiplication and division without using rules:

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Graham Fletcher folded me in on the conversation and I haven’t been able to stop thinking about it. I haven’t been able to think of a quick way to sum it all up, either.  So, here we are.

A few people commented on this with some good ideas and I’ll highlight those here.

As I mentioned, I haven’t been able to stop thinking about this.  The more I thought of it, the deeper I dove into it.  I decided to focus on multiplication first and after the past few days of thinking (this is just the beginning), I think I this is involves 2 big ideas:

  1. What are integers and how can I use what I know about them (absolute value, other operations, and the negative symbol “-“, etc.) to make sense of multiplying these numbers?
  2. What does multiplication really mean and how can we use what we know about it to apply it to a new system of numbers?

When I’ve thought about this in the past with my own students, I focused only on the first idea.  But the idea of what multiplication really means plays a huge role in making sense of integer multiplication (and later division).


The whole numbers and their opposites have several ideas associated with them and all of them really need to be developed deeply and conceptually in order to get to the point of making sense of operating with them.  That said, here are what I see as the big ideas  that stand out for the sake of this discussion (blue bold highlights):

6.NS.5 – Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.NS.6 – Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

7.NS.1A – Describe situations in which opposite quantities combine to make 0For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

7.NS.2A – Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.


As briefly as possible, here’s what multiplication really means (I’m taking this straight from the standards, to be as transparent as possible here:

3.OA.1 – Interpret products of whole numbers
e.g. Interpret 5×7 as the total number of objects in 5 groups of 7 objects each

So, when reading a multiplication problem out of context:

____ x ____ = ____

really means:

____ groups of ____ things in each group = ____ total things

This idea is important for three reasons:

  1. In the absence of context, students contextualize the “naked math.” Giving meaning to numbers in order to work with them is one piece of the number sense puzzle.
  2. This builds on an idea of grouping and sharing from Kindergarten, so it is accessible to all students.  Building connections!
  3. Representing multiplication with pictures, arrays, and/or number line diagrams makes more sense when students think of multiplication in this way.

Making Sense of Integer Multiplication






Blogarithm Posts

Last year I had the honor of being asked to write four posts for NCTM’s Math Teaching in the Middle School Blog: Blogarithm (one of the coolest math blog names out there).  They were posted every two weeks from November through the end of December (which just shows that I can post more frequently if someone is reminding me every other week that my next post is due (thanks Clayton).

Pythagorean Decanomial

The four posts are a reflection of a lesson I taught with a 6th grade teacher, in September of last year, who was worried (and rightfully so) that her students didn’t know their multiplication facts.  After a long conference, we decided to teach a lesson together.  I modeled some pedagogical ideas and she supported students by asking questions (certain restrictions may have applied).

Links to the four posts are below.

  1. Building Multiplication Fluency in Middle School
  2. Building Multiplication Fluency in Middle School Part 2
  3. Building Multiplication Fluency in Middle School Part 3
  4. Building Multiplication Fluency in Middle School Part 4

While you’re at the Blogarithm site check out some other guest bloggers’ posts.  Cathy Yenca has some great posts on Formative Feedback, Vertical Value Part 1 and Part 2, and 3-Act Tasks


The Best Part about Blogging

This is super exciting!  I love it when teachers keep thinking – especially when I stop!  What you’re about to read is truly the best part of blogging!

Readers of Under the Dome have been terrific commenters and questioners of my posts over the last 2 years and you all just keep getting better.  Recently, Sharon Wagner, a teacher I met during a three-day summer institute in June visited my blog and reached out to share her ideas about the Olympic Cola Display 3-act task.

Sharon’s words:

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Through the course of a few emails over the summer and a lot of my time spent doing things outside of the MTBoS (my lovely wife got some of her honey-do’s completed and I got some of my Mike-do’s finished) I have Sharon’s extension and am now posting it with her blessing!  Please take a look.  Her idea is a natural extension and allows students to design their own display using the colors of Coca-Cola twelve packs (which she most helpfully added to her document).  Any Pepsi fans out there?

Sharon’s idea also ups the rigor by providing an audience (the merchant).  This, again, is a part of that natural extension (of course someone designs these displays for the merchants).  As for the Standards for Mathematical Practice . . . let’s just say your students will be engaging in multiple SMPs.

Again, this is super exciting.  I love to share my ideas here, but when someone else takes it and makes it better – in this case by adding to it – everyone wins.  Especially the students in our classrooms.

Thank you Sharon.

Sharon’s Display Extension:

coca cola display project extension