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:
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:
- 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?
- 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 0. For 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 ____ = ____
____ groups of ____ things in each group = ____ total things
This idea is important for three reasons:
- 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.
- This builds on an idea of grouping and sharing from Kindergarten, so it is accessible to all students. Building connections!
- 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