Relational Thinking Strategies: Multiplication
When we know the strategies we want students to uncover, we become more strategic with the problems we pose, the numbers we choose, and the way we facilitate student discussions.
Students (who are allowed to solve in any way that make sense to them) often implicitly use the properties of operations and equality to simplify the problem. When we look and listen closely to student thinking, we can facilitate class conversations that make these relationships explicit.
Keep in mind:
- If you show students how a strategy works you’re doing the thinking for them. You can use problem types, number choices, and discussion facilitation to put students into a position to collaboratively construct these relationships.
- It takes an incredibly long time to construct these strategies (and more importantly, the ideas behind them). Be patient with your students and yourself.
- Arrays are abstract for students. I’m using them here for the purposes of our discussion, but I do not recommend showing these animations to students. For each strategy you’ll see one way a student might model and use relationships.
Let’s take a look at the strategies: Partial Products, Over and Subtract, 5 is Half of 10, and Factoring.
Partial Products
The distributive property of multiplication over addition allows us to break a multiplication problem into chunks. Often that will be by place value, but don’t get too hung up on that (for example, I can multiply by 27s by breaking them up into 25s and 2s, since 25s are usually convenient to multiply).
Many different number choices will help surface Partial Products. What really matters (in my opinion) is keeping it rooted in the context. Notice how easy it is to understand the distributive property when you talk about boxes and chocolates:
Breaking up a box of chocolates into 10 and 8 chocolates makes sense. Counting 10 chocolates from each box, then 8 chocolates from each box, then adding them together makes sense. Even though it might not be initially apparent to us, directly modeling the problem (by 1s, shown below) can give students access to the properties of operations.
We won’t know how the student solved this problem until we ask. Maybe they counted by 2s (which connects to the associative property of multiplication) or considered the 10 and 8 in each group (which connects to the distributive property of multiplication).
A student who has modeled and counted by 1s has grounded themselves in the context and the magnitude of the quantities in a way that will allow them to make sense of peers’ ideas. They are primed for a “light bulb moment,” whether it is the chocolates modeled with the numeral “18” or the idea of breaking up 18 into 10s and 8s.
Over and Subtract
Like seeing negative space in art, you can shift your perspective to consider what is missing in a multiplication problem and compensate. You might be inclined to call it the distributive property of multiplication over subtraction, but if you consider subtraction to be “the addition of the opposite,” we can see it as the distributive property of multiplication over addition.
To nudge this conversation, I like to pose a multiplication problem with two number sets and ask students to solve both. Or, I might do the problems back to back within the week.
Sometimes it is hard to remember what to subtract after you’ve adjusted. When thinking about 20×5 to help you with 18×5, are we subtracting 5, 20, 18, or something else? Here again, the context makes it easy to see that we’re subtracting 2 boxes of 5 chocolates.
5 is Half of 10
Why does 5 get its own category? Although the example here is 5×18, this idea shows up in problems like 16×23 (more on that below).
Depending on how a student describes their thought process this strategy could utilize the distributive property of multiplication over addition (solving 10×18, then imagining splitting it into two chunks) or the associative property of multiplication (thinking about 10×18 as 2x5x18).
To surface this discussion, you might offer contexts where the amount in each group stays the same and students consider the relationship between 10 groups and 5 groups.
Half the boxes means half the chocolate, meanwhile the number in each box didn’t change. Context!
Factoring
When you create groups of groups or break something into equal sized chunks, you’re likely using the associative property of multiplication. Halving and doubling (or tripling and thirding, quadrupling and fourthing, etc) relies on this property.
Students often start combining equal groups in ways that make sense to them. When we see this, we can bring it to the whole class and have them discuss the ways it transforms the problem.
I can imagine counting 2 boxes at a time as 10 chocolates rather than 1 box as 5 chocolates. The more comfortable students are explaining the relationship between the two units (boxes and chocolates, for example), the easier ratios and proportions will be!
Mixed Strategies
We’ve looked at 4 categories of strategies, now let’s see how they might combine to help you solve a problem like 16×23. Keep in mind there are way more than 2 ways to solve this problem.
And one more:
Pam Harris says the goal is “fewer, bigger chunks.” If students consider the numbers before solving, they’re less likely to automatically break up problems into four chunks by place value (which is only a good strategy if you have other options to choose from).
The better we know the strategies (and the underlying relationships and ideas), the easier it will be to put students into a position to construct those strategies with their peers using meaningful contexts, carefully sequenced problems, and discussion.
After you’ve engaged in problem solving with context, you might enjoy the multiplication Math Flips decks, which get students talking about the important relationships in multiplication.
Questions to consider:
- Do you (as a mathematician yourself) gravitate toward one of these strategies more often than the others? Is there one that you want to work on?
- What is the role of context in connecting student’s informal and intuitive understandings with more formal ideas (like properties of operations)?
- How can a series of related problems and number sets help students generate relationships and uncover strategies for themselves (rather than being shown how to solve)? Check out this post where Liz Romero facilitates a discussion that nudges her students to discuss the distributive property of multiplication.
- What does this mean for your practice? Who might you talk with about this?