Did you know that there are five meanings of multiplication? When I first learned about them several years ago, I couldn’t believe I hadn’t learned about them in school.

The first thing that seemed crazy to me was that there could be *multiple meanings* of multiplication. But what seemed stranger was that every type of multiplication could be put into one of five groups.

Now that I’m familiar with the meanings of multiplication, I see them everywhere. Whenever I come across a word problem, my first thought is to identify which type of multiplication I’m dealing with.

For students to be successful in math, I strongly believe that they need to be exposed to the meanings of multiplication. They start with one or two in early elementary, and by 8th grade they should be fluent in all five.

But when I bring them up in my workshops, I always hear the same thing: “Why have I never heard about this before?”

If we’re not teaching math educators about the five meanings, how can we expect students to learn them? So if you’re a math educator, or you know one, read this article to the bottom. And please share.

## The Meanings of Multiplication Are Not Strategies

As soon as I hear about a “new” rule in math, I’m skeptical. I can already imagine all the talented math educators who are thinking, “Hold on! There are more than five ways to multiply.”

And that’s exactly right. I try to avoid saying “five types” of multiplication, even though it’s easier on the ears. There are many ways to categorize multiplication, such as hard vs easy, procedural vs conceptual, or visual vs symbolic.

But the meanings of multiplication aren’t *ways to multiply –*– better to think of them as *why we multiply.*

The beauty of inquiry-based math is that students can discover their own strategies. And there are countless ways to *perform* multiplication. There are visual strategies, mental math, algebraic reasoning, standard and non-standard algorithms, and so on.

But when it comes to meanings, there really are only five.

## What Are the Five Meanings of Multiplication?

The five meanings of multiplication are the *reasons* we multiply. Try to imagine a time before multiplication existed. You’d have to go back *at least* to the ancient Babylonians, who are said to have invented times tables over 4,000 years ago.

But wait just a minute! There is evidence that someone in sub-Saharan Africa actually discovered multiplication **over 20,000 years ago**! A tool called the Ishango Bone, found on the border of present-day Uganda and the Congo, works as a combination calendar and slide rule.

So we may not be able to know exactly who invented multiplication. But I like to picture a woman counting boxes of dates in a dimly lit room with clay walls. Each wooden box holds 24 dates. The boxes are stacked 5 high, 4 across and 6 deep.

To find out how many dates she has, this poor woman would have to add 24, plus 24, plus 24, plus 24, on and on and on. (There was even a time before the discovery of addition, when she would have had to count each individual date).

Suddenly, she has a Eureeka moment: “Wait a minute! Each box has the same number of dates. What if there was a shortcut for counting a large number of equal groups?” And multiplication was born.

‘Equal groups’ is the first meaning of multiplication. Later, we realized that the same strategies for counting equal groups could be used to find area, volume, and much more.

Eventually we ended up with five unique purposes for multiplication. (It should be noted that each meaning of multiplication also applies to division – anything we do with multiplication, we can undo with division.)

### Meaning One: Equal Groups

Equal groups is how we typically think of multiplying. It’s the classic problem of counting boxes of dates. Students tend to have the easiest time recognizing these problems in context.

Dividing by equal groups is trickier. Most students see division as ‘splitting into a set number of groups.’ For example, if I have 8 cookies to divide among four friends, how many does each friend get? This is called partitive division.

But division can also mean splitting into ‘groups of a set size.’ So if I divide those same 8 cookies into *groups of 2*, how many friends can I share with? This is quotative division (or repeated subtraction).

To illustrate how important the difference is, I use the example of a shoe factory manager leaving for the day. She points to a pile of 100 shoes, telling her employee to ‘divide them by two and put into boxes.’ The next day, she’s shocked to find 50 shoes shoved into one box and 50 into another.

The idea that the divisor can mean the number of groups, OR the size of each group is a critical mathematical concept. When the divisor is the size of a group, the quotient is the number of groups. When the divisor is the number of groups, the quotient is the size of each group.

While students can rely on partitive division at first, they hit a wall with fraction operations. What does it mean to divide 3 into 1/3 of a group? But repeatedly subtracting 1/3 is easy to understand.

Quotative division is even behind long division and partial quotients.

### Meaning Two: Rates

Rates are similar to equal groups, but with an extra layer of complexity.

I could have 4 boxes of 24 dates, an equal groups problem. Or I could eat 24 dates an hour for 4 hours, making it a rate problem.

The difference is the “per hour,” which is actually an embedded division problem, or fraction:

4 hours x 24 dates/hour = 96 dates

As in most rate problems, the denominator of our rate is ‘1.’ (Making it a *unit rate). *We eat 24 dates for every 1 hour. Since we’re dividing by 1, most adults simply ignore the division.

But this can create confusion for students. The division is important, because it affects our units. It’s not correct to say “4 hours times 24 dates.” That would make our result “96 date-hours.”

Instead, the hours in the denominator of the rate cancel the hours in “4 hours,” giving us 96 regular dates.

Students may also be asked to find a rate. In this case, we take the total amount and divide by the period. If we knew that we ate 96 dates over 4 hours, our rate would be 96 dates per 4 hours. Or we could simplify to find the unit rate:

96 dates ÷ 4 hours = 24 dates/hour

Rates are crucial for much of later math, including functions, slope, and pretty much everything in calculus.

The nice thing about teaching rate is that it is relatively easy to connect to skip counting and repeated addition. “If I eat 3 cookies each hour, let’s count up to find how many I’ve eaten after 5 hours” (Or 25 hours, 1,011 hours, or 2.6 hours).

### Meaning Three: Multiplicative Comparison

Multiplicative comparison is another area that many students find difficult. These are situations where one thing is x many times as big as something else. I may be twice as old as Jermaine’s sister, or have one third as many cookies as Sonja.

One reason these problems are challenging is because students tend to think of operations as something we ‘do to’ a number. So when I multiply 5 apples by 7, I’m adding a bunch of extra apples. But with multiplicative comparison, I’m describing differences that already exist.

Another reason for difficulty is that students confuse these problems with additive comparison. Many will simply add four when asked to find ‘four times as many.’ Other students may not be strong enough with additive comparison (they understand addition as combining, but not comparing).

Students need an understanding of multiplicative comparison to handle middle school standards, such as ratios and geometric transformations.

When teaching multiplicative comparison, provide students examples of additive comparison and multiplicative comparison. This ensures that they understand both concepts, and that they can tell the difference between them.

### Meaning Four: Rectangular Array

While I love all the meanings of multiplication, if I *had* to pick one, I’d pick the rectangular array.

Rectangular arrays lend themselves beautifully to visual models, including both arrays and area models.

*An Array and an Area Model, both Examples of the Rectangular Array meaning of multiplication*

A simple array is a set of objects organized into rows and columns. It provides a foundation to help students understand area and area models, which are measurements of two-dimensional space.

Many of our students only understand *number as quantity*, or distinct objects. Arrays and area models can help them connect this idea that they already know to the idea of number as a *measure of space*.

When we teach area, we usually teach students that area is a formula: “l x w,” “½ b x h,” or even “????r².”

But area is so much more than that. It’s an entirely different meaning of multiplication, and of number. I teach students that we live in a 3-dimensional world, and we can measure in all three-dimensions.

Length is a measure of 1 dimension, area is a measurement of two, and volume is a measurement in three dimensions. Theoretical mathematicians keep going into the 4th dimension and beyond.

Today, we tend to think of measures of space (geometry) as less important than quantities. The geometry unit is always at the back of the textbook, and we rarely see geometry standards in the “focus standards” for any grade level. But to the ancient Greeks, it was the complete opposite.

By using rectangular arrays, we can help our students go beyond seeing area as a formula. This can help them to stop confusing area and perimeter. They’ll also have an easier time seeing volume as an extension of area, rather than as a completely new concept.

### Meaning Five: Cartesian Products

Cartesian products are the least common, most understood meaning of multiplication.

A Cartesian product is another way to describe combinations. These problems often include matching shirts and pants to make different outfits, or different ways to stack an ice cream cone.

To find a Cartesian product, pair each element from the first group with each element from the second group. This can be represented visually, in a table. The total number of possible pairings is your product.

Multiplying to find combinations is a foundation for probability. In the above example, if you are randomly assigned a cone, the probability it has only one flavor is 3 out of 9, or ⅓.

The idea of matching values together also introduces ordered pairs, hence the name “Cartesian” product, as in Cartesian plane.

Cartesian products can be introduced through games and activities, such as dice and cards. I used to hold Roshambo tournaments (rock paper scissors) to teach about combinations and introduce game theory.

## Why Teach the Five Meanings of Multiplication?

Students who learn the different meanings of multiplication solidify their understanding of, and fluency with, basic operations. It helps them see how multiplication is a repetition of addition and an inverse of division.

This doesn’t take the place of times tables or algorithms. But it certainly reinforces them. It also helps students connect math to the outside world and help them to make sense of word problems.

I first learned about the five meanings of multiplication in Chapter 4 of *Math Matters* by Suzanne Chapin and Art Johnson. If you want more depth on the topic, I highly recommend reading this book.

## Bringing the Five Meanings to Your Classroom

It should be noted that students don’t actually need to perform each type of multiplication differently. We should be careful to emphasize this when we introduce them to our classes.

The beauty of multiplication (and the other operations) is that they are abstract processes that we can apply to a range of real-world scenarios.

One way to help students understand the meanings of multiplication is to demonstrate them with visual models and manipulatives. As students become comfortable with the multiple meanings, they will begin to recognize them in the context of word problems.

You can help them to see equal groups and rectangular arrays using tools such as a visual model organizer. For problems involving rates, try a Representing Functions organizer. The idea is to help them see that word problems, visual models, and symbols are all just different representations of the same idea.

Eventually, students should be able to apply their understanding of multiplication to abstract situations. Number sentence proofs are a great activity for this purpose. These have students go beyond solving equations to actually explain why equations are true, false, or “sometimes true.” Number sentence proofs can be used with or without variables. They can reinforce students’ understanding of fractions, integers, exponents, and more.

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**About the Author**

Jeff Lisciandrello is the founder of Room to Discover and an education consultant specializing in student-centered learning. His 3-Bridges Design for Learning helps schools explore innovative practices within traditional settings. He enjoys helping educators embrace inquiry-based and personalized approaches to instruction. You can connect with him via Twitter @EdTechJeff