The distributive property is a common theme throughout math learning. While it has long held precedence in algebra as the way to multiply expressions, it’s now being taught in the early days of multiplication as a way to build numeracy, estimation skills, and mental mathematics, as well as making sense of multi-digit multiplication.
Technically, this property as it is most commonly used is the distributive property of multiplication over addition. Or at least that’s its official name. In this space, I propose that we begin shifting that language to distributive property of multiplication across division.
This was originally published on mathteacherbarbie.com. If you are reading it elsewhere, you are reading a stolen version.
The Distributive Property involving Multiplication and Addition
Though it is not the only distributive property, the “distributive property” that most students see says we can distribute multiplication across multiple additive (or subtractive) terms. In elementary school, it will appear in a context to build number sense and mental arithmetic involving “friendly numbers” like 5, 10, 20, 100, etc. For example, students might learn to multiply $13\times 5$ by thinking of $13=10+3$ and then $13\times 5=(10+3)\times 5=10\times 5 + 3\times 5$, allowing them to use mental math to conclude $50+15$ or $65$. In this case, the $5$ is distributed across the number bond that totals 13 but consists of a combination of a “friendly” number (10) and a small number (3).
Students may then use the distributive property across subtraction to do such problems as $17\times 5$: since $17=20-3$, then $17\times 5=(20-3)\times 5= 20\times 5 – 3\times 5$, allowing for them to conclude that $17\times 5 = 100-15=85$ with less cognitive load and greater number sense than the traditional algorithm would require.
In previous generations, pre-algebra has been where the Distributive Property (short for Distributive Property of Multiplication over Addition) has been introduced. Once we introduce variables, the distributive property lays a foundation for several ways of “simplifying” or rewriting and reframing equivalent expressions. For example: $3(x+6)$ becomes $3x+18$ with the distributive property since $3\cdot x=3x$ and $3\cdot 6=18$.
However, there are other “distributive properties.” Most students eventually encounter one of them, but by a different name: the distributive property of exponentiation across multiplication is simply referred to as “one of the exponent properties.” Others are only encountered in more advanced math classes taken exclusively by STEM majors. I do wonder if a greater inclusion of the “multiplication across addition” portion of this distributive property might help students avoid common pitfalls in the use of the distributive property, as well as the application of the similar exponent rule. But that’s a topic for a different article.
The problem with “Multiplication Over Addition”
In math class, we’ve developed a bad habit of some casual language. In this case, the word “over” has been implicated in comparing numerators and denominators in fraction notation and its correlated division. We often say “three over four” to mean three-fourths or $\frac{3}{4}$, for example.
In fact, so many of us use this casual language so often that it’s become automatic; neither we nor our students notice most of the time we use it. In some sense, this is great because automatic language reduces our cognitive load when we’re trying to work problems or communicate ideas. However, because this casual language has developed this automaticity, that word “over” can easily trigger ideas and frameworks of fractions and divisions, confusing and complicating the soup of ideas that our students have to wade through to find what we’re trying to teach.
In this case, not only is the soup more complicated, but it often has extra cortisol and other brain stress chemicals due to negative associations and feeling around fractions and division — two math topics that frequently induce self doubt.
When I’m introducing how to use the distributive property, I don’t want my students wading through fractions and division until it’s time to do so. I want to keep that soup as simple and cortisol-free as possible to start.
Furthermore, the word “over” in standard English usage indicates a direction and/or relationship in space between two or more items. (In fact, for most of us, this spatial relationship is what led to using the word over when talking about numerators and denominators!) In the distributive property, however, whether we write it symbolically or model it with visual or tactile representations, nothing involved really has this same spatial relationship of “over.” So, not only can students not rely on their math understandings of this word, but also they can’t rely on their general-use understandings of this word over.
Why “Multiplication Across Addition” is better
First, “across” has not (that I know of anyway) taken on any special meanings in math. When we do use the word “across” in math classes, we usually use it in the same spatial-relationship ways that we would use the term in general language. There are no confusing math-specific paradigms that this word brings up just because we say it in math class.
Further, when we look at the distributive property symbolically, we do typically see the spatial relationship of the multiplication going across the addition, and partnering with each term along the way. Thus, this meaning of the term across is preserved across (pun intended) the typical presentations of this distributive property.
Even when we work with physical or drawn models, we can still watch the “across” happen in the same spatial way. For example, in this image modeling $3(x+6)$ from my article Factoring out the GCF in algebra, we see the algebra tiles moving across the page, separating into two groups that are each 3 units tall. We also see the 3 trailing and “replicating” itself across the groups to illustrate the distributive property.
I acknowledge I am one small content creator, and one sole math instructor. However, I claim here that across in this context better illustrates what we visually see and physically do when we distribute multiplication across addition. It is not charged with the same negative emotions and stress chemicals that the word over can unintentionally evoke in a math context. I propose that we change the name to the
Distributive Property of Multiplication Across Addition
It’s far better on multiple levels.
What do you think?