There are two basic conceptual ways that multiplication is explained: Repeated addition and the area model. Many people who are adept at multiplication (including teachers) take for granted what many students have trouble connecting; in the case of multiplication, it’s not immediately obvious why repeated addition and the area of a region ought to result in the same value.
Of the two, seeing multiplication as repeated addition is more restrictive: It requires at least one of the numbers being multiplied be a positive integer. For younger students, though, it’s easier to understand: \(4\times 3=4+4+4=3+3+3+3\). If four people each have three apples, or if three friends each have four apples, then between them they have twelve apples.
This also allows for a fairly intuitive understanding of division: If you have twelve apples and need to distribute them evenly among four people, each person will get three apples. In this model, multiplication is repeated addition and division is repeated subtraction.
The area model is more flexible in that neither number has to be an integer, but they both have to be positive to make full sense. Indeed, my own brain tends to set negative signs aside when performing multiplication and division, and then using the parity of signs (even or odd) to determine whether the result is positive or negative.
In the area model, we imagine (or even draw) a rectangle with a given width and height, and then determine its area. So if we want to know \(3\times 4\), we can create a diagram and count the squares inside it. In the diagram, there are twelve squares, and so \(3\times4=12\).
On one level, this is a modification on repeated addition, in a way that may not be immediately obvious to learners: There are four rows of three squares. It can also be confusing to students who want to count intersections (which are visible) rather than white-space squares.
Here’s how we can used the area model to evaluate \(3.2\times4.4\). Now we have the same 12 large squares, but we can also count the smaller squares, of which there are 52. In this diagram, each small square is 1/25th of a large one, and 25 small squares make a large one. This means that those 52 small squares count for 2 large ones, plus two left over. Hence \(3.2\times4.4=14\frac2{25}=14.08\).
The problem with counting squares is that it’s tedious, especially when counting small squares. It also requires graph paper, and is problematic with larger numbers. This has led to several scaffolded variations on it.
One of these is the box method, which is useful for both numbers and polynomials. Let us say we are multiplying 235 by 30. Create a table with each place value on the top and the side, as shown here:
In each cell, place the product of the numbers on the top and the side. Then, add up all the cells to find the value: 6000 + 900 + 150 + 800 + 120 + 20 = 7990.
While this feels needlessly time consuming for numbers, it can be a major boon for multiplying polynomials. It makes it easier to find mistakes and to make sure that all pairings have been completed. Also, if both polynomials are complete (that is, quadratics have three terms, cubics have four terms, etc.), then the like terms will line up along diagonals. The example shown demonstrates that \((x^2+2x+3)(x+5)=x^3+7x^2+13x+15\).
A common variation of the box method for numbers is the lattice method. This leverages the idea that diagonals will have like terms, but is less transparent than the box method. Unfortunately, it is often taught without the original scaffolding, and so it can look like a trick or like magic to many students. This time, we’re going to write the individual digits on the top and side, and then put the two-digit results in each cell, on either side of a diagonal line.
To find the result, add along the diagonals for each digit, going from right to left: 0, then 9, then 19, then 6. If any sum is greater than 9, take its tens place and add it to the next number (that is, “carry” it). This yields 7990.
In my experience, some students love the lattice method, but a greater number hate it. And I doubt many of them, in either case, can explain why it works. For those students who love it, I think it’s a decent enough method, but (as an Algebra II teacher) I don’t actively teach it.
The next two common methods are mechanical shortcuts that seem more attached to repeated addition, but are at least similar to the area model in the numbers being used. They are also similar to each other.
With partial products, we pair every digit of each multiplicand, keeping zeroes in mind:
The result is then determined by adding these products. Notice that these are the same values as in the box method: It’s using the box method strategy without drawing the related picture.
The version I think most older people think of when doing this sort of multiplication conflates one direction of the pairing into a single step, which leads to less writing but more potential for confusion, especially if we leave out the zero place holders:
In this version, we mark “carries” above the individual digits. This can be confusing especially if the second multiplicand has multiple digits. A (slightly) less confusing variant of this apparently taught in other countries is to put the “carries” below, with a separate line for each digit of the multiplicand, as shown here:
Nonetheless, students understandably struggle with this method. I’ve even seen math teachers express frustration with understanding how this works.
In 2023, it’s a fair question to ask: Why do we need to know how to do this at all? Everyone has extremely powerful calculators in their possession nearly all the time, why bother with calculating by hand?
One answer to this suggests a shift in pedagogy: The box method is time-consuming if the goal is mechanical processing, but crucial if the goal is number sense and understanding. And the latter should be the goal of mathematics education in this century.
Indeed, it might be even better if the size of the boxes reinforced that hundreds are larger than tens, which are larger than ones. It wouldn’t be practical to make them to scale, but a visual remind such as what’s shown here would reinforce the concept.
So what’s the best? The method that demonstrates what we want to demonstrate. I think we put far too much emphasis on the ability to perform the mechanics of multiplication in an era where calculators can evaluate the product of two ten-digit numbers faster and more reliably than most people can evaluate the product of two two-digit numbers. More number sense, less mechanical processing.