Multiplication and Division
Developing the meaning of multiplication
In Think Math!, intersections are introduced very early because they lead in so many other directions as well as multiplication: map skills, mathematical reading, systematic list-making and discrete mathematics, multiplication, coordinates. Some of these don’t need to mature until much later on, but some, like mathematical reading, are so essential that they must be developed quite early.
In Grade 1 (Chapter 10), children are on the floor, laying out a tiny town with streets and avenues, describing how to get from one location to another, learning map skills and language in a way that foreshadows coordinate grids (no order required in their naming of intersections, but clear awareness that each intersection has a "name" that depends on which street and avenue cross there). At this point, they are not learning multiplication at all -- that is, they are not associating the numbers of streets and avenues with the number of intersections -- but they are developing the language and attention around knowing which of several related questions has been asked: how many streets, how many avenues, how many roads of any kind (adding), how many intersections, how far from one to another (attention to intervals between intersections rather than the intersections themselves)… In all cases, their method of answering is just what first graders can do: mostly counting and some adding, but listening to and knowing what to count.
In Grade 2, children do begin to study multiplication itself. Quite early (Chapter 1, lesson 11), they see a model for finding the pairings of objects from two sets (in the context of phonics!) The multiplication content of this isn’t pursued until chapter 13, but the arrays of dots continue to appear, in other contexts, as does the idea of combining things. Over the course of the year, they will use the intersection model for pairing shirts and pants, building two-story Lego buildings, counting how many sandwiches with various choices of bread and filling… (See multiplication.) The several chapters on multiplication in Think Math! focus on using multiplication as an easy way to "count" objects arranged in a rectangular array.
Through grade 3, the focus is on what multiplication and division are, on developing facts and images, and on showing some of the uses (e.g., pairing pants and shirts, flavors of ice-cream, etc.).
In Grades 4 and 5, we use the same images to develop the algorithms for multiplication and division. See, for example, Think Math! Grade 4, Ch 2. The intersection dots and the equivalent area squares fade -- too much of a nuisance for large numbers -- so that by Grade 5, Ch 5 and 8, what started as arrays become “open” area models.
Developing the multiplication and division algorithms
The area model
We can picture 3x4 as an array and count the tiles. In principle, we can also draw a 26x48 array and count the tiles, but counting one by one becomes impractical: too tedious, too unreliable.
We can "sketch" the idea and count more efficiently. Here is a picture that shows, in fact, every tile, but the heavy lines and the colors help group regions that can be counted in simpler ways. The blue region is made up of large 10x10 squares, so some mathematical understanding saves us from counting the 100 tiles in each square; we can just count the eight hundreds. The red and yellow regions contain rectangles one of whose dimensions is 10; again, we can use mathematical knowledge (multiplication by 10) to "count" the tiles in these regions. And the green region requires that we know the basic number fact 6x8. (We can, of course, figure that one out, too, but it is much more freeing to have the "fact" knowledge in our head.)
Without the heavy lines, we can still describe each colored region with a "simple" multiplication -- one that involves only the knowledge of a "basic fact" and an understanding of how to derive, say, 6x40 from 6x4.
Using the cross number puzzles format to summarize
What remains, then, is to add all these results -- the number of tiles in each region -- to get the total. The current common American algorithm computes what amounts to a subtotal of the regions. In the illustration below, the blue and red regions are subtotaled to 960 and the yellow and green regions are subtotaled to 288; these are exactly the numbers that the standard algorithm produces! (See cross number puzzles for more about how this format is developed and used.)
If, instead, we had subtotaled the columns instead of the rows, we'd get blue + yellow = 1040, and red + green = 208. Again, these are exactly the numbers that the standard algorithm produces!
In either case, adding the subtotals gives the total.
But, we need not bother with the subtotals. We can just write down the result of every multiplication we must do -- in this case, four multiplications are required, no matter what algorithm one chooses -- instead of dissecting a result, writing down part of it, carrying the other part, and risking making some error because we don't know which part to write down and which to carry, or we don't know where to carry it, or we don't know what to do next.
When we write down each product as we get it -- not using the "carry" step -- we get not only greater clarity and less opportunity for confusion and error, but we also see a structure that more clearly resembles the one we will need to build for algebraic multiplications. In algebra, multiplications like (20 + 6)(40 + 8) become more general: (a + b)(c + d), where any numbers might be placed in the spots marked by a, b, c, and d. There is no shortcut here, no subtotals; the result is the sum of four separate products (like the 800 + 240 + 160 + 48 shown above). Why not encourage students to do that from the start?
Common errors with the multiplication algorithm
Any set of rules, if they are not understood, are vulnerable to decay. We forget parts, or mix up the order, or confuse the rule with some other rule that we've learned.
Treating multiplication like addition
One common error that children make when using the standard algorithm (or any of its variants) for multiplication is that they multiply the ones by the ones and the tens by the tens, and forget the other two multiplications. That is, in multiplying 47 ×53, they’ll multiply 50 × 40 and 3 × 7, but forget to multiply 3 × 40 and 50 × 7.
The four multiplications that need to be done can be summed up this way:
The same imagery will develop multiplication of fractions (Grade 5, Ch7 L11), and the same multiplication “rule” can be used for multiplying mixed numbers.
- ↑ A comment on mathematical reading: English reading proceeds left to right. Even the top-to-bottom is just an accommodation to the fact that it’s easier to write on arectangular sheet than on a long thin strip of paper. Mathematical reading is different. Most of it is “two-dimensional.” On a table or chart, the meaning of each cell depends both on the row and the column it is in; on a bar graph, the meaning of each bar depends on which bar (horizontal location) and its height (vertical location); on a coordinate grid, each point is identified by horizontal and vertical location. And there’s more. Encountering this early, often, and in age-appropriate ways helps the whole enterprise for years.