Jeff Suzuki teaches mathematics at Brooklyn College, and is one of the founders of the Mid-Hudson Valley Math Teachers Circle, a group of teachers, professors, and mathematics aficionados working to promote mathematics education in the Hudson Valley.
Dear Jeff, What is an "area model"?
and 5th grade, students are introduced to the multiplication of multi-digit
the more effective means of finding these products involves an area model,
which connects to the 3rd grade result that the area of a rectangle is the
product of its width and length.
example, you can find the product 23 × 37 by finding the area of a rectangle
with a width of 23 and a length of 37:
there is little gained if you simply multiply 23 × 37! What makes the area
model useful is that area is a conserved quantity. In other words, you
can break one large area (in this case, that of a 23 × 37 rectangle) into
several pieces; find the areas of the pieces individually; then add to get the
area of the whole.
is where the early exposure to number bonds is useful: a student who
thinks of a number like 12 as something that can be broken apart into 10 + 2
(or 7 + 5 or any other combination) will have no difficulty seeing a number
like 23 as 20 + 3, and 37 as 30 + 7. This allows them to divide the rectangle
breaking 37 and 23 apart in this fashion, we have four rectangles whose areas
are easy to find, because their lengths and widths are numbers with a single
non-zero digit. Thus the rectangle in the upper left has width 30 and length
20, for area 30 × 20 = 600; the rectangle in the upper right has width 7 and
length 20, for area 7 × 20 = 140; the rectangle in the lower right has area 30
× 3 = 90; and the rectangle in the lower left has area 7 × 3 = 21. Adding these
four areas together gives us the product: 851.
larger products can be found using the area model. For example, 356 × 48 can be
found by writing 356 as 300 + 50 + 6 and 48 as 40 + 8 (again, retaining the
feature that each term has only a single non-zero digit), then finding the
areas of the six regions.
point out one useful feature of the area model, I’ve introduced an error in the
consider the standard algorithm (also introduced in the 5th
grade) for this product; again, I’ve introduced an error in the computation:
3 5 6
× 4 8
2 8 4 8
1 4 7 4
1 7 5 8 8
the standard algorithm takes up less space on the page, the steps are just
steps: they have no justification beyond “Do as you’re told,” so it’s easy for
students to lose track of what they’re doing and make mistakes, and hard to
find and fix any mistakes that they make. In contrast, the area model is much
more transparent, so it’s easier for students to keep track of what they’re
doing, they’re less likely to make mistakes, and it’s easier for them to find
and fix mistakes.
might try your hand at finding the error in the area model computation, and the
error in the standard algorithm computation; then seeing how easy it is to fix
these errors in the respective algorithms.
READ MORE: Answers to Common Core questions