Common Core @ Home: Math - What's an area model?



Part Seven of a continuing series

common core math hudson valley new york




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"?

In 4th and 5th grade, students are introduced to the multiplication of multi-digit numbers.

One of 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.

For 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:


Of course, 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.

This 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 as:

By 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.


Even 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.

To point out one useful feature of the area model, I’ve introduced an error in the computation:


Now 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

 

While 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.

You 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.

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