Understanding the Area Model for Multiplication
I have noticed that it is common to see the Area Model used to illustrate multiplication and the Distributive Property of Multiplication over Addition. However, the Area Model should only be used for problems involving area. It is very important to use the proper model for the given problem.
Let’s review a Common Core Standard that lists the Area Model as a method to use for multiplication, 4.NBT 5.
4.NBT: Number and Operations in Base Ten.
Multiply a whole number of up to four digits by a one-digit number, and multiply two, two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular array, an/or area models.
If a person is skimming the Common Core Standards or focuses only on one part of the standards, it would be easy to misinterpret the meaning of a standard and assume that an Area Model is a good way to show multiplication of whole numbers, for example, 24 x 36.
Sometimes it is correct to use an Area Model to show 24 x 36. For example, it is appropriate to use the Area Model to calculate the number of square units in a rectangle that is 24 units long and 36 units wide. In this case, the Area Model illustrates 24 units x 36 units = 864 units2. Other times an Area Model is not appropriate. Let’s consider the problem, “Calculate the number of eggs in 24 baskets where there are 36 eggs in each basket.” The correct calculation would be 24 baskets x 36 eggs in each basket = 864 eggs in 24 baskets. If a student follows the example of finding area using the Area Model, then a student would be confused how to state the units for the answer. The student could ask, “Is the answer stated as 864 baskets2 or 864 eggs2 or 864 basket-eggs?” Each of these is incorrect for this problem.
If we delve into the Common Core Standards further, we notice that the standard 3. MD (# 5, 6, 7a and 7b) gives more insight into when the Area Model is appropriate. Pay close attention to the wording.
3. MD: Geometric measurement: understand concepts of area and relate area to multiplication and to addition.
Recognize area as an attribute of plane figures and understand concepts of area measurement.
A square with side length 1 unit, called “a unit square” is said to have “one square unit” of area, and can be used to measure area.
A plane figure which can be covered without gaps or overlaps by n unit squares is said to have an area of n square units.
Measure areas by counting unit squares (square cm, square m, square in, square ft, and improvised units)
Relate area to the operations of multiplication and addition.
Find the area of a rectangle with whole-number side lengths by tiling it, and show that the area is the same as would be found by multiplying the side lengths.
Multiply side lengths to find areas of rectangles with whole-number side lengths in the context of solving real world and mathematical problems, and represent whole-number products as rectangular areas in mathematical reasoning.
The standard expects the lesson to demonstrate with deep understanding that an array of square tiles can be thought of as a-units x b-units = a x b square units. In other words, it develops the concept of the Area Model. Thus, it is providing reason that the product of two unit-lengths is square-units. An Area Model is a simplified Array Model that should only be used to find area.
I have noticed that two of the common models used to show multiplication of whole numbers, the Area Model and Array Model, are two models that are often used interchangeably and that sometimes there is no distinction made between the models.
In other words, a model will be shown and named Area Model when the appropriate model is really an Array Model. These models are similar but there are differences that we need to explore to improve our students understanding of concepts rather than mindlessly use a model as a tool to get an answer.
For example, if someone is showing 24 x 36 with a rectangular math drawing, they often label that drawing “Area Model” when in reality they should label it an “Array Model.”
You may ask, “What is the big deal? Aren’t you making a mountain out of a mole hill?”
Remember that it is very important to develop students’ understanding and not just teach them to memorize a process that “always works”. Finding the area of a rectangle and counting objects in an array are usually based on different concepts. Area is restricted to a 2-dimensional measurement and calculates the size of a shape in square-units, but an Array Model calculates the total number of objects in B groups with C objects per group.
Let’s compare the Array and the Area models. In Figure 1, an Array Model is used to show 4 groups with 3 eggs in each group. We have 4 baskets x 3 eggs in each basket is 12 eggs, not 12 eggs2. In contrast, an Area Model is used to calculate the area of a rectangle that is 4 inches in width and 3 inches in length by calculating 4 in x 3 in = 12 in2 as shown in Figure 2. What is the justification for this area calculation? Why does this method of calculating the area of the rectangle make sense? We will explore this below.
The Area Model justifies why the formula A = L x W gives us the total number of square units in a rectangle. Why can we just multiply 4 in and 3 in and then say the product is 12 in2? Could we do the same process for all other problems involving multiplication of whole numbers? For example, does 4 baskets x 3 eggs per basket = 12 basket-eggs?
When we calculate the area of a rectangle that is 4 inches in width and 3 inches in length we are really calculating how many square inches are in the rectangle. We can see this idea being presented in Common Core Standard 3.MD 7a because the standard mentions “tiling”. We can view the area of the rectangle as 4 groups of 3 in2 per group for a total of 12 in2 as shown in Figure 3A and Figure 3B below.
If my length is 4 inches, I can think of that length as being broken into 4 equal parts where each part is 1 inch in length. Then I can subdivide my rectangle horizontally into 4 rows. I can also subdivide my rectangle vertically into 3 columns that are each an inch in width. When I subdivide my rectangle both horizontally and vertically, I created several 1-inch-by-1-inch squares in my rectangle. This is defined as a square unit in the CC Standard 3MD #5. So now my rectangle is made up of 4 rows with 3 square-inches in each row as shown in Figure 3A below. So, I can now think of the area of my rectangle as 4 groups with 3 square inches in each group for a total of 12 square inches. Do you see that the Area Model is a special case of the Array Model?
Sure it is easy to calculate the area by calculating length times width, but just memorizing this formula without understanding will cause confusion down the road for students. We want to prepare students to be good thinkers and problem–solvers and not just use memorized tricks.
Consider this scenario: A landscaper must figure out the area of a rectangle with length = 4 yards and width = 3 feet. If the landscaper uses the formula A = L x W without understanding, then the landscaper might say that the area of that rectangle is 12 yard- feet, or 12 yd2 or 12 ft2. Do you see the disconnect?
Conversely, a person who understands that if you want square units when you subdivide the rectangle horizontally and vertically, then both length and width must be measured in the same units so that the rectangle will be made up of square units. It is appropriate for the landscaper to use the area formula to find the area, but with an understanding of the Area Model the landscaper would know that both the length and width of the rectangle must be measured in the same units.
On the flip side, another issue concerning modeling multiplication of whole numbers is using an Area Model when an Array Model should be used. Let’s say we want to find how many total eggs we have if we have 24 baskets with 36 eggs in each basket, and we draw the Area Model shown in Figure 4 below.
Do you see that we should use an Array Model instead since our objects in each group are not square units but eggs? So, our model should show 24 rows with 36 eggs in each row, not 24 rows of 36 square units per row and not 24 x 36 egg2. Each row in the array would show the number of eggs in each basket. Do you see how students could get confused if we say we are using an Area Model, but the units are not square-units?
In conclusion, if we look more deeply into the Common Core Standards 3.MD 7a, we can conclude that the Area Model is only appropriately used when calculating areas of rectangles.
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