- Area model can be used from beginning multiplication to algebra
- Helpful for division as well as multiplication
- The model works for fractions and decimal multiplication
Are you looking for a way to teach multiplication before your students learn the standard algorithm? The area model is a great tool that you can introduce your students to in third grade. One great thing about this representation is that it can be used with multiple skills. I’ve also included some ways that you can use Cuisenaire rods while developing these skills.
Building Arrays With Tiles Connects to the Area Model
When students first begin working with arrays, they often build them with cubes or counters. This work can help them see they can use repeated addition to find the total number of objects in the array. Arrays that are built with tiles or cubes, such as Cuisenaire 1 rods, can be pushed together into a rectangle. Students can begin making connections between the 3 by 4 array that they built and the rectangle they made with a length of 4 tiles and a width of 3 tiles. They will also see that the sum they found for the array using repeated addition will be the same as the product or area of the 3 by 4 rectangle. Of course, when students are drawing arrays, dots make more sense, but tiles are a great concrete tool to help students connect arrays and multiplication with area.
Students Make a Connection to a Representational Model With Tiles or Cuisenaire Rods
Students can also make a connection between the arrays they are building and the representational model. For example, four dots in a row can be represented by a rectangle with length of four units and width of one unit. Take the time to show the representation each time students build a rectangle with tiles or Cuisenaire rods. Soon students will be able to do it on their own. Once students can draw these representations, they can move beyond drawing and counting squares. As we all have probably seen, student arrays and drawings of tiles can get a little uneven. The drawings will make more sense to students once they understand the connection between the dots or tiles and the side lengths (factors).
Students Understand Area as it Relates to Repeated Addition and Multiplication
Students learn to find the quantity in an array with repeated addition or with multiplication. Then they recognize the area of a rectangle that’s been partitioned into squares can be found in the same way. I’ve been working with a third-grade class that learned about area right after learning multiplication.
I’ve noticed two things. First, many of the students saw the connection to repeated addition and multiplication right away. This made moving to rectangles with just measurements instead of squares to count fairly quick. While not all students had their multiplication facts memorized, they understood that counting individual squares was not necessary. Second, there was no confusion of area with perimeter. The students connected area to multiplication. In the past, I’ve experienced, and you likely have, too, that students confuse area and perimeter. We try to give them catchy ways to remember which is which. I always think back to a conference I attended where Greg Tang said if we teach students to use the area model for multiplication, they will never confuse area and perimeter.
The Area Model Creates a Great Visual for the Distributive Property of Multiplication
Once students have an understanding of the area model for multiplication, they can use it to show the distributive property which can be helpful when multiplying larger facts and multi-digit numbers. The program that my district uses teaches students that breaking apart one factor can help when multiplying numbers greater than five. As instructed by the program, students draw arrays and then divide the arrays. While that is certainly one way to go, it can require students to draw a lot of objects. This can be messy and students can miscount.
I taught students to build the rectangle with Cuisenaire rods and physically break it apart (concrete). Most students have mastered twos and fives and can skip-count for facts they don’t know yet. Next, they drew the area model represented by the rods (representational). Teaching this in third grade gives students a tool that many find helpful for multiplying larger numbers in fourth grade and beyond.
The Area Model Can Be Used for Division
Students can combine what they know about the area model and their understanding of the relationship between multiplication and division to use it for division. Cuisenaire rods are a great concrete manipulative for representing division problems using the area model. To find the quotient, use rods the length of the divisor to create a rectangle with an area that is the same as the dividend. For example, in the equation 24 ÷ 8 = ? students would make a rectangle with an area of 24 using the 8 (brown) rods. The length of the rectangle would be 8. The width would be 3 since there are three of the 8 rods. So 24 ÷ 8 = 3.
When students draw the rectangle with the area and given side length, they can extend this concrete model to the representational stage. If the sides shown below with dotted lines are erased, students can begin to transfer this model to the traditional algorithm for division.
What about division problems that result in a remainder?
Yes, you can use the area model and Cuisenaire rods for these problems, too. First, make the largest rectangle possible with the rod the length of the divisor. For example, for the problem 40 ÷ 7 use five of the black 7 rods. If they are not strong with multiplication facts, students can skip-count by sevens until they get as close to 40 as possible without going over. They will then see that they have a rectangle with an area of 35 with one yellow 5 rod left to make the starting dividend of 40. The visual shows that 40 ÷ 7 = 5 R 5.
Using the Area Model to Divide Larger Numbers
Understanding how to use this for division can be helpful for students who are not quite ready for the standard algorithm. Here are two examples of using it with larger numbers. I was in a fourth-grade classroom where a student was using the area model to divide 135 by 5. He set up his problem as shown below, with only the 135 and the 5 in the lower, right corner in place.
Then, he told me he would multiply 5 x 11 and see how many times he could take that out of the total, an area model to show taking out partial quotients. Finally, he added the partial quotients, shown in orange, to show that 135÷5=27.
Here’s another approach. The dividend has been decomposed into numbers that are easily divisible by five, 100+30+5. Each is in a separate section of the rectangle. As with the other example, each partial quotient is written above the appropriate section. The partial quotients are then added to find the quotient, 27.
But what if it’s not such a neatly divisible problem?
Use the same process. Instead of decomposing the dividend into the expanded form of the number, as shown above, find numbers that are multiples of the divisor. A student could start with 180 or just use 60 depending on their comfort level. For this example, I started with 120.
The Area Model Can be used to Multiply Fractions and Decimals
Extend the area model to multiply fractions. Students will already be familiar with placing one factor along the width and one factor along the length of a rectangle. When multiplying larger numbers, they learned to decompose them into their place values. They will apply that concept and their early work with decomposing fractions to set up the area model for multiplying fractions. In the picture on the left, colored lines show the product of 3/4×1/2. The blue lines shade one-half of the rectangle while the red lines shade three-fourths of the rectangle in the opposite direction. You’ll notice that this results in three-fourths of the one-half or three-eighths of the rectangle being marked with both colors.
The other rectangle shows how to use the area model to multiply mixed numbers. Decompose each factor into a whole number and a fraction. Then multiply each part. Finally, add the partial products to find the total product.
Multiplying Decimals
This model is helpful when multiplying decimals, too. Students will decompose each factor and multiply each part in the same way they did with whole numbers. Then, they will add each partial product to get the product of the two decimals.
Beyond Elementary School, Using the Area Model in Algebra
If students have the area model in their tool bag they will have it as an option in algebra. The model works when multiplying binomials like (a + 3)(a + 2). In the picture below a Cuisenaire ten rod represents a and the one rod represents 1. A base-ten hundred flat replaces 10 ten rods.
Check out this area model simulation from the University of Colorado for some games to practice this skill.
I hope you’ll agree that the area model is a great tool for students to have.
If you’d like to read more about using Cuisenaire rods, check out this post on Introducing Cuisenaire Rods.
I also have a few Cuisenaire rod-related products that may be of interest to you.
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