# MathBait™ Multiplication

# The Area Model

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Ready to introduce multi-digit multiplication to your students? In this article we breakdown the Area Model providing key information on the mathematics behind the method, when and how to introduce the Area Model to Students, and the pros and cons teachers and parents must consider.

## Details

Resource Type

Method

Primary Topic

Multiplication Methods

Unit

4

Activity

1

of

6

This model of teaching multiplications is particularly useful as a first step in multiplying larger values. Clearly leaning on the third interpretation of multiplication, the Area Model can be completed in three steps. For our example we will examine multiplying 38Ã—46.

#### Step 1: Draw a rectangle

Students begin by drawing a rectangle in which each side represents one factor they are multiplying. In this case, we can assign one side to represent 38 and the second to represent 46.

#### Step 2: Partition each side

This is the step that matters. Students practice manipulating numbers, which is a big theme in*Â ** Marco the Great and the History of Numberville*. Seeing numbers as malleable is an incredible power when it comes to math. Students may turn 46 into 40+6 or 20+26 or 10+36 or even 10+10+10+10+6. We see this as the great equalizer in mathematics. It allows students to play to their strengths as they can select values they are comfortable with multiplying. As a rule of thumb, it is generally easiest to partition the rectangle into tens and ones, so we'll follow this convention.

#### Step 3: Find the area of each partition

Next, students find the area of their partitions. Instead of having to multiply large values like 46 and 38, the problem is reduced to smaller and easier computations. Here, 30Ã—40 can be computed as (3Ã—10)Ã—(4Ã—10)=(3Ã—4)Ã—100=1200. Similarly, 6Ã—30 can be quickly determined as 180 and 40Ã—8 as 320. The final computation gives us 8Ã—6=48.Â

#### Step 4: Sum the smaller areas

To find the product of 46Ã—38, students sum the areas of the smaller rectangles to determine the total area of the shape. In this case we have 1200+320+180+48=1748.

While the Area Model has many benefits and is a great choice for introducing the multiplication of larger products, it has some downfalls as well. However, the math behind the area model makes for an excellent extension to the Distributive Property. Students will find themselves reliant on this property well into college math.

We introduce the Distributive Property in MathBaitâ„¢ Multiplication Part 3.

Not only is the Area Model and the Distributive Property the great equalizer, it continues to build students' ability to manipulate numbers which is one of the most impactful skills a student can have. This will support their work in prime factorization which in turn contributes to a multitude of topics in Prealgebra, Algebra, and Geometry. For this reason, we highly recommend students learn the area model.

#### Overall Assessment of the Area Model

##### PROS

Helps students develop a strong number sense and uncover hidden patterns in numbers

Allows for flexible thinking as students can build their fences in whatever way they'd like

Excellent introduction to the Distributive Property

Introduces students to the Area Addition Postulate, beneficial in application problems and Geometry

Connects to multiplying binomials in Algebra and having a result of 4 terms helps avoid common errors

##### CONS

Drawing out squares and rectangles is tedious and time consuming

If students do not understand why the model works it is not beneficial in strengthening their foundation

#### Our Rating:

#### Our Recommendation

The Area Model earns an A+ with us. If your students are just learning to multiply multi-digit values, this model is a great place to start. It allows students to lean on their existing knowledge and make sense of place value. If your students are struggling with the standard algorithm, this is the method you want to pivot to. It allows students to better organize their work and minimize errors.

We recommend all students learn the area model. Not only is it beneficial for basic multiplication, it can also support students in Algebra, as they identify patterns (such as finding perfect squares and completing the square) and in multiplying binomials or expressions with variables.

To transition your student to the area model, try this mini-lesson.

##### Transitioning to the Area Model

Revisit the activity "What's That Called?" from MathBaitâ„¢ Multiplication Part 3. Next, provide students with an empty multiplication table and allow them to select a product (preferably not a square). Instruct students to draw their rectangle on the chart.

Previously, they noticed the number of 1Ã—1 squares matched their product (here we have 12 1Ã—1 squares in our 3Ã—4 rectangle).

Now ask students to draw along one of the grid lines inside their rectangle from top to bottom and another from left to right. Students may pick any grid line within their rectangle.

Invite students to share their thoughts. Why would we do this? How is this helpful? Highlight that we have now created 4 smaller rectangles. Ask students how many 1Ã—1 squares are in the large rectangle (this value hasn't changed with our markings, in our drawing we still have 12).

Next, have students count how many small 1Ã—1 squares are in each of the four smaller rectangles. Here we have 4, 4, 2, and 2. Allow students to sum these values to confirm together they still have their total.

Finally, ask students to write a multiplication statement for each smaller rectangle they have created. In our example we have 2Ã—2, 2Ã—2, 1Ã—2, and 1Ã—2. Compute the products, what do they notice?

Ask students what in their life they might connect to the rectangles. Modeling can be helpful. For instance, if I start with 12 LEGO and break them into four smaller piles, I still have the same number of blocks because I haven't added or taken any away, just rearranged them. After sharing, explain that we can use this idea to make more complex questions easier.

Provide students with graph paper and the expression 22Ã—13. Allow students to work in pairs as they attempt to solve the product using rectangles. Encourage students to draw out the large rectangle and break it into smaller pieces as they previously did.

After having time to generate their solution, ask different groups to share their findings. Highlight particularly useful strategies such as breaking the rectangle into 20+2 and 10+3. This allows us to lean on the work completed in the *Distribution*Â section of MathBaitâ„¢ Multiplication Part 3 and carry over our knowledge of one-digit products to multiples of 10.

Conclude by explaining to students they are now able to complete multiplication of larger values! They can change any multiplication question into smaller questions they already know how to find. Allow students time to practice this new skill.

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