One of the first things children learn when they start learning how to multiply numbers is to make patterns with objects in an array. The children chart the manipulatives to discover they have a length and a width. Counting all the manipulatives, they find a total. Through this early experience, students begin to build a skill that will continue to develop all the way through high school algebra.
Common Core and school curriculums began emphasizing non-standard algorithms over the traditional methods. The purpose of the area model is the mathematical development of students. The purpose of developing methods is to gain a lasting understanding of the mechanics of math rather than finding an answer to a quick math problem. While the standard algorithm is often the most efficient way to solve a problem, it hides the reasoning behind the math from students learning to do more complicated math at a younger and younger age. Yes, the area model looks very different from the math many of us did as children, but the mechanics remain the same.
Area models and arrays are based on a simple concept: the length of a rectangle times the width equals the total area. Area models that students typically use are simple physical arrays.
The basic model lays the groundwork for learning that will continue throughout high school. How can this model be used to help young students better understand? The visual difference between what adding looks like compared to multiplication is the most important use of this model. It makes it more clear how different 7 + 3 is from 7 x 3. Students will need to know this distinction when studying the order of operations. When students have mastered multiplication facts, they move on to two-digit multiplication. It is here that the model takes the turn that many adults begin to be uncomfortable with math!
What is an Area Model?
In mathematics, an area model is a model or rectangular diagram used to solve multiplication and division problems, where the factors or the quotient and division determine the length and width of the rectangle.
By using number bonds, we can break one large area of the rectangle into several smaller boxes, making calculation easier. Next, we get the area of the whole, thereby determining the product or quotient.
You can multiply two 2-digit numbers with the area model by following the steps below:
Write the multiplicands in expanded form as tens and ones.
For example, 35 as 30 and 5 and 27 as 20 and 7.
Draw a 2 × 2 grid, or simply a box with 2 rows and 2 columns.
On the top of the grid (box), write the term for one of the multiplicands. One on the top of each cell.
Write the terms of the other multiplicand on the left side of the grid. One on the side of each cell.
In the first cell, write the product of the number and the tens. In the second and third cells, write the product of the tens and ones. In the fourth cell, write the product of the ones.
In order to determine the final product, add up all the partial products.
Here, for example, the area model has been used to multiply 27 and 35.
27 x 35 = ?
27 x 35 = (20 + 7) x (30 + 5)
Here we see how to find the product of a 3-digit number by a 2-digit number using the area model.
374 x 43 = ?
374 = 300 + 70 + 4
43 = 40 + 3
Hence, 374 x 43 = (300 + 70 + 4) x (40 + 3)
As mentioned earlier, we can use the area model for the division. Let us divide 825 by 5.
825 ÷ 5 = ?
825 ÷ 5 = 100 + 60 + 5 = 165
Another example to help you understand better
630 ÷ 18 = ?
630 ÷ 18 = 30 + 5 = 35
Some Interesting facts about the area model
- The area model is also known as the box model.
- Finding the area of a rectangle is the basis of the area model of solving multiplication and division problems. Area of a rectangle = Length × Width.
- The area model of multiplication uses the distributive law of addition.
- Expanded forms can be used to multiply numbers with more than 2 digits.
- The division is incomplete if the difference is greater than the divisor. In the case where the difference is less than the divisor, then it is the remainder.
- Is it possible to start the process with another length of a rectangle? Yes, for example, in 555 ÷ 15, if we take into consideration the sub-rectangle of length 5 units then the area becomes 15 ⨉ 5 or 75 sq. units and the rest of the area becomes 555 − 75 = 480 sq. units. In a similar manner, the procedure can be continued.
Benefits using the Area Model
The “open-ended” approach provides entry points for students to begin solving larger division problems and multiplying large numbers.
Students will relate more by using and illustrating the “boxes” for the area/rectangular model. The rectangles represent an actual box or group of something and are symbolic.
In order to improve students’ understanding of the model and performance, you should encourage them to solve the problem “in a certain way”.
Usage of Area Models
As we introduce multiplication in primary, arrays and area models assist not only in supporting the development of proportional reasoning, but also teach us how to develop strategies that build number flexibility and the automaticity of math facts.
Arrays and area models can be used to illustrate many big ideas in mathematics including, but not limited to:
- Distributive Property with Whole Numbers
- Factoring (Common, Simple/Complex Trinomials)
- Completing the Square
- Multiplying a Binomial by Monomial
- Multiplying a Binomial by Binomial (aka FOIL)
- Finding Area with Whole Number Dimensions
- Perfect Squares & Square Roots
The term “array” may not be familiar to many people. Fortunately, the definition is fairly straightforward:
Mathematics defines an array as a group of objects ordered in rows and columns.
Despite their simplicity, they are extremely helpful when teaching multiplication and building conceptual understanding for more abstract ideas that require fluency with procedures.
Area Model of Multiplication
The area model of multiplication is often regarded as the most conceptually understandable multiplication strategy for young learners. In addition to the fact that this strategy builds upon the learner’s spatial reasoning (more than half of all young children follow this as a preferred learning technique), it also enables learners to isolate the partial products of multiplication problems, an essential step in learning the strategy for multiplication.
At the start of their maths journey, students are taught to:
- Use a variety of tools and strategies to relate multiplication of one-digit numbers and division by one-digit divisors to real-life situations (e.g., place objects in equal groups, write repeated addition or subtraction sentences, use arrays, etc);
- Apply a variety of mental strategies to multiply 6 x 6 and divide to 36 ÷ 6, (e.g., skip counting doubles, doubles plus another set);
- Determine, through investigation, the properties of zero and one in multiplication (i.e., any number when multiplied by zero equals zero; any number when multiplied by 1 equals the original number) (Example problem: Using tiles create arrays that represent 3 x 3, 3 x 2, 3 x 1, and 3 x 0. Explain what you think will happen when you multiply any number by 1, and when you multiply any number by 0.);
- Use the area model to visualize products of two numbers;
- Consider using the area model to understand both the decomposition of numbers and the distributive property of multiplication,
Understanding the area model as a computational tool requires understanding the following:
- numbers can be decomposed into the sum of smaller parts; and
- the distributive property can be used to break down one large multiplication problem into several smaller ones.
Students should keep this in mind before starting the area model.
Consider 3 x 2, or “3 groups of 2”:
3 x 2
“3 groups of 2”
This may very well seem like a simple and or even unnecessary representation, but having a visual representation of the multiplication of the two numbers will always yield a rectangular array is an important concept. The unit not only shows the connections between Number Sense and Numeration and Measurement but also implicitly provides students with insight into how more abstract mathematics works later in grades 9 and 10.
To be able to skip count more fluently, students should continue to practice counting and quantity, which includes unitizing. The use of arrays allows students to continue developing their ability to unitize and work with composing and decomposing numbers.
All models, though different, deal with numeric length and width. It is possible to simplify algebraic expressions by using area models which do not require numeric values. A manipulative called algebra tiles are usually used to build algebraic area models.
Once students get a grasp of the basics, they can progress to more advanced use of the model that serves as a great precursor to encourage students to begin creating their own algorithms and gradually, connect to the standard algorithm.
Multiplying Polynomials using an Area Model
Multiplying polynomials is really about using the distributive process, but very often using it multiple times over each polynomial gets multiplied by everything else it turns out to be a bit tricky.
As you know, a rectangle’s area is equal to its width times its length. Let us illustrate this with an example.
Say we have this rectangle here and we state the top is 7 and the side is 6; the area would equal 7×6 or 42.
The same calculation would apply if we break this rectangle into four pieces, instead of seven we will break it into four and three, which is seven, and on the side here will make it one and five, which equals 42.
This is what we mean. 4×1 gives me this little area, 4×1 is 4, 3×1 is 3, 4×5 will be 20, 3×5 is 15, we divided this into four separate pieces and when we add them up, we will get 42. 4+3 is 7, 20 is 27, and 15 more is 42.
Therefore, we can use this idea of broken rectangles to multiply binomials. Here’s how.
We could do the product x+3 times 2x+1 by writing its rectangle. We are going to call this x+3, 2x+1 is going to be the side lengths, so now when we multiply each of these four things and add them together we will get the same answer.
So x times 2x is 2x squared, 3 times 2x is 6x, x times 1 is x, 3 times 1 is 3, so when we add all those together we will get 2x squared plus, 6x plus x is 7x plus 3 3. That is the answer for this product. This is the same answer you would have gotten using FOIL.
This method makes sure that each term gets multiplied by every other term. The x gets multiplied by 2x and 1 and the 3 gets multiplied by 2x and 1.
This area model is also really really useful when you get into big polynomials,
Area Model of Division
In division problems, the rectangular diagram or model is used to solve the factors or quotients and the divisor determines the rectangle’s length and width.
Let us take the example of a rectangle, where we find the area by multiplying the width and length. So if the length is 32 units and the width of 23 units we determine the area by multiplying 32 by 23. In other words, the product 32 x 23 can be represented geometrically as the area of a rectangle with a length of 32 units and a width of 23 units.
Similarly to this in a division problem, for eg. 555 ÷ 15 can be represented geometrically as the missing dimension of a rectangle with an area of 555 sq. units and having a length of 15 units on one side.
The rectangle can be further divided into smaller rectangles by measuring the length of each smaller rectangle again and again. In order to get the desired length, add these lengths together.
Consider a smaller rectangle of a height of 15 units and a length of 20 units as a starting point. This means the rectangle’s area is 300 square units, and the rest of the rectangle is 555 – 300 or 255 square units.
Now we have a sub-division area of 255 square units. Since 15 x 10 = 150, a new rectangle can be drawn with a height of 15 units and a length of 10 units.
Finally, 15 x 7 = 105 is found. As a result, the rectangle has a height of 15 units and a length of 7 units, totaling 105 square units.
The rectangle’s length is therefore 20 + 10 + 7 units, or 37 units.
To conclude, 555 ÷ 15 = 37.
Area Model Division With Remainders
As already discussed, arrays are objects that are grouped into columns and rows. The columns are vertical and the rows are horizontal. The dividend is the number being divided, and the divisor is how many numbers are in each group. When you divide something, it does not always divide evenly, and some numbers are leftover. These numbers are remainders. When splitting with arrays, you can see the remainder, which will help you visualize the math.
Commonly used Terms
- Distributive property: A property of multiplication that can be used to simplify problems, for example, 6 fours = 5 fours + 1 four or 6 × 4 = (5 × 4) + (1 × 4).
- Long division: A method of solving division problems; also known as the standard algorithm for division.
- Quotient: An answer that is obtained by taking one number and dividing it by another. For example, in
28 ÷ 4 = 7, the number 7 is the quotient.
- Remainder: The number that is left over when a whole number is divided by a whole number, for example, 25 ÷ 6 = 4 with a remainder of 1.
- Standard algorithm: The standard steps that are followed to solve a specific type of problem. For example, the process of long division is a standard algorithm.
Question 1: Find the area using an area model.
First, factor in and substitute the given dimensions into the formula for length and width.
In this case A=(16.2 mm)(2.3 mm)
Then, use the area model representation to find the answer. Change 16.2 and 2.3 to whole numbers and break up the numbers according to place value.
16.2 → 162 → 100 + 60 + 22.3 → 23 → 20 + 3
Next, find the areas of the smaller rectangles.
Add up the areas.
2,000 + 1,200 + 300 + 180 + 40 + 6=3726
Finally, include the decimal point into the sum. Calculate the number of decimal places in 16.2 and 2.3. There are a total of two decimal places. Move the decimal point two places to the left.
A = 37.26 mm2
The area is 37.26 square millimeters.
Question 2: Find the product using an area model.
In the first step, represent 1.5 horizontally and 2.5 vertically on the same area model.
Change 1.5 and 2.5 to quantities of hundredths.
Complete the rectangle by filling in the area.
Then, add up the number of units in each section.
The product of 1.5 times 2.5 is 3.75.
The area model helps students develop a rich understanding of multiplication and division through a variety of problem contexts and methods that elicit multiplicative thinking.
In truth, daily life presents us with various contexts that are multiplicative in nature and teachers must use appropriate strategies and models that resonate with children’s intuitions as they engage in these concepts.
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