Using the Area Model to Solve Math Problems

The area model is one way of organizing our place-value calculations while completing a multi-step multiplication problem. Just like any other multiplication “method” or “algorithm,” the area model helps us be methodical and organized to ensure we multiply each place value in one number against each place value in the other number exactly once and no more than once. Thus, the area model will be most helpful in problems that involve multiplying numbers that have more than one digit.

To solve problems with the area model, first identify the multi-digit numbers to multiply. Write each number in expanded form with the ones, tens, hundreds, etc. Set up an area model rectangle and multiply. Finally, check and report your answer.

Identify the multi-digit numbers to be multiplied

Which numbers in the problem have a multiplicative relationship? Will multiplying them get you closer to the answer to the question?

There are actually two parts to this. First is to understand what the question is that you want or need to answer. Can you write a phrase or sentence response to the question in natural language, leaving blanks where the numbers will eventually go? If not, look for the question or the problem you’re trying to solve. Knowing where you’re going is half the battle in making sense of word problems. If you need more guidance with this, check out the section on prewriting answers in my blog post on word problems.

Once you know where you’re going, are there two numbers in the problem that make sense to multiply together and will get you closer to your goal? Is at least one of them more than one digit, and is the other 3 or fewer digits? If so, then the area model is probably a great choice for organizing the multiplication! (The area model will still work with different numbers of digits, but sometimes other tools are easier, as discussed in this post.) But how do you know what numbers should be in multiplication relationships? As my post Add, Subtract, Multiply, Divide in Word Problems? Beyond Key Words discusses, common multiplication relationships are ones that can be viewed as repeated groups of the same size (eg, 3 groups of 4 children each) or as scaling up and scaling down (such as when multiplying by a fraction or percentage, or when dealing with proportions and scaling).

Write each number in expanded form

Expanded form rewrites each digit as the total amount it represents, with addition signs to combine the values back into the whole.

For example, the number 4,672 would be read “four thousand, six hundred seventy-two”. We can follow either those words or the place values to expand 4,672 into $4000+600+70+2$.

What to do about zeros in place values? For some students, the model will make more sense to expand with the zeros in place. This will create either whole columns or whole rows of zeros in the final steps, but children and adults who thrive on making sense of a problem and/or consistency and routine, this may be the way to go. For these individuals, 3,070 would expand to $3000+0+70+0$ or even $3000+000+70+0$. For children and adults who prefer to minimize effort they may see as “wasted,” expanding without the zeros may work well. For these individuals, they might expand 3,070 to $3000+70$ and proceed with the area model.

More details about this and the next step are found in my post Setting Up and Drawing an Area Model.

Set Up the Area Model and Carry Out Computations

If you’re not familiar with the area model, I recommend you check out What Does an Area Model Represent and Setting Up and Drawing an Area Model. If you’re at least a little familiar with the area model, I’ll summarize how to compute with it here.

Start with a rectangle on your page. It should be large enough to split up into smaller boxes that can each hold a readable multi-digit number. Along the top of the rectangle, write the expanded form of your first number, stretching out the spaces between the numbers and addition signs so that the expanded form uses the whole length. Along the left side, write the expanded form of the second number, again spacing it out so that it takes up the whole height of the rectangle.

Just below or beside each addition symbol, draw a line either down or across to split the rectangle into smaller boxes. Each smaller box should have an “address” made up of one number at the top and one along the side.

Multiply the number directly above that smaller box with the number directly to the left of that box, and write the result in the box. Repeat with each of the smaller boxes.

Add all the numbers in the boxes together. Both the multiplying and the adding should be fairly easy if you correctly expanded the numbers according to place values at the beginning.