Multiplication Models: Area Models and Partial Products

How do you write a multiplication sentence for a model?

Identify the two factors first. For an array with 3 rows and 4 objects in each row, write 3 × 4 = 12. The factors describe the rows and the objects in each row; the product is the total number of objects. Four rows of 3 also contain 12 objects, although the arrangement is different.

For a rectangle, use its two side lengths as the factors. A rectangle that is 3 units tall and 4 units wide has an area of 12 square units. When a rectangle is divided into smaller parts, add the lengths along each complete side before writing the multiplication sentence. Do not add all the side labels together to find the area.

Area model example: 1,234 × 56

We will use the same place-value idea with a four-digit factor. If this feels like a large first example, begin with two-digit area models, then return to this one.

1. Expand both factors

1,234 = 1,000 + 200 + 30 + 4
56 = 50 + 6

Keep the value of each digit: the 2 represents 200, and the 5 in 56 represents 50. Thinking of them as 2 and 5 would change the calculation.

2. Split the rectangle into rows and columns

Draw four columns for 1,000, 200, 30, and 4. Draw two rows for 50 and 6. Four columns and two rows make 4 × 2 = 8 smaller rectangles. Every column must meet every row.

3. Find the eight partial products

Multiply the row label by the column label for each box. For example, the box where 50 meets 200 contains 50 × 200 = 10,000.

The top row represents 1,234 × 50. The bottom row represents 1,234 × 6. These are partial products because each one accounts for part of the complete rectangle.

Add the partial products and check the answer

Add each row first to keep the calculation organized:

50,000 + 10,000 + 1,500 + 200 = 61,700
6,000 + 1,200 + 180 + 24 = 7,404
61,700 + 7,404 = 69,104

Therefore, 1,234 × 56 = 69,104. The model works because multiplying by 50 + 6 is the same as multiplying by 50 and by 6 separately, then adding. This is the distributive property.

As a rough check, 1,200 × 60 = 72,000. Our exact answer is reasonably close. An estimate can catch a misplaced zero or a missing large partial product, but it does not prove that every calculation is correct.

Common mistakes to check

• Missing a box: check that you have eight products, including 50 × 4 and 6 × 30.
• Losing place value: 50 × 200 is 10,000, not 1,000 or 10.
• Changing a factor: the full problem is 1,234 × 56, not 1,234 × 5.
• Adding labels instead of products: add the numbers inside the boxes to find the product.

Try an area model yourself

Use a four-column, two-row model to calculate 2,103 × 24. Write 2,103 as 2,000 + 100 + 0 + 3, and 24 as 20 + 4. Keep a zero column if it helps you track the places, or leave it out once you understand why its products are zero.

Need a smaller step first? Try one-digit multiplication using partial products. For a wider review of whole-number multiplication and place value, use the free Grade 5 Math Online Center.

Watch the multiplication-model lesson