Introduction: Representing Multiplication with an Area Model
Representing multiplication with an area model introduces Grade 5 students to visualizing multi-digit products as the area of a rectangle. To multiply 23 × 347, students decompose 23 into 20 + 3 and 347 into 300 + 40 + 7, then compute six partial products (20×300=6000, 20×40=800, 20×7=140, 3×300=900, 3×40=120, 3×7=21) and sum them. For a 3×3-digit problem like 256×347, there are nine partial products. This Pengi Math (Grade 5) Chapter 2 skill builds the conceptual foundation for the standard multiplication algorithm.
Key Concepts
Property An area model represents a multi digit multiplication problem as the area of a big rectangle. First, break both multi digit numbers into their place value parts. The total area of the big rectangle (which is the final product) is the sum of the areas of all the small rectangles made from these parts. These small areas are called partial products. When you multiply two broken down numbers like $(a 1 + a 2) \times (b 1 + b 2 + b 3)$, you can find the total product by adding up all the partial products: $$(a 1 \times b 1) + (a 1 \times b 2) + (a 1 \times b 3) + (a 2 \times b 1) + (a 2 \times b 2) + (a 2 \times b 3)$$.
Examples To model $\boldsymbol{23 \times 347}$, you draw a rectangle with side lengths $23$ and $347$. Decompose $23$ into $20 + 3$, and decompose $347$ into $300 + 40 + 7$. The partial products are the areas of the smaller rectangles: $20 \times 300 = 6000$, $20 \times 40 = 800$, $20 \times 7 = 140$, $3 \times 300 = 900$, $3 \times 40 = 120$, and $3 \times 7 = 21$.
To model $\boldsymbol{256 \times 347}$, you draw a rectangle with side lengths $256$ and $347$. Decompose $256$ into $200 + 50 + 6$, and decompose $347$ into $300 + 40 + 7$. The partial products are the areas of the smaller rectangles: $200 \times 300 = 60000$, $200 \times 40 = 8000$, $200 \times 7 = 1400$, $50 \times 300 = 15000$, $50 \times 40 = 2000$, $50 \times 7 = 350$, $6 \times 300 = 1800$, $6 \times 40 = 240$, and $6 \times 7 = 42$.
Common Questions
How does an area model work for multiplication?
Draw a rectangle with both factors as its dimensions. Decompose each factor by place value, creating smaller rectangles. Each smaller rectangle's area is a partial product; sum all partial products for the final answer.
How do you multiply 23 × 347 using an area model?
Decompose 23 = 20 + 3 and 347 = 300 + 40 + 7. Calculate six partial products: 6000 + 800 + 140 + 900 + 120 + 21 = 7,981.
What is a partial product in multiplication?
A partial product is the result of multiplying one place-value component of a number by one component of the other number. All partial products are added together to get the total product.
How many partial products does a 2-digit × 3-digit area model have?
Six partial products, since you multiply 2 parts (tens and ones) by 3 parts (hundreds, tens, ones) for a total of 2 × 3 = 6 smaller rectangles.
How does the area model connect to the standard multiplication algorithm?
Each row in the standard algorithm represents a group of partial products summed together. The area model makes these hidden steps visible.
What grade and chapter introduces the area model for multiplication?
Grade 5, Chapter 2: Multi-Digit Multiplication and Division with Place Value in Pengi Math.