An area model is a rectangle used to multiply or divide, with each side broken into place-value parts. To find 23 × 14, split the sides into 20 + 3 and 10 + 4, find the area of each smaller box (200, 30, 80, 12) and add these partial products: 322.
In Spanish
modelo de área
Also called
box method, area model multiplication, partial products model
Split each factor by place value, multiply the parts, and add the four partial products: 23 × 14 = 322.
Formal definition and notation
An area model represents a product a × b as the area of an a by b rectangle. Writing a = a1 + a2 and b = b1 + b2 (usually by place value) splits the rectangle into four smaller rectangles, and by the distributive propertya × b = a1b1 + a1b2 + a2b1 + a2b2. For division, the total area and one side are known, and the model finds the missing side in parts.
Area is the measurement of the space inside a shape. An area model borrows the idea as a tool for multiplying any two numbers.
Quick check: 3 questions
Question 1 of 3
An area model for 54 × 3 has which two boxes?
Show answer and explanation
Answer: A. Split 54 into 50 + 4: boxes 50 × 3 = 150 and 4 × 3 = 12, total 162. B drops place value, C multiplies the parts of 54 together, and D is legal but is not the place-value split.
Question 2 of 3
The four boxes for 32 × 17 are 300, 210, 20 and 14. What is the product?
Show answer and explanation
Answer: C.300 + 210 + 20 + 14 = 544, and 32 × 17 = 544. The other choices are addition slips.
Question 3 of 3
Which expression matches an area model with sides (x + 5) and (x + 2)?
Show answer and explanation
Answer: D. The boxes are x2, 2x, 5x and 10, which add to x2 + 7x + 10. A and B leave out the middle boxes, and C adds the sides instead of multiplying.
Area Model: frequently asked questions
Why do schools teach the area model?
The area model shows where each part of a product comes from. In 23 × 14, students see four separate products, 200, 30, 80 and 12, instead of digits that seem to slide around. That makes place value visible and prepares students for the standard algorithm and, later, for multiplying expressions like (x + 4)(x + 1).
Is the area model the same as the box method?
Yes, mostly. “Box method” usually means the same grid of partial products, often drawn without trying to show the sizes. Some teachers keep “area model” for a sketch where the box sizes roughly match the numbers. Both rely on the distributive property and give the same answer.
Place ValuePlace value is the value of a digit based on its position in a number. In 3,507 the 5 is worth 500, because it sits in the hundreds place. Each place is worth 10 times the place to its right: ones, tens, hundreds, thousands, and after the decimal point, tenths and hundredths.Number System · Grade 3, Grade 4, Grade 5Distributive PropertyThe distributive property says that multiplying a number by a sum gives the same result as multiplying it by each term and then adding: a(b + c) = ab + ac. It multiplies a factor by each term inside parentheses, so 3(x + 4) = 3x + 12. It works with subtraction too, and using it backward is factoring.Algebra · Grade 6, Grade 7, Grade 8DivisionDivision is the operation of splitting a number into equal groups. In 12 ÷ 3 = 4, the dividend 12 is shared into 3 equal groups (the divisor), and each group gets 4 (the quotient). Division undoes multiplication, because 3 × 4 = 12. If the groups do not come out even, the amount left over is the remainder.Operations · Grade 3, Grade 4, Grade 5SideA side is one of the straight line segments that form the outline of a polygon. Two sides meet at each corner, or vertex. A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2PolynomialA polynomial is an expression built from variables and coefficients using only addition, subtraction, and multiplication, with whole-number exponents. Examples are 3x2 + 5x − 7 and x + 9. Its degree is the highest exponent it contains. A variable in a denominator or under a square root usually means the expression is not a polynomial.Algebra · Algebra 1, Algebra 2Standard AlgorithmThe standard algorithm is the usual written column method for adding, subtracting, multiplying or dividing multi-digit numbers. You line up digits by place value, work one place at a time, and regroup (carry or borrow) when a place overflows or runs short. For 478 + 365 it gives 843 in three short steps.Operations · Grade 4, Grade 5, Grade 6SquareA square is a quadrilateral with four equal sides and four right angles. It is a special rectangle and a special rhombus at the same time. For a square with side s, the perimeter is 4s, the area is s2, and each diagonal is s√2.Geometry · Kindergarten, Grade 2, Grade 3LengthLength is how long something is: the distance from one end of an object to the other. You measure it by counting equal units laid end to end with no gaps or overlaps, such as inches, feet, centimeters or meters. A new pencil is about 7 inches long.Measurement · Grade 1, Grade 2, Grade 3ArrayAn array is an arrangement of objects in equal rows and equal columns, forming a rectangle. A 4 by 6 array has 4 rows of 6, so it holds 4 × 6 = 24 objects. Arrays show multiplication as equal groups, and turning one sideways shows that 4 × 6 = 6 × 4.Operations · Grade 2, Grade 3, Grade 4NumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2AreaArea is the amount of surface a flat shape covers, measured in square units. It counts how many unit squares fit inside the shape, which is why the units are squared. A rectangle’s area is length times width, so a 5 by 3 rectangle has area 15 square units.Geometry · Grade 3, Grade 4, Grade 5Long DivisionLong division is a step-by-step written method for dividing large numbers. You work through the dividend one digit at a time from the left, repeating four steps: divide, multiply, subtract, bring down. For 872 ÷ 4, the steps give the quotient 218 with a remainder of 0.Operations · Grade 4, Grade 5, Grade 6BinomialA binomial is a polynomial with exactly two terms, joined by a plus or minus sign. Examples are x + 7, 3m − 4 and a2 − 9. Combine like terms before counting: 3x + 5x is 8x, one term. A polynomial with one term is a monomial; three terms make a trinomial.Algebra · Algebra 1, Algebra 2Expanded FormExpanded form is a way of writing a number as the sum of the values of its digits. Each digit is multiplied by its place value, so 4072 = 4000 + 70 + 2, or 4 × 1000 + 7 × 10 + 2 × 1. It shows what every digit is worth. Zeros add nothing, so they are usually left out.Number System · Grade 2, Grade 3, Grade 4