An 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.
Read an array by rows or by columns: 4 sixes and 6 fours are both 24. The rows must all be the same length.
Formal definition and notation
An r by c rectangular array is an arrangement of objects in r rows and c columns, with exactly c objects in every row and r in every column. It contains r × c objects. Each row models an equal group, so the array represents the product r × c; reading by columns represents c × r. (In computing and in matrices, “array” also names an ordered table of values.)
Symbol
Read it as
Meaning
r × c
r by c
an array with r rows and c columns; rows are usually said first
Key facts
Every row has the same number of objects, and every column has the same number.
Rows go across; columns go up and down. A 3 by 5 array has 3 rows and 5 columns.
Rotating an array swaps rows and columns but keeps the total, which shows the commutative property.
The different arrays you can make with a number of objects match its factor pairs: 12 objects make 1 by 12, 2 by 6 and 3 by 4 arrays.
A number of objects that forms only a 1-row array (besides turning it) is a prime number.
Splitting an array into two smaller arrays shows the distributive property: 4 × 6 = 4 × 4 + 4 × 2.
Try it: build an array
Interactive
Each green square is one object in the array. Drag the corner to build 3 by 8, then 4 by 6 and 2 by 12: all hold 24. The area readout is the total number of objects.
This interactive needs JavaScript. The worked examples below show the same ideas step by step.
Array examples, step by step
Example 1: Count an array
Warm-up
A muffin tray has 3 rows with 4 muffins in each row. How many muffins are there?
The tray is a 3 by 4 array.
Add the rows: 4 + 4 + 4 = 12.
Or multiply: 3 × 4 = 12.
Answer: There are 12 muffins.
Example 2: All arrays for 18
Core
List every rectangular array you can make with 18 chairs (count 2 by 9 and 9 by 2 as one shape).
Find factor pairs of 18: 1 × 18, 2 × 9, 3 × 6.
4 and 5 do not divide 18; after 3 the pairs repeat.
So there are 3 array shapes.
Answer: 3 shapes: 1 by 18, 2 by 9 and 3 by 6.
Example 3: Split an array
Test-style
A garden has 7 rows of 13 plants. Split the array into a 7 by 10 part and a 7 by 3 part to find the total.
7 by 10 part: 7 × 10 = 70.
7 by 3 part: 7 × 3 = 21.
Total: 70 + 21 = 91.
Answer: The garden has 91 plants.
Common mistakes
✗ MistakeCalling 3 rows of 5, 5 and 4 dots a 3 by 5 array of 15.
✓ InsteadIt has 14 dots and unequal rows, so it is not an array.
An array shows separate objects you can count. An area model is a solid rectangle split by place value, used for bigger products like 23 × 14.
Arrayvs.Matrix
A matrix is a rectangular table of numbers used in algebra; an array in elementary math is a picture of equal rows of objects.
Quick check: 3 questions
Question 1 of 3
How many objects are in a 5 by 8 array?
Show answer and explanation
Answer: B. 5 rows of 8 is 5 × 8 = 40. 13 adds the sides, 58 just writes the digits together, and 45 is 5 × 9.
Question 2 of 3
Which number of tiles can be arranged ONLY as a single row (or column)?
Show answer and explanation
Answer: D. 13 is prime, so its only array is 1 by 13. 15 makes 3 by 5, 21 makes 3 by 7, and 9 makes 3 by 3.
Question 3 of 3
Which equation matches an array with 6 rows of 3?
Show answer and explanation
Answer: B. Six rows of 3 is six addends of 3, which is 18. 6 + 3 adds the dimensions, and the others use 6 as the row size.
Array: frequently asked questions
What is an array in multiplication?
In multiplication, an array is a picture of a product: objects in equal rows and columns. The number of rows and the number in each row are the factors, and the total is the product. Eggs in a carton (2 rows of 6) show 2 × 6 = 12. Arrays help students see why multiplication works and why order does not matter.
How do arrays help with division?
Division asks for a missing side of an array. To find 28 ÷ 4, arrange 28 objects in 4 equal rows and count how many land in each row: 7. If objects are left over and cannot complete a row, the leftovers are the remainder, as with 30 objects in 4 rows: 7 in each row, 2 left.
MultiplicationMultiplication is the operation that combines equal groups to find a total. The fact 4 × 6 = 24 means 4 groups of 6, or 6 + 6 + 6 + 6. The numbers you multiply are called factors, and the result is the product. Order does not matter: 6 × 4 is also 24.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 2Commutative PropertyThe commutative property says that changing the order of the numbers does not change the result of adding or multiplying them: a + b = b + a and a × b = b × a. For example, 4 + 9 = 9 + 4 and 3 × 5 = 5 × 3. Subtraction and division are not commutative.Operations · Grade 1, Grade 3, Grade 6Factor PairA factor pair is two whole numbers that multiply to give a certain number. The factor pairs of 12 are 1 × 12, 2 × 6 and 3 × 4. Listing factor pairs is a sure way to find every factor of a number: start at 1 and stop once the pairs begin to repeat.Number System · Grade 4, Grade 5, Grade 6Prime NumberA prime number is a whole number greater than 1 with exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11 and 13. The number 1 is not prime, and 2 is the only even prime number. Whole numbers greater than 1 that are not prime are composite.Number System · Grade 4, Grade 5, Grade 6Distributive 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 8LengthLength 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 3Area ModelAn 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.Operations · Grade 3, Grade 4, Grade 5Place 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 5MatrixA matrix is a rectangular array of numbers arranged in rows and columns, written inside square brackets. A matrix with m rows and n columns has dimensions m × n, read “m by n”. Each number is an entry, named by its row and then its column: a23 is in row 2, column 3. The plural is matrices.Algebra · Algebra 2, PrecalculusDivisionDivision 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 2RemainderA remainder is the amount left over when one whole number does not divide another evenly. In 23 ÷ 5, four groups of 5 make 20 and 3 are left, so 23 ÷ 5 = 4 R 3. The remainder is always smaller than the divisor. A remainder of 0 means the division is exact.Operations · Grade 4, Grade 5, Grade 6Repeated AdditionRepeated addition is adding the same number again and again. It is one way to understand multiplication: 3 + 3 + 3 + 3 = 4 × 3 = 12. The number of times you add is one factor, and the number being added is the other. It only works when all the addends are equal.Operations · Grade 2, Grade 3AreaArea 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 5Square NumberA square number is the result of multiplying a whole number by itself. The first square numbers are 1, 4, 9, 16, 25 and 36, because 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9 and so on. That many dots fill a square exactly, like 16 dots in a 4 by 4 square.Operations · Grade 3, Grade 4, Grade 5ProductThe product is the answer you get when you multiply two or more numbers. In 4 × 6 = 24, the product is 24, and 4 and 6 are the factors. In word problems, the words “product of” always mean multiply, so “the product of 3 and 7” is 21.Operations · Grade 3, Grade 4, Grade 5