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What Is an Array in Math?
An array is a set of objects arranged in equal rows and equal columns. Four rows of six counters is a 4 by 6 array holding 24 counters, and the total is always rows times columns. This guide covers the definition, how an array turns multiplication and division into something you can count, why rotating one proves 4 × 6 = 6 × 4, the way a number’s possible arrays are exactly its factor pairs, and how an array differs from a matrix and an area model.
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What is an array in math?
An array is a set of objects arranged in equal rows and equal columns. Four rows of six counters is a 4 by 6 array, and it holds 24 counters.
The word describes the arrangement, not the objects. Dots, counters, squares and tiles all make arrays; what matters is that every row has the same number of items and every column does too. A ragged arrangement is not an array.
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A 4 × 6 array. Four rows, six columns, 24 dots. Rows are named first by convention, so this is “four by six” and not “six by four”.
Arrays are taught early because they turn multiplication from a rule to be memorised into a picture that can be counted. The total is the number of rows times the number of columns, and you can check it by counting the dots.
How an array shows multiplication
Every array states a multiplication fact twice over, once by rows and once by columns.
- Count the rows. Four.
- Count the columns. Six.
- Multiply. 4 × 6 = 24.
- Turn the array a quarter turn and it becomes six rows of four. Still 24 dots; not one has been added or removed.
That last step is a proof rather than a demonstration. Rotating the picture cannot change how many dots are in it, so 4 × 6 and 6 × 4 must be equal. This is the commutative property of multiplication, and the array is the reason it is obvious rather than something to take on trust.
The same picture also explains division. Twenty-four dots dealt into four equal rows gives six per row, so 24 ÷ 4 = 6. The array does not change; only the question asked of it does.
Arrays and factors are the same list
Ask how many different arrays a number can make and you have asked for its factor pairs.
| Array | Total | Factor pair |
|---|---|---|
| 1 × 24 | 24 | 1 and 24 |
| 2 × 12 | 24 | 2 and 12 |
| 3 × 8 | 24 | 3 and 8 |
| 4 × 6 | 24 | 4 and 6 |
Four arrays, four factor pairs, and the same four read backwards give the rotations. Listing arrays and listing factors are one activity, which is why classrooms use counters for both. The guides to factors and to products cover the same relationship in words rather than dots. The factor calculator produces the same list without the counters.
This gives a picture of what a prime number is. Seven objects can be arranged as 1 × 7 and in no other way, because no other whole numbers multiply to 7. A prime is a number with only the one boring array. A square number is the opposite kind of special case: 16 makes a 4 × 4 array, and being able to form a genuine square is what the name records.
Array vs matrix vs area model
Three closely related ideas that arrive at different points in a curriculum.
| What it holds | What you do with it | |
|---|---|---|
| Array | Objects, dots or squares | Count them; read off a multiplication fact |
| Area model | A rectangle, not separate objects | Multiply length by width, including decimals |
| Matrix | Numbers, in named positions | Add, multiply and transform with them |
An array is the concrete version, an area model is the continuous version, and a matrix is the algebraic version. A 2 × 3 matrix has two rows and three columns and so holds six entries, counted exactly as an array of dots would be. The matrix calculator operates on that grown-up form.
In programming the word means something related but looser: an array there is an ordered list, and a list of lists is called a two-dimensional array precisely because it is laid out in rows and columns like this one. Same picture, different subject.
Worked examples
- Find the total. An array has 4 rows and 6 columns. Total is 4 × 6 = 24.
- Find a missing side. An array of 24 objects has 3 rows. Columns are 24 ÷ 3 = 8.
- Count the possible arrays. For 24 there are four distinct shapes: 1×24, 2×12, 3×8, 4×6. For 7 there is only 1×7, because 7 is prime.
- Recognise a square. 16 objects make a 4 × 4 array, so 16 is a square number.
The second of those is the one that carries forward. Being handed a total and one side and asked for the other is division dressed as arrangement, and it is the same question a rectangle area problem asks later on.
Common mistakes
- Counting one by one. The point of an array is that you do not have to. Rows times columns.
- Naming columns before rows. The convention is rows first, so a 4 by 6 array has four rows.
- Calling an uneven arrangement an array. Every row must hold the same number, and so must every column.
- Treating 4 × 6 and 6 × 4 as different totals. They are the same array turned a quarter turn.
- Confusing an array with a matrix. An array holds objects to be counted; a matrix holds numbers to be operated on.
Quick summary
- An array is objects in equal rows and equal columns.
- Rows are named first: a 4 by 6 array has 4 rows, 6 columns and 24 objects.
- Total = rows × columns, which is why an array models multiplication.
- Rotating an array proves 4 × 6 = 6 × 4.
- The arrays a number can form are exactly its factor pairs; a prime has only one.
- A matrix is the same layout holding numbers instead of objects.
The factor calculator lists every factor pair, which is the same list as every array a number can form, and the matrix calculator works with the algebraic version of the same grid.