What Is an Array in Math?

An array is a set of objects arranged in equal rows, and in math it is how children see multiplication: 3 rows of 4 dots is 4 + 4 + 4, which is 3 × 4 = 12. Because every row holds the same number, a child can count the rows instead of every dot. Second graders use arrays to add equal rows, up to 5 rows of 5. Third graders write the multiplication sentence and use arrays to learn their facts, which they are still expected to know from memory by the end of third grade. In computer programming the word array means something else; this guide is about the grade-school one.

The most useful thing to do at home is to count by rows, not by ones: 4, 8, 12. Three free array worksheets with answer keys, from adding the rows to drawing your own arrays, are at the end of this guide.

What is an array in math?

An array is a group of objects arranged in rows and columns, with the same number of objects in every row. A row goes across and a column goes down. Four cookies in one row and three in the next are not an array, because the rows are not equal. An array is named by its rows and then its columns, so “3 by 4” means 3 rows with 4 in each row.

A child already knows plenty of arrays and can count every one of them:

• An egg carton is 2 rows of 6, which makes 12.
• A 12-cup muffin tin is 3 rows of 4.
• A window of 6 panes is often 2 rows of 3.
• A chocolate bar scored into squares is rows of equal pieces.
• Chairs set out for a show stand in rows of the same length.
• A ten frame, from kindergarten math, is 2 rows of 5, which makes 10.

The array is one of the models on your child’s math homework, and one of the few that the Common Core standards name: second grade’s 2.OA.C.4 says “rectangular arrays”. A matrix in algebra or an array in a computer program uses rows and columns for a different job; a second grader’s array is a picture of equal groups.

How do arrays help with multiplication?

An array helps with multiplication because every row is one equal group, so the picture shows the adding, the multiplying, and the shortcuts at once. Take 3 rows of 4 dots through four steps:

1. A row is a group. Each row of 4 is one group, so 3 rows is 3 groups of 4. The Common Core Standards Writing Team’s progressions document reads an array as equal groups “if each row or column is considered as a group”.
2. Add the rows. Adding them gives 4 + 4 + 4 = 12. Writing the running total beside each row, 4, 8, 12, turns the adding into counting by fours, a support the same document describes for arrays up to 5 by 5. Counting by fives on a hundreds chart is the same kind of count.
3. Write it as multiplication. The sentence is 3 × 4 = 12, read “3 groups of 4”. An array for 8 × 4 is 8 rows of 4 dots, 32 in all. In the United States, 3 × 4 usually means 3 groups of 4; the progressions document notes that many other countries read it as 4 groups of 3, and both readings give 12.
4. Turn it, and split it. An array turned a quarter turn swaps its rows and columns, so 3 rows of 4 become 4 rows of 3 and 3 × 4 = 4 × 3, the commutative property. An array cut between two rows breaks a hard fact into two easy ones: 7 × 6 is 5 rows of 6 and 2 rows of 6, so 30 + 12 = 42. That is the distributive property, and third-grade standard 3.OA.B.5 gives its own example, 8 × 7 as 8 × 5 plus 8 × 2, which is 40 + 16 = 56.

The same array answers a division question read the other way. If 12 dots sit in 3 equal rows, each row holds 4, so 12 ÷ 3 = 4, which is how third graders first meet division as sharing (3.OA.A.2).

Draw the dots as squares touching edge to edge, and the array becomes the area of a rectangle: 3 rows of 4 squares cover 12 square units. Third-grade standard 3.MD.C.7 relates that area to multiplying the side lengths. The area model is that picture for bigger numbers. On a number line, three equal jumps of 4 land on 12, the same equal groups shown as distance.

What grade learns arrays?

Children add the rows of arrays in second grade and use arrays to multiply in third grade. Common Core standard 2.OA.C.4 asks second graders to “use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns”, with no × sign. Multiplication starts in third grade with 3.OA.A.1, “interpret products of whole numbers”, and 3.OA.A.3 names arrays in word problems.

Grade	What children do with arrays	Common Core	Texas (TEKS)
Kindergarten–1st grade	Count objects laid out in rows; no array standard	None (K.CC.B.5 counts up to 20 things “arranged in a line, a rectangular array, or a circle”)	None
2nd grade	Add equal rows to find the total, up to 5 rows and 5 columns, and write the sum of equal addends	2.OA.C.4	2.6(A): join equal sets of objects (arrays not named)
3rd grade	Read an array as multiplication, solve problems with arrays, use them to learn facts to 10 × 10, then area	3.OA.A.1, 3.OA.A.3, 3.OA.B.5, 3.OA.C.7, 3.MD.C.7	3.4(D) arrays up to 10 by 10, 3.4(E), 3.4(K), 3.5(B), 3.6(C)

In Texas, second graders join equal groups of real objects, and arrays appear by name in third grade, “arranged in arrays up to 10 by 10” (TEKS grade 3, 3.4(D)). The grade 2 expectation, 2.6(A), is to join “equivalent sets of concrete objects” (TEKS grade 2).

A second grader who adds the rows and has never seen the × sign is doing exactly what second grade asks.

Why do schools teach arrays instead of just memorizing times tables?

Schools teach arrays so the times tables mean something first, and children still memorize them. Common Core standard 3.OA.C.7 ends: “By the end of Grade 3, know from memory all products of two one-digit numbers.” Texas asks third graders to “recall facts to multiply up to 10 by 10 with automaticity” (3.4(F)). The array comes before the memorizing, not instead of it.

The problem	The way most parents learned it	With an array
6 × 7	Recite “six sevens are forty-two”	6 rows of 7. Not known yet? 5 rows of 7 is 35, and one more row of 7 makes 42
7 × 6	A separate fact to memorize	The same array turned: also 42

The Common Core’s writing team says in its progressions document that fluency with these facts is not a matter of “instilling facts divorced from their meanings, but rather the outcome of a carefully designed learning process that heavily involves the interplay of practice and reasoning”. The turnaround also cuts the memorizing: the 100 facts from 1 × 1 to 10 × 10 are only 55 different facts once a child knows that 3 × 8 and 8 × 3 are the same.

The What Works Clearinghouse practice guide Assisting Students Struggling with Mathematics (2021) lists arrays, with strip diagrams, tables and number lines, among the simple visual representations that “may help students organize information”. The progressions document also says third graders can be encouraged to move “as early as possible” from drawing every dot to quicker diagrams. The array is a step on the way, not where multiplication ends.

The array is for meaning and for working out a fact your child has not memorized yet, and practice is for speed; third grade expects both. Why the rest of school math looks different is in new math explained for parents.

How to explain arrays to your child

Explain arrays at home with real objects first, in the order of the steps schools use: rows, then adding, then multiplying, then turning and splitting. Each step takes a few minutes, and a handful of dried beans or an egg carton is enough.

1. 1

   Build one with real things

   Lay out beans, coins, or LEGO bricks in 3 rows of 4 on the table, or open an egg carton. Ask your child to check that every row has the same number in it. If one row is longer, it is not an array yet.

2. 2

   Say it in rows

   Say "3 rows, 4 in each row." Row and column words are hard for children, because 3 rows of 4 has 4 columns, so keep saying "in each row" until your child says it too.

3. 3

   Count by rows

   Point to each row in turn and count 4, 8, 12, writing the running total beside each row. Then write the addition: 4 + 4 + 4 = 12. This is where second graders stop.

4. 4

   Write the multiplication sentence

   For a third grader, write 3 × 4 = 12 under the addition and read it as "3 rows of 4 is 12." Point out that the 3 counts the rows and the 4 counts what is in each row.

5. 5

   Turn it

   Turn the mat a quarter turn so the rows run the other way. Now there are 4 rows of 3. Ask how many there are before your child counts: still 12, so 4 × 3 = 3 × 4.

6. 6

   Split it

   Slide one row a little way from the others: 2 rows of 4 and 1 row of 4 make 8 + 4 = 12. Then build a fact your child does not know yet, such as 7 rows of 6, and split it into 5 rows and 2 rows: 30 + 12 = 42.

Then go on an array hunt. Look around the house for things in equal rows. Have your child say each one as “rows of”: a carton of 2 rows of 6, a window, a tray of cupcakes. Homemade counters, such as buttons or pasta shapes, work as well as anything bought.

Common array mistakes

Four of the five array mistakes below come from not seeing the rows, and each one has a fix you can make on the spot.

• Counting every dot by ones. Researchers led by Michael Battista found that “many students do not see row-by-column structure” in arrays of squares. Count by rows instead: 4, 8, 12.
• Mixing up rows and columns. In 3 rows of 4 there are 4 columns, and the Common Core progressions document calls row and column language “difficult”. Say “4 in each row” until the words stick.
• Counting the corner dot twice. A child who counts a row and then a column counts the shared corner dot in both. The same document says a corner object is counted “as part of a row or as a part of a column but not both”. Count rows only.
• Drawing rows of different lengths. A drawn array with 4 dots in one row and 5 in the next gives a wrong total. Check each row before counting.
• Thinking 3 × 4 and 4 × 3 are different answers. Turn the array a quarter turn and count again: 12 both times.

A study of primary children using arrays for multiplication, by Patrick Barmby and colleagues, found both “ways in which the array representation helped children, and also problems that children had with using the array”. A child who finds arrays hard at first is not unusual, and going back to real objects is the fix.

Array games and activities

Array games need only dice, a deck of cards, or a handful of beans, and each one ends with your child saying the array aloud.

• Roll and build. Roll two dice. The first is the number of rows, the second how many in each row. Build it with beans and say the total: “3 rows of 5 is 15.”
• Array war. Each player turns over two cards from ace (1) to 10, builds or draws that array, and the larger total wins both. A third grader says the multiplication sentence: “6 × 4 is 24.”
• How many ways? Give your child 12 beans and ask how many different arrays they can make: 1 row of 12, 2 rows of 6, 3 rows of 4, and the same three turned. The What Works Clearinghouse guide uses the same activity with 24 counters to teach factors; at home, 12 is enough.
• Square tiles. Lay sticky notes or square tiles edge to edge in rows and count the squares: 2 rows of 5 tiles cover 10 squares. That is the start of area.

Free array worksheets

Our free array worksheets come as three PDFs, one task each, in the order a child uses them: add the rows, write the multiplication, then draw your own.

• Arrays: Add the Rows, for second grade. Arrays from 2 rows of 2 up to 5 rows of 5; under each, your child writes how many rows, how many in each, and the addition.
• Arrays: Write the Multiplication, for third grade. Arrays up to 10 by 10, four to a page, each with its multiplication sentence and the turnaround.
• Draw the Array, for either grade. Your child draws each array on a light dot grid, from words such as “3 rows of 4” and then from a multiplication such as 4 × 6.

Each PDF has an answer key and ends with a page of instructions for grown-ups. The arrays are not numbered, so the key matches answers by position, and every sheet prints in black and white with no account and no email.

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Free printable array worksheets

Three free PDFs in black and white, with no account and no email, in teaching order: add the rows, then write the multiplication, then draw the array. Start with the one your child can already do. Each has an answer key, and every PDF ends with a page of instructions for grown-ups on why, what for, and how to use it.

1. 1Arrays: Add the Rows (PDF)24 arrays · answer key included · 6 pagesDownload
2. 2Arrays: Write the Multiplication (PDF)16 arrays · answer key included · 6 pagesDownload
3. 3Draw the Array (PDF)24 arrays · answer key included · 6 pagesDownload