What are multiplication arrays?
An array is a set of objects arranged in equal rows: every row has the same number. Three rows of 4 dots is an array for 3 × 4. Arrays are how most children first see why multiplication works, because you can count the dots one by one, add the rows (4 + 4 + 4 = 12), or multiply (3 × 4 = 12), and all three give the same answer.
Arrays also show two rules that make the times tables half as long. Turn the array a quarter turn and 3 rows of 4 become 4 rows of 3, still 12 dots, so 3 × 4 = 4 × 3. And the same picture read the other way is division: 12 dots shared into 3 equal rows gives 4 in each row, so 12 ÷ 3 = 4. The Turn it button in the array maker shows the first rule, and the facts list under the array shows both.
How to teach multiplication with arrays
- Build real arrays first. Egg cartons, muffin trays, chairs in rows and window panes are all arrays. Ask how many rows and how many in each row.
- Add the rows. Second graders write the total as repeated addition: 5 + 5 + 5 for three rows of 5. Skip counting by the row size is the same idea; see our skip counting charts.
- Write the multiplication sentence. Rows first, then the number in each row: 3 rows of 5 is 3 × 5 = 15.
- Draw arrays for facts. Drawing 6 × 7 as an array makes a hard fact countable and shows it is 6 × 5 plus 6 × 2.
- Read arrays as division. Once children know the array, ask how many are in each row when the total is shared out.
Array worksheets for 2nd and 3rd grade
| Grade | Level | Sheets | Common Core |
|---|---|---|---|
| 2nd grade | Up to 5 × 5 | Repeated addition, write the sentence | 2.OA.C.4 (arrays up to 5 rows and 5 columns) |
| 3rd grade | Up to 5 rows of 10, then 10 × 10 | Write the sentence, draw the array, division arrays | 3.OA.A.1, 3.OA.A.3, 3.OA.B.5 |
Of the multiplication array worksheets 2nd grade classes use, the repeated addition sheets come first: each problem shows an array with a blank for every row and for the total, then the same total as a multiplication. Array worksheets 3rd grade classes move to writing the multiplication sentence straight from the array, drawing an array for a fact such as 4 × 7, and division array worksheets where the total is shared into equal rows. The 10 × 10 level fits multiplication arrays grade 3 practice before the times tables are memorized.
Every sheet is new each time you press New worksheet, and no array repeats on a page. Choose dots, stars, squares or hearts, color or black and white, US Letter or A4. These printable multiplication arrays worksheets print on one page, with an answer key that fills every blank in red. For the times tables themselves, print a multiplication chart, and for addition facts try fact family houses.
How we tested this tool
We checked the tool by script in Google Chrome on Windows across 2,304 sheets: all four sheet types, all three levels, 4 to 10 problems, all four pictures, with and without answers, in color and black and white, on Letter and A4. In 14,976 problems with 274,176 pictures drawn, every array has exactly the rows and columns asked for, stays inside its box and clear of the writing lines, and no array repeats on a page; on the answer keys all 2,418 red answers sit on their blanks and match the right numbers (rows, row size, product, or the number in each row). In the array maker we built every array from 1 × 1 to 12 × 12 and checked the picture count and the multiplication and division facts. Testing caught three problems we fixed: a long title ran into the web address on A4, the numbers in the fill-in sentences could touch the signs next to them, and at 12 problems per page the dots shrank to about 2 mm, so the levels now stop at 10 problems (6 for 10 × 10) and the smallest dot is 3.3 mm. We printed sheets to PDF with Microsoft Edge (one US Letter portrait page) and tested the page at a 375 px phone width.
Frequently asked questions
What is an array in math?
A set of objects in equal rows, such as 3 rows of 4 dots. The rows times the number in each row gives the total: 3 × 4 = 12. Arrays are the usual first picture of multiplication in 2nd and 3rd grade.
How do arrays show repeated addition?
Each row is one group, so you add the rows: 3 rows of 4 is 4 + 4 + 4 = 12. The repeated addition sheet has a blank for every row and for the total, then the same total as a multiplication.
How do arrays show division?
Read the array the other way: 12 dots shared into 3 equal rows gives 4 in each row, so 12 ÷ 3 = 4. The division array worksheets give the total and the number of rows and ask how many are in each row.
Why is 3 × 4 the same as 4 × 3?
Turn a 3 by 4 array a quarter turn and it becomes 4 rows of 3 with the same 12 dots. This is the commutative property, and it means children only need to learn about half of the times tables.
Which level should I choose?
For 2nd grade use Up to 5 × 5, which matches the Common Core limit of 5 rows and 5 columns. For 3rd grade use Up to 5 rows of 10, then Up to 10 × 10.
Is there an answer key?
Yes. Set Answers to Show and print again: every blank is filled in red, and Draw the array sheets show the array drawn in.