Multiplication arrays

What a multiplication array is

An array is objects arranged in equal rows. Three rows of 4 dots is a picture of 3 × 4, and the total, 12, is the product. Arrays turn multiplication from a memorized fact into something you can see and count, which is why teachers use them before and alongside times tables.

On this page you drag the corner of the grid or use the steppers to set any number of rows and columns from 1 to 12. A readout shows the rows, the columns and the total as the array grows. Choose dots for early counting, or squares when you want to connect multiplication to area.

Why arrays help

Arrays make three big ideas obvious. First, multiplication is repeated addition of equal groups: 4 rows of 6 is 6 + 6 + 6 + 6. Second, order does not matter: turn 3 rows of 7 on its side and it becomes 7 rows of 3 with the same number of dots. Third, a big array can be split into smaller ones, which is the idea behind mental strategies and, later, long multiplication.

That third idea is the one children use most. A child who does not know 7 × 8 can build it, then picture it as 5 × 8 and 2 × 8 stacked together: 40 + 16 = 56. Common Core standard 2.OA.C.4 starts this work in grade 2, asking children to find the total in arrays of up to 5 rows and 5 columns using repeated addition.

Ideas by grade

• Grade 2: arrays up to 5 by 5. Count by rows using repeated addition, such as 4 + 4 + 4 for 3 rows of 4.
• Grade 3: times tables to 10 by 10. Build each fact and its turnaround, and use known facts to find new ones.
• Grade 4: factors. Find all the arrays for a number like 36. A number whose only array is a single row, such as 7 or 11, is prime.
• Grade 5: link to area. Use squares to show that the area of a rectangle is length times width.
• Times tables to 12 by 12: the 11 and 12 tables are easier when 12 rows of 7 is pictured as 10 rows of 7 plus 2 rows of 7.

A 10-minute arrays session

• Choose one times table the child is working on, such as the 6s.
• Build 1 row of 6, then 2 rows, then 3, and say the totals aloud: 6, 12, 18. That is skip counting by 6.
• Ask for a harder fact, such as 6 × 7. Build 5 rows of 7 first and ask how many more rows are needed.
• Turn the array: build 7 rows of 6 and confirm the total is the same.
• Write the two facts the child found, 6 × 7 = 42 and 7 × 6 = 42, and ask them to say how the array proves it.

Short and regular beats long and rare. Ten minutes a few times a week, focused on one table, is far more useful than an hour on all of them at once.

From arrays to fluent facts

Arrays build understanding, and fluency comes from practice. Eurekami's Flashcards include spaced review so facts come back just before they fade, and the free game demos let a child practice tables with no sign-up. Lessons on multiplication draw arrays live from the numbers in each question.