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Multiplication Arrays

Primary schoolMathematicsAn array of equal rows: rows × columns counts every counter exactly once

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About this experiment

An array is a set of counters laid out in equal rows and columns, like eggs in a box or seats in a cinema. Arrays make multiplication something you can see. Build any array up to 12 × 12, count it row by row in jumps, turn it round to discover that 3 × 4 and 4 × 3 are the same, and split it into two smaller arrays to work out a hard times table from easier ones.

Learning goals

  • Describe an array as equal rows and equal columns
  • Write a multiplication sentence for an array and read it as "rows of"
  • Use repeated addition and counting in steps to find the total
  • Show with a turned array that multiplication can be done in any order
  • Split an array to break a multiplication into two easier ones

How to use it

  1. 1Make an array with 3 rows and 4 columns. Say it out loud: 3 rows of 4.
  2. 2Press Count the rows and count along with the jumps: 4, 8, 12.
  3. 3Choose Turn it and press Turn. How many rows and columns are there now? Did the total change?
  4. 4Make 6 × 7. Choose Split it and split after column 5. Use 6 × 5 and 6 × 2 to find 6 × 7.
  5. 5Look at Other arrays with the same number of counters. Find every array that makes 24.
  6. 6Find a number that can only be made as one long row. What do you notice about it?

The model

Multiplication is repeated addition of equal groups
Turning an array does not change the number of counters (commutative law)
Splitting the columns breaks one multiplication into two smaller ones (distributive law)

Assumptions

  • Rows run across the page and columns run down, so 3 × 4 is read as 3 rows of 4. Some countries read it the other way, as 4 taken 3 times. Both give the same total.
  • Arrays go up to 12 × 12 to cover the times tables learned in primary school.

Try this in class

  1. Q1An egg box has 2 rows of 6. Write two multiplication sentences for it.
  2. Q2Why is 5 × 8 the same as 8 × 5? Use an array to explain.
  3. Q3How could knowing 10 × 7 help you work out 9 × 7?
  4. Q4Which numbers up to 30 can be made as a square array?
  5. Q5A gardener has 18 plants and wants equal rows. List all the ways to plant them.

Frequently asked questions

Why use arrays to teach multiplication?

An array shows equal groups all at once, so children can see the rows to add, count in steps, and discover rules like turning and splitting without having to memorise them first. The same picture later becomes the area model for long multiplication and for algebra.

Is 3 × 4 three fours or four threes?

Conventions differ between countries and textbooks. In this simulation it means 3 rows of 4. What matters most is that children can see both readings give 12, which is exactly what the turn shows.

What are factor pairs?

Two whole numbers that multiply to make a number. Each factor pair of 12, such as 3 and 4, is a different array with 12 counters.