How to teach multiplication: groups, arrays, then facts

How to teach multiplication is mostly a question of order: start with equal groups of real things, connect them to repeated addition, skip counting and arrays, and leave the facts for last. Three plates with 4 crackers each make 4 + 4 + 4 = 12, which we write as 3 × 4 = 12.

How to teach multiplication: what are equal groups?

Multiplication finds the total when every group has the same number in it. Put 3 plates on the table and 4 crackers on each plate. How many crackers? Your child can count them all and get 12, and that is multiplication already, before any times sign.

The Grade 3 standards describe it this way: interpret 5 × 7 as “the total number of objects in 5 groups of 7 objects each.” So the first number says how many groups and the second says how many in each. Read 3 × 4 as “3 groups of 4.”

Here are the steps to try.

1. Make groups together. “Put 4 crackers on each of 3 plates.” Let your child do the placing.
2. Ask for the total before counting: “How many crackers do you think there are?” Then count all of them.
3. Change one thing. Keep 3 plates and put 5 on each, or keep 4 on each and use 2 plates.
4. Say the sentence each time: “3 groups of 4 is 12.”
5. Only then show how to write it: 3 × 4 = 12.

Use dry cereal or counters if crackers run out, and keep small items away from children under three.

How is multiplication like repeated addition?

Repeated addition means adding the same number again and again: 4 + 4 + 4 = 12. It is a bridge from adding, which children already do. Say it out loud while you point at each plate: “4, and 4 more is 8, and 4 more is 12.”

The Grade 2 standards build toward multiplication this way. They ask children to use addition to find the total in a rectangular array, and to “write an equation to express the total as a sum of equal addends.” In England, Year 2 children solve problems using materials, arrays, repeated addition and mental methods. Repeated addition gets slow with big numbers, which is a good reason to move on to skip counting and facts later. Our guide to how to teach addition covers the adding that this relies on.

How does skip counting lead to multiplication?

Skip counting is repeated addition said as a chant. Touch each plate and count by 4: “4, 8, 12.” The last number you say is the total, and the number of touches is the number of groups.

The Grade 2 standards ask children to skip-count by 5s, 10s and 100s. Counting by 5s over 4 hands gives 5, 10, 15, 20, which is 4 groups of 5. Our skip counting guide shows a hundred chart and ways to teach the 2s, 5s and 10s.

What is an array in math?

An array is objects in equal rows and equal columns. A muffin tin, an egg carton and a chocolate bar are arrays, and so are the windows on many buildings. An array shows a multiplication fact as a picture your child can count.

1. Build an array with 12 counters: 3 rows of 4.
2. Ask, “How many in each row?” and “How many rows?”
3. Find the total by adding the rows: 4 + 4 + 4.
4. Build 4 rows of 3 from the same 12 counters, then 2 rows of 6 and 6 rows of 2. The total stays 12 each time.

The Grade 2 standards use arrays of up to 5 rows and up to 5 columns, so most of your first arrays can stay that small. A long array such as 2 rows of 6 is fine once 3 by 4 feels easy.

Why is 3 × 4 the same as 4 × 3?

Turn the array a quarter turn. 3 rows of 4 becomes 4 rows of 3, and the counters have not changed. The Grade 3 standards use this as an example: if 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. England’s Year 2 requirements say children should “show that multiplication of 2 numbers can be done in any order.”

This cuts down the number of facts to learn: if your child knows 2 × 7 = 14, they also know 7 × 2 = 14. Let them find this out by turning the array before you tell them.

Which facts should we learn first: 2s, 5s or 10s?

Start with the 2s, 5s and 10s. England’s Year 2 requirements name these three tables, and they match the skip counts by 2s, 5s and 10s. Here is what each looks like.

• 10s: 4 × 10 is 4 tens, 40. Link it to place value: 4 groups of ten make forty. Our guide to place value shows the bundles.
• 5s: count by 5s, or think of fingers and nickels. 6 × 5 is 5, 10, 15, 20, 25, 30.
• 2s: doubles. 2 × 7 is 7 + 7. Pairs of socks and shoes help, and so does our guide to even and odd numbers.

Practice in short bursts, mixed in with building arrays and groups. Ask “how do you know?” more often than “what is it?”

What comes after the 2s, 5s and 10s?

Build the other facts from the ones your child knows. The Grade 3 standards give an example: knowing 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 40 + 16 = 56. Children can split a harder fact into two easy ones, and an array can be cut into two smaller arrays to show it. The standards also describe division as an unknown-factor problem: find 32 ÷ 8 by asking what number makes 32 when multiplied by 8.

When do schools teach multiplication?

In the US, Grade 2 builds the foundations: odd and even numbers and rectangular arrays. Grade 3 introduces multiplication itself, with word problems in situations involving equal groups and arrays, and fluency within 100. The standards say that by the end of Grade 3 children should know all products of two one-digit numbers from memory. In England, Year 1 children solve one-step multiplication and division problems using concrete objects, pictures and arrays, with the teacher’s support, and Year 2 children recall the 2, 5 and 10 tables.

Our curriculum pages put some of this earlier than Common Core does. The first grade curriculum page (ages 6 to 7) lists arrays, repeated addition and multiplying and dividing by 2, 5 and 10, and the second grade curriculum page (ages 7 to 8) lists the 3s, 4s and 8s. Children vary, so go by what your child can do with groups and arrays, not by the grade.

Which games help with multiplication?

Equal Groups is a free game with three ways to play, for ages 6 to 8. In Share, your child deals cookies onto plates so each plate has the same number, which is division. In Groups, they fill plates with the number asked for, like 4 plates of 3. In Arrays, they tap squares to make rows and columns, like 3 rows of 5. If they build 5 rows of 3 instead, the game says so and asks them to turn it around, which is a good moment to ask why the total is the same. Each round uses 2 to 5 plates (or rows) of 2 to 5 cookies, so the totals go up to 5 × 5 = 25. After building, your child picks how many: on each plate, or in all. More games for this age are on the second grade games page.

If your child finds equal groups hard after many weeks of calm practice, talk to their teacher.