*What multiplication arrays are, why they’re so helpful for teaching the concept of multiplication, and how to use a multiplication array to help your child master the times tables. Includes a free printable multiplication array that you can use to model multiplication problems from 1×1 up to 10×10. *

Hands-on manipulatives help children make sense of math. They make math concrete and help children understand math at a deeper level.

But, when it comes to memorizing the multiplication facts, hands-on materials can become pretty cumbersome. After all, counting out 8 groups of 7 may take your whole math lesson! And, all that time spent counting won’t really help your child remember that 8 times 7 equals 56, either.

That’s why multiplication arrays are so helpful. Instead of counting out piles and piles of little plastic objects, you can model multiplication problems by simply sliding a piece of paper. Even better, multiplication arrays provide an simple visual model that not only helps kids understand the concept of multiplication–**it helps kids memorize the times tables faster**, too.

## What’s a multiplication array?

An array is any arrangement in rows or columns. Cards laid out into rows to play Memory, seats arranged in rows for a recital, or numbers arranged in an Excel spreadsheet are all examples of arrays.

A multiplication array is simply an arrangement of rows or columns that matches a multiplication equation. You can make arrays out of objects or pictures, and you can use any sort of shape. For example, here are 3 different arrays that all show 3 × 4.

(Generally, the first number refers to the number of rows, and the second number refers to the number of columns. So, all of the above arrays are considered 3 × 4 arrays, not 4 × 3, even though the total number (12) would be the same either way.)

## What are the benefits of a multiplication array?

### 1. Multiplication arrays make it easy to visualize multiplication problems.

Hands-on objects are great for introducing multiplication, but they can be a bit of a pain when you’re doing a lot of problems or working with larger numbers. With a paper dot array, you can slide an L-shaped cover over the top of the array and show any multiplication fact you want from 1×1 up to 10×10. Here’s what the dot array and L-cover look like.

Here’s how you use them. For example, let’s say we wanted to help your child understand what 6 × 8 looks like. 6 × 8 means “6 groups of 8,” so slide the L-cover so that the dot array looks like this.

Each of the 6 rows has 8 dots, so there are 6 groups of 8 dots. So, the total number of dots in the array is the answer to 6 × 8. There are 48 dots showing, so 6 × 8 = 48.

### 2. Multiplication arrays help kids use strategies–not rote memorization–to find answers.

Let’s take 6 × 8. It’s one of the toughest facts for kids to memorize, but most children find it quite easy when they use 5 × 8 as a stepping stone.

Here’s how you teach your child to do this:

5 × 8 is 40. (5 × 8 makes a nice stepping stone, since children’s familiarity with 5s from their early years of arithmetic usually make the ×5 facts easy to learn.)

6 × 8 is just one more group of 8 than 5 × 8.

So, you can just add 40 + 8 to find the answer: 6 × 8 = 48.

This strategy works for all of the ×6 facts. And the good news is that there are similar strategies for all the multiplication facts!

### 3. Multiplication arrays make it easy for kids to see the commutative property in action.

The commutative property says that you can multiply numbers in any order and still get the same answer. For example, 2 × 7 and 7 ×2 both equal the same answer: 14.

To show your child this fact, use the dot array to show a multiplication fact. Then, turn the array 90 degrees. Now, the dot array shows the related multiplication fact–but the total number of dots didn’t change.

Very helpful so much.

This is wonderful. Thank you

Hi Kate,

I was wondering if you have any thoughts on how RightStart teaches multiplication as arrays. It teaches the opposite, that 3 x 4 should be represented as 4 rows of 3 apples. This seems different from the way that every other program teaches. I understand Dr. Cotter’s reasoning, but it bothers me that it deviates from the generally agreed on way of teaching arrays. Do you recommend teaching arrays the way you have them here, or sticking with RightStart if that’s the program we’re using? I don’t know how long we will be homeschooling so I worry about confusion if we transition to public school down the road.

Hi Grace,

Like you, I respect Dr. Cotter’s reasoning, but I find it a curious decision. (In Third Grade Math with Confidence, I’ll be using the more usual definition.) But I don’t think it’s anything to worry about, either. The beauty of arrays is that they show children that you can interchange the factors and still get the same product, so I’d suggest simply emphasizing that rather than the specific RS definition. Even if your child switches to a bricks-and-mortar school next year, I don’t expect it would cause much confusion. Kids are pretty adaptable, and any standardized test wouldn’t assume that kids know one definition rather than another for defining arrays.

Happy Math!

Kate

Thank you for your quick reply! Truthfully I wish I could switch her to Math With Confidence. I like that it has fewer manipulatives, that it’s mastery-oriented, and that it still has the strong focus on conceptual understanding. She is in 2nd grade now so she is right in between what is available currently and what will be released in the spring. Is there any possible way to get my hands on the 2nd grade level early?

Wish I could get it to you, but I’m not able to distribute it while it’s in production. Sorry that the timing doesn’t work out for your daughter!

That makes sense. I just wanted to share with you that since I left these comments I purchased First Grade Math With Confidence for my younger daughter and it’s going great! She is having a blast with math and it’s so parent-friendly. Thank you for writing such an awesome curriculum!

Glad to hear that it’s going well, Grace!

Hello Kate,

Thank you so much! I have a 8 yr old High Functioning Autistic child and this method is AWESOME and he’s getting it!! Thank you, thank you very much for what you are doing to help others. Can’t wait to get my PDF by email.

That’s wonderful that it’s clicking for him! Thanks for taking the time to let me know, and happy math!

Kate

Thank you for sharing…very helpful

thank you so much

Thank you