Understanding Multiplication: math lesson, Grade 3 – download the PDF

Imagine you are setting up chairs for a school play. You could count them one by one, but there is a much faster way: multiplication. In this chapter you will learn what multiplication really means, how to picture it with groups, arrays, and number lines, and how to use a few clever rules to multiply with confidence.

1. Equal groups

Multiplication starts with equal groups. Equal groups means every group has the same number of items. Four boxes with three crayons in each box are equal groups. Four boxes with 3, 2, 5, and 3 crayons are not.

In the picture there are 4 groups and 3 dots in each group. We say “4 groups of 3.” To find the total, we can count every dot: there are 12.

2. Repeated addition

Because every group is the same size, you can add the same number again and again. This is called repeated addition.

\[ 3 + 3 + 3 + 3 = 12 \qquad \text{is the same as} \qquad 4 \times 3 = 12 \]

The number you add is the size of one group. The number of times you add it is the number of groups. Multiplication is a shortcut: instead of writing a long addition, you write one short equation.

3. Multiplication equations

A multiplication equation has three important parts. The numbers you multiply are called factors. The answer is called the product.

\[ \underbrace{4}_{\text{factor}} \times \underbrace{3}_{\text{factor}} = \underbrace{12}_{\text{product}} \]

4. Arrays

An array shows objects in equal rows and equal columns. Rows go across, and columns go up and down. Egg cartons, muffin pans, window panes, and the seats in a movie theater are all arrays.

This array has 3 rows with 5 dots in each row. It shows \( 3 \times 5 = 15 \). Each row is one group, so an array is just equal groups lined up neatly.

5. Skip counting and number line models

Skip counting means counting forward by the same number each time. If you skip count by 4, you say 4, 8, 12, 16, and so on. Each number you say is one more group of 4.

You can also show multiplication on a number line. Start at 0. Make equal jumps, one jump for each group. The number of jumps is the number of groups, and the length of each jump is the size of a group. Where you land is the product.

Here, 3 jumps of 4 land on 12, so \( 3 \times 4 = 12 \).

6. The commutative property

Look at these two arrays. One has 3 rows of 5, and the other has 5 rows of 3. If you turn the first array on its side, you get the second one. Both show 15 dots.

This is a big help. If you know \( 2 \times 9 = 18 \), then you also know \( 9 \times 2 = 18 \). You only need to learn about half of the facts.

7. Multiplying by 0 and by 1

Two special numbers make multiplication easy.

Why? \( 8 \times 1 \) means 8 groups with 1 item in each group, so there are 8 items. And \( 8 \times 0 \) means 8 groups with nothing in them, so there is nothing at all. By the commutative property, \( 1 \times 8 = 8 \) and \( 0 \times 8 = 0 \) too.

8. Choosing the best model

You now have many ways to think about the same product. Take \( 4 \times 6 = 24 \):

• Equal groups: 4 baskets with 6 apples in each basket.
• Repeated addition: \( 6 + 6 + 6 + 6 = 24 \).
• Array: 4 rows of 6 chairs.
• Skip counting: 6, 12, 18, 24.
• Number line: 4 jumps of 6 from 0 to 24.

Pick the model that makes the problem clearest. Use an array for things in rows, a number line for jumps and distances, and equal groups for items in bags, boxes, or plates.

Key takeaways

• Multiplication finds the total of equal groups: \( 4 \times 3 \) means 4 groups of 3.
• It is a shortcut for repeated addition: \( 3 + 3 + 3 + 3 = 4 \times 3 \).
• The numbers multiplied are factors, and the answer is the product.
• An array has equal rows and columns. A number line uses equal jumps from 0.
• Skip counting by the group size gives the product.
• Commutative property: \( a \times b = b \times a \).
• Any number times 1 is itself. Any number times 0 is 0.

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