Arrays, repeated addition, foundational multiplication
Elementary Foundations • K-5
Multiplication is a faster way to add equal groups. Instead of adding 3 + 3 + 3 + 3, you can multiply: 4 groups of 3, written as 4 x 3 = 12. Let's discover how multiplication works and why it is so useful.
Whenever you have equal groups, you can multiply. The multiplication symbol (x) means "groups of."
We call the numbers being multiplied factors, and the result is the product.
An array is a set of objects arranged in equal rows and columns. Arrays make multiplication visible.
3 rows with 4 in each row:
Count them: 4 + 4 + 4 = 12, or simply 3 x 4 = 12.
Just like addition, multiplication is commutative: 3 x 4 = 4 x 3. If you turn the 3-by-4 array on its side, you get a 4-by-3 array -- still 12 dots! This means you only need to learn half the multiplication table.
| Rule | Example | Why It Works |
|---|---|---|
| Anything x 0 = 0 | 5 x 0 = 0 | Zero groups of anything is nothing |
| Anything x 1 = itself | 7 x 1 = 7 | One group of 7 is just 7 |
| x 2 = double | 6 x 2 = 12 | Two groups is the same as adding to itself |
| x 5 = skip count by 5 | 4 x 5 = 20 | 5, 10, 15, 20 |
| x 10 = add a zero | 3 x 10 = 30 | The digits shift one place left |
Skip count by 3 a total of 6 times:
The 6th number is 18, so 6 x 3 = 18.
If you do not remember 7 x 6, break one factor into easier parts:
Students sometimes confuse multiplication with addition. Remember: 3 x 4 means 3 groups of 4 (which equals 12), not 3 + 4 (which equals 7). If you see the x symbol, think "groups of."
1. Write 5 x 3 as repeated addition and find the product.
5 x 3 = 3 + 3 + 3 + 3 + 3 = 15.
2. Draw an array for 2 x 6 and find the product.
2 rows of 6:
● ● ● ● ● ●
● ● ● ● ● ●
2 x 6 = 12.
3. Find 8 x 5 using skip counting.
Count by 5s eight times: 5, 10, 15, 20, 25, 30, 35, 40. So 8 x 5 = 40.
4. Use breaking apart to solve 8 x 7.
Break 8 into 5 + 3. Then 5 x 7 = 35 and 3 x 7 = 21. Add: 35 + 21 = 56.
5. There are 4 bags with 9 marbles in each bag. How many marbles in all?
4 x 9 = 36 marbles. (You can think: 4 x 10 = 40, minus 4 x 1 = 4, so 40 - 4 = 36.)
Summary: Multiplication is repeated addition of equal groups. Arrays help you see multiplication as rows and columns. The order of factors does not change the product. Key strategies include skip counting, using known facts (x0, x1, x2, x5, x10), and breaking numbers apart.