Making Sense of Multiplication

The idea in one sentence

Multiplication is a fast way to count equal groups.

Picture an array

For $4 \times 3$, draw 4 rows with 3 dots in each row:

\[3 + 3 + 3 + 3 = 12\]

The array shows why $4 \times 3 = 12$. It also shows why $3 \times 4$ has the same total: turn the array sideways.

Break a hard fact into easy facts

Suppose $7 \times 8$ is hard to remember. Split 7 into 5 and 2:

\[7 \times 8 = (5 \times 8) + (2 \times 8) = 40 + 16 = 56\]

This is the distributive property, but the name is less important than the move: break one group into friendlier chunks.

Useful patterns

• Multiplying by 2 means doubling.
• Multiplying by 5 gives an answer ending in 0 or 5.
• Multiplying by 10 shifts every digit one place left and adds a zero for whole numbers.
• Multiplying by 9 is the same as multiplying by 10 and subtracting one group.

Common mix-ups

• Counting uneven groups as multiplication. The groups must be equal.
• Forgetting what each number represents. In 4 rows of 3, 4 counts the rows and 3 counts each row.
• Believing order changes the answer. It changes the picture, but not the total.

Check yourself

Use a friendly split to solve $6 \times 7$.

One way: $(5 \times 7) + (1 \times 7) = 35 + 7 = 42$.