Multiplying Two-Digit by Two-Digit Numbers

Multiplying two-digit numbers might seem tricky at first, but it becomes easy when you break the numbers down into smaller parts. There are three main ways to solve problems like $34 \times 56$: the area model, partial products, and the standard algorithm.

Method 1: The Area Model

The area model uses a grid to break numbers into their tens and ones (expanded form). Let's solve $34 \times 56$.

1. Break the numbers apart: $34$ becomes $30 + 4$, and $56$ becomes $50 + 6$.
2. Draw a grid: Draw a $2 \times 2$ box. Write $30$ and $4$ across the top, and $50$ and $6$ down the side.
3. Multiply the parts: Fill in each box by multiplying the top number by the side number:
   • $30 \times 50 = 1500$
   • $4 \times 50 = 200$
   • $30 \times 6 = 180$
   • $4 \times 6 = 24$
4. Add them all up: $1500 + 200 + 180 + 24 = 1904$

So, $34 \times 56 = 1904$.

Method 2: Partial Products

This method is very similar to the area model, but you write the numbers stacked vertically instead of drawing a box. You multiply each part of the bottom number by each part of the top number.

Let's do $34 \times 56$ again:

• Multiply the ones: $6 \times 4 = 24$
• Multiply the ones by the tens: $6 \times 30 = 180$
• Multiply the tens by the ones: $50 \times 4 = 200$
• Multiply the tens by the tens: $50 \times 30 = 1500$

Add all the partial products together: $24 + 180 + 200 + 1500 = 1904$

Method 3: The Standard Algorithm

The standard algorithm is the fastest way to multiply on paper. Let's solve $45 \times 23$.

Step 1: Multiply by the ones digit. Multiply $45$ by $3$ (the ones digit of $23$):

• $3 \times 5 = 15$. Write down the $5$ and carry the $1$.
• $3 \times 4 = 12$. Add the carried $1$ to get $13$.
• Your first row is $135$.

Step 2: Multiply by the tens digit. Because the $2$ in $23$ actually stands for $20$, you must put a placeholder zero on the next line before you start multiplying.

• Place a $0$ in the ones column.
• $2 \times 5 = 10$. Write down the $0$ and carry the $1$.
• $2 \times 4 = 8$. Add the carried $1$ to get $9$.
• Your second row is $900$.

Step 3: Add the rows. $135 + 900 = 1035$

So, $45 \times 23 = 1035$.

Example Problem

Let's try one more: $99 \times 11$.

Using the standard algorithm:

1. Multiply $99$ by the $1$ in the ones place: $99$
2. Put down your placeholder zero: $0$
3. Multiply $99$ by the $1$ in the tens place: $990$
4. Add the two rows: $99 + 990 = 1089$

So, $99 \times 11 = 1089$.