How Many Times? - Multiplication of 2-Digit Numbers

Multiplication as Repeated Addition: Multiplication is a quick way to add the same number multiple times. For example, $4 \times 3$ means adding $4$ three times ($4 + 4 + 4 = 12$). Visually, imagine $3$ baskets, and each basket contains $4$ apples. To find the total, we multiply the number of items in each group by the number of groups.

Multiplying by Tens: When multiplying a number by $10, 20, 30, \dots$, multiply the non-zero digits first and then place a zero at the end of the product. For example, $5 \times 20$ is calculated as $5 \times 2 = 10$, then add the zero to get $100$. Visually, $3 \times 10$ can be seen as $3$ bundles of $10$ sticks each.

The Box Method (Grid Multiplication): This method involves breaking 2-digit numbers into expanded form (tens and ones) and placing them on a grid. To multiply $23 \times 15$, draw a $2 \times 2$ box. Write $20$ and $3$ at the top, and $10$ and $5$ on the side. Calculate the area of each smaller box: $20 \times 10$, $3 \times 10$, $20 \times 5$, and $3 \times 5$. Adding these four products together gives the final answer.

Order Property (Commutative Property): Changing the order of the numbers does not change the result of the multiplication. For example, $6 \times 4 = 24$ and $4 \times 6 = 24$. Visually, an array of dots with $6$ rows and $4$ columns contains the same number of dots as an array with $4$ rows and $6$ columns when rotated.

Multiplication by Zero and One: Any number multiplied by $0$ is always $0$. For example, $15 \times 0 = 0$. Visually, this is like having $15$ empty boxes. Any number multiplied by $1$ stays the same. For example, $25 \times 1 = 25$. Visually, this is like having $1$ box with $25$ items inside.

Patterns in Multiplication: Multiplication tables often follow patterns. For example, in the table of $5$, the products always end in $0$ or $5$. Visually, if you look at a number line and skip-count by $5$, you will always land on numbers ending in these digits.

Doubling: Multiplying a number by $2$ is the same as doubling the number ($n + n$). Visually, imagine looking at a set of objects in a mirror; the total number of objects you see is the original set multiplied by $2$.