Multiplication - Multiplication by 2 and 3-digit numbers

Multiplication is the process of repeated addition. For example, $15 \times 3$ is the same as $15 + 15 + 15$. In the equation $24 \times 5 = 120$, $24$ is called the Multiplicand, $5$ is the Multiplier, and $120$ is the Product. Visually, this is arranged in vertical columns where the multiplier is placed under the multiplicand.

When multiplying by $10, 100,$ or $1000$, we use a shortcut. Simply write the multiplicand and append the number of zeros present in the multiplier. For instance, $45 \times 100 = 4500$. In a place value chart, this looks like shifting the digits of the original number to the left and filling the empty units and tens places with zeros.

The Standard Algorithm for 2 and 3-digit multiplication involves 'Partial Products'. When multiplying $123 \times 25$, you first multiply $123$ by $5$ (ones place) and then by $20$ (tens place). These two results are called partial products and are added together at the end. Visually, these are written in separate rows below the main problem.

The 'Place Holder Zero' is crucial when multiplying by the tens or hundreds digit. When you move to multiply by the tens digit of the multiplier, you must place a $0$ in the ones column of that row because you are actually multiplying by a multiple of $10$. If multiplying by the hundreds digit, you place two zeros ($00$) as placeholders.

Regrouping or 'Carrying Over' occurs when the product of digits in one column exceeds $9$. The tens digit of that product is carried over to the next place value column (to the left) and is added after the next multiplication step. Visually, these carried numbers are written in a smaller font size above the multiplicand digits.

The Distributive Property allows us to break down large numbers to make multiplication easier. For example, $35 \times 12$ can be visually represented as $35 \times (10 + 2)$, which equals $(35 \times 10) + (35 \times 2)$. This can be imagined as splitting a large grid into two smaller, manageable sections.

Properties of Multiplication include the Commutative Property (changing the order of numbers does not change the product: $12 \times 5 = 5 \times 12$) and the Zero Property (any number multiplied by $0$ is $0$). Visually, the commutative property is like rotating an array of dots by 90 degrees; the total number of dots remains the same.