Ways to Multiply and Divide - Multi-digit Multiplication Algorithms

Multiplication as Repeated Addition: Multiplication is a shortcut for adding the same number multiple times. For example, $5 \\times 3$ means $5 + 5 + 5 = 15$. Visually, this can be represented as an array or a grid of dots with $3$ rows and $5$ columns, totaling $15$ dots.

The Standard Algorithm (Column Method): This is the most common method for multi-digit multiplication. You write the numbers vertically, aligning their place values. You multiply the top number by each digit of the bottom number (the multiplier) starting from the ones place, and then you add the results together.

Place Value and Zero Placeholders: When multiplying by the tens digit of the multiplier, you must place a '0' in the ones column of that partial product line to act as a placeholder. Visually, this shifts the entire row of numbers one position to the left, reflecting that you are multiplying by a multiple of $10$ (e.g., $20$ instead of just $2$).

Area Model (Box Method): This visual approach breaks numbers into their expanded forms (like $24$ into $20 + 4$) and places them along the sides of a grid or 'box'. You multiply the values for each section of the grid and then add all the internal products together. Visually, it looks like a large rectangle divided into four or more smaller rectangular sections, each representing a part of the total product.

Distributive Property: This property states that multiplying a number by a sum is the same as doing each multiplication separately and then adding them: $a \\times (b + c) = (a \\times b) + (a \\times c)$. For example, $8 \\times 102$ can be thought of as $8 \\times (100 + 2)$, which is $800 + 16 = 816$.

Multiplying by Multiples of 10, 100, and 1000: When you multiply a whole number by $10$, $100$, or $1000$, the digits shift to the left, and you append the same number of zeros to the right. For example, $36 \\times 100 = 3600$. On a place value chart, the number $3$ moves from the tens place to the thousands place.

Partial Products: Every time you multiply a digit from the multiplier by the entire multiplicand, you create a 'partial product'. For a $2$-digit multiplier, there will be two partial products that must be added to find the final answer. Visually, these are the separate lines of numbers you see before the final addition step in the column method.