Number Sense and Operations - Place Value to Millions

Understanding the Millions Period: Place value is organized into periods of three digits separated by commas. For numbers up to millions, we have the ones period (ones, tens, hundreds), the thousands period (thousands, ten thousands, hundred thousands), and the millions period. Imagine a table where each column header represents a value, such as $1,000,000$ for the leftmost column, followed by $100,000$, $10,000$, $1,000$, $100$, $10$, and $1$.

The Base-10 Relationship: Each place value position is exactly $10$ times greater than the position to its immediate right. For example, $1$ ten thousand is equal to $10$ thousands. Visually, you can picture this as a group of $10$ small cubes being bundled together to form a rod, and $10$ rods forming a flat square, continuing up to a large cube representing $1,000,000$.

Standard, Word, and Expanded Forms: Numbers can be represented in three ways. Standard form uses digits ($5,230,400$), word form uses language (five million, two hundred thirty thousand, four hundred), and expanded form breaks the number down by the value of each digit ($5,000,000 + 200,000 + 30,000 + 400$).

Digit Value vs. Place Value: The place value is the position of a digit (e.g., the hundred thousands place), whereas the value is the result of the digit multiplied by its place (e.g., $4$ in that place has a value of $400,000$). Visually, this is like a seat in a stadium having a specific label, but the 'value' depends on who is sitting in it.

Comparing and Ordering Numbers: To compare large numbers, align them by their place values starting from the highest period (millions). If the digits in the highest place are the same, move one position to the right until you find a difference. We use the symbols $>$ (greater than), $<$ (less than), and $=$ (equal to). A number line can help visualize this, where numbers further to the right are always larger.

Rounding to the Nearest Million or Hundred Thousand: To round a number, identify the target place value and look at the digit to its immediate right. If that digit is $5$ or greater, 'round up' by adding $1$ to the target digit; if it is $4$ or less, the target digit stays the same. All digits to the right of the target place become zeros. Imagine a hill where the numbers $0, 1, 2, 3, 4$ are on the left slope and $5, 6, 7, 8, 9$ are on the right slope sliding forward to the next ten.