## Compare, Order, and Round Whole Numbers

An important application of place value is its use in comparing numbers. When comparing 51,432 and 9,567, students will note that the first number has five digits and the second has four digits. So 51,432 > 9,567. This method is justified because for whole numbers the least 2-digit number (10) is greater than the greatest 1-digit number (9); the least 3-digit number (100) is greater than the greatest 2-digit number (99); the least 4-digit number (1,000) is greater than the greatest 3-digit number (999); and so on.

To compare two whole numbers with the same number of digits, students begin by comparing digits in each place, starting from the left. The number with the greater digit in that place is the greater number. For example, when comparing 612 and 487 students note that 6 > 4. Since the digit in the hundreds place is greater in 612 than in 487, the digits in the tens and ones places will not affect the comparison. So 612 > 487. When comparing 4,371 and 4,358, students note that both numbers have the same digits in the thousands place and in the hundreds place. Since 7 > 5, 4,371 > 4,358. At this grade level, students should also realize that 7 > 5 can be written as 5 < 7, which implies that 4,358 < 4,371.

The concept of place value is also applied when rounding numbers. When rounding a number n to the nearest hundred, students find the number closest to n that has all zeros to the right of the hundreds place. For example, 5,378 rounded to the nearest hundred will be either 5,300 or 5,400. Since 5,400 = 5,378 + 22 and 5,300 = 5,378 − 78, 5,378 is closer to 5,400 than to 5,300. When both differences are equal, the rounding rule calls for choosing the greater value. Thus, 5,378 rounded to the nearest hundred is 5,400.

The number line makes it easy to visualize why 5,378 rounds to 5,400: the point representing 5,378 is closer to the tick mark at 5,400 than to the tick mark at 5,300.

When a number lies halfway between two tick marks, it is rounded to the value represented by the tick mark at its right.

Teaching Model 2.3: Round Two-Digit and Three-Digit Numbers