## Compare, Order, and Round Whole Numbers and Money

An important application of place value is its use in the comparison of numbers. When comparing 51,432 and 9,567, students will note that the first number has five digits and the second number has four digits. They can thus conclude that 51,432 > 9,567. This method is justified by the fact that 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 with the same place value, starting from the left. The number that has the greater digit in that place is the greater number. For example, when comparing 21,487 and 21,612, students note that the digits in the ten thousands and thousands places are the same but the digits in the hundreds place are different. The fact that 6 > 4 implies that 21,612 > 21,487. At this grade level, students should also realize that 6 > 4 can be written as 4 < 6, which implies that 21,487 < 21,612.

The concept of place value is also applied to rounding. When rounding a number n to the nearest hundred, students determine 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 5,378 = 5,300 + 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 to its right.

Money
Money provides a natural introduction to decimal notation. The fact that the notation for whole numbers can be extended to represent rational numbers has many important consequences. Key to such an extension is the use of a decimal point to the right of the digit in the ones place. The place-value rule that each digit has a place value equal to 10 times that of the digit to its right can be extended to digits to the right of the decimal point and restated as “Each digit has a place value equal to one tenth that of the digit to its left.” This can be represented as follows.

 Ones Tenths Hundredths 1 1 × 1 , or 0.1 × , or 0.01 6 . 0 5

In this chapter, decimals are shown as money amounts.

 Dollars Dimes Pennies 6 0 5
This amount is written as \$6.05.

When writing money amounts, a zero is placed in the hundredths place when there are no hundredths. Thus, three dollars and fifty cents is written as \$3.50 rather than \$3.5.

Comparing money amounts is similar to comparing whole numbers. Starting at the left, the numbers in each place are compared. When different digits occur, the greater digit indicates the greater amount.

\$25.34 > \$15.34
\$25.54 > \$25.34
\$25.34 > \$24.29

Encourage students to compare place values, not coins or bills. For example, suppose students are comparing these two groups of coins.

Students may assume that because 3 quarters > 1 quarter, or the first group has more coins, the first group of coins has a greater total value than the second group. However, the values are \$0.87 and \$0.90, respectively. Since 9 > 8 and \$0.90 > \$0.87, the second group of coins has the greater total value.

The usual method of making change is to start with the amount of the purchase and then count on, starting with the coin of the smallest denomination, until the amount tendered, or given, is reached. The objective is to use the least number of coins and bills possible when making change. This may involve some analysis. Suppose an item that costs \$2.08 is paid for with a \$5 bill. The illustration shows how a student might count on to find the amount of change that is due.

The total amount of change can be found by adding the values of the coins and bills given as change. This amount can also be found by subtracting the total cost from the amount tendered, or given.

Teaching Model 2.1: Compare Numbers