## Money

Because it is based on the decimal system, money can serve as a valuable tool in conveying arithmetic concepts. Conversely, students need arithmetic skills in order to be able to handle money.

Money provides a natural introduction to decimal notation. The fact that the decimal 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 which states that each digit has a place value equal to ten 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.

When arithmetic lessons involve money concepts, students may find it tempting to write “2 nickels = 1 dime.” While this says something about the values of the coins involved, it does not represent a mathematical use of the equal sign. In relating money to mathematics, students must distinguish between a particular coin and the number that represents the coin's value. Once this distinction has been established, the fact that “2 nickels have the same value as 1 dime” corresponds to the addition fact “5 + 5 = 10.”

When using words to express a whole number, students begin by counting the number of digits to determine the place value of the left-most digit. Then they read the value of the number from left to right. For example, 5,703 is read five thousand, seven hundred three. Similar rules apply to finding the value of a mixed collection of coins. Students start with the highest denomination coin and work down to the lowest denomination. Finding the value of a coin collection in cents corresponds to an addition problem in which the addends are 100, 50, 25, 10, 5, and 1. Skip counting can be used to combine the values of several coins of the same denomination.

In making change we sometimes use a process called “counting on,” which is based on every subtraction problem being equivalent to an addition problem. That is,

ab = c is equivalent to a = b + c.

If a \$1 bill is used to pay for an item that costs 83¢, the problem of making change corresponds to the subtraction problem

100 − 83 = 17.

This is equivalent to 100 = 83 + 17. When dealing with money, we may circumvent the subtraction problem by “counting on” from 83 to 100. If only pennies are available, we count on as follows.

83 + 1 = 84; 84 + 1 = 85; 85 + 1 = 86; …; 99 + 1 = 100

Here, 17 successive additions of 1 are required to get from 83 to 100. However, if coins of higher denominations are used, fewer additions will be needed. Another way to make the change 17¢ is

83 + 1 = 84; 84 + 1 = 85; 85 + 5 = 90; 90 + 10 = 100.

The numbers that are added to 83 are 1, 1, 5, and 10, corresponding to two pennies, one nickel, and one dime. As expected, 1 + 1 + 5 + 10 = 17.

Teaching Model 3.3: Make Change