Comparing and       Ordering Fractions,       Mixed Numbers, and       Decimals   Introducing the Concept   Developing the Concept       Adding and       Subtracting Decimals   Introducing the Concept   Developing the Concept

## Comparing and Ordering Fractions, Mixed Numbers, and Decimals

Decimal numbers are another way to write fractions or mixed numbers. You can calculate with decimals as you do with whole numbers because they follow the same place-value rules as whole numbers. A digit in any place has a value 10 times the value of the digit one place to the right.

The 1 in the hundreds place has a value of 100—ten times the value of the 1 in the tens place, which has a value of 10. The 1 in the ones place has a value of 1—ten times the value of the 1 in the tenths place, which has a value of 0.1, or . The 1 in the hundredths place has a value of 0.01, or .

Students may have trouble reading a decimal number. The biggest confusion seems to come with the use of the word and. It's best to reserve this word to denote the decimal point, trying not to use it when reading a whole number.

Read 125 as "one hundred twenty-five."

Read 100.25 as "one hundred and twenty-five hundredths."

Read 0.12 as "twelve hundredths."

It's also common to hear the decimal point read as point, followed by a listing of the digits in order: zero point one two. Try not to do this when you're teaching students about decimals and fractions as it undermines your efforts to establish the relationship between the two forms.

A good model for decimal numbers is a square grid. Use 10 10 grids to represent one whole. Each unit in the grid represents 0.01 and each column in the grid represents 0.10 or 0.1. If students fill in the grids always starting at the top left and working top to bottom, left to right, they can use them to compare and order any decimal numbers.

This grid represents one whole (1) or 100 of 100 equal parts, .

This grid represents 1 hundredth (0.01). As a fraction, 1 part of 100 equal parts is .

This grid represents 10 hundredths or 1 tenth (0.10 = 0.1). As a fraction, 10 parts of 100 equal parts is , or .

These grids represent 1 and 25 hundredths, 1.25, and 1, or 1.

Students will soon see that they don't need to depend on the grids to compare decimal numbers because they can compare and order decimals just as they compare and order whole numbers: Line up the digits. Begin in the greatest place. Find the place where the digits are different. Compare. Just as 8 ones is greater than 6 ones, 8 tenths is greater than 6 tenths. The only time this method might lead to confusion is when the decimal numbers, like 0.15 and 0.6, don't have the same number of digits. If you try to align the digits on the right, you might think that 0.15 is greater than 0.6. It's extremely important for students to understand that when they line up whole numbers for comparison or computation, they're lining up the ones places. When they compare decimals, if they always line up the decimal points, they won't get confused. A place-value chart can help them keep things straight. Students who still have difficulty can use zeros as placeholders.

As with whole numbers, start comparing the digits in the greatest place. Both numbers have the same number of ones, so compare tenths. Since 6 tenths is greater than 1 tenth, 0.6 > 0.15.

A number line is also a good tool for decimals, and mixed numbers.

This model is especially useful when comparing decimals and a mix of fractions with different denominators. As you move from left to right on a number line, numbers—whether they're fractions, decimals, or mixed numbers—increase or become greater. As you move from right to left, numbers decrease or become less.

Another method is to use a place-value chart. For example, order these numbers from least to greatest: 1.40, 1, 1.

First, change the fractions to decimals. Write the decimals in hundredths. To compare, begin in the greatest place. Compare the ones. They are the same. Compare the tenths: 5 tenths are greater than 4 tenths, 4 tenths are greater than 2 tenths, so the final order from least to greatest is 1, 1.40, 1.

Because decimal numbers follow similar rules as whole numbers, you add and subtract decimal numbers like whole numbers. To be sure the place values line up, first line up the decimal points. You may wish to write zeros as placeholders if needed. Then add or subtract as you would with whole numbers. Write the decimal point in the answer. Estimate to check addition; add to check subtraction.

Rounding and estimating with decimal numbers follow similar procedures to rounding and estimating with whole numbers. When rounding, find the place you want to round to. Look at the digit to the right. Round as you do with whole numbers. Rounding to the nearest whole number:

15.5 rounds to 16
15.4 rounds to 15

Rounding to the nearest tenth:

1.55 rounds to 1.6
1.54 rounds to 1.5

To estimate sums and differences with decimal numbers, have students first round each decimal to the nearest whole number and then add the rounded numbers.

• To add 1.235 to 3.74, round to the nearest whole number: 1.23 + 3.74 is about 1 + 4. Then add: 1 + 4 = 5.
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