Using the State Curriculum: Mathematics, Grade 4

The decimal place-value system uses digits to represent any number, no matter how large or how small. In this system, all whole numbers can be expressed as a sum of the products of the digit placeholders and a power of ten. For example, 345 can be written as (3×100) + (4×10) + (5×1). Each place value is ten times the place value to its right.

The decimal place-value system provides a way to represent fractions with denominators of 10, 100, 1000, etc. — all powers of ten The decimal point is an invention to show the position of the unit's or one's place. The place value to the left of the decimal point is always the one's place. The decimal place-value system can be extended to the right to include values less than one. As students develop the concepts of fractions with base-ten denominators using models, each of those base-ten fractions has a place on the place-value chart. Decimal Squares provide excellent concrete models of fractions and decimals:

1. The unit square represents one whole.

   

2. Decimal squares show how the unit square can be divided into 10 tenths.

   

3. It takes 10 tenths to represent 1 whole.

   

4. Decimal squares also model how the unit square can be divided into 100 hundredths.

   

5. It takes 10 hundredths to represent 1 tenth and 100 hundredths to represent 1 whole.

   
   

6. The decimal place-value system is expanded to include tenths and hundredths to the right of the decimal point. Note that the decimal point shows that the unit's or one's place is to the left of the decimal point.
7. This chart shows the place values in decimal notation.

8. Using the decimal squares, would be represented by 7 parts of the decimal square divided into tenths and that is equivalent to 0.7.

   

9. To represent 0.05, show as 5 parts of the decimal square divided into hundredths.

   

10. The blocks provide the opportunity to model numbers in different decimal forms using the place values to the right and left of the decimal point.

    

    one whole	one tenth	one hundredth
    1
    1	.1	.01

    2.35 can be expressed as = 2 + + = 2 + 0.3 + 0.05.

On the place-value chart:

It is important to understand that the base-ten place-value system extends indefinitely in two directions. The study of decimals should always be developed in tandem with fractions. These are not two separate topics. Decimals are another way of writing fractions. Decimals can be thought of as fractions with base-ten denominators. In other words, all the denominators are 10, 100, 1000, etc. All fractions can be renamed in decimal form.

Decimals should also be linked to "friendly" fractions. When representing , think of how 100 would be divided into 4 equal parts. Each part would have 25 parts. Then focus on the 3 parts of the numerator. If you count the number of equal parts, you will find that there are 75 equal parts or 0.75 or seventy-five hundredths.

To read a decimal:

1. Read the whole number part
2. Read the decimal point as and.
3. Read the number to the right of the decimals point as you would a whole number.
4. Read the place value of the last digit.

For example: 0.5

Would be read as zero and 5 tenths.

For example: 96.34

Would be read as ninety-six and thirty-four hundredths.

Decimals can be written in many forms, such as standard form, word form, and expanded form.

Standard form	27.32
Word form	twenty-seven and thirty-two hundredths
Expanded form (decimal)	20 + 7 + 0.3 + 0.02
Expanded form (fraction)	20 + 7 + +

A real world connection to decimals is money. Students can understand that 10 pennies can be traded for a dime. Use this to connect the place value system and decimals. A penny is one tenth of a dime and a hundredth of a dollar. A dime is one tenth of a dollar.