## Operations With Decimals and Powers of Ten: When Students Ask

• How can I remember the names of the different place values?
Most students can remember the first three whole-number place values: ones, tens, and hundreds. Use the place-value chart to point out to students that the pattern of ones, tens, and hundreds is repeated in the thousands, millions, and billions groups. These groups are called periods. For decimal places, have students relate the six decimal places in the chart with the powers of ten: 10-1 means tenths, 10-2 means hundredths, and 10-3 means thousandths. They can relate the exponent to the number of zeros to the written word.
• What is an exponent?
When you multiply 3 and 8, the numbers 3 and 8 are called factors of the product, 24. When a number is multiplied by itself several times, you can make the notation simpler by using an exponent. For example, 10 x 10 x 10 x 10 x 10 can be written 105. In the expression 105, the 10 is called the base and 5 is called the exponent. The exponent tells you how many times the base is used as a factor.

34 means 3 x 3 x 3 x 3.
10² means 10 x 10.
53 means 5 x 5 x 5.

Exponents are often used to write very large or very small numbers. For example, 1011 is easier to write than 100,000,000,000, and 10-9 is easier to write than .0000000001.

• Where do I put the decimal point in a product?
There is one important rule to remember when multiplying decimals: The problem and the answer must have the same number of decimal places. Stated mathematically, the sum of the number of decimal places in the factors must equal the number of decimal places in the product.

Sometimes it is necessary to add zeros as placeholders to place the decimal point correctly in the product.

• Where do I put the decimal point in a quotient?
When you divide by a whole number, place the decimal point in the quotient directly above the decimal point in the dividend.

When you divide by a decimal, first multiply both the divisor and the dividend by a power of 10, so the divisor is a whole number. Then follow the rule above.
Remind students to write their problems neatly and keep the digits in the quotient lined up with the appropriate digits in the dividend. Students who have trouble writing their quotients correctly can practice decimal division on graph paper to help align their problems.

• How do I know if my answer is reasonable?
Students can use estimation to decide if a decimal answer seems reasonable. For example, rounding can be used to find a reasonable estimate of the product of 4.2 x 252.31.

Since 4 x 250 = 1,000, a reasonable answer would be about 1,000. If a student multiplied and got 10,597.02, a quick check of the decimal-point placement would reveal that the product should be 1,059.702

• Is there a visual way to see multiplication of decimals?
Show your students that you can model decimal multiplication with graph paper and colored pencils. Use a 10 x 10 grid to represent a decimal model of 1. To show 0.5 x 0.3, display the following model.

You can use different-colored pencils to represent each decimal factor and product. Five rows of the grid are colored blue to represent 0.5. Three columns of the grid are colored yellow to represent 0.3. The intersection of the colors, the squares that are green, represents the product. Since 15 out of 100 squares are green, the product is or 0.15 .