Decimal Fractions. A certain number is expressed by symbols called digits. How many symbols or digits do we use? 10 (0,1,2,3,4,5,6,7,8,9)

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1 Decimal Fractions Review Place Value A certain number is expressed by symbols called digits. How many symbols or digits do we use? 10 (0,1,2,3,4,5,6,7,8,9) digits: symbols used to write numbers in standard form Naming or recording numbers with symbols is called numeration. Our numeration system is called the "decimal" or "base-10" system because it groups by tens to name numbers. In a place-value system the value of each digit is determined by the place or position that it occupies in a numeral. Each digit in the base-10 system can have many values. Example = 4,444 (4,000)+(400)+(40)+(4) not =16 In the decimal system of numeration, the value of any place is 10 times the value of the place on its right and one-tenth the value of the place on its left. Because of these relationships, each place value in a numeral is called a "power of 10." value of each digit: is determined by multiplying the digit times the value of the place or position it occupies See place value chart on page 22. example: 6,381 value of 6 = or 6,000 value of 3 = or 300 value of 8 = or 80 value of 1 = or 1 Decimal point is used to separate the one's and the tenth's place. Decimal comes from the Latin stem decem, which means 10. (decade, decagon, December-the 10 th month of the Old Roman Calendar) Procedure: when reading a mixed decimal, the decimal point is read "and" first read the whole number, then read the word "and" to show the position of the decimal point, and then read the decimal fraction (eight thousand, nine hundred two and fifty-four thousand, nine-hundred eighty-seven hundred-thousandths)

2 six and twenty-five thousandths Procedure: To write a mixed decimal, write the whole number, place the decimal point after the whole number, and then write the decimal fraction Practice: To give the students practice in reading numbers with decimal points, write a series of digits on the chalkboard. Place a decimal point between any two digits, and have the students read the number. When they have read it correctly, erase the decimal point, place it between two other digits, and have them read the new number. Continue in this manner, forming as many numbers as possible from the digits you have written. Review Expanded Notation expanded notation: a representation of a number as a sum of powers of ten (expanded form) Review: value of a digit: equals the digit times its place value value of a numeral: equals the sum of the values of its digits Examples: 405 = ( ) + ( ) + ( ) = ( ) + ( ) + ( ) + ( ) + (8 1/10 1 ) + (6 1/10 2 ) or 10-1 or 10-2 *********** Another was of writing fractions and mixed numbers is using decimal fractions and mixed decimals which we call "decimals" for short. When we write decimals, we do not use numerators or denominators. The place of a digit tells us the value of the decimal fraction. We use a period (.) to write a decimal. We call this period a decimal point. The digits to the left of the decimal point are whole numbers. The digits to the right of the decimal point are fractions. To change a decimal to a common fraction: (1) The digits of the decimal, without the decimal point, become the numerator. (2) The number of places we have, tells us the denominator. (3) Reduce. To change a fraction (with a denominator of a power of 10) to a decimal: (1) Write down the numerator. (2) To determine the number of decimal places read the denominator.

3 10 - one decimal place 1,000 - three decimal places two decimal places 10,000 - four decimal places Fractions to Decimals Review changing decimals to fractions. Changing a fraction to a decimal: (review if the denominator is a power of ten) If the denominator is not a power of ten: (1) If possible, change fraction to an equivalent fraction with a denominator of a power of ten. 4/5=?/10 4/5 = 8/10 =.8 If the denominator cannot be changed to a power of ten:: A fraction indicates a division problem (the fraction bar means "divided by."). (2) Procedure: divide the denominator into the numerator. (When dividing remember to align the digits in the quotient with the appropriate digits in the dividend.) Mixed numbers: the whole number is still written as a whole number when a mixed number is changed to decimal form example: 5 3/5 = 5.6 Terminating & Repeating Decimals terminating decimal: a decimal with a finite number of digits repeating decimal (nonterminating): a decimal with an infinite number of digits where one or more of the digits repeat in a pattern In a repeating decimal a bar is drawn over the repeating digit or digits. (could also use 3 dots) The division must be carried out far enough to determine where the pattern starts to repeat.

4 Adding & Subtracting Mixed Numbers Adding mixed numbers that contain like fractions: Find the sum of the fractions and add this sum to the total of the whole numbers. 15 1/ /8 17 4/8 = 17 1/2 Adding mixed numbers that contain unlike fractions: Change the fractions to like fractions and follow the above procedure. (Renaming involves only the fractional parts. The whole numbers are not changed.) 3 1/4 = 3 2/8 4 3/8 = 4 3/ /2 = + 5 4/8 12 9/8 = 13 1/8 (The answer is not complete until it is written in lowest terms.) Multiplying Decimals When multiplying with decimals: Multiply the numbers as if they were whole numbers and then place the decimal point. (The number of decimal places in the product should equal the total number of decimal places in the factors.) NOTE: sometimes zeros may need to be inserted in the answer. or "is approximately equal to" Dividing Decimals by whole numbers Dividing a decimal by a whole number is just about the same as dividing a whole number by a whole number. There is one important difference: we write the decimal point in the

5 quotient directly over the decimal point in the dividend. Then we divide as we do with whole number. Sometimes you must annex zeros in order to solve a problem. (Remember writing zeros after a decimal does not change the value of the number.) example: 0.2 / 4 or 3 / Dividing by a Decimal Example:.56.4 To divide, we must ask how many.4's there are in.56. Trying to do this is hard. We need an easier way to divide when the divisor is a decimal. The easier way is to make the divisor into a whole number. Then we can divide as we always do. Procedure: To divide when the divisor is a decimal: (1) Move the decimal point in the divisor to the right until you have a whole number. You must do this in the divisor first. (2) Move the decimal point in the dividend the same number of places you moved the decimal point in the divisor. (3) Divide as you would a problem with the divisor as a whole number. Why we can move the decimal point to the right? = (10/10 is one) Multiplying & Dividing by Powers of 10 Put a decimal on the board and have the students multiply by 10, 100, 1000, etc. and see

6 what the pattern is. Then use 10 1, 10 2, 10 3, etc. Procedure: Multiply Decimals by Powers of 10: (1) Move the decimal point to the right as many places as we have zeros in the multiplier. (or the exponent) (2) When we have to move the decimal point more places than we have digits in the answer, we write a zero for each place we need. Put a decimal on the board and have the students, divide by 10, 100, 1000, etc. and see what the pattern is. Then use 10 1, 10 2, 10 3, etc. Since dividing is the opposite of multiplying, you move the decimal point to the LEFT as many places as we have zeros in the divisor. We write zeros when necessary. Rounding Off of Decimals When you are told to round off to a given place, you must divide to one place more than the place you need. Then you round off according to the rules of rounding off. Procedures for Rounding: (1) Mentally point out the place (named place) to which you are going to round the number. (2) Look at the digit to the right of the named place a) If it is greater than or equal to 5, increase the digit in the named place by 1. ("rounding up") b) If it is less than 5, the named place is not changed. (3) a) If a whole number, replace each digit to the right of the named place with a zero. b) If a decimal place, drop the rest of the digits to the right of the named place.