HW: page 168 (12-24 evens, 25-28) Extra Credit # 29 & 31

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1 Lesson 5-1 Rational Numbers pages Review our number system and real numbers. Our Number System Real Complex Rational Irrational # Integers # Whole # Natural Rational Numbers the word "rational" comes from ratio, meaning the quotient of 2 #'s fractions are used to name rational numbers A number is called a rational number if and only if it can be represented by a fraction whose numerator is an integer and whose denominator is a nonzero integer. "Q" was chosen to represent the set of rational numbers because "R" is standard for the set of real numbers. "Q" may stand for the quotient of two integers. Review negative fractions and where the sign can go _ Review comparing fractions. Use cross multiplication a/b = c/d if ad = bc fractions are = if ad > bc fractions are > if ad < bc fractions are < Go over graphing fractions and comparing fractions using a number line. (see page 167, examples 4 & 5) Procedure for Graphing a Number on a Number Line: (1) Draw a line. (2) Mark the line with segments of equal length. (3) Label the line with an appropriate unit of measure. (4) Place a dot at the indicated number on the number line. < > /2-1/2 0 1/ /2 2 A number is less than another number when it lies to the left of the other number on a number line; a number is larger than another number when it lies to the right of the other number on a number line HW: page 168 (12-24 evens, 25-28) Extra Credit # 29 & 31

2 Lesson 5-2 Writing Decimals as Fractions pages The decimal system was named from the Latin word for tenth, decimus. Other words that came from decimus: decimeter, decigram, deciliter, December (Old Roman calendar - 10 th month). Two Types of Decimals (1) TERMINATING - can be written as a fraction with a denominator of 10, 100, 1000, etc. (a decimal with a finite number of digits) (2) REPEATING - goes on forever; (nonterminating): a decimal with an infinite number of digits where one or more of the digits repeat in a pattern use ellipsis or bar notation _ Another way 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. The decimal point is used to separate the one's and the tenth's place. How to Read a Decimal 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 hundredthousandths) How to Translate from an English phrase to a decimal 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

3 Practice: (1).057 (2) (3) (4) (5) 2, (6) (7) nine hundred twenty-five thousandths) (8) one hundred twenty-eight and two-tenths (9) two hundred six and twenty-six hundredths (10) thirty-four thousand three hundred seventeen and twenty seven thousandths (11) two hundred four and four thousandths (12) one hundred forty and thirty-eight ten-thousandths How 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.

4 How to change a repeating decimal to a fraction (see page 170, examples 3 & 4) (1) Let a variable equal the number with a repeating decimal. n =.555 (2) Multiply the repeating decimal by a power of ten. 10 if one number repeats 100 if two numbers repeat 1000 if three numbers repeat, etc. Example:.5 Let n represent the repeating decimal n =.5 _ Since one decimal repeats multiply each side of the equation by 10 10n = 5.5 (3) Subtract the smaller equation from the larger equation. _ 10n = 5.5-1n =.5 9n = 5 (4) Solve for n. 9n = 5 9n/9 = 5/9 n = 5/9 HW: page 171 (20-52 evens, 55-63)

5 Lesson 5-3 Rounding Whole Numbers & Decimals pages 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 (up to the decimal point) 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. HW: page 173 (15-22, 30-32) Extra Credit: 34

6 Lesson 5-4 Rationals: Adding & Subtracting Decimals pages (Put an addition and subtraction problem with whole numbers on the board. Have students solve each. Then insert decimal points in each number, in the same place. Ask students if and how the addition and subtraction will be changed.) Do several and ask what conclusions can be drawn? Ask the students if they think it would have mattered if I had not placed the decimal point consistently in the numbers? What do you think? (1) What does the decimal point do in decimal numbers? (it keeps track of the place values of the digits in the numbers) (2) Why is it important that the decimal points be lined up in addition and subtraction? (because only like things can be added or subtracted; the like things here are like units.) (3) When zeros are added to the right of the last decimal digit, how is the decimal changed? (it isn't changed in value, only in appearance) NOTE: if no decimal point is showing, it is after the last digit Solving equations: x = r = t = Simplifying algebraic expressions: x x -9.7y y 62.91a (21.08a a) 2.67b ( )b 0.97e f + 5.7e HW: page 176 (14-36 evens, 38-43, 45) Extra Credit: 46

7 Lesson 5-5 Rationals: Adding & Subtracting Like Fractions pages mixed number: the sum of a whole number and a fraction All fractions must have the same denominator (name) if they are to be added or subtracted. Adding Fractions that have the same denominator ("like fractions"): (1) Find the sum of the numerators (2) Write the sum over the common denominator (remember the denominator is the name) (3) Reduce the answer Subtracting is the same as addition only you find the difference instead of the sum. 5/ /12 = x 5/12 11/12 = a 5/12 + 7/12 = y 5/12 7/12 = b 5/12 + 4/12 = z 5/12 4 /12 = c 6 5/ /12 = d 6 5/ /12 = h 6 5/ /12 = e 6 5/12 4 7/12 = i 6 5/ /12 = f 6 5/ /12 = j 6 5/ /12 = g 6 5/12 9 4/12 = k /9 = p /9 = q 7 5 7/9 = r /9 = s 2¾ - 12¼ + 5¾ = m -8 1/5 6 4/5 3/ /5 = n HW: pages (12-28 evens, 30-33) Extra Credit: 34-36

8 Lesson 5-6 Rationals: Adding & Subtracting Unlike Fractions pages Can you add yards and feet together? Adding Fractions that have different denominators: (1) change fractions to the same denominator (LCD is LCM) (can always find a common denominator by multiplying the two numbers together) (Using LCD is usually easier and less time consuming) (2) Find the sum of the numerator (3) Write the sum over the LCD (4) Reduce the answer Subtracting is the same as addition only you find the difference instead of the sum. HW: page 182 (16-28 evens, 30-35) Extra Credit: 36

9 Lesson 5-7 Solving Equations and Inequalities pages Review: Procedure for Solving Equations: (1) Determine the operation performed on the variable in the equation. (2) Perform the opposite operation on both sides of the equation. (3) Simplify both sides of the equation. (4) Check your answer. Main goal in solving an equation is to get the variable by itself on one side of the equation and a number on the other side. Inequalities with addition or subtraction are done the same way - just remember to put the inequality sign in the answer (such as: x > 5 3/4). x - 2 5/6 = -5 1/6 a /6 > -5 1/ /6 +2 5/6 a /6 > -5 1/6 x = -2 1/3-2 5/6-2 5/6 a > = y < b = y 8.08 < b HW: pages (14-36 evens, 40-42, 49-52) Extra Credit: 53

10 Lesson 5-10 Adding and Subtracting Measures pages What would it be like to have no system of measurement? If we are to measure something, we need a unit of measure. standard unit of measure: one that people have agreed to use length - inches, feet, yards, miles 1 foot (ft) = 12 inches (in) 1 yard (yd) = 3 feet (36 inches) 1 mile (mi) = 1760 yards (5,280 feet) weight - ounces, pounds, tons 1 pound (lb) = 16 ounces (oz) 1 ton (T) = 2,000 pounds liquid volume - ounces, cup, pint, quart, gallon 1 cup (c) = 8 fluid ounces (fl oz) 1 pint (pt) = 2 c (16 fl. oz.) 1 quart (qt) = 2 pints (4 c) (32 fl. oz.) 1 gallon (gal) = 4 qt (8 pt) (16 c) (128 fl.oz.) Is there a pattern in the way that the units are related to one another? No (because of this it has some disadvantages) Converting from one measure to another: (1) from a larger unit to a smaller unit first determine the number of times the smaller unit is contained in the larger unit then multiply the given measure by this number example: 14 yards into feet 3 ft = 1 yd 14 x 3 = 42 feet (2) from a smaller unit to a larger unit first determine the number of times the smaller unit is contained in the larger unit then divide the given measure by this number example: 72 in =? yd 36 in = 1 yd 72/36 = 2 yd Adding measures: Procedure: To add measures, arrange them in vertical columns. Be careful to place like units of measure in the same column. Add each column separately. If possible, simplify the result by converting smaller units to larger ones. 10 lb. 8 oz. 8 yd. 2 ft. + 5 lb. 2 oz. + 5 yd. 2 ft. 15 lb. 10 oz. 13 yd. 4 ft. = 14 yd. 1 ft. Subtracting measure: 1. Place the subtrahend beneath the minuend. Be careful to place like units of measure in the same column. 2. Starting with the column that has the smallest unit of measure, subtract each column separately. 3. When in any column the number in the subtrahend is larger than the number in the minuend, change 1 of the next larger unit in the minuend to its equivalent number of smaller units. 10 lb. = 9 lb. 16 oz. 8 yd. 1 ft. = 7 yd. 4 ft. - 5 lb. 2 oz. - 5 lb. 2 oz. - 5 yd. 2 ft. - 5 yd. 2 ft. 4 lb. 14 oz. 2 yd. 2 ft. HW: pages (6-24 evens, 25-31) Extra Credit: 32

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