Notes #1: Operations on Decimals, Converting Decimals to Fractions A. What are Decimals?

Notes #1: Operations on Decimals, Converting Decimals to Fractions A. What are Decimals? Give some examples of decimals: Where do you see decimals outside of math class? When would you need to know how to add, subtract, multiply, or divide decimals? B. Adding and Subtracting Decimals To add or subtract decimals, follow these steps: 1. Align the numbers. Make sure that the are lined up, use as place holders 3. Add/Subtract as you do with integers 4. Make sure your answer has a and that your answer makes sense. Estimate the following sums/differences. Then evaluate. 1.) 14.65 + 9.588.) 14.08 56. Estimate: Estimate: 3.) 3 1.04 Estimate: Solution: Solution: Solution: C. Multiplying Decimals To multiply decimals, follow these steps: 1. Align the numbers, with the shorter number on the. Multiply as you do with integers 3. In the original problem, count the number of digits to the of the decimals points. 4. In your solution, move the decimal this many places to the 5. Make sure your answer has a and that your answer makes sense. 6. Estimate the following products. Then evaluate. 4.) (4.5)(.07) Estimate: 5.) (0.03)(1.64) Estimate: Solution: Solution: 1

D. Dividing Decimals To divide decimals, follow these steps: 1. The number before the division sign, or the number on the numerator of a fraction, is placed the division house.. The number after the division sign, or the number on the denominator of a fraction, is placed to the left of and of the division house. 3. Move decimals to the right in both numbers until the divisor (on the of the house ) is a whole number 4. Bring the decimal up top 5. Divide as you would with whole numbers; don t worry about the decimal point. Estimate the following division problems. Then evaluate. 6.) 33.54 4.3 Estimate: 7.) 40.3 1.4 Estimate: Solution: Solution: Convert the following fractions into decimals by writing as a division problem. 8.) 7 5 9.) 8 6 Estimate: Estimate: Solution: Solution: E. Converting Decimals to Fractions Let s first identify the place values in a decimal:

To convert a decimal to a fraction, follow these steps: 1. Read the number out loud. The number to the left of the decimal point is a whole number.. When you get to the decimal point, say and and start writing your fraction. 3. Read the number to the right of the decimal point and the place value of the final digit. 4. This place value represents the denominator of the fraction (either 10, 100, 1000 etc) 5. As always, reduce your fraction to simplest form. Example: 45.678 is read as forty five and six hundred seventy eight thousandths Convert the following decimals into simplified fractions: 10.) 3.8 11.) 10.06 Notes #: Adding, Subtracting, Multiplying, and Dividing Fractions A. Adding/Subtracting Fractions and Mixed Numbers with Like Denominators To add fractions with like denominators, follow these steps: 1. Add/Subtract the of the fractions (check to see if you have to borrow). Keep the the same 3. Reduce your answer 4. Check to see if you need to carry (if the fraction is greater than ) Add the fractions; leave in simplified form: 7 1 7 1.).) 4 4 4 1 3 3 11 8 8 3.) 5 1 3 1 4.) 1 1 5 7 1 9 9 3

5.) 1 5 7 6.) 6 6 3 15 3 17 17 B. The LCM (Least Common Multiple) of a Set of Numbers The LCM of a set of numbers is the smallest number that all of the given numbers can divide into evenly with no remainder. Find the LCM of each set of numbers: 7.), 3, 6 8.), 4, 5 9.) 8, 1 10.) 6, 10 C. Adding/Subtracting Fractions and Mixed Numbers with Unlike Denominators To add/subtract fractions with unlike denominators, follow these steps: 1. Find the LCM of the denominators. un-reduce each fraction so that it now has this new LCM as a denominator 3. Add/subtract as before (watch for borrowing, carrying, and reducing) Add the fractions; leave in simplified form: 5 7 11.) 1.) 8 1 1 1 3 8 13.) 4 5 7 1 3 14.) 5 1 10 3 10 3 3 4 4

Multiplying and Dividing Fractions A. Multiplying Fractions To multiply fractions, follow these steps: 1. Convert all mixed numbers to improper fractions. Multiply across the and across the (shortcut: cross- first!) 3. Reduce your answer. You may leave the solution as an improper fraction or mixed number Multiply the fractions; leave in simplified form: 5 3 1.).) 3 1 1 10 3 7 4 9 3.) 1 3 B. Dividing Fractions To divide fractions, follow these steps: 1. Convert all mixed numbers to improper fractions. the second fraction over and turn it into a multiplication problem 3. Remember, first! 4. Reduce your answer. You may leave the solution as an improper fraction or mixed number. Divide the fractions; leave in simplified form: 6 1 1 4.) 5.) 3 5 5 3 4 6.) 3 8 15 Notes #3: Sections 1.1 and 1. Section 1.1: Translating Expressions and Writing Rules A. Writing Algebraic Expressions Write whether each expression implies addition, subtraction, multiplication, or division: 1.) sum.) difference 3.) more than 4.) quotient 5.) product 6.) twice 7.) added to 8.) less than 5

9.) total 10.) minus 11.) plus 1.) three times Look for these key words and write an algebraic expression for each phrase: (if no variable is defined, define your own) 13.) the sum of m and 4 14.) twice y added to 17 15.) the quotient of 3 and the sum of x and 5 16.) the product of 9 and 3 less than p 17.) a number divided by 7 18.) three times the sum of 11 and a number Define variables and write an equation to model each situation: 19.) The total cost of gas is the number of gallons times $4.19 0.) The area of a rectangle is equal to the length of the base times the length of the height. 1.) Number Total Pay of Hours $15.00 4 $30.00 6 $45.00 8 $60.00 Section 1.: Evaluating Exponents and Order of Operations A. Evaluating Exponents An exponent is indicates how many times to multiply a number by itself. Evaluate each expression: 1.) 3.) m when m = 8 3.) a 4 when a = 3 B. Order of Operations When there are multiple operations to evaluate, follow this order: P E M D A S 6

Evaluate each expression: 4.) 4 + 3(15 3 ) 5.) 4[4(8 ) + 5] 6.) 14 + 6 3 8 7.) 5 + 4 8 3 8.) 4(8 3) 3 9.) 6 30 1 9 3 C. Evaluating Algebraic Expressions Be sure to follow PEMDAS when evaluating algebraic expressions. When you substitute a value for a variable, use. 10.) a b 5, for a = 1, b= 11.) a b a, for a =, b = 6 3 ( ) 1.) t s 3 for s = 3, t=9 13.) Complete the table: x x 5 3 4 Notes #4: Sections 1.3 and 1.4 A. Classifying Real Numbers We organize Real numbers into the following categories: Real Numbers Rational Numbers Integers Whole Numbers Natural Numbers 7

State the set(s) of numbers to which each number belongs: 1.).5.) -7 3.) 15 3 4.) 0 True or false? If false, provide a counterexample (an example that proves it s false) 5.) All natural numbers are integers. 6.) All integers are natural numbers. B. Comparing Real Numbers If you are comparing fractions, make sure that they have the same denominator first! Compare the numbers using >, <, or = 7.) 1.45 1.44 8.) 4 5 5 6 Order the numbers in each group from least to greatest 3 7 3 5 9.),, 10.),,.7 4 8 3 4 8 Section 1.4: Adding Rational Numbers A. Adding Rational Numbers If the signs are the same, the numbers and keep the the same. If the signs are opposite, the numbers and keep the of the number. *The same rules apply whether you are adding integers, fractions, or decimals* Add. Illustrate the addition for #1-3 on a number line. 1.) -3 + 6.) 7 + (-4) 3.) - + (-6) 8

Add. 4.) - + (-13) 5.) 7 + (-14) 6.) -1. + 31.9 7.) 9 + (-7.8) 8.) -3.4 + (-1.) 9.) 7 1 1 6 10.) 3 1 3 1 11.) 4 1 1 3 1 4 1.) 1 1 1 3 4 13.) a + (-3.4) for a = 9.8 14.) - + [7 + (-3)] 15.) 5 + [-7 + 4] Notes #5: Sections 1.5 and 1.6 A. Subtracting Real Numbers Always turn a subtraction problem into an addition problem:, Then you can use the rules you already know from addition. Convert subtraction into an addition problem. Evaluate. 1.) 14 6.) -18 (-3) 5 3.) (-19) 4.) -6 (-) (-4) 1 + 3 5.) -4. (-3.6) 8. 6.) 3 (-15) 9

7.) 3 1 1 8.) 4 4 7 1 5 3 1 6 4 9.) 7 4 10 15 10.) 1 1 3 4 11.) 1 7 4 5 10 1.) (a b) for a = -4 and b = 7 B. Absolute Value Absolute value describes how far a number is from zero on a number line. A distance is always. Evaluate. 13.) 7 3 14.) 11 4 15.) 8 5 16.) -q r + p for p = -1.5, q = -3, r = C. Multiplying and Dividing Real Numbers (negative) (negative) = (negative) (positive) = (negative) (negative) = (negative) (positive) = Evaluate: 17.) (-3)(5)(-) 18.) (-) 4 19.) - 4 0.) 4 3 ( 5) 3 1.) 3 (-7 + 5) 3.) (a + b) for a = 6, b = -8 10

3.) 4s (-3t) for s = -6, t = - 4.) xy + z, for x = -4, y = 3, z = -3 Notes #6: Sections 1.7 and 1.8 A. The Distributive Property The distributive property describes multiplying across an addition or subtraction problem. Simplify each expression. 1.) (x + 6).) -5(8 b) 3.) (5c 7)(-3) 4.) -(4 b) 5.) 0.4(3m 8) 6.) (4 6 p) 3 B. Combining Like Terms When adding or subtracting algebraic expressions, look for buddies (terms that have the same ). Add/subtract the coefficients but leave the variables the same. Simplify each expression. 7.) 4m 3p (-m) + p 8.) 17k 4w + w (-5k) 9.) 5 x y 1 x 4 y 6 3 5 10.) (3h + ) 4h 11.) 5(t 3) t 1.) 4( f 3 g) 0.5(6 g f) 11

C. Properties of Numbers It is important to know the many rules that you use when you add and multiply numbers. Multiplication Property of Zero: Identity Property of Addition: of Multiplication: Inverse Property of Addition: of Multiplication: Commutative Property of Addition: of Multiplication: Associative Property of Addition: of Multiplication: Name the property that each equation illustrates: 13.) 83 + 6 = 6 + 83 14.) (1)(4y) = 4y 15.) 15x + 15y = 15(x + y) 16.) (8 7) 6 = 8 (7 6) 17.) (8 7) 6 = (7 8) 6 18.) 0 + (-5m) = -5m 19.) 0 = 0 18 0.) -xy + xy = 0 1.) 1.5 1 3 1

D. Algebraic Proof You use many of these properties when simplifying algebraic expressions. Next to each step, name the property illustrated:.) a. 4c + 3( + c) = 4c + 6 + 3c a. b. = 4c + 3c + 6 b. c. = (4c + 3c) + 6 c. d. = (4 + 3)c + 6 d. e. = 7c + 6 e. 3.) a. 8w 4(7 w) = 8w 8 + 4w a. b. = 8w + (-8) + 4w b. c. = 8w + 4w + (-8) c. d. = (8 + 4)w + (-8) d. e. = 1w + (-8) e. f. = 1w 8 f. Notes#7 Review: Solving One-Step Equations A. Using Addition and Subtraction To undo a positive number (or addition), you must do the opposite: To undo a negative number (or subtraction), you must to the opposite: Remember, whatever you do to one side of the equation, you must do to the!! Addition Property of Equality Subtraction Property of Equality Solve each equation. Show your check step where indicated. 1.) x + 7 =.) 5 4 a 3.) n 0.6 = 4 (check) : (check) : (check) : 13

4.) c + 4 = -.5 5.) 1 8 r 6.) 3 3 5 7 x 6 8 7.) 4 1 a 8.) 5 10 3 1 x 8 4 B. Using Multiplication and Divison To undo a coefficient (multiplication), you must do the opposite: To undo a denominator (division), you must to the opposite: For problems involving a variable multiplied by a fraction (i.e. 3 4 x ), multiply by the Remember, whatever you do to one side of the equation, you must do to the!! Multiplication Property of Equality Division Property of Equality Solve each equation. Show your check step where indicated. 1.) 5x = 5.) 3y = -4 3.) 1 4 3 y (check): (check): 14

4.) 5 6 3 x 5.) x x 6 6.) 10 5 m 7.) 1 3 8.) t 6 9.) y 4 4 5 15 (check): (check): 10.) 5 x 10 11.) 6.3x 44.1 1.) 3.1y 1.7 7 14 15

Algebra: Chapter 1 Study Guide HW # 8 Name: Please show ALL of your work (this means no calculators). NO WORK = NO CREDIT Estimate an answer. Then, add/subtract/multiply/divide the decimals. Use your estimate to check your work. 1.) 41.36 73.99.) 8.6 14.7 Estimate: Solution: 3.) (3.5)(0.04) Estimate: Solution: 4.) 39.78.6 Estimate: Solution: Estimate: Solution: Convert as directed: 5.) Convert to a fraction in simplest form:.16 6.) Convert to a decimal: 3 3 8 Add/subtract/multiply/divide the fractions; leave in simplest form. 3 5 3 5 7.) 5 8 6 8.) 7 10 6 9.) 1 3 5 3 10.) 1 4 5 Write an expression for each phrase: 11.) twice the sum of 9 and p 1.) 4.3 less than the product of y and 3 13.) the quotient of m and the difference of w and 14.) the sum of three times j and twice b 16

Write a phrase for each expression: 15.) 3k 6 16.) 4(d + 3) 17.) 5 4 p Simplify each expression: 18.) (37) [4 4 (1 ( 1))] 19.) 1 3[8 (1 5)] 0.) 9 + [4 (10 9) ] 3 Evaluate each expression for a = -1, b =, c = -3: 1.) b a + 3c.) -c (a + b) 3.) 1 c ab Name the set(s) of numbers to which each number belongs: 4.) 4.1 5.) 0 Order the numbers from least to greatest: 5 1 6.),, 6 3 Compare the numbers using >, <, or =: 5 7 7.) 8.) 6 8 1 0.333 3 Simplify each expression: 9.) 6 ( 3) 4 9 30.) 5 ( 3) 4 31.) 4 1 3 5 3.) 64 3 33.) 3 1 4 1 34.) (3a b) 4(b + a) 17

1 4 3 36.) 35.) z 5 1 m3 n( m n) 1 Which property does each equation illustrate? 35.) 3z + 0 = 3z 36.) 3a + (b + c) = (3a + b) + c 37.) 3 3 0 38.) 3 1 3 39.) (x + 3y)(4z) = (4z)(x + 3y) 40.) -3(m n) = -6m + 3n Give a reason to justify each step: 41.) 4[m (m + 3)] = 4[m (4m + 6)] = 4[m + (-4m 6)] = 4[(m + (-4m)) 6] = 4[(-3m) 6] = -1m 4 Complete the table 4.) c 1 3 c Evaluate for x = -, y = 3, z = -1 43.) x xy z 44.) xy 3z 5 18

Solve each equation. 45.) x 8 14 46.) y 7..9 47.) z 4.5.8 48.) 8 m n 49.) 4 50.).5 4 7 c 3 5 51.) k 19.6.3 5.) 1 a 3 6 53.) 1 7 3 x 19