Arithmetic and Algebra Worksheets, Study notes of Algebra

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Essentials to Mathematics Arithmetic and Algebra Worksheets Shirleen Luttrell 2012 circle.adventist.org Luttrell 2012 2 Contents Chapter 1 Number System ............................................................................................................................................ 5 1a- Translating Mathematical Symbols ..................................................................................................................... 6 1b – Number Systems ................................................................................................................................................ 7 1c - Number Systems ................................................................................................................................................. 8 1d - Which Number is Bigger? .................................................................................................................................. 9 1e-Adding & Subtracting Integers ........................................................................................................................... 10 1f- more Adding & Subtracting Integers ................................................................................................................. 11 1g-Multiplying & Dividing Integers ........................................................................................................................ 12 1h-Expanding Numbers ........................................................................................................................................... 13 Test REVIEW: Integers ........................................................................................................................................... 14 Chapter 1 Test .......................................................................................................................................................... 15 Chapter 2 Fractions ...................................................................................................................................................... 16 2a-Finding Fractions ................................................................................................................................................ 17 2b-Proper and Improper Fractions ........................................................................................................................... 18 2c-Adding & Subtracting Fractions of Different Denominators .............................................................................. 19 2d-Multiplying & Dividing Fractions ...................................................................................................................... 20 2e-Adding Fractions of Like Denominator .............................................................................................................. 21 2f - Like Terms ........................................................................................................................................................ 22 2g-Solving Basic Algebraic Equations (one step) ................................................................................................... 23 2h-Solving Basic Algebraic Equations (two steps).................................................................................................. 24 2i-More Solving Basic Algebraic Equations ........................................................................................................... 25 2j-Solving for a Variable (multiple steps) ............................................................................................................... 26 2k-Solving for a Variable (Multiple Like Terms) .................................................................................................... 27 2L-More Practice Solving Equations ....................................................................................................................... 28 2m-Basic Algebraic Equations with Perimeter ........................................................................................................ 29 2n-Basic Algebraic Equations with Area ................................................................................................................. 30 2o-Basic Algebraic Equations with Fractions .......................................................................................................... 31 2p-Basic Algebraic Equations with Fractions .......................................................................................................... 32 2q-Basic Algebraic Equations with Fractions .......................................................................................................... 33 2r- More Fractions with Perimeter and Area ........................................................................................................... 34 Chapter 2 Test .......................................................................................................................................................... 35 Chapter 3 Decimals ..................................................................................................................................................... 37 3a-Decimal Notation ................................................................................................................................................ 38 3b-Switching Between Fractions and Decimals ...................................................................................................... 39 3c- Operations with Decimals.................................................................................................................................. 40 3d-Adding and Subtracting Decimals ...................................................................................................................... 41 3e-Multiplying & Dividing Decimals ...................................................................................................................... 42 3f-Decimals with Area and Perimeter ..................................................................................................................... 43 3g-Multiplying & Dividing Decimals ...................................................................................................................... 44 3h-More Multiplying & Dividing Decimals ............................................................................................................ 45 3i-Circumference with Decimals ............................................................................................................................. 46 3j-Area with Decimals ............................................................................................................................................. 47 3k-Surface Area & Volume with Decimals ............................................................................................................. 48 3L- Similar Shapes and their Surface Area & Volume ............................................................................................ 49 3m-Surface Area & Volume of Pyramids & Spheres .............................................................................................. 50 3n-Mixed Review of Shapes and Objects ................................................................................................................ 51 3o-Mixed Review of Volume & Surface Area ........................................................................................................ 52 3p-Mixed Review of Volume & Surface Area ........................................................................................................ 53 Chapter 3 Test .......................................................................................................................................................... 54 Luttrell 2012 5 Chapter 1 Number System Prior Skills: • Convert fractions to decimal for sheet 1c • Time measurements for sheet 1c • Basic understanding of decimal and fractions for sheet 1d Luttrell 2012 6 Name: ______________________ Date: _____ 1a- Translating Mathematical Symbols For each question, translate the equation and then solve by mental math. No calculator! Example: 3x = 21. Translation: Three times a number is 21. Answer: x = 7 1. x - 4 = 13 2. x + 5 = 8 3. 8 - x = 5 4. 4x = 12 5. 2x = 6 6. T + 7 = 10 7. 14 - t = 5 8. 21 - x = 13 9. Y ÷ 3 = 6 10. 9 ÷ P = 1 11. 8×P = 32 12. 6 × R = 54 Luttrell 2012 7 Name: ______________________ Date: _____ 1b – Number Systems Complex Numbers - All numbers are complex. Their form is a + bi. These numbers will be taught later! Real Numbers – numbers found on the “number line”. If written as a complex number, they would look like a+0i. Imaginary numbers - points not on the standard number line. If written as complex, they would have form 0+bi. Zero - It is both real and imaginary. Rational Numbers – Real numbers that can be expressed as a ratio of two integers. If written as a decimal, they would be terminating or repeating. Irrational Numbers - reals that CANNOT be expressed as a ratio of integers. If written as a decimal, they would be nonterminating and nonrepeating decimals. Transcendental Numbers - irrational numbers that can NOT be solved by algebraic methods Integers - whole numbers and their opposites Non-integers - another name for a reduced fraction where 1 is NOT in the denominator. Whole numbers - 0, 1, 2, 3… Natural Numbers (counting numbers) - 1, 2, 3… Digits - whole numbers from 0 to 9, those numbers which make up our numerals Even - integers divisible by 2 Odd - integers that are NOT divisible by 2 Positive - reals greater than 0 Negative - reals less than 0 Answer the following about numbers: 1. On a separate piece of paper, create a hierarchy for the number systems above. For each branch, list three examples of the number system. 2. Which of the following is not a rational number? 3.1 3.01 3.111... 3.1234322344523... 3½ 3. Which of the following is not a rational number? 3.4 -3.4 3.444... -3.444 3.040040004... 4. Which is not an integer? 2 -2 0 ½ 4 2 5. What type of number is this: (rational, irrational, integer, real...) A. -3.4 B. 5 C. 12 D. 0 6. Explain which decimals are rational numbers? How can you tell them from an irrational number? Luttrell 2012 10 Name: ______________________ Date: _____ 1e-Adding & Subtracting Integers Perform the operations without a calculator. Show work by plotting the operations on a number line. 1. There are several ways to add or subtract integers. Some think of money debts, others think of protons versus electrons. The following example is showing how addition is about gaining and subtraction about losing in terms of the real number line. -4 + 9 7 - 5 -3 - 3 1 - 7 1 + 3 = 5 = 2 = -6 = -6 = 4 -4 0 1 2 3 4 5 -6 -5 -4 -3 1 2 3 4 Simplify the following by doing the indicated operation: 2. - 4 - 9 3. -7 – 5 4. -3 + 3 5. -1 – 7 6. 1 - 3 7. -5 – 9 8. 8 – 3 9. 53 – 42 10. 31 – 82 11. - 44 + 53 12. - 35 + 35 13. 23 – 17 14. 2 – 4 – 6 15. 2 + (- 4) – 6 16. 0 – 2 + 6 Luttrell 2012 11 Name: ______________________ Date: _____ 1f- more Adding & Subtracting Integers You may use your calculator only to check your answers. Simplify the expressions. Find the result. 1. 7 + 3 2. -6 – 3 3. -8 – 6 4. 6 + (-3) 5. (+6) + (-3) 6. -7 + (-8) 7. - 4 + (+2) 8. 4 + (-2) 9. 5 – 8 10. -78 – 21 11. -32 – 21 12. -55 – 44 13. 34 – 43 14. -34 + 68 15. 54 – 59 16. -90 + 90 17. 3 – 6 18. –4 + 5 19. 4 – 5 20. 6 – 5 21. 7 – 17 22. 10 – 15 23. 0 – 1 24. -3 + 4 25. -14 + 25 26. -5 + 10 27. -1 + 8 28. -7 + 23 29. 8 – 3 30. 3 – 6 31. 10 – 6 32. 4 – 7 33. -1 + 3 34. -10 + 6 35. -7 + 4 36. -7 + 8 Translate the following expression and find the integer that represents the overall change. 37. The temperature starts at -15°C, drops 10°C, rises 5°C and rises 8°C. 38. A person starts with $50, earns $12, spends $15, earns $18, and spends $22. 39. A submarine starts at sea level, dives down 125 m, dives another 72 m, and rises 42 m. 40. An elevator starts on the seventh floor, descends 5 floors and ascends 9 floors. Luttrell 2012 12 Name: ______________________ Date: _____ 1g-Multiplying & Dividing Integers You may use your calculator only to check your answers. 1. (-8)(-3) 2. -6(- 4) 3. 5(-9) 4. 10(-3) 5. -6(-3) 6. -2(5) 7. 15(- 4) 8. 16(-3) 9. 17×(-5) 10. (-8)(-9) 11. (-7)(31) 12. 90(100) 13. 8 × (-3) 14. -3 × (-2) 15. -5 × -14 16. 12×12 17. -9 ÷ -3 18. -18 ÷ (-9) 19. 20 ÷ (-5) 20. -72 ÷ (-8) 21. -100 ÷ (-10) 22. -35 ÷ 7 23. 36 ÷ 4 24. 81 ÷ (-3) 25. 95 ÷ (-19) 26. -32 ÷ 4 27. 64 ÷ 8 28. 42 ÷ (-6) 29. -(-9) 30. -0 31. (-1)(-1)(-1)(-1)(-1) 32. 2(-3)(4)(-5)(6) Luttrell 2012 15 Chapter 1 Test Name: _____________________ Date: _________ SHOW WORK. A calculator is allowed on this test, but work still must be shown for full credit. Only questions regarding interpreting the directions are allowed – no talking to anyone but the teacher until all have finished and have submitted their test. 1. Put the following numbers in order from smallest to biggest: -9, 9, -1, 1, 7, -7, 0, 2, -3, -5, 6 2. Write each number in standard form: a. 500,000 + 1,000 + 70 + 2 b. 20,000 + 300 + 4 c. 4,000,000 + 800,000 + 2,000 + 900 3. Rewrite using algebraic symbols: a. three less than a number B. Five more than twice a number is thirteen. 4. Fill in the blanks with either standard form or expanded form. a. 62,723 = ( ) _ _ _ _ _ ( ) _ _ _ _ _ _ ( )6 10 7 10 3 104 2 0× + + × + + × b. ________________= ( ) ( )3 10 9 102 5× + × 5. Solve: a. x - 2 = 5 b. x -(-3) = 7 6. Circle each of the following types of numbers that best describes 4.5. a. Real b. Rational c. Complex d. Transcendental e. Integer Bonus: Fill in the next three numbers that continues the sequence: 1, 3, 6, 10,__, __, __ 5 5 5 5 5 5+0 Luttrell 2012 16 Chapter 2 Fractions Prior skills: • For sheet 2a, know perimeter • For sheet 2n, know area An asterisk (*) next to a question, such as question 17 & 18 on sheet 2i implies that the student may find the question challenging. The questions may have come from an activity we did in class prior to the worksheet. If you using the worksheets without other resources, just beware that the students may have difficulty with asterisk questions. Luttrell 2012 17 Name: ______________________ Date: _____ 2a-Finding Fractions For each question, translate the equation and then solve by mental math. 1. Darcy decides to eat only ⅓ of a candy bar. Draw a candy bar and shade in what was eaten. 2. Students at SLA walked 20 laps to help the Terry Fox Foundation. Some walked only ¾ the laps. Make 20 squares to represent the laps and shade in the amount some only walked. 3. The perimeter of the park is about 4 miles. Someone walked only ¼ of it. Draw a circle and color the fraction of the circle walked. 4. Nicole bought four apples. One was eaten this morning. What fraction of apples are left? 5. There are 150 days of school. If students have been in school for 15 days, what fraction of the school year is left? 6. The perimeter of the building is 400 feet and is getting a new coat of paint, what fraction of the building is left to paint if only 100 feet got painted? Draw a picture of the outline of the building and where it’s painted. Does your drawing look like others? Luttrell 2012 20 Name: ______________________ Date: _____ 2d-Multiplying & Dividing Fractions Use your calculator only to check your answer. Leave answer as simplified proper fractions. 1. 3 4 8 9⋅ 2. 4 5 10 17× 3. 5 6 3 10× 4. 6 7 14 15× 5. 8 9 27 28× 6. 3 7 21 10( ) 7. 2 43 4 2 5× 8. − 4 5 15 24( ) 9. 3 4 3× 10. 4 5 15( )− 11. 3 2 3( )− 12. 2 31 4 1 3× 13. 3 42 3 1 11× 14. 2 31 17 2 5× 15. 5 71 3 1 2× 16. 2 1 2 5 6÷ 17. 4 113 5 1 2÷ 18. 5 7 10 21÷ 19. 8 9 1 93÷ 20. 4 ½ ∙ 8 ⅔ 21. (8⅔) / (4½) Question 21 is an example why there are many types of parentheses and division symbols. Some symbols make the question cluttered and hard to read. Luttrell 2012 21 Name: ______________________ Date: _____ 2e-Adding Fractions of Like Denominator For each question, translate the equation and then solve by showing your algebraic steps. 1. The perimeter of the room is 248 inches. If two walls of the rectangular room are 80 ½ long, how long is each of the other walls? 2. A rectangular room has perimeter of 320 ½ inches. One wall is 70¼. Find the dimensions of the other three walls. 3. A triangle has sides 1⅜, 2⅝, and 2⅞. Find its perimeter. 4. A rectangle has dimensions 6⅔” by 7⅓”. Find its perimeter. 5. A rectangle has perimeter 8⅛ and one known side of 2⅞. What are the dimensions of the rectangle? 6. A hexagon (6-sided shape) has sides 2⅞, 1⅜, 4, 5⅛, 4½ and a perimeter of 20. How long is the missing side? 7. Find the perimeter of an octagon with equal sides of 4⅝. An octagon has 8 sides. 8. A pentagon has perimeter of 30⅔. If four sides are known to be 4⅓, 5⅔, 7⅔, and 6⅔, how long is the remaining side? Luttrell 2012 22 Name: ______________________ Date: _____ 2f - Like Terms Constant: A number that doesn’t change. In the expression 2x + 3, the constant is 3. The 2 is a coefficient. Variable: A number that may change, usually is represented by a letter. In the expression 2x + 3, the variable is x. Term: Any constant or variable that is being added or subtracted. In the expression 2x + 3y, the terms are 2x and 3y. Expression: A collection of terms that together represents a number. Equation: When two expressions are equal. Usually the goal is to find the value of the variable that makes the equation true (equal). Like Terms: Two or more terms having the same variables with the same exponents. The variables do not need to be in the same order; 2wz and 3zw are like terms. Coefficients are ignored since they refer to the amount of this term you have. Combining like terms: When the terms are alike, you add or subtract (depends on signs) the coefficients. Adding 2wz and 3zw would give you 5wz. Subtracting 2wz from 3zw would result in 1wz (better written as wz). 1. Write a like term for each of the following: a. -16y b. -5 c. 4xy2z d. 5h 2. Determine which of the following sets are made up of like terms: a. {4x, 3x2, 3x3} b. {xyz, -3xyz, 5yz} c. {2xy, -3xy, -8xy} For the following exercises, combine like terms. 3. 5x2 - 6x2 + 4xy + 3y2 - 2y2 4. (8y2 + 6y - ½) + (-y2 - 2y + ¾) 5. (3x2 + 4x + 4) + (x2 - 2x - 2) 6. Add x2 - ¼ and -x2 - 5x + ⅞. 7. (5x3 - 10x2 + 3x) - (-3x3 + 5x2 - 8x0) 8. (5x-3) - (- 4x2 + 3x - 1) 9. Subtract -3x3 - 2x2 - 6x0 from -2x3 - 3x2 - 6x. 10. (2 - x) - (4 + 5x) 11. Subtract 2x3 - 3x2 + 4x from -2x3 - 3x2 + 5x +4x0. 12. (3 + 3x) + (1 - 2x) Luttrell 2012 25 Name: ______________________ Date: _____ 2i-More Solving Basic Algebraic Equations Use your calculator to check your answer. Leave answer as simplified proper fractions. 1. 3x + 2 = 14 2. x − + =3 4 10 3. x + 3 = 3 ¾ 4. ½ x - 8 = 16 5. 5x - 10 = 5 6. x 2 3 7+ = 7. x − − =7 5 15 8. ¼ x = 32 9. x 6 8 10− = 10. 12 58= +x 11. 5x - 3 = 17 12. 4x + 7 = -1 13. 2 3 8 6x − = 14. − + = −3 4 4 8x 15. − + =x 3 1 4 16. 5x + 3 = -12 *17. 2x +8 = 4x + 2 *18. 3x – 1 = x + 5 Luttrell 2012 26 Name: ______________________ Date: _____ 2j-Solving for a Variable (multiple steps) When solving for a variable, reverse the order of operations. The objective is to isolate that variable by getting it to one side of the equation and all the constants to the other side. If faced with an equation with a nested variable (see example C), eliminate the outer portion before messing with the interior of an expression. To eliminate terms or coefficients, you will need to apply the inverse operations. Examples: A. 3x = 24 B. 3x + 1 = 25 C. (3x-1) ÷ 5 = 3 (3x)÷3 = 24÷3 3x+1 -1 = 25 - 1 (3x-1)÷5×5 = 3×5 x = 8 3x = 24 3x - 1 = 15 3 3 24 3 x = 3x -1 + 1 = 15 + 1 x = 8 3x = 16 3 3 16 3 x = x = 5⅓ Solve for the variable in the following equations: 1. 3x - 4 = 11 2. 5x + 7 = - 3 3. -3x + 2 = 17 4. 2 3 3 5 x + = 5. 2 5 2 3 x + = 6. 2(2x - 1) = 8 *7. 2 5 2 3 3 x + − = *8. 3 2 3 7 6 x + + = *9. 3(x+5) - 2 = 7 Luttrell 2012 27 Name: ______________________ Date: _____ 2k-Solving for a Variable (Multiple Like Terms) When faced with an equation that has multiple terms of the same variable you need to solve, get those terms together on one side of the equation. It doesn’t matter what side of the equation you move the terms to, as shown below. You may have to simplify before you can move terms around. There are many alternate methods of solving for a variable; valid methods require the use of the Field Axioms and PEMDAS. Example: 2(3w + 2) - 12 = 3w - 11 6w + 4 - 12 = 3w - 11 6w - 8 = 3w - 11 6w - 3w - 8 = 3w - 3w - 11 or 6w - 6w - 8 = 3w - 6w - 11 3w - 8 = - 11 -8 = -3w - 11 3w - 8 + 8 = 8 - 11 11 - 8 = -3w -11 + 11 3w = -3 3 = -3w 3w÷3 = -3÷3 3÷(-3) = -3w÷(-3) w = -1 -1 = w Solve for the variable: 1. 8x - 2(x - 8) + 4x = - 4 2. 4x - 2(x+3) - 4x = -3 3. - 3(x + 5) = 10 - 2x 4. 12z - 3(z - 7) = -(5z + 7) 5. -6(w - 3) = 3w - 9 6. 16x - 4(x - 8) + 8x = -8 *7. 5y - [7 - (2y - 1)] = 3(y - 5) + 4(y + 3) 8. x x x 4 2 3 13 6+ + = Luttrell 2012 30 Name: ______________________ Date: _____ 2n-Basic Algebraic Equations with Area For each question, translate the equation and then solve by showing your algebraic steps. Leave answers as proper fractions. 1. A rectangular room is 8 feet by 10 feet. Draw the layout and label. Find the area of the room. 2. A rectangular kitchen is 12 feet by 9 feet. Draw its layout and find the area of the room. 3. A rectangle is 9 ½ by 10 ¼. Find its area. 4. A triangle has a base of 4 ½ and a height of 5¾. Find its area. 5. A rectangle is 3⅝ by 4⅛. Find its area. 6. A square has a side of 3⅓. Find its area. 7. A bedroom is to be carpeted. Its dimensions are 9⅓’ by 11⅞’. How much square feet does it have? How much square yardage does it have? (3 ft = 1 yd, 9 ft² = 1 yd²) Luttrell 2012 31 Name: ______________________ Date: _____ 2o-Basic Algebraic Equations with Fractions For each question, translate the equation and then solve by showing your algebraic steps. Leave answers as proper fractions. 1. 2x - 3 = 5 2. 3x - 1 = 11 3. 4x - 1 = 15 4. 3x - ½ = 5 ½ 5. 4x - 1¼ = 3¾ 6. 6x - 1⅛ = 4⅜ 7. 3x - 2 = 5 8. 4x + 5 = 16 9. 5x + 30 = 3 10. -2x = 9 11. -3x + 1 = 6 12. -6x + 4 = -9 13. 3 2 4 5x + = − 14. 5 3 4 6x − = 15. 4 5 3 4 1 81 10x − = Luttrell 2012 32 Name: ______________________ Date: _____ 2p-Basic Algebraic Equations with Fractions Solve for x by showing your algebraic steps. Leave answers as proper fractions. 1. The area of a rectangle is length times width. If the area of a rectangle is 9⅜ cm² and its width is 4¼ cm, what is its length? 2. If a rectangle’s area is 100¾ cm² and its length is 9½ cm, what is its width? 3. Area of a triangle is half its height times base. If the area of a triangle is 10 cm² and has base of 3 ½ cm, what is its height? 4. If the area of a triangle is 4 ½ and its height is 3¼, what is its base? 5. 3 2 27 44x = 6. 4 5 6 25x = 7. 4 9 30 21x = 8. 3 4 3 4 11 12x − = 9. 8 9 1 2 7 83 10x + = 10. 4 1 111 3 3 4 1 8x + = Luttrell 2012 35 Name: ___________________________ Date: __________ Chapter 2 Test SHOW WORK. A calculator is NOT allowed on this test. You must work alone. Questions regarding interpreting the directions are allowed. Simplify your fractions! 1. Put in order from smallest to greatest. a. 1 2 1 3 1 5 1 4 1 1, , , , B. - ½, ¼, - ⅝, ⅔ 2. A. Write as an improper fraction: 6 3 4 . B. Write as a mixed number: 23 8 . 3. Add the fractions, leaving answer as a proper fraction: a. 3 8 7 8+ B. 4 5 2 33+ 4. Subtract the fractions, leaving answer as a proper fraction: a. 17 58 9 4 9− B. 6 31 2 7 8− 5. Multiply and simplify to proper fractions: a. 5 6 9 10× B. 5 2 3 9 17× 6. Divide and simplify to proper fractions: a. 3 4 7 8÷ B. 8 41 2 ÷ 5 5 5 5 5 5 Luttrell 2012 36 Chapter 2 Test, continued 7. Match the terms _____ a. Vinculum A. Top part of a fraction _____ b. Denominator B. Bar separating parts of fraction _____ c. Numerator C. Bottom part of a fraction _____ d. Decimal D. Dot separating whole from parts _____ e. Improper fraction E. 41 5 _____ f. Proper fraction F. 3 2 5 _____ g. 0.06 G. 6 thousandths _____ h. 600 H. 6 hundredths _____ I. 0.006 I. 6 hundreds _____ j. 0.6 J. 6 tens K. 6 tenths 8. Find the perimeter of the rectangle whose dimensions are 1¾ cm by 3 ⅓ cm. 9. Find the area of the rectangle whose dimensions are 5 ½” by 10". Bonus: (2 pts) Let N = 20×30×50×70×110×130. What is the smallest positive prime number which is NOT a factor of N? (3 pts) What is the value of x which satisfies the equation 5 20 7 28 9 36 11 44 80 1+ + + + =x ? 0 5 5 10 Luttrell 2012 37 Chapter 3 Decimals Prior Skills: • Fractions • Area • Perimeter • For sheet 3k, volume and surface area • For sheet 3m and 3o, Pythagorean Theorem • For chapter 3 test, order of operations and definition of addend Luttrell 2012 40 Name: ______________________ Date: _____ 3c- Operations with Decimals Only use your calculator to check your answer! Order the decimals from least to greatest. 1. 9.33, 9.4, 9.44, 9.45, 9.446 2. 2.11, 2.111, 2.121, 2.112 3. 3.4, 3.43, 3.424, 3.4509, 3.43509 4. 1.2, 1.302, 1.3002, 1.30002 Simplify the expressions. 5. 8.275 – 5.857 6. 18.93 + 149.42 7. 87.944 – 6.58 8. 14.923 + 1.8 9. 13.245 + 1.4467 10. 12.3 + 1.43 + 1.5607 11. 2.3 – 3.12 12. 0.45 + 45 13. 0.45 – 45 14. 4.01 + 0.034 15. -0.475 – 3.78 16. -7.2 + 10.56 17. 1.8 + 2.401 + 1.05 18. 3.45 + 2.356 Luttrell 2012 41 Name: ______________________ Date: _____ 3d-Adding and Subtracting Decimals Use your calculator only to check your answer. 1. Find the perimeter of the room with dimensions: 8.35' by 11.24'. 2. Find the perimeter of a triangle whose sides are 4.3", 6.71", and 5.901". 3. Find the perimeter of the room with dimensions 9.12' by 10.8'. 4. Find the perimeter of the room with dimensions 7.02' by 11.76'. 5. If the perimeter of a room is 48' and the length is known to be 10.8', what is the other dimension? 6. If the perimeter of a room is 65' and the length is known to be 12.5', what is the other dimension? 7. The perimeter of an equilateral (all sides equal) triangle is 34.8". What is the side length? 8. The perimeter of a square is 38". What is the side length? 9. The perimeter of a regular hexagon (6-sided shape with equal sides) is 39'. What is the side length? Luttrell 2012 42 Name: ______________________ Date: _____ 3e-Multiplying & Dividing Decimals Only use your calculator to check your answer. 1. (-14.3)×(-2.1) 2. (-0.05)×(-0.05)×(-0.05) 3. -0.003 × 16.1 4. 15.2 × 5.2 5. -18.34(-1.02) 6. 0.0003(-1.2) 7. -18.46 – (-18.79) – 18.46 8. -15.37 – 14.35 + 6.2 9. -83.26 – (-15.6) +(-18.2) 10. 9.3 • 1.23 11. 0.0023 • 1.57 12. 3.14 • 2.718 Solve the following problems with the aid of a calculator. 13. Your bank account started off with $40. Over the course of the week you either put money or took money out of your account. Here are the pile of ATM transactions: 15.50, - 4.75, -20, 55.75. How much did you have at the end of the week? 14. Your room has dimensions of 11.23 feet by 9.56 feet. What is its perimeter and area? 15. The monkey cage at a laboratory is 5.3 feet by 5.3 feet. What is the perimeter of the cage and it’s area? 16. A zoologist recommended that captive monkeys should have AT LEAST a cage 10 feet by 20 feet. What is its perimeter and area? 17. What is the difference in areas between the two cages in problems #15 and #16? Luttrell 2012 45 Name: ______________________ Date: _____ 3h-More Multiplying & Dividing Decimals Simplify without the use of a calculator. 1. 2.9 × 3.1 2. 4.06 × 1.2 3. 27 ÷ 100 4. 34.45 ÷ 1000 5. 0.45 × 100 6. 53.4 × 10,000 7. 3.2 ÷ 0.4 8. 0.45 ÷ 0.9 Solve without the use of a calculator. 9. If 42 people are going to the beach, and cars seat at most 5, how many cars are needed? (We are omitting vans and other modes of transportation.) 10. If you spent $39.75 on three pair of jeans, what is the average price of one? 11. If you drive 500 miles in a car that uses 22 gallons, what is your average mile per gallon? 12. If a can weighs 10 grams, then how much is it in kilograms? How much in milligrams? 13. A teacher wants to show a video in her 35 minute class. If the video lasts 112 minutes, how many classes must she reserve for its use? 14. A lawn of 162 ft by 210 ft will have a house built on it. If the house will be 51 ft by 81 ft, how much lawn will remain? Put answer in units of feet and yard. Luttrell 2012 46 Name: ______________________ Date: _____ 3i-Circumference with Decimals No calculator is allowed. Use 3.14 when approximating π. Round to the nearest hundredth. 1. Calculate the circumference of the circle with radius 9". 2. Calculate the circumference of the circle with diameter 2". 3. Calculate the circumference of the circle with radius 3". 4. Find the radius of a circle with circumference of 31.41 inches. 5. Find the diameter of a circle with circumference of 6.28 inches. 6. Find the radius of a circle with circumference of 9.42 inches. 7. Find the diameter of a circle with circumference of 4.71 inches. Luttrell 2012 47 Name: ______________________ Date: _____ 3j-Area with Decimals No calculator is allowed. Use 3.14 when approximating π. Round to the nearest tenth. 1. Calculate the area of the circle with radius 9". 2. Calculate the area of the circle with diameter 2". 3. Calculate the area of the circle with radius 3". 4. Calculate the area of the circle with diameter 3". 5. Calculate the radius of a circle with area of 28.26 squared inches. 6. Calculate the radius of the circle with area of 50.24 squared inches. 7. Calculate the diameter of the circle with area of 314 squared inches. Luttrell 2012 50 Name: ______________________ Date: _____ 3m-Surface Area & Volume of Pyramids & Spheres A calculator is allowed. If your calculator does not have a π button, use 3.14 to approximate π. 1. A pyramid is twice as big as a similar pyramid with volume of 25 in3. Find the volume of the other pyramid. 2. A pyramid has a square base of sides 5 inches and a height of 3 inches. What’s the volume of the pyramid? 3. A triangular pyramid is made up of equilateral triangles with a side of 2 feet. What is the pyramid’s surface area? (Use Pythagorean Theorem to find the height of a triangle.) 4. A pyramid has a triangular base with area of 15 squared inches and a height of 6 inches. What’s the volume of the pyramid. 5. A man decides to make a playhouse that’s a triangular pyramid like the one in question #4, but three times as large. What would be the new volume? 6. Find the volume of a sphere that has a radius of 9 inches. 7. Find the volume of a sphere that has a radius of 4 inches. 8. Find the surface area of a sphere that has a radius of 9 inches. 9. Find the surface area of a sphere that has 4 inches. 10. A. Finding the amount of leather to cover a basketball is an example of (area, volume). B. Finding the amount of air contained in a basketball is an example of (area, volume). Luttrell 2012 51 Name: ______________________ Date: _____ 3n-Mixed Review of Shapes and Objects A calculator is allowed. If your calculator does not have a π button, use 3.14 to approximate π. 1. The area of a square is 625 cm2. Find the length of the sides. 2. The area of a rectangle is 600 cm2. Find the width of the rectangle if the length is 40 cm. 3. The area of a triangle is 45 cm2. Find the base of the triangle if the height is 9 cm. 4. The area of the circle is 8π cm2. Find the circumference of the circle. 5. The area of a square is 400 cm2. Find the perimeter of the square. 6. The perimeter of a rectangle is 30 cm with the length being 8 cm. Find the area of the rectangle. 7. If the surface area of a box is known to be 300 cm2 and have a base of 5 cm by 4 cm, what is the height of the box? 8. If the surface area of a box is known to be 320 cm2 and have a height of 36 cm and a width of 2 cm, what is the depth of the box? 9. If the surface area of a sphere is 400π cm2, what is its volume? 10. If the surface area of a sphere is 100π cm2, what is its volume? 11. If the surface area of a cube is 600 cm2, what is its volume? 12. If the surface area of a cube is 96 cm2, what is its volume? 13. If the volume of a cube is 27 cm3, then what is its surface area? 14. If a square pyramid has height of 10 cm and a volume of 60 cm3, what is the dimension of its base? 15. If a square pyramid has a height of 5 cm and a base side of 6 cm, what is its volume? Luttrell 2012 52 Name: ______________________ Date: _____ 3o-Mixed Review of Volume & Surface Area A calculator is allowed. Leave answers in terms of π. 1. A cube has surface area of 72 in². 2. A tetrahedron has surface area of What is the area of each face? of 27.7128 in². What is the length of each e d g e ? 3. A prism has a base of 4" by 8" and surface area 4. A prism has dimensions of 112 in². What is its height? of 3" by 4" by 5". What’s its surface area? 5. A cylinder has radius of 2" and 6. A cylinder has diameter of 2" height of 6". What’s the surface area? and height of 3". What’s the surface area? Luttrell 2012 55 Chapter 3 Test, continued 7. Find the perimeter of the rectangle whose dimensions are 2.45 cm by 3.1 cm. 8. Find the area of the rectangle whose dimensions are 4.03” by 11". 9. Simplify a. 4(3 - 4)2 - 5 ÷ 10 b. [3(2-1)×5- 4] ÷ 10 + 1 10. Find the volume and surface area of a cylinder whose diameter is 10 cm and its height ‘ is 20 cm. Bonus: (3 pts) The sum of ten positive odd numbers is 20. What is the largest number which can be used as an addend in this sum? Bonus: (3 pts) Two 5×5 squares overlap to form a 5×7 rectangle, as shown. What is the area of the region in which the two squares overlap? 5 5 10 5 10 Luttrell 2012 56 Chapter 4 Percents and Proportions Prior Skills: • For sheet 4f, placement values • For sheet 4g, powers of ten • For sheet 4h, an introduction to other bases would be good for students to get a historical sense of mathematics. • For sheet 4i, decimal representations of fractions • For sheet 4k, practice understanding and solving with percents • For sheet 4m, practice with understanding and solving proportions Luttrell 2012 57 Name: ______________________ Date: _____ 4a- Field Axioms Simplifying any expression or solving any equation requires the use of the following axioms: Closure: A set of numbers is closed if the unique sum or product of an operation is also in the same set. For example: 3+8 = 11, so the set of reals would be closed, but not digits. Commutativity: x+ y = y + x or xy = yx. The order to how the sum or product is obtain is not important, e.g. 2 + 1 = 1 + 2. Associativity: How the sum or product is grouped isn’t important. Examples can be seen in x + (y + z) = (x + y) + z or 2(3×4) = (2×3)×4. Distribution of Multiplication Over Addition: x(y + z) = xy + xz Identity: Zero plus any number remains that number. One times any number remains that number. For example: 0+2 = 2, 1×2 = 2. Identity is about leaving the value unchanged! Inverse: To obtain zero, sum the number and its opposite. To obtain a one, multiply the number and its reciprocal. The Inverse defines subtraction and division! For example: 2 + (-2) = 0 which is the same as 2 - 2 = 0. You can see it with 2(½) = 1 which is 2/2 = 1. 1. What is the additive inverse of ⅝? 2. Write an example of the distribution property. 3. What is the multiplicative inverse of ⅓? 4. What is the multiplicative inverse of x? 5. Rewrite the following using addition: 5 - 5 = 0. 6. Rewrite the following using multiplication: 5/3. 7. Fill in the justifications (axioms) used in the proof of the Multiplicative Property of Zero: a. 0 = 0 Reflexive Property b. 0 + 0 = 0 ________________ c. x(0 + 0) = x(0) Multiplication Property of Equality d. x(0+0) = 0 + x(0) ________________ e. x(0) + x(0) = 0 + x(0) ________________ f. x(0) = 0 Add/Subtraction Property of Equality 8. Fill in the justifications to the following problem: a. 4x - 3 = 5 Given b. 4x - 3 + 3 = 5 + 3 _______________, Add Property of Equality c. 4x + 0 = 8 _______________ d. 4x = 8 Add Property of Zero e. (¼)(4x) = (¼)(8) _______________ f. (¼*4)x = 2 _______________ g. 1*x = 2 _______________ h. x = 2 Identity Luttrell 2012 60 Name: ______________________ Date: _____ 4d- Using Absolute Values within the Order of Operations The American way to remember the order is to say “Please Excuse My Dear Aunt Sally” where the beginning of each word represents the operation: Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Since division is definition a multiplication of fractions, whenever you see them together, you go left to right. This can explain why the Canadian expression “Bed Mas” is still valid in spite of them listing division first. Again when multiplication and division are together, the order is the left operator goes first. The same can be said of addition and subtraction. 1. 2 – 3(4+5) +3∙5 2. 2∙3 + 3(2-1) 3. 4(3-22)-3∙7+8∙4 4. 3÷9 ⋅6–42 +5(3-32) 5. 5 – 22 + 34 6. –4*2*6 – 12/3 + 33 7. 2 3 4 3 2 6 82+ × − ÷ +( ) 8. ( )( )3 3 4 3 52− − × 9. 3 4 6 5 2× ÷ + × There are various ways to write a parenthesis: { }, ( ), [ ], ---- , | |. The absolute value is a special type of parenthesis because it makes the expression contained within a positive value. The first two questions have been done for you. 1. | -3 | = 3 2. | 1-4 | = | -3| = 3 3. –2 | 3+8-10 | 4. | 2-6 |∙3 5. | -½ | 6. (-3+4 2)(-6) 7. –3(23 + 4) ∙5 8. 1-5 | 5 – (9+1)| 9. -2|3-(7-3)| 10. [3 - (2 - 4)][3 + | 2 - 4 |] 11. 4 - | 1 - 7 | 12. 8 - 3| 5- 42 + 1| Luttrell 2012 61 Name: ______________________ Date: _____ 4e - Evaluating Most times you need to determine the value of an expression (the number an expression represents). In order to find the value, you need to know what the variables represent. By substituting the value of the variables and simplifying, the value of the expression will be found. Note the vertical alignment of the work; try to show your work in similar manner. Example A: Evaluate the expression when y = 2: 3y + 2y2 - 6(y - 4) + 8y3. Step I - substitute value of y. 3(2) + 2(2)2 - 6(2 - 4) + 8(2)3. Step II - simplify. 6 + 2(4) - 6(-2) + 8(8) 6 + 8 + 12 + 64 90 Example B: Evaluate 3 2 1 2 3 6 ( ) ( )x x x + − − + when x = -3 3 2 3 1 2 3 3 3 6 ( ( ) ) ( )− + − − − − + 3 6 1 2 6 3 ( ) ( )− + − − 3 5 12 3 ( )− + − = − 3 3 1 Evaluate when a = -1. 1. (4a + 3)a - (2 + 2a) 2. 12a2 - 3a + 2 3. 3a4 + 5a3 - 6a2 - 2a Evaluate when x = 1, y = 2, and w = 2 4. 2 3 5 3x x+ + − 5. -(-x) 6. x - (x - {x - y}) 7. 4w + 7 + 3w -2 + 2w 8. 1 3 3 1 w w w w w w w + × + × − 9. 3 2 5 ( ) ( )x y x y x y + − − + Luttrell 2012 62 Name: ______________________ Date: _____ 4f- Expanded Form Do without the aid of a calculator. Check your answers with a calculator at the end. Fill in the missing parts by writing out the numeric value. 1. 8,375 = eight thousand, three hundred __________________ 2. 312, 435, 600 = three hundred twelve million, ___________________________, six hundred 3. 412 = ____________________________ twelve 4. 8,300,567 = eight million, ____________________________, five hundred sixty-seven Identify the numeric value of the number indicated: Example 567,890 5 represents the _______hundred thousands______ 5. 320,045 2 represents the ____________________________ 6. 1030 3 represents the ____________________________ 7. 321 3 represents the ____________________________ Write out in expanded form: Example: 431.24 4(100) + 3(10) + 1(1) + 2(1/10) + 4(1/100) 8. 3, 243 9. 345,645 10. -23,120 11. 304.125 12. 300,561, 100 Write out in standard form: Example: 3(1000) + 4(10) + 8(1) + 9(1/10) = 3,048.9 13. 4(1,000,000)+8(10,000)+3(1,000)+7(100) = 14. 2(10,000) + 8(100)+3(10)+4(1/10) = Luttrell 2012 65 Name: ______________________ Date: _____ 4i-More Practice with Standard and Non-Standard Forms Exercise A. Write the following numbers in expanded form: Example: 835 = 8(102) + 3(10 1) + 5(10 0) 1. 435,800 2. 12,300,506 3. 1000 4. 15,000 5. 4,000,000,000,000,000 6. 5,000,000,000,000 Exercise B. Write the following numbers in expanded form, using the example as a guide: Example: 1.2345 = 1 2 3 4 51 10 1 100 1 1000 1 10 000+ + + +( ) ( ) ( ) ( ), 7. 12.005 8. 0.0004 9. 0.100304 10. 4.500005 Exercise C. Write the following numbers in expanded form, using the example as a guide: Example: 1.2345 = 1 2 10 3 10 4 10 5 101 2 3 4+ + + +− − − −( ) ( ) ( ) ( ) 11. 12.005 12. 0.0004 13. 0.100304 14. 4.500005 Exercise D. Write the following numbers in standard form. 15. 8(106) + 3(10 5) + 5(10 4) + 1(10 0) = 16. 4(1013) + 8(10 10) + 5(10 5) = 17. 3 10 5 10 4 10 5 103 0 1 2( ) ( ) ( ) ( )+ + +− − = Luttrell 2012 66 Name: ______________________ Date: _____ 4j - Scientific Notation Since mathematicians and scientists have tried to find ways of simplifying written expressions, scientific notation is common for writing VERY large or small numbers. This is very useful when talking about the distance between stars, light years, the number of molecules in an object, the people on the planet, etc. Note how this large number is rewritten: 64000000 = 6.4∙107 The 6.4 is the mantissa and is always a number between 1 and 9, inclusively. The exponent 7 is the characteristic and says what power of 10 you are multiplying by. Some people just remember that is how many spaces to move the decimal point. A negative moves it left and a positive moves the decimal right. A small decimal like 0.000007 is rewritten as 0.000007 = 7∙10-6 . 1. 0.000123 2. 2,340,000,000 3. 0.0000000345 4. 435000000000000 5. 76,000,000,000 6. 0.0000308 Express each number into scientific notation. For some you may have to round to three significant digits in the mantissa. 7. 23,000,000,000,000 7. ______________________ 8. 134,000,000,000 8. ______________________ 9. 456,000,340,000,000 9. ______________________ 10. 788,999,213,543,345,532,345 10. ______________________ Express each number in standard form. 11. 2 34 104. × 11. ______________________ 12. 3 4 1014. × 12. ______________________ 13. 6 7 107. × 13. _______________________ 14. 2 6 109. × 14. _______________________ 15. 3.4 × 10 -5 15. _______________________ 16. 1 × 10 -10 16. ______________________ 17. 3.25 × 10 -6 17. ______________________ Fill in the missing spots of the equation. 18. 0 0000234 2 34 10. .= × 19. 31 10 000000315. × =− 20. 0.0045 = 45∙10-3 21. 6.1 ∙ 10 = 610,000 Luttrell 2012 67 Name: ______________________ Date: _____ 4k - Percents Write the equivalent percent for each of the following: 1. 0.04 2. 0.12 3. 0.45 4. 1.23 5. 1 Convert the percent into standard decimal form: 6. 1% 7. 23% 8. 123% 9. 0.3% 10. 56% Solve using percents: 11. What is 10% of $32? 12. What is 5% of $32? 13. What is 15% of $32? 14. What is 5% of $42? 15. What is 10% of $42? 16. What is 20% of $42? 17. What is 15% of $42? 18. What is 8% of $40? 19. What is 110% of 80? 20. A suit at JcPenny’s has a $185 tag. But it’s on sale for 30% off. How much will it cost without tax? If the Nova Scotia government has a provincial tax of 15%, how much will the suit cost? 21. If you spent $42.75 on 3 CDs, how much does each one cost? What percent of the total cost is one CD? 22. A dress has a price tag of $90. The dress is on sale for 20% off and there is a 6% sales tax. What is the total cost of the dress? How much money do you save? 23. A dress has a price tag of $86. a. If it is on sale for 20% off, what will be the cost? (Ignore tax) b. What is the price of the dress on sale with a 6% tax? c. How much would the dress cost when not on sale? Luttrell 2012 70 Name: ______________________ Date: _____ 4n – Mixed Review Solve the following. Show your work! Calculator is allowed. 1. If you drive 650 miles in a car uses 30 gallons, what is your average mpg? 2. If a house is 2.3 km away, how far is it in meters? Millimeters? 3. A teacher wants to show a movie in her 50 minute class. If the movie is 210 minutes long, how many classes must she need in order to watch the entire movie? 4. A lawn of 100 ft by 150 ft has a house built on it. The house is 50 ft by 30 ft. What percent of the lawn is left to landscape? 5. Amy used to work 60 hours each week. Now she works 45 hours a week. What is the percent decrease? Luttrell 2012 71 Chapter 4 Test Name: ___________________________ Date: _____________ SHOW WORK for full credit!!! You must work alone. Only a calculator is allowed. 1. Determine which ratio is larger. Use the symbols <, >, or = to fill the blank: a. 5 6 ____ 8 9 b. 11 15 ____ 23 27 c. 15 18 ____ 5 6 2. Write the ratios: a. There are fourteen P-3 students, fourteen 4-6th grade students, six 7-8th graders, fourteen10th graders and nine 11-12th grade students at SLA. Write the ratio of the 4-6th to the 7-8th graders. b. Ethan and Ben played checkers. Ethan won 8 games and Ben won 12. Write the ratio, in simplest form that compares Ethan’s score to Ben’s. 3. Write each rate in simplest form, and give its units: a. Kelsey skied downhill twice in 30 minutes. b. The Morash family visits Martock 20 times in five months. C. My granny bought 4 tickets to the Tattoo for $160. 4. Solve: a. 4 24 54x = b. 40 8 5n = C. 12 13 5= x 5. Solve these word problems: a. The ratio of the boy’s shadow to the flag pole’s shadow is 1:10. If the boy is 160 cm tall, how high is the pole? b. On a map 2.5 cm represents 300 km. What distance would a 3.5 cm line represent? 6. A. Which is cheaper: $3.20 for 8 L or $2.40 for 4 L? b. Paul’s family travelled 190 km in 3 hours. At this speed, how far do they go in 5 hours? 5 5 5 5 5 5 Luttrell 2012 72 Chapter 4 Test, continued 7. Answer each part: a. What is 40% of 70? b. What is 25% of 40? c. Fifteen is 10% of what number? d. Sixteen is 25% of what number? e. Write as a percentage: 7 girls in a class of 18 students. 8. A city recycles 78% of the newspapers sold there. The Chronicle Herald has a readership of 316,700. How many of the Chronicle Herald gets recycled? 9. It costs you $25 to make a sweater and want to sell it for profit at $45. What percent markup do you have (percent increase)? 10. SLA has 73 students and is praying for 100 next year. What is the percent increase? 11. Answer the following short questions: a. Write as standard form: 2.34×105. b. Write as a sum of powers: 300.04. c. Write as standard form: 6(103) + 3(102) + 4(100) d. Simplify: 3-3×23∕4 – 3 e. Evaluate 3x2 – 4x + 1 when x = -1. Bonus: (5pts) Two identical jars are filled with equal number of marbles. The marbles are colored red or white. The ratio of red to white in jar I is 7:1 and 9:1 in jar II. If there are 90 white marbles all together, how many red marbles are in Jar II? 10 5 5 5 5 Luttrell 2012 75 Name: ______________________ Date: _____ 5b – More Practice Translating Algebra was founded by people who needed to find answers to problems. For centuries, people would try to solve problems without the use of variables. It wasn’t until Fibonacci in the 1400s made using symbols (mathematical operators and variables) popular. Note the difference between the two equations below: Tom’s age plus 4 equals Sally’s age T + 4 = S Students typically have problems translating sentences into expressions or equations. Once they get the equation written, they no longer have as much difficulty. Translate the following into expressions or equations. Define your variables: 1. Sally is five years less than twice Tom’s age. 2. One less than Sally’s hourly wage. 3. Three times as many rocks 4. Thrice as many hours 5. Sally’s and Tom’s wages sum to be $50,000. 6. Four less pounds 7. The sum of two numbers 8. A number and 3 9. My height is 4 inches more than yours. 10. Five less than a number 11. The height of the room is half the length. 12. Twice the age of Sally. 13. Tom earns 1.25 times as much as Sally. 14. Four more pounds Solve: 15. I ate ½ my daily peanut butter sandwiches for lunch. Had I eaten one more sandwich, I would have eaten ⅝ my daily sandwiches. How many sandwiches do I eat daily? 16. Tom wanted Sally’s telephone number. Knowing she lived in Berrien Springs, with an exchange number of 473, Tom just needed the last four digits. Sally slyly stated that 45 added to his age of 30 equaled to 1000 less than half her telephone number (ignoring the exchange). Luttrell 2012 76 Name: ______________________ Date: _____ 5c – Number Line Real numbers can be plotted on a line from left to right with the numbers in ascending order. That means negatives are placed on the left end and positives on the right. Remember -100 is smaller than -1, so -100 would be further left than -1. For equalities, a solid dark dot on the number line indicates the value of the variable that makes the equation true. For example, an equation is found that x = 4. The graph of x = 4 is shown below: | | | | | | | | -2 -1 0 1 2 3 4 5 6 Below are examples of inequalities and their solutions graphed. Remember that the inequality symbols switch when multiplying (dividing) by a negative. Also an open circle or ) shows a strict inequality, as in the constant is not included in the solution set. On the other hand, a closed circle or ] shows that the constant is included. Example: 4x - 1 < 7 -x - 2 ≥ 2 4x < 8 -x ≥ 4 x < 2 x ≤- 4 | | | | ◯ | | | | | | | | | | | -2 -1 0 1 2 3 4 5 6 -6 -5 - 4 -3 -2 -1 0 1 Solve and graph the solution on a number line: 1. 3x - 2 = 13 2. 2x + 2 = 12 3. (4 - 2)x = 12 4. - 4 ≤ x - 1 ≤1 5. -3w < 12 6. 6(1 - x) - 3x ≤ 12 7. -2z > 12 8. 2y ≥-11 9. 3x + 4 > 5 Luttrell 2012 77 Name: ______________________ Date: _____ 5d – Absolute Values On a real number line, what is the distance between: 5 and 12, -3 and 5, or 4 and 7? We find the distance by subtracting the smaller value from the larger: 12-5, 5 - (-3), 7 - 4. So what is the distance between 0 and x? It would be written as x - 0. But what if we knew x was 5 units from zero? The number could be five less or five more than zero. Then symbolically it would be written as |x - 0| = 5, then x is either -5 or 5. The absolute value, magnitude, of x - 0 (or x) gives the distance, without specifying direction. The equation | x - 1| = 5, can be thought as the distance x from 1 is 5 units. So starting on the real number line at 1, you would count to the right or left 5 units, getting two answers: 6 and - 4. Expressions with absolute values can get more complicated, so you may want to remember a certain rule: |xy| = |x|⋅|y|. For example: |-x| = 3 can be written as |-1|⋅|x| = 3. Then |-1| really is 1 since the absolute value is asking for the magnitude of -1. So the equation really is 1⋅|x| = 3, which gives the answer of -3 and 3. Simplify each expression and graph the solution. 1. The distance between 4 and 6 is 2. 2. The distance between 5 and 11 is 6. 3. The distance between 7 and -3 is 10. 4. The distance between x and 3 is 5. 5. The distance between x and -2 is 4. 6. The distance between x and 3 is more than 6. 7. The distance between x and -2 is less than 4. Solve each equation by translating its symbolic meaning first. Graph the solution. 8. | x - 4 | = 2 9. | x - 8| = 3 10. | x + 3| = 5 11. |x + 2| = 6 Luttrell 2012 80 Name: ______________________ Date: _____ 5g – Cartesian Coordinate Plane Rene Descartes came up with a way to systematize giving directions. He took two real number lines and had them intersecting at zero to form perpendicular angles. At each integer, you can draw a vertical or horizontal line. After a while you will have formed a grid, with each line intersecting at integer coordinates (lattice points). The horizontal real number line is commonly referred to as the x-axis and the vertical number line is called the y-axis. The point (x,y) can be found by moving left/right along the x-axis and then from that new point, moving up/down y- spaces. For example the ordered pair (2,-1) would be found by starting at the origin (0,0) and moving right 2 spaces and down 1 space. There are other coordinates, such as polar coordinates, but those are for a later course. 1. Draw a coordinate system, label the integers from -10 to 10 on both axes. 2. Draw a dot and label the points R(-2, 4), E(1, 5), S(4, -3) and T(-7,-2). 3. Draw a coordinate system, label the integers from -10 to 10 on both axes. 4. Label the points A(-2,4), B(-1, 2), and C(3, -6). Connect the dots. What does figure does it look like? How could you tell that it really is that shape? Luttrell 2012 81 Name: ______________________ Date: _____ 5h – Polynomials and Variation Polynomials are algebraic expressions that involve only the operations of addition, subtraction, and multiplication of variables. Examples of polynomials are 3, 3x, 3x + 1, 3x2 + 6x+1, and x3 + x. Polynomials are described by the number of terms in the expression. Special names are used for one-term through three-termed polynomials: monomial, binomial, trinomial. Polynomials are also described by the highest-degree term in the expression: constant (zero- degree), linear (first-degree), quadratic, cubic, quartic, quintic, hextic (sextic), heptic (septic), any higher degrees are labeled by the ordinal value. Match the following polynomials. If it isn’t a polynomial, explain wy: ___ A. 3x2 + 6x+1 1. Quadratic trinomial ___ B. 3x2+ 8x4 2. Quintic monomial ___ C. 3 4x + 3. Non-polynomial; since ________________ ___ D. 3x5 4. Non-polynomial, since division by variable ___ E. 5 5. Hextic binomial ___ F. 23x5 + x4 + 3 6. Quartic binomial ___ G. x6 + x4 7. Constant ___ H. 8 4x + 8. Quintic trinomial Ever had such a conversation? “This recipe calls for eggs and flour.” “How much flour?” “One egg is needed for every cup of flour.” The first statement tells what is needed, a simple list of ingredients. The third statement gives the proportion of each amount. Expressing how variables relate in an equation is very similar. In the following exercises, you’ll get a chance to symbolically express that relationship of variables. In the first couple of examples, you see the use of k. This letter represent the proportion; its value is of secondary importance to the actually relationship of the variables. As noted in the dialogue, the most important was getting the ingredients and then the proportion. 9. y varies directly with x. (answer: y = kx) 10. y varies indirectly with x. (y = k/x) 11. y varies with the reciprocal of x. 12. y varies with the cube of x. 13. y varies with the square root of x. 14. y is a constant. 15. y varies linearly with x. 16. y varies inversely with x. 17. Solve: The number of pounds you weigh is directly proportional to the number of kilograms you are. Sally steps onto a scale calibrated in kilograms and finds that she is 60 kilograms. Her bathroom scale says 140 pounds. Write the equation expressing pounds in terms of kilograms. How much would she weigh if she was 25 kilograms? Luttrell 2012 82 Name: ______________________ Date: _____ 5i – Identifying Functional Relations Identify which of the following are functions. Determine the domain and range of each relation. 1. 2. 3. 4. 5. 6. 7. 8. 9. Luttrell 2012 85 Name: ______________________ Date: _____ 6a – Linear Equations Polynomials with x and y to the first power and are expressed either as Ax + By = C or y = mx + b are linear equations. They are called linear equations because when graphically represented, the solutions form the shape of a line. There is a pattern between the solutions that makes it easy to distinguish linear equations. In the spaces below, fill in the missing pieces to the pattern: Equations: A. B. C. y = 2x - 1 y = -3x + 2 2x + 3y = 5 (-3, -7) (0, 2) (-5, 5) (0, -1) (1, -1) (1, 1) (3, 5) (2, - 4) (7, -3) (6, 11) (3, -7) (13, -7) In equation A, as x increased by___, y values increased by___. In equation B, x increased by___ each time y decreased by___. In equation C, increasing x by___ made y decrease by___. It is necessary to find the solutions to a linear equation. Some people make a t-chart, where they choose values of x (or y) and find the corresponding value of y (or x). 1. Fill in the t-chart for x - 2y = 6. x -2 1 2 y -3.5 -3 -2 2. Fill in the t-chart for the following data: St. Joseph’s ice arena has a $3 admittance per person. Let x represent the number of people in a group and y be the admittance price. x 1 3 5 y 21 42 3. Let x represent a bag of apples that are being sold at $1.99 a bag. If y represents cost, fill in the chart below: x 4 12 y 11.94 19.90 Luttrell 2012 86 Name: ______________________ Date: _____ 6b – Graphs of Linear Equations Some of the solutions to 2x - y = 1 are (-2,-5), (0,-1), and (1, 1). Recognizing 2x-y = 1 as a linear equation, the conclusion can be made that there is a pattern between the ordered pairs. Between the first two pairs, x increases by 2. But between the last two pairs, it increases by 1. With such inconsistency, it will give some difficulty for writing an equation when only given coordinates. So to circumvent this problem, slope was defined. Slope is the ratio of the change between y- values to the change between x-values. As any ratio can be reduced to simplest terms, the ratio 4/2 between the first two ordered pairs reduces to 2, which is the ratio between the last two pairs. Algebraically, slope (m) is defined as m y y x x = − − 1 2 1 2 . Other expressions are rise run y x = ∆ ∆ . Slope can be found by taking the ratio between two ordered pairs, or by looking at the equation when it is in the form y = mx + b (slope-intercept form). Solving for y in the example gives y = 2x - 1. If x = 0, the equation gives y = -1, just like the -1 in the equation. The y-intercept (b) is represented by the -1. Go ahead and plot the point on the graph. By evaluating more values of x, more ordered pairs are obtained: (1,1) and (2, 3). The same can be done by starting at the y- intercept and moving right 1 and up 2 and make a dot. Notice the slope was 2 and there is a x- coefficient of 2 in the equation. So graphing can be done by knowing your m’s and b’s. Graph the following: 1. 2x + y = 3 2. 6x + 2y = 4 3. -6x + 3y = 9 4. y = x - 3 5. y = 5x - 2 6. y = (½)x - 3 7. y = -2x + 1 8. -x - y = 3 9. y = 3x Luttrell 2012 87 Name: ______________________ Date: _____ 6c –Writing Linear Equations A needed skill is being able to write a model to represent values on a graph. A model is an equation that represents the data. It may be an approximation, especially when the values do not form a true shape of a line, parabola, etc. There are four equations to represent a line, and each are useful when depending on a set of questions or given data. These equations are: slope- intercept (y = mx + b), point-slope (y - y1 = m(x - x1)), standard form (Ax + By = C, where A, B, and C are integers), and double-intercept form ( x/a + y/b = 1). To write the equation of a line, you need either two points or a point and the slope. Let’s follow two methods below with the points (1,-1) and (3,5). First find slope: 5 1 3 1 6 2 3 − − − = = ( ) . Method 1: Method 2: y = mx + b y - y1 = m(x - x1) y = 3x + b y - 5 = 3(x - 3) 5 = 3(3) + b y - 5 = 3x - 9 5 = 9 + b y = 3x - 4 - 4 = b y = 3x - 4 1. Write the equation that has slope 2. Write the equation that has m = 3 of ⅔ and y-intercept of -7. and passes through (0,-2). 3. Write the equation that passes 4. Write the equation that passes through (0,2) and (3,1). through (0,3) and (2,1). 5. Write the equation that passes 6. Write the equation that passes through (2,1) and (4,-3). through (1,2) and (10,2). 7. Write the equation that passes through (-6, 3) and (3,7). Luttrell 2012 90 Name: ______________________ Date: _____ 6f – Perpendicular Lines Lines that form at right angles are said to be perpendicular or orthogonal. If one line has a slope of m, then the other has the slope of -1/m. The product between the slopes of perpendicular lines is always -1. Another way of expressing the slopes is to say the slopes are negative reciprocals of each other. Example 1: Determine if the line 2x - y = 5 is perpendicular to x + 2y = 3. Solution: The equations can be written as y = 2x - 5 and y = (-½)x + 1.5. Since 2(-½) = -1, the lines are perpendicular. Example 2: Determine if the lines 2x - 3y = 5 and 6x + 4y = 3 are perpendicular. Solution: The equations in slope-intercept form are y = (⅔)x-5/3 and y = (-6/4)x + ¾. Since -6/4 × ⅔ = -1, the lines are perpendicular. Note: -6/4 = -3/2 which is the negative reciprocal of ⅔. Example 3: Write the linear equation perpendicular to 2x - y = 5 which passes through (- 4,2). Solution: The given line has slope of 2, so the perpendicular line must have m = -½. Substitute the point and slope into the point-slope form to get y - 2 = -½(x + 4). Simplify into slope-intercept form, y = (-½)x, or standard form, x + 2y = 0. There is a pattern in the examples that make finding perpendicular lines easier. Look out! 1. Determine which of the following lines are perpendicular. Show your work. A. 3x - 2y = 5 B. 6x + 9y = 1 C. 6x - 4y = 4 D. 9x + 6y = 1 2. Which of the following is perpendicular to -3x - y = 4? Show work. A. x + 3y = 2 B. 9x + 3y = 3 C. 3x - y = 3 D. x - 3y = 5 3. Write the linear equation perpendicular to 5x - 3y = 1 that passes through (4, 2). 4. Write the linear equation in standard form that is perpendicular to 4x - 3y = 7 at (1,-1). Luttrell 2012 91 Name: ______________________ Date: _____ 6g – Linear Inequalities The solution to a linear inequality is a set of points that make the inequality true. In the case of the inequality, the solutions include points that lie on one side of the line. The points on the line, the boundary to half-planar solutions, are solutions only if the inequality includes an =. Use the following example as a guide. Example: 6x + 3y > 12 3y > 12 - 6x y > 4 - 2x Plot some points on the line y = -2x + 4. Connect the dots with a dashed line since the line itself contain no solution to the inequality. Now pick a point that does NOT lie on the line. (0,0) is an easy choice. Since 0 + 0 > 12 is false, then the other side of the dashed line must contain the solutions. Shade the solution side. Graph the solutions. 1. -2x - y > 4 2. 3x + y > 5 3. x - y > 4 4. -2x + y ≤4 5. x + 2y ≥ 5 6. 4x - 3y < 9 Luttrell 2012 92 Name: ______________________ Date: _____ 6h – Systems of Equations- Substitution System is to equation as a paragraph is to a sentence. Since all the sentences in the paragraph relate to the same idea, all the equations in a system relate to the same variables. The solution to a system is the ordered pair(s) that makes all the equations true. One way of finding the solution is to graph each equation and see where the graphs intersect. Unfortunately if the graphs intersect anywhere but a lattice point, it is hard to determine the exact values of the ordered pair(s). Instead most mathematicians either do a substitution or an elimination process. Example: solve the system of 2x + y = 3 and 3x - 5y = -2. Step I: choose one variable in one equation to solve y = 3 - 2x Step II: Substitute the expression into other equation 3x - 5(3-2x) = -2 Step III: Simplify and solve for remaining variable 3x - 15 + 10x = -2 13x - 15 = -2 13x = 13 x = 1 Step IV: Substitute known variable into either equation 2(1) + y = 3 2 + y = 3 y = 1 Step V: write answer Ans: (1,1) Solve. Sketch a graph to confirm results. 1. x + y = 3 2. 5x - y = 6 3. y = -x + 1 x - y = 2 3x - 2y = -2 y = -3x + 5 4. y = 3x + 5 5. x - 4y = 5 6. 4x - 2y = 2 y = 2x + 2 3x - 4y = -1 -3x + 2y = 1 Luttrell 2012 95 Name: ______________________ Date: _____ 6k – Linear Word Problems You can identify a linear equation even if it is written in the jargon of a word problem. The biggest clue is to identify a constant change in the values. If you read the cost for a pencil is $0.50, then two pencils should cost $1.00. Then identify your variables, choosing those not easily mistaken for a number. Convert given values into ordered pairs. Find the slope and then write the equation. Once you have an equation, you can predict almost anything. Solve: 1. Pencils are sold at the bookstore for 49 cents each. How much would 75 pencils cost? 2. Some reception halls have a flat fee for use of the hall and then a fee of $25 or more for each person they will be serving. For a wedding reception, Black Forest Inn charges about $64 per person plus $5500 for the use of their inn. How much will it cost my brother to host a wedding of 250 people? 3. An auto repair shop charges $25 an hour. They say your muffler needs to be replaced for $65. It takes them 2.5 hours to fix your exhaust system. How much will the bill be? 4. To frame an oil painting, framers charge a rate based on the perimeter of the painting. You choose a polished wooden frame that has a price tag of $8 a foot. Will the price of framing exceed your $125 budget if your painting is 3.5' by 2.5'? 5. A Nissan Sentra sold in 1996 for $17,000. Ten years later its worth is $3,400. How much would it be worth in another five years, if the depreciation followed a linear model? (In real life, cars depreciate exponentially.) Luttrell 2012 96 Name: ______________________ Date: _____ 6L – More Word Problems (Mixed Review) Solve the word problems: 1. When water freezes, it is 0̊C (or 32̊F). When water reaches its boiling point, it is 100̊C (or 212̊F). What is the temperature in Fahrenheit if the water is 72̊C? 2. Machinery is purchased for $450, but five years later is worth $0. What would be the worth of the machinery after two years? (Assume linear depreciation.) 3. A room is twice as long as the width. If the area of the room is 72 square feet, what is the perimeter? 4. A lawn is 1000 feet around. If the length is three times the width, how much square feet is the lawn? 5. A recipe wants 4 cups of flour and 1 cup of oats. How much flour is needed for ⅔ cups of oats? 6. A recipe wants 3 cups of flour and ½ cup of butter. How much butter would you use for 8 cups of flour? 7. Keith is 16 years older than Shirleen and three times as old as Rachel. If the sum of their ages is 96, how old are they? Luttrell 2012 97 Chapter 6 Test Name: ___________________________ Date: ____________ SHOW WORK. A calculator is allowed on this test. Attach any scratch paper that’s used. 1. Solve for the variable: A. -3(n + 2) = 21 B. 3 4 3 8p = C. 1 35− =x 2. An airplane travels eight times as fast as a car. The difference in their speeds is 420 km/h. How fast is each vehicle travelling? 3. Graph the following lines. Label intercepts. a. y = -3 B. x = 4 C. y = 2x - 3 4. Graph the solutions: a. 2y < -6 b. y < 2x - 3 c. 2x - y ≥5 5. Solve the system: 3x - 2y = 6 x + 2y = 10 5 5 10 10 5