Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Name: ___________________________________ This packet is designed to help you retain the information you learned in 7th grade and realize what skills are essential for you to have as you enter 8th grade. The most important topics to review for next year are INTEGERS (know your rules!) and ALGEBRA. Use websites and online games to help you strengthen your skills in these areas! (ex. www.math.com or www.mathguide.com/lessons/Integers.html) Have a Wonderful Summer! Your eighth grade teachers look forward to working with you next year. Topic: Integers Examples: Addition Same signs: Add & keep sign +6 + +5 = +11 -8 + -2 = -10 Subtraction +10 -5 Different signs: Subtract & take sign of larger value +9 + -5 = +4 -6 + +1 = -5 Multiplication Keep–Change-Opposite - - 8 = +10 + +8 = 18 Division Same signs: Positive product (+7) (+8) = +56 (-2) (-6) = +12 Same signs: Positive quotient +42 +6 = +7 -24 -8 = +3 Different signs: Negative product (+3) (-9) = - 27 (-5) (+4) = - 20 Different signs: Negative quotient +56 -7 = - 8 -50 +2 = - 25 – +12 = -5 + -12 = -17 -20 - -8 = -20 + +8 = - 12 Recall the order of operations: 1 – Parentheses (or grouping symbols) 2 - Exponents 3 - Multiplication / Division (left to right) 4 - Addition/Subtraction (left to right) Find each answer. 1. - 12 + - 7 = ______ 2. - 25 + 18 = ____ 3. 2 + - 25 = ____ 4. - 28 – - 8 = _____ 5. 11 – - 5 = ____ 6. - 21 – 4 = ____ 7. (- 9) (- 8) = _____ 8. ( 2 ) ( - 12) = _____ 9. - 35 - 7 = _____ 10. - 48 + 8 = _____ 11. (- 2) ( + 6) (- 5) = _____ 12. - 30 + 24 6 . - 2 = _____ 13. 16 4 + 2 . - 8 = _____ 14. - 3 (1 – 8) + 23 = _____ ** Practice your INTEGER RULES using websites and on-line games!! You really MUST know these!! ** page 1 Scientific Notation A number written as a number that is at least 1, but less than 10 multiplied by a power of 10. Ex. 7.16 105 716,000 in standard form 9.2 103 .0092 in standard form Write each of the following in standard form. Answers: 15. 8.2 105 15. ___________________ 16. 2.45 104 16. ___________________ 17. 7.28103 17. ___________________ 18. 9.1102 18. ___________________ Write each of the following in scientific notation. 19. 25,900 19. ___________________ 20. .039 20. ___________________ 21. .0007 21. ___________________ 22. 3,207,000,000 22. ___________________ Rounding Rounding a numerical value means replacing it by another value that is approximately equal but has a shorter, simpler, or more explicit representation. We round numbers to a specific place value. UNDERLINE the place value you're rounding to. Then check the place to the right and decide whether to keep it the same or round up. Round each of the following decimals to the nearest tenth. Write your answer on the blank. 23. 87.46 _____________ 24. 8.862 _____________ 25. 13.4395 _____________ 26. 654.839 _____________ 27. 23.648 _____________ 28. 32.971 _____________ Round each of the following decimals to the nearest hundredth. Write your answer on the blank. 29. 26.879 _____________ 30. 429.76492 _____________ 31. 675.495132 _____________ 32. 3.8961 _____________ page 2 Topic: Square roots The square of 5 is 25. 5 ∙ 5 = 52 = 25 5 5 The square root of 25 is 5 because 5∙5 = 25 OR 52 = 25 The square of an integer is called a perfect square. The square root of a perfect square is an integer. Complete the T –Chart below. You must know these perfect squares. Square Root of a Perfect Square Integer Value 1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 page 3 To approximate square roots that are not perfect squares, determine the perfect squares which the radicand lies between. Example: To approximate 32 , think of the perfect squares 32 is between (25 and 36). 25 < 5 < 32 < 36 < 32 6 therefore 32 is between 5 and 6. (Because 32 is closer to 36 than 25, we may estimate that 32 ≈ 5.7.) Find each square root. DO NOT USE A CALCULATOR. 1. 16 = 4. 225 = ____ ____ 2. 36 = 5. 169 = ____ ____ 3. 121 = 6. 81 = ____ ____ State which integers the square root lies between. Show your work. DO NOT USE A CALCULATOR. Ex. : 81 < 85 < 100 therefore 85 is between the integers 9 and 10 . 7. _______ < 37 < _______ therefore 37 is between the integers ______ and ______. 8. _______ < 26 < _______ therefore 26 is between the integers ______ and ______. 9. _______ < 61 < _______ therefore 10. _______ < 119 < _______ 11. _______ < 68 < _______ therefore 12. _______ < 50 < _______ therefore 50 is between the integers ______ and ______. 13. _______ < 40 < _______ therefore 40 is between the integers ______ and ______. 14. _______ < 18 < _______ therefore 18 is between the integers ______ and ______. 15. _______ < 32 < _______ therefore 32 is between the integers ______ and ______. page 4 therefore 61 is between the integers ______ and ______. 119 is between the integers ______ and ______. 68 is between the integers ______ and ______. Topic: Polynomials page 4 Multiplying & Dividing Monomials (powers with the same base) Recall that exponents are used to show repeated multiplication. You can use the definition of exponent to help you understand how to multiply or divide powers with the same base. Examples: Multiplication: 1. 23 24 (2 2 2) (2 2 2 2) 27 (Note: the values are: 8 2. x3 16, which is equal to 128, which is 2 7 ) x 2 ( x x x) ( x x) x 5 Product of Powers rule: You can multiply powers with the same base by adding their exponents. Division: 1. 26 2222 2 2 2 2 2 2 24 2 2 2 2 2. x10 xxx x x x x x x x x x x x3 7 x x x x x x x x Quotient of Powers rule: You can divide powers with the same base by subtracting their exponents. Using the power rules above, simplify each expression. Write your answer using exponents. Answers: 1. 72 75 2. 38 36 1. ___________ 6 3. 64 4. x 2 x6 2. ___________ 3. ___________ 4. ___________ 5. n 6 n 4 6. y 3 y 6 5. ___________ 6. ___________ 7. m8 m7 8. x 7 x 7. ___________ 8. ___________ 9. 38 33 10. 1010 106 9. ___________ 10. __________ 11. __________ 11. x11 x7 12. 47 4 13. __________ 14. __________ Find the value of n in each equation. 13. 54 5n 512 12. __________ 14. 127 12n 125 page 5 Combining like terms and applying the Distributive Property In algebraic expressions, like terms are terms that contain the same variables raised to the same power. Only the coefficients of like terms may be different. In order to combine like terms, we add or subtract the numerical coefficients of the like terms using the Distributive Property: ax + bx = (a + b)x . Examples: 1. 2x + 9x = (2 + 9) x = 11x 2. 12y - 7y = (12 – 7) y = 5y 3. 5x + 8 - 2x + 7 = 3x + 15 Here, the like terms are: 5x and -2x = 3x and: 8 + 7 = 15 The Distributive Property of multiplication over addition/subtraction is frequently used in Algebra: Examples: 1. 7 (2x + 9) = 7 2x + 7 9 = 14x + 63 2. 4 (6 – 5x) = 4 (6) - 4(5x) = 24 - 20x Answers: Simplify each expression by combining like terms. 15. 8y + 2y 15. ______________ 16. 10 – 6y + 4y + 9 = 16. ______________ 17. 3x + 7 – 2x = 17. ______________ 18. 8n – 7y – 12n + 5 – 3y = 18. ______________ Apply the distributive property and write your answer in simplest form. 19. 7 (x – 4) = 19. ______________ 20. 5 (4n – 3) = 20. ______________ 21. - 6 (3y + 5) = 21. ______________ 22. - 4 (8 – 9x) = 22. ______________ 23. 8 (3n + 7) – 10n = 23. ______________ 24. – 4 (5 + 7y) + 6 (2y – 9) = 24. ______________ page 6 Topic: Algebra Solving equations by using the Addition, Subtraction or Multiplication Property of Equality. Check the solution. x 5 9 2 5 5 2 x 4 2 1 2 x 8 Ex 1: Check: Ex 2: 7x – 6 = 11x – 14 7 x 6 11 x 14 7x 7x 6 4 x 14 + 14 + 14 8 4x x 5 9 2 4 4 2 = x 8 5 9 2 4 + 5 = 9 9= 9 Check: 7x - 6 = 11x - 14 7(2) - 6 = 11 ( 2) - 14 14 - 6 = 22 - 14 8 = 8 Translate and evaluate the following equations. Ex 3: The product of 4 and a number is 28. Example: 2 4 n 28 Ex 4. The quotient of a number and 3 is 15. n 15 3 n 45 4n 28 4 4 n 7 Key words: Addition: sum, more than , increased by Subtraction: difference, less than Multiplication: product quotient Information: Hints, topics, definitions. Any kind of Division: helpful information. Solve the following equations. Show your work and check your solution. 1. 2 x 5 17 2. x 9 12 3. 5x + 8 = - 12 3 Check: Check: Check: *** Get more practice SOLVING 2-STEP EQUATIONS using websites and on-line games!! ** page 7 (Apply the distributive property first.) 4. 4 x 8 32 5. Check: 7. 8x – 5 = 6x + 7 Check: x 8 20 4 Check: 6. 2 (x – 7) = 8 Check: 8. 10x + 3 = 4x – 15 Check: 9. 6x – 3 = 8x – 11 Check: Translate each sentence to an algebraic equation. Then use mental math to find the solution. Equation Solution 10. One-half of a number is -12. _____________________ _____________ 11. 6 more than 7 times a number is 41. _____________________ _____________ 12. 5 less than three times a number is 10. _____________________ _____________ 13. 16 increased by twice a number is – 24. _____________________ _____________ page 8 Topic: Angle Relationships… and Algebra Notation: m means the " measure of angle " means congruent or equal in measure Vertical Angles Complementary Angles Supplementary Angles 1 1 2 4 3 2 Angles that are opposite each other across two intersecting lines. Two angles whose sum is 90o. m 1 m 3 and m 1 m 2 900 m 2 m 4 1 2 Two angles whose sum is 180o. m 1 m 2 1800 FOR ALL STUDENTS: State how the angle labeled x is related to the angle with the given measurement. Find the value of x in each figure. 1) xº 49º 3 1) _________________________ x = ___________ 2) _________________________ 2) x = ___________ xº 62º page 9 3) 3) _________________________ xº 123º x = ___________ Find the measures of each of the angles indicated. 4) 4) 1 2 3 m 1 = ___________ m 2 = ___________ 50º m 3 = ___________ ENRICHMENT: Required for students entering the Enriched Algebra Course 5) 5) Relationship _____________________ (11x - 4)° 2 (3x + 10)° →1 x = ________________ 3 m 1 = ___________ m 2 = ___________ m 3 = ___________ page 10 Topic: Coordinate Geometry Recall that we can graph coordinates (x, y) on the coordinate plane. The x-axis is the horizontal axis, and the y-axis is the vertical axis. To graph a line: 1) make a table of values for the equation 2) choose approx. 5 values for x and solve each for y 3) plot the coordinates on the coordinate plane 4) draw a line through the points (use arrows) and label the line with the equation Example. Graph the line y = 2x + 1 x 2x +1 y (x, y) -1 2(-1) +1 -1 (-1, -1) 0 2(0) +1 1 (0, 1) 1 2(1) +1 3 (1, 3) 2 2(2) +1 5 (2, 5) y = 2x + 1 FOR ALL STUDENTS: Complete the tables then graph the equations. 1. y = 3x - 1 x 3x - 1 y (x, y) y (x, y) -2 -1 0 1 3 2. y = -2x+3 x - 2x + 3 -2 -1 0 1 3 page 11 Complete the tables then graph the equations. 3. y=½x x ½x y (x, y) 4x – 5 y (x, y) -4 -3 0 2 6 4. y = 4x – 5 x ENRICHMENT: Required for students entering the Enriched Algebra Course 5. Look for a pattern to help you fill in the missing values in the table. Then try to write an equation that relates x and y. a) b) x y x y -2 -11 -2 5 -1 -7 -1 3 0 -3 0 1 1 1 1 -1 2 2 3 3 ___________________________ ___________________________ page 12 Topic: Factors Factors are numbers that are multiplied to get a product. Factors are numbers which a given number is divisible by. Ex. List the factors of 20: 1, 2, 4, 5, 10, 20 20 has 6 different factors The Greatest Common Factor (GCF) of 2 or more numbers is the largest number that is a divisor of the given numbers. Ex. The GCF of 30 and 24 is 6. Common factors or 30 and 24 are 1, 2, 3 and 6. The Least Common Multiple (LCM) of 2 or more numbers is the smallest number which is divisible by each of the given numbers. The LCM is the same as the least common denominator of 2 fractions. Ex. The LCM of 6 and 10 is 30. Note: Recall, you also had a method for finding the GCF and LCM by making a Venn diagram with the prime factorization of the numbers. List all the factors of each of the following numbers. Then count how many different factors the number has. Number of factors: 1. Factors of 28: ______________________________________________ _________ 2. Factors of 40: ______________________________________________ _________ 3. Factors of 45: ______________________________________________ _________ 4. Factors of 36: ______________________________________________ _________ 5. Factors of 100: ______________________________________________ _________ Find the Greatest Common Factor (GCF) for each pair of numbers. (The GCF is the greatest divisor of the numbers.) 6. GCF of 14 and 21 = __________ 9. GCF of 60 and 20 = __________ 7. GCF of 24 and 16 = __________ 10. GCF of 6 and 25 = __________ 8. GCF of 45 and 30 = __________ 11. GCF of 8 and 60 = __________ Find the Least Common Multiple (LCM) for each pair of numbers. 12. LCM of 4 and 6 = __________ 15. LCM of 5 and 7 = __________ 13. LCM of 9 and 12 = __________ 16. LCM of 20 and 80 = __________ 14. LCM of 6 and 10= __________ 17. LCM of 40 and 32 = __________ 18. Write the prime factorization of each number using exponents (remember… make a factor tree!). 40 72 45 700 page 13 Topic: Geometry You should know the following formulas and be able to use them to find the area or perimeter of a geometric figure. Perimeter of a polygon = the sum of the sides Rectangle: Square: Parallelogram: Triangle: Trapezoid: Circle: P = 2l + 2w P = 4s P = s1 + s2 + s3 + s4 P = s1 + s2 + s3 P = s1 + s2 + s3 + s4 Circumference = πd A = lw A = s2 A = bh A = ½ bh A = ½ (b1 + b2)h A = πr2 Find the perimeter and area of each figure. Use π ≈ 3.14 and show your work. 1. 2. 5 cm 4.5 cm 7 cm 3. 4. 7 ½ in 8 cm 10 cm 5 in 4 in 20 cm 9 in 5. 6. 7 cm (Write answers in terms of pi and as a decimal.) 5 ft 5 cm 4 cm 5 cm 11 cm page 14