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Academir Charter School Middle Rh RT CE , 2014 Summer Math Packet )( MOE Academir Charter School Middle is committed to fostering continuous student learning, development and achievement. This commitment extends through the summer as to prevent the decay of learning. Research indicates that the summer break may have detrimental learning effects for many students. On average, all students lose skills, particularly in mathematics. It is crucial that parents are diligent during the summer to ensure their students' educational growth and success. Given the connection between the importance of sustaining academic skills, summer learning loss and parental involvement, it is reasonable to assume that structured academic activities during the summer would help mitigate this loss and may even produce gains. In an effort to promote and reinforce mathematics learning and retention during the summer months, Academir Charter School Middle is implementing a Summer Math Packet for all grade levels. The Summer Math Packet is a series of worksheets designed to review basic math skills and keep students thinking mathematically over the summer break. The skills addressed in the packet are important to your student moving to the next level of mathematics. Please have your student follow all directions in the packet and have him/her show all work on an organized and labeled notebook paper, as needed. The students are to bring the completed packet and notebook paper showcasing their calculations of problems within the packet with them on the first day of school. Academir teachers will review the packet during the first few days, which will be followed with a diagnostic assessment to gauge students mathematical knowledge and level. Basic Operations There are four basic operations in mathematics: + (addition) the sum of 8 and 2 is 10 as in 8 + 2 = 10 — (subtraction) the difference of 8 and 2 is 6 as in 8 — 2 = 6 x (multiplication) the product of 8 and 2 is 16 as in 8 x 2 = 16 ÷ or F(division) the quotient of 8 and 2 is 4 as in 2)8 or 8 ÷ 2 = 4 the quotient of 2 divided into 8 is 4 as in 2)8 = 4 the quotient of 8 divided by 4 is 2 as in 8 ÷ 4 = 2 There are several other words that are used: total (+), remainder (—), times (x), divide up (÷). In working these problems, read carefully and rewrite the problems so that you can do them correctly. Example: Find the sum of 15 and 27. 15 + 27 Do the indicated operation. Write the answer on the line. 1. Find the sum of 21 and 35 2. Find the product of 5 and 12 3. Find the difference of 23 and 17 4. Find the sum of 567 and 197 5. Find the quotient of 1,900 and 25 6. Find the product of 31 and 306 7. Find the difference of 321 and 176 8. Find the quotient of 7,004 and 68 9. Find the total of 57, 73, and 81 10. What is 19 times 87? 11. What is the remainder of 694 minus 405? 12. What would you have if you divided up 256 into 8 equal shares? Exponents Recall that multiplication is repeated addition. So 6 times 2 means six 2's added together or 2 + 2 + 2 + 2 + 2 + 2 = 12. An exponent is making repeated multiplication happen, or three to the fourth power multiplied together would be 3 x 3 x 3 x 3 = 81; thus, exponents makes a fifth operation. An exponent is a superscript number written to the upper right of a number. If 6 is the base and the 2 is the exponent, we say "6 to the second power," which is the area of a square with sides of 6 units. So some people will say, "six squared." A problem would look like 6 2 or 6^2, which appears on computers and some calculators. The third power of a number such as four to the third power would look like this-43. So people will say "4 cubed," since the answer would be the area of a cube with sides of 4 units. Example: 82 = 8 x 8 = 64 122 = 12 x 12 = 144 10^2 = 10 x 10 = 100 Do the indicated operation. Write your answer on the line. 1. 22 = 6. 102 = 11. 122 = 2. 32 = 7.82 = 12. 152 = 3. 5 2 = 8. 63 = 13. 105 = 4. 24 = 9. 73 = 14 23 x 32 = 5. 33 = 10. 104 = 15. 103 x 104 = AWL Order of Operations The biggest rule in mathematics is the "Order of Operations." Everyone must have the same rules so that the answers will match. Everyone uses these rules: businesses, engineers, and scientists. Computers and calculators use the rules as well. The rules are four steps: 1. Do the operations with parentheses first. 2. Do exponents. 3. Perform multiplication or division in order from left to right. 4. Finally, do addition and subtraction from left to right. Examples: 1. 5 + (6 x 3) = 5 + 18 = 23 2. 4 + 5 x 8 = 4 + 40 = 44 3. 9 x 5 +3 = 45 + 3 = 15 Do the indicated operations. Write your answer on the line. 1. 7 x 4 —(4 + 7). 9.(15 — 9)x 92 = 2.9 x 6 + 4 = 10. (37 — 21)x(11 + 4)= 3. 15 + 3 4 = 11.12 — 3 + 4 x 8 + 5 — 9 = 4. 5(7 — 3)= 12.2 x 9 + 15 - 3 — 18 = 5.35 — 3 x 7 = 13.2 + 3 — 4 + 5 — 6 + 7 — 8 + 9 x 2 = 6.5 + 7 — 8 = 7. 7 + (8 x 10) — 15= 8. (87 — +3 = 14. 9 — 8 + 7 — 6 + 5 — 4 + 3 — 2 x 2 = Prime Numbers it Prime numbers are natural numbers greater than one that have exactly two factors, one and itself. Composite numbers are numbers that have more than two factors. The Fundamental Theorem of Arithmetic: Every natural number greater than one is either a prime or a composite. A composite can be expressed uniquely as a product of primes. Prime factorization is needed in many problems. To find the prime factorization of a number, express the number in terms of prime numbers. A list of the first ten prime numbers is: 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29. Remember the list never stops; there is no end, and the number one is not a prime number since it only has one factor, itself. Example: Find the prime factorization of 12. 12 ÷ 2 = 6 6 4- 2 = 3 3 is prime, so the prime factorization will be 12 = 2 x 2 x 3 or 2 2 x 3 The identity property tells you that the product of any number multiplied by 1 is that number. Find the prime factorization of the following numbers. Write the answer on the line. 1. 14 S. 36 9. 48 2. 18 6. 31 10. 45 3. 8 7. 125 11. 56 4. 27 8. 96 12. 121 Fractions: Adding and Subtracting Fractions are found on all standardized tests and in all types of applications. Fractions are how the world deals with parts of a whole. Fractions may be proper fractions, improper fractions, or mixed numbers. The power of the number one is the secret of fractions. The number one may take on many different appearances. The equivalent fractions of one have the same numerator and denominator. So there is a series of fractions all equal to one that looks like: 1 = = 3 = = = 8 =... To add or subtract fractions, you must have common denominators. The numerators are what you add or subtract. Therefore, the denominators are just labels that must be the same. After having common denominators, add (or subtract) the numerators. That answer is this number over the common denominator. Examples: To add: 2 4 i 3 X 4 X 12 3 "5 3 ____ ). 1 +qX 12 17 12 = 5 To subtract: - _5_ 1 12 4 2 6 X 12 4X3 12 7 12 Find the sum or difference and reduce to lowest terms. Write the answer on the line. 1. + L1,4- = 5. 2.3+2 = 6. i +4 = 3. 2 7. ai — 1 14 = 11. 253 —214= 4.214+2 = 8. 51 — = 12. 36— 2i= 9. 10i = g-= 10. 5 — = 1 5g - 4 = Fractions: Multiplying and Dividing One way of thinking about multiplication is that the answer is the area of a rectangle. When fractions are multiplied, the answer is part of a rectangle with sides of the two factors. (Picture of When multiplying 2 by 4, the answer is 8 . a square 1 by 1, with the shading of of 3x3 a a Example: 4 X 8 = 4 x 8 — 32 To divide fractions, take the reciprocal of the divisor and multiply. This algorithm will give you an answer, but what does it mean? There are several ways to look at the answer. A square of that pizza. 4 = :11 ; there pizza is cut into 4 rows and 4 columns, so one piece would be of was only of a pizza and you eat a quarter of it. You had 3 pieces. 4 Example: 3 2 3_ 3 8 ÷ 3 = 8 A 2 16 With mixed numbers, change them to improper numbers and follow the Order of Operations. Examples: A 4+2 x n 41 2 4 3.§ 7-- 2 x 3 = 6 = _5_ = 3 X 7 = 21 = 3 =1 3 Find the product or quotient and reduce to lowest terms. Write the answer on the line. 1. 2 X 4 = 2. 4. x 5 .3=s= = = 10.15=4= 7.12x3= 11.122=13= 8.25 x 1 12- = 12. et ÷ 1 14 = 13. What is the area of a poster that is 1 feet by 22 feet? 9.14x5= 14. What is the area of a desktop that is 22 feet by 5 feet? Integers Integers are whole numbers and their opposites. Most of us would say they are positive and negative numbers. Remember that zero is also in the group of integers. Zero is not positive or negative; it is between the positive and negative numbers. Ordering integers from smallest to largest (or least to greatest) will take some thinking. Placing integers on a number line will help. A real number is any number that can be shown on a number line. Usually these numbers would be fractions and decimals. Numbers that cannot be shown as fractions or decimals are called irrational numbers. An irrational number is a number that cannot be represented by a ratio of two integers. That is because this type of number does not terminate. It does not repeat the way rational numbers do. Since these numbers are hard to represent, some are shown with symbols like it or 4-2-. Write the numbers listed below in order from least to greatest. If the numbers are already in order, indicate that by writing correct. Write the answer on the line. 1. 7, -9, 15, 3 4. 38, -49, -23, 34 2. 345, 241, -256, 265 5. 289, -321, 192, 305 3. - 15, -29, 38, 57 6. 21, 28, -15, -18 - - - - Write the numbers listed below in order from least to greatest. Then put the following sets of numbers on the number line provided. Write the answer on the line. ■- 121110-9 -8-7-6 -5 4 -3 -2 1 01+2 434+5 467+8+9101112 7. - 11, 5, 12 8. 51, -4, i 9.64, - 10i, 514, 2 10. - 3i, 53, 41, -12, 11 - 11. -2i, - 2i, 11, 34 12. - 818-, 9i, 3i, 2i, 18 - - Adding Integers When adding two numbers together, a popular model to explain this operation is movement on a number line. Addition of positive numbers moves to the right, and addition of negative numbers moves to the left. There are two basic rules for sign operations: 1) When adding two integers with the same sign, add the absolute values, and use that sign for the answer. 2) When adding two integers with different signs, subtract the two numbers—the lesser absolute value from the greater absolute value. Use the sign of the greater absolute value for the answer. Examples: A. Add -5 + (-7) = Absolute value of -5 is 5 and of -7 is 7, so 5 + 7 = 12 and same signs (negative) = - 12 B. Add 3 + (-7) = Absolute value of 3 is 3 and of -7 is 7 different signs, so 7 - 3 = 4 and using the sign of the greater absolute value = -4 Determine the sum of each of the following. Write the answer on the line. 9. 573 + (-336) = 1. 7 + (-9) = 2.19 + 23 = 10. -96 + 69 = 3. -6 + (-31) = 11. 62 + (-48) = 4. -25 + 42 = 12. -1693 + (-2791) = 5. 108 + (-24) = 13. 871 + (-25) + 531 = 6. - 121 + (-25) = 14. 25 + (-91) + 39 = 7. 150 + (-63) = 15. 17 + (-34) + 51 = 8. 524 + 378 = 16. - 19 + 38 + (-57) = e Subtracting Integers Subtraction is the inverse operation of addition. That means the rules will be similar to addition. If two numbers are added together and the answer is 0, the numbers are called additive inverses or opposites of each other. So, 5 is the opposite of -5, because their sum is 0. Using symbols to show this: x+ (-x) = 0. The rule for subtraction will be to subtract a number, you add the opposite. A problem like 7 — 4 = 3 could be changed to 7 + ( -4) = 3 and have the same answer. To subtract a number, add its opposite. For integers a and b, a — b = a + (-b). Examples: B. 6 — 11 = 6 + (-11)= -5 D. -12 — (-4) = -12 + 4 = -8 A.5 —(-8). 5 +8 = 13 C. -7 —(-9)= -7 + 9=2 Determine the sum of each of the following. Write the answer on the line. 1. 7 — (-9) = 9. 59 — 79 = 2. 47 — 32 = 10. 128 — 278 = 3. 8 — 15 = 11. 378 — (- 125) = 4. 13 — 34 = 12. - 52 — 52 = 5. -23 — 7 = 13. -168 — 168 = 6. -47 — 57 = 14. 321 — 752 = 7. 17 — (-39) = 15. 5 — 6 — 7 = 8. 28 — (-13) = 16. -12 — (- 14) — (- 16) = 61 Multiplying and Dividing Integers In multiplying and dividing integers, there are two rules. First, the product of two integers that have the same sign is positive. Second, the product of two integers that have different signs is negative. Examples: A. (-6) x (-8) = 48 B. (- 36) ÷ 9 = -4 Determine the solution for each of the following. Write the answer on the line. 1. 7 x (-9) = 9. 30 ÷ 15 = 2. (-5) x 12 = 10. (- 120) ÷ 3 = 3. 6 x 18 = 11. 3 x (-4) + (-2) x (-8) = 4. (-7) x 11 = 12.8 x 3 +(-6)÷2 = 5. 10 x (- 31) = 13.10 x(-5)+ 12 ÷ 4 = 6. 288 ÷ 32 = 14. 4 x (- 11)- 3 x (- 12). 7. 60 ÷ (-30) = 15. (-5) x (-20) 8. 25 ÷ (-5) = 16. 24 - (-3) - 30(-3) = - - 36 ÷ 6 = ( 1 = (99-) - 1-9 '01 = (91-4 (6-) T(9t7-) 'oz = = (617-) + CZ. '6 (9-) ÷ 9Z '6L = LE = (V-) x (9-) - 01--11 - t71 - LL =CV+61-1 = Pl. x (6-) '91 = (L-) - 91. '9 = Z x (1,-) EZ = = 13. - XLL- *SL =9- = (9-) x 91- '1, 1 = (E-) = 'FL 't? + CZ- = 9- x = (9-) - 6- t = 9-x 8-'LL = (94 + •ouil oqi uo JOIRSUe oipm •stuamoad 6upuolioi ay; alndiuo) •Aoamooe pue peeds punq 01 Ail pue oRewinpe qu !Nemo eq !sem., pue sempeocud aye melneu •suolieledo ae6elui pue suogeiedo lo Jew) eqj Jeqie6o1 64upq Si silo(); eq `SeSpi8X8 &won eq u! mpeid salaam zsgimmon Variables A variable may be a letter or symbol. A variable written with a number is an expression. Any operation may be used with a variable. A coefficient is the number that is written next to a variable. The coefficient is multiplied by whatever the variable is. In the expression: 2x + 3y, 2 is the coefficient of x and 3 is the coefficient of y. In an expression, a constant is a quantity that stays the same, a number that is added to or subtracted from a variable. Writing algebraic expressions from verbal expressions is a needed skill. Examples: the sum of a number and 5, would be x + 5 the product of a number and 7, would be 7x 6 less than a number, would be x — 6 a number divided by 2, would be Using x as the variable, write an algebraic expression for each on the line. 1. the product of a number and 12 2. the quotient of a number and 5 3. the total of a number and 16 4. a number decreased by 15 5. a number to the fourth power 6. 3 less than a number 7. twice a number decreased by 12 8. three-fourths of a number plus 25 Write a verbal expression for each on the line. 9. x + 9 10. x — 12 11. 2x — 5 12. 2 + 18 13. x2 + 9 14. 5x — 13 Distributive Property In the multiplication of any two numbers, 6 x 15 for example, there are different ways to determine the answer. We know that 5 x 9 = 45, but 5 x (3 + 6) = 15 + 30 = 45. This is an example of the Distributive Property. The Distributive Property states that for any three numbers a, b, and c, a x (b + c) = (ab) + (ac) or a x (b — c) = (ab) — (ac). A term is a number, a variable, or a product or quotient of numbers and variables. In simplifying an expression, you add or subtract like terms together. In adding or subtracting, you add or subtract the coefficients together. (You do not add exponents together!) Example: x x (3 + 5) = 3x + 5x = 8x; 3x and 5x are the like terms. Since they are like terms, the coefficients can be added or subtracted. Remove parentheses and add like terms, if possible. Simplify each expression. Write the answer on the line. 1.15 x(2 +3)= 9.y x (7 + 13) = 2.11 x(3 + 7)= 10.3 x(x+ 5). 3. 6 x (12 + 9) = 11. 15 x (4+x). 4. 8 x (10 + 12) = 12. 5(x + y) = 5. 8 x (20 — 7) = 13. x(15 + 8) = 6. 9 x (30 — 3) = 14. y(10 7. 12 x (30 — 5) = 15. x(x + 9) = 8. x(4 + 8) = 16. x(2x — — y) = 3) = - Solving One-step Equations If you are asked to solve an equation for a variable, you need to find a value for the variable that makes the equation true. One-step equations require you to reverse the operation of the problem to find the value of the variable. Examples: Addition Property of Equality: You can add or subtract the same number from both sides of an equation and the equation will stay balanced. A. x + 3=9 B. x - 5 = 12 x+3-3=9-3 x - 5 + 5 = 12 + 5 (subtract 3 from both sides) (add 5 to both sides) x=6 x = 17 Multiplication Property of Equality: You can multiply or divide both sides of an equation by the same number and the equation will stay balanced. C. 5x = 25 D. 4=8 5x _- 25 4x4=8x4 5 5 (divide both sides by 5) (multiply both sides by 4) x =5 x = 32 Solve the following problems using the Properties of Equality. Show each step. Write the answer on the line. 1. x + 9 = 12 9. 6x = 18 2. x + 15 = 28 10. 8x = 56 3. x + 5 = 8 11. 15x = -45 4. x + 25 = - 15 12. 3x = 36 5. x - 15 = 21 13. A. = 8 5 6. x - 23 = 17 14. A. = 8 8 7. x - 5 = 8 15. 8. x - 12 = - 17 16. - - - '4 ( 4) = 24 = - 10 111. Solving TWo-step Equations In two-step problems, you work the equation backward. You have the two Properties of Equality, one for addition/subtraction and one for multiplication/division. Since this is backward, the Order of Operations is reversed. Examples: 4x + 6 = 18 A. x B. a - 7 = -5 x a - 7 + 7 = -5 + 7 4x + 6 — 6 = 18 — 6 (subtract 6, both sides) (add 7, both sides) x 4x = 12 4x 12 4=4 8=2 ax x 8 = 2 x 8 (divide by 4, both sides) (multiply by 8, both sides) x =3 x = 16 Solve the following problems using the Properties of Equality. Show each step. Write the answer on the line. 1. 4x 7 = 9 2. 7x + 4 = 32 3. 5x + 3 = 18 9. (4) + 6 = 13 10. 11. 2 — 5 = 23 15 — 3 = -2 4. 2x — 6 = 14 12. (1-) + 12 = 7 5. 6x + 11 = 35 13. x + 3 = 19 6 6. -4x + 3 = 31 14. 12x + 9 = 12 4 7. -3x — 5 = 28 8.6 + 15 = 16 15. 9 —5 =8 16.x + 5 = - 17 +5 3 - v(5 Solving Equations—Variable on TWo Sides 0111.001.0, 4044•Muff, r.11.1PMC, ...10i , s.....1* You may find complicated equations with variables on both sides. To solve these equations, use the Addition Property of Equality that will remove a variable from one side. Then solve for the variable. Examples: A. 2x + 3 = x — 3 2x+ 3 —x=x— 3 —x (subtract x, both sides) x + 3 = -3 x + 3 — 3 = -3 — 3 (subtract 3, both sides) x = -6 B. 5x— 10 = 3x+ 4 5x-10 —3x= 3x+ 4 — 3x (subtract 3x, both sides) 2x — 10 = 4 2x— 10 + 10 = 4 + 10 (add 10, both sides) 2x = 14 2x _ 14 2 2 (divide by 2, both sides) x=7 Solve the following problems using the Properties of Equality. Show each step. Write the answer on the line. 1.3x+ 2 = 2x+ 6 9. 6x + 7 = 8x-13 2.5x+ 3 = 4x+ 10 10. 1 + 2x= 5 —2x 3. 4x — 7 = 2x — 3 11. 13 + 3x= 13 + 2x 4. 10x — 15 = 6x— 7 12. 5 — 2x=x— 7 S. 2x+ 25 =x+ 11 13. - 12 +x= 15 — 2x 6.13x+ 15 = 7x+ 27 14. -3 — x = 24 — 4x 7. 3x — 1 = 4x + 5 15. x + 3 = x + 6 8. 2x+ 9 =5x+ 3 16. -5x— 10 =-5x+2 ANSWER KEY Page 7 +41 ei rade 1. -2 2. 42 3. -37 4. 17 5. 84 6. - 146 7. 87 8. 902 9. 237 10. -27 11. 14 12. -4,484 13. 1,377 14. -27 15. 34 16. -38 Page q 1. 16 2. 15 3. -7 4. - 21 5. -30 6. - 104 7. 56 8. 41 9. -20 10. 150 11. 503 12. - 104 13. - 336 14. -431 15. -8 16. 18 1 - paqe I 1. 56/. 60 3. 6 4. 764 5. 76 6. 9,486 7. 145 8. 103 9. 211 10. 1,653 11. 289 12. 32 .t the Page 10 1. - 63 2. - 60 3. 108 4. -77 5. -310 6. 9 7. -2 8. -5 9. -2 10. -40 11. 4 12. 21 13. -47 14. -8 15. 94 16. 82 PoT l. 4 2. 9 3. 25 4. 16 5. 27 6. 100 7. 64 8. 216 9. 343 10. 10,000 11. 144 12. 225 13. 100,000 14. 72 15. 10,000,000 n Page S 1. 17 2. 58 3. 1 4. 20 5. 14 6. 4 7. 72 8. 27 9. 552 10. 240 11. 37 12. 5 13. 17 14. 2 • Page 11 1. - 14 2. -4 3. - 11 4. -21 5. 11 6. 25 7. 24 8. -51 9. 24 10. 119 11. 40 12. 18 13. -4 14. -90 15. 847 16. - 126 17. 1 18. 10 19. --20 20. - 10 Page 4- 1.2x 72.2 x 3 x 33.2 x 2 x 2 4. 3x 3 x 35.2 x 2 x 3 x 3 6.1 x 317.5 x 5 x58. 2x2x2x2x2x 39.2x 2x2 x 2 x 3 10. 3 x 3x 511.2 x 2 x 2 x 712.11 x Page . 1. 5 / 8 2. 5/6 3. 1 1/6 (7/6) 4. 2 3/4 5. 1/12 6. 1/3 7. 2 1/12 (25/12) 8. 1 3/4 (7/4) 9. 7 1/12 10. 11 1/6 11. 3 7/12 12. 13/24 Page (C) 1. 3/8 2. 1/6 3. 9/20 4. 1 1/2 (3/2) 5. 1 7/9 (1619) 6. 1 3/5 (8/5) 7. 1 8. 4 9. 3/4 10. 20 11. 7 1/2 12. 7 13. 3 3/4 sq ft 14. 12 1/2 sq ft Page 1. 12x 2. x/5 3. x + 16 4. x - 15 5. x4 6. x - 3 7. 2x 12 8. 3/4x + 25 Examples of answers: 9. a number plus 9 10. a number decreased by 12 11. twice a number minus 5 12. half of a number plus 18 13. a number squared plus 9 14. five times a number decreased by 13 - Page 13 1. 75 2. 110 3. 126 4. 176 5. 104 6. 243 7. 300 8. 12x 9. 20y 10. 3x + 15 11. 60 + 15x 12. 5x + Sy 13. 23x 14. lOy -y2 15. x2 + 9x 16. 2x2 - 3x Page 14- 1. 3 2. 13 3. - 13 4. -40 5. 36 6. 40 7. -3 8. -5 9. 3 10. 7 11. -3 12. - 12 13. 40 14. 64 15. -96 16. 90 ' Page 15 1. 4 2. 4 3. 34. 10 5. 4 6. -7 7. - 11 8. 6 9. -28 10. 56 11. 15 12. 15 13. 48 14. 1 15. 39 16. 18 Page lip 1. 4 2. 7 3. 2 4. 2 5. - 14 6. 2 7. -6 8. 2 9. 10 10. 1 11. 0 12. 4 13. 9 14. 9 15. no solution 16. no solution Page 7 1. -9, 3, 7, 15 2. -345, - 256, 241, 265 3. - 29, - 15, 38, 57 4. -49, -38, -23, 34 5. -321, -305, -289, 192 6. -18, -15, 21, 28 7. - 11, 5, 12 8. -4 1/27, 3/4, 5 1/2 9. -10 3/4, -8 1/4, 1/2, 5 1/4 10. - 12, -5 2/3, -3 1/3, 4 1/2, 11 11. -2 3/4, -2 1/4, 1 1/2, 3 1/4 12. -9 7/8, -8 1/8, -3 3/8, 1 7/8, 2 5/8 Writing Answers will vary.