ࡱ> ~q`  `bjbjqPqP <::g7...$RP*>LR@M:.*> LLLLLLL$QhdT|LE.L:LAAA.:LALAA> .D: Mios@:;HTM0@MATATtDT.DDJ|AtdJJJLLjJJJ@MRRRD~D8RRRRRR  Northwest High Schools Bridge to Algebra 2  Summer Review Packet 2009 DUE Monday, August 31, 2009 Student Name ________________________________________ This packet has been designed to help you review various mathematical topics that will be necessary for your success in Bridge to Algebra 2. Instructions: DO ALL PROBLEMS WITHOUT USING A CALCULATOR. Show all work that leads you to each correct solution. Additional copies of this packet may be obtained from the Northwest High School website:  HYPERLINK "https://northwesths.net" https://northwesths.net ALL work should be completed and ready to turn in. This packet will count as part of your first quarter grade. Due Date: Monday, August 31, 2009 Deadline: Friday, September 4, 2009 If you have any questions regarding the summer math packet, please call Vic Weinstein at (301) 601-4651. ENJOY YOUR SUMMER! WE ARE LOOKING FORWARD TO SEEING YOU IN THE FALL.  Table of Contents Fractions page 1 Order of Operations page 2 Integers page 3 Rounding Numbers page 5 Evaluating Expressions page 6 Combining Like terms page 7 Graphing page 8 Solving Equations page 12 Exponents page 13 10. Polynomials page 14 11. Factoring page 15 Fractions To simplify a fraction, divide numerator and denominator by a common factor. Ex.-  EMBED Equation.3  To add or subtract fractions, rewrite the fractions using a common denominator, then add or subtract the numerators. Ex.-  EMBED Equation.3  To multiply fractions, multiply numerator times numerator and denominator times denominator. To divide fractions, multiply by the reciprocal. Simplify answers as needed. Ex.-  EMBED Equation.3  Ex.-  EMBED Equation.3  Simplify the following fractions: 1.  EMBED Equation.3  = 2.  EMBED Equation.3 = 3.  EMBED Equation.3  = Perform the following operations and simplify if necessary: 4.  EMBED Equation.3  = 5.  EMBED Equation.3 = 6.  EMBED Equation.3 = 7.  EMBED Equation.3  = 8.  EMBED Equation.3  = 9.  EMBED Equation.3  = 10.  EMBED Equation.3  = 11.  EMBED Equation.3  = 12.  EMBED Equation.3  = 13.  EMBED Equation.3  = 14.  EMBED Equation.3  = 15.  EMBED Equation.3  = 16.  EMBED Equation.3  = 17.  EMBED Equation.3  = 18.  EMBED Equation.3  = Order of Operations To avoid having different results for the same problem, mathematicians have agreed on an order of operations when simplifying expressions that contain multiple operations. 1. Perform any operation(s) inside grouping symbols. (Parentheses, brackets above or below a fraction bar) 2. Simplify any term with exponents. 3. Multiply and divide in order from left to right. 4. Add and subtract in order from left to right. One easy way to remember the order of operations process is to remember the acronym PEMDAS or the old saying, Please Excuse My Dear Aunt Sally. P - Perform operations in grouping symbols E - Simplify exponents M - Perform multiplication and division in order from left to right D A - Perform addition and subtraction in order from left to right S Example 1 Example 2 2 32 + (6 + 3 x 2) -7 + 4 + (23 8 -4) 2 32 + (6 + 6) -7 + 4 + (8 8 -4) 2 32 + 12 -7 + 4 + (8 - -2) 2 9 + 12 -7 + 4 + 10 -7 + 12 -3 + 10 = 5 = 7 Evaluate each expression. Remember your order of operations process (PEMDAS). 1. 6 + 4  2 " 3 = 2. 15 3 " 5  4 = 3. 20  7 " 4 = 4. 3(2 + 7)  9 " 7 = 5. 12 " 5 + 6 6 = 6.  EMBED Equation.3  6 " 2 = 7. 16 2 " 5 " 3 6 = 8. 10 " (3  62) + 8 2 = 9. 32 [16 (8 2)] = 10. 180 [2 + (12 3)] = 11. 5 + [30  (8  1)2] = 12. 5(14  39 3) + 4 " 1/4 = 11 - 2 Operations with Integers Adding Integers Like SignsDifferent Signs Add the numbers & carry the sign Subtract the numbers & carry the sign of the larger number ( + ) + ( + ) = + ( +3 ) + ( +4 ) = +7 ( + ) + ( ) = ? ( +3 ) + (2 ) = +1 ( ) + ( ) = ( 2 ) + ( 3 ) = ( 5 ) ( ) + ( + ) = ? ( 5 ) + ( + 3 ) = 2 Subtracting Integers Dont subtract! Change the problem to addition and change the sign of the second number. Then use the addition rules. ( +9 ) ( +12 ) = ( +9 ) + ( 12) ( +4 ) (3 ) = ( +4 ) + ( +3 )  ( 5 ) ( +3 ) = ( 5 ) + ( 3 ) ( 1 ) ( 5 ) = ( 1 ) + (+5)  Simplify. Do not use a calculator for this section 1. 9 + (-14) 2. 20 (-6) 3. -18 + 7 4. -14 16 5. -6 (-7) 6. -30 + (-9) 7. 14 20 8. -8 17 9. -12 + 11 10. 1 (-12) Multiplying and Dividing Integers If the signs are the same, If the signs are different, the answer is positive the answer is negative Like SignsDifferent Signs (+ ) ( + ) = + ( +3 ) ( +4 ) = +12  ( + ) ( ) = ( +2 ) ( 3 ) = 6 ( ) ( ) = + ( 5 ) ( 3 ) = + 15 ( ) ( + ) = ( 7 ) ( +1 ) = 7  (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12  ( + ) / ( ) = ( +2 ) / ( 3 ) = 6 (+ ) / ( + ) = + ( +3 ) / ( +4 ) = +12  ( ) / ( + ) = ( 7 ) / ( +1 ) = 7  Simplify. Do not use a calculator for this section. (-25)(-3) = 2. -200 = -1 (-11)(-5) = 4. (-2)(5)(-7) = 5. -66 = 6. (30) (-4) = 2 (2)(14)(-1) = 8. 48 = -4 9. -120 = 10. -168 = -4 8 Rounding Numbers Step 1: Underline the place value in which you want to round. Step 2: Look at the number to the right of that place value you want to round. Step 3: If the number to the right of the place value you want to round is less than 5, keep the number the same and drop the rest of the numbers. If the number to the right of the place value you want to round is 5 or more, round up and drop the rest of the numbers. Example: Round the following numbers to two decimal places (hundredths).  Hundredths 23.1246 23.12  2. 64.2685 64.27   3. 83.7951 83.80  Round the following numbers to the hundredths place. 1. 18.6231 _____________ 6. 0.2658 ______________ 2. 25.0543 _____________ 7. 100.9158 _____________ 3. 3.9215 _____________ 8. 19.9816 ______________ 4. 36.9913 _____________ 9. 17.1083 ______________ 5. 15.9199 _____________ 10. 0. 6701 ______________ Evaluating Expressions Example Evaluate the following expression when x = 5 Rewrite the expression substituting 5 for the x and simplify. 5x = 5(5)= 25 -2x = -2(5) = -10 x + 25 = 5 + 25 = 30 5x - 15 = 5(5) 15 = 25 15 = 10 3x + 4 = 3(5) + 4 = 19 Evaluate each expression given that: x = 5 y = -4 z = 6 1. 3x 5. y + 4 2. 2x2 6. 5z 6 3. 3x2 + y 7. xy + z 4. 2 (x + z) y 8. 2x + 3y z 9. 5x (y + 2z) 13. 2x(y + z) 10. x2 + y2 + z2 14.  EMBED Equation.3  Combining Like Terms What is a term? The parts of an algebraic expression that are separated by an addition or subtraction sign are called terms. The expression 4x + 2y 3 has 3 terms. What are like terms? Terms with the same variable factors are called like terms. 2n and 3n are like terms, but 4x and 3y are not like terms because their variable factors x and y are different. To simplify an expression, you must combine the like terms.  Simplify 1. 5x + 8x 2. 3y 6y 5x + 8x = (5 + 8)x = 13x 3y 6y = (3 6)y = -3y 3. 3x + 4 2x + 3 4. 2b + 5c + 3b 6c 3x 2x + 4 + 3 = (3 2)x + 4 + 3 = x + 7 2b + 3b + 5c 6c = (2+3)b + (5 6)c = 5b c Practice: Simplify each expression 6n + 5n 2. 37z + 4z x 5x 4. 3n + 1 2n + 8 7t + 9 4t + 3 6. 4r + 3r + 6y 2y 8g + 9h 4g 5h 8. 2m + 3n 4m + 5n 6x + 5x + 4x + 10x 10. 3x + 5xy + xy + 4y Graphing Points in a plane are named using 2 numbers, called a coordinate pair. The first number is called the x-coordinate. The x-coordinate is positive if the point is to the right of the origin and negative if the point is to the left of the origin. The second number is called the y-coordinate. The y-coordinate is positive if the point is above the origin and negative if the point is below the origin. The x-y plane is divided into 4 quadrants (4 sections) as described below.    All points in Quadrant 1 has a positive x-coordinate and a positive y-coordinate (+ x, + y). All points in Quadrant 2 has a negative x-coordinate and a positive y-coordinate (- x, + y). All points in Quadrant 3 has a negative x-coordinate and a negative y-coordinate (- x, - y). All points in Quadrant 4 has a positive x-coordinate and a negative y-coordinate (+ x, - y). Plot each point on the graph below. Remember, coordinate pairs are labeled (x, y). Label each point on the graph with the letter given. 1. A(3, 4) 2. B(4, 0) 3. C(-4, 2) 4. D(-3, -1) 5. E(0, 7)  Example: F(-6, 2)      Determine the coordinates for each point below: Example. ( 2 , 3 ) 6. (____, ____) 7. (____, ____)    8. (____, ____) 9. (____, ____) 10. (____, ____)  Complete the following tables. Then graph the data on the grid provided.  Example: y = -2x - 3  XY -33-21-1-10-3 11. y = x + 2  XY 012 12. y = 2x  XY 012 13. y = -x XY-3-113 14. y =  EMBED Equation.3 x - 1   15. y = - EMBED Equation.3 x + 1 XY-303 Solving Equations To solve an equation means to find the value of the variable. We solve equations by isolating the variable using opposite operations.  Example: Solve. 3x 2 = 10  + 2 + 2 Isolate 3x by adding 2 to each side. 3x = 12 Simplify 3 3 Isolate x by dividing each side by 3.   x = 4 Simplify Check your answer. 3 (4) 2 = 10 Substitute the value in for the variable. 12 2 = 10 Simplify 10 = 10 Is the equation true? If yes, you solved it correctly! Solve the following equations. 1. 7x 17 = 60 2. 5y 13 = 37 3.  EMBED Equation.3  4. 3(x+2) = 18 5. -2 + 10x = 8x -1 6. 2(a 3) + 5 = 3(a 1) 7. -4y + 3y 8 = 24 8.  EMBED Equation.3  9. -4r + 5 6r = -32 10. 6x + (-3) = -12 11. 6y 14 3y = 8(7 (-2)) 12. 4c + 5c 8c = 13 + 6 Exponents  Exercises: Simplify using the Rules of Exponents. 1.  EMBED Equation.3   EMBED Equation.3  2.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3.  EMBED Equation.3   EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  7.  EMBED Equation.3  8.  EMBED Equation.3  9.  EMBED Equation.3  10.  EMBED Equation.3  11.  EMBED Equation.3  12.  EMBED Equation.3  Express using a positive exponent. 13.  EMBED Equation.3  14.  EMBED Equation.3  Addition, Subtraction and Multiplication of Polynomials Only like terms can be added or subtracted. Like terms have the same variables with the same exponents. Only the coefficients (numbers) are added or subtracted. A subtraction sign in front of the parentheses changes each term in the parentheses to the opposite. Multiply the coefficients and use the rules of exponents for the variables. Remember: FOIL F first O outers I inners L last OR Box Method Examples: 1) Add the polynomials. 2) Subtract the polynomials.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  3) Multiply the polynomials.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  Exercises: Add, subtract, or multiply the polynomials. Show all work. 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  7.  EMBED Equation.3  8.  EMBED Equation.3  9.  EMBED Equation.3  10.  EMBED Equation.3  Factor Polynomials Always look for the greatest common factor first. Dont forget to include the variable in the common factor. Factor into two parentheses, if possible. Check your answer by multiplying. Examples: Factor  EMBED Equation.3  Question: What factor is common to the coefficients of 15, 12, 27, and 3? Answer: 3 Question: What exponent is common to variables of  EMBED Equation.3 ,  EMBED Equation.3 ,  EMBED Equation.3 , and  EMBED Equation.3 ? Answer:  EMBED Equation.3  Pairs of FactorsSums of Factors-1, 2423-2, 1210-3, 85-4, 62=  EMBED Equation.3  Factor EMBED Equation.3  Think: What multiplies to -24 and adds to +5? = EMBED Equation.3  Exercises: Factor the polynomials. Show all work. 1.  EMBED Equation.3  2.  EMBED Equation.3  3.  EMBED Equation.3  4.  EMBED Equation.3  5.  EMBED Equation.3  6.  EMBED Equation.3  7.  EMBED Equation.3  8.  EMBED Equation.3  9.  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