Summer Review For Students Entering Algebra 2
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1 Summer Review For Students Entering Algebra Teachers and administrators at Tuscarora High School activel encourage parents and communit members to engage in children s learning. This Summer Review For Students Entering Algebra has been developed to provide friends, famil, and communit members a summer resource to help students reach their full potential. The packet has been designed to provide a review of Algebra 1 skills that are essential for student success in Algebra. The Tuscarora Mathematics Department recommends that students complete this booklet during the summer and bring it, with work shown, to school on the first da of Algebra. Assistance with the packet will be provided at the beginning of the school ear. Completion of this packet over the summer before beginning Algebra will be of great value to helping students successfull meet the academic challenges awaiting them in Algebra and beond. These challenges include: Algebra SOL College placement and advanced placement (AP) tests. Resources included in this packet are the following: A listing of the Algebra 1 objectives students need to have mastered to be successful in Algebra, A set of problems emphasizing important objectives from the Algebra 1 curriculum that will help students prepare for Algebra, Directions: Students are requested to work in pencil and show their work in the packet or on lined paper to accompan the packet. Round answers to the nearest hundredth, if necessar. A calculator ma be used on an of the problems in this packet. Families are encouraged to use the man resources available at the following websites: has video lessons and practice problems for all topics listed in this packet -- offers assistance on various algebra topics
2 Algebra 1 Skills Needed to be Successful in Algebra A. Simplifing Polnomial Epressions Objectives: The student will be able to: Appl the appropriate arithmetic operations and algebraic properties needed to simplif an algebraic epression. Simplif polnomial epressions using addition and subtraction. Multipl a monomial and polnomial. B. Solving Equations Objectives: The student will be able to: Solve multi-step equations. Solve a literal equation for a specific variable, and use formulas to solve problems. C. Binomial Multiplication Objectives: The student will be able to: Multipl two binomials. D. Factoring Objectives: The student will be able to: Identif the greatest common factor of the terms of a polnomial epression. Epress a polnomial as a product of a monomial and a polnomial. Find all factors of the quadratic epression a + b + c b factoring and graphing. E. Radicals Objectives: The student will be able to: Simplif radical epressions. F. Graphing Lines Objectives: The student will be able to: Identif and calculate the slope of a line. Graph linear equations using a variet of methods. Determine the equation of a line. 3
3 A. Simplifing Polnomial Epressions I. Combining Like Terms - You can add or subtract terms that are considered "like", or terms that have the same variable(s) with the same eponent(s). E. 1: E. : -8h + 10h 3-1h - 15h 3-8h + 10h 3-1h - 15h 3-0h - 5h 3 II. Appling the Distributive Propert - Ever term inside the parentheses is multiplied b the term outside of the parentheses. E.1: 3(9 " 4) 3# 9 " 3# 4 7 "1 E. : 4 ( ) 4 " " III. Combining Like Terms AND the Distributive Propert (Problems with a Mi!) - Sometimes problems will require ou to distribute AND combine like terms!! E.1: 3(4 " ) +13 3# 4 " 3# " " 6 E. : 3(1 " 5) " 9("7 +10) 3#1 " 3# 5" 9("7) " 9(10) 36 "15+ 63" 90 "
4 PRACTICE SET 1 Simplif ! 15 3.! (11b! 3) 3.! ( 5! 6) 4. ( 8c + 3) + 1(4c! 10) 5.! (! ) + 6(5 + 7) 5
5 I. Solving Two-Step Equations B. Solving Equations A couple of hints: 1. To solve an equation, UNDO the order of operations and work in the reverse order.. REMEMBER! Addition is undone b subtraction, and vice versa. Multiplication is undone b division, and vice versa. E.1: 4 " = = = 8 E. : 87 = " " 1 " 1 66 = "11 "11 "11 " 6 = II. Solving Multi-step Equations With Variables on Both Sides of the Equal Sign - When solving equations with variables on both sides of the equal sign, be sure to get all terms with variables on one side and all the terms without variables on the other side. E. 3: = " 4 " 4 8 = " 4 " 4 4 = = 6 III. Solving Equations that need to be simplified first - In some equations, ou will need to combine like terms and/or use the distributive propert to simplif each side of the equation, and then begin to solve it. E. 4 : 5(4 " 7) = " 35 = "10 "10 10 " 35 = = = 8 6
6 PRACTICE SET Solve each equation. You must show all work. 1. 5! = (3! 4) = = 4(1! 9) 4.! 131 =! 5(3! 8) ! 15 =! (3! 8) IV. Solving Literal Equations - A literal equation is an equation that contains more than one variable. - You can solve a literal equation for one of the variables b getting that variable b itself (isolating the specified variable). E.1: 3 =18, Solve for. 3 3 = 18 3 = 6 E. : 5a "10b = 0, Solve for a. +10b =+10b 5a = 0 +10b 5a 5 = b 5 a = 4 + b 7
7 PRACTICE SET 3 Solve each equation for the specified variable. 1. Y + V = W, for V. 9wr = 81, for w 3. d 3f = 9, for f 4. d + t = 10, for 5. P = (g 9)180, for g h = 10 + u, for 8
8 I. Reviewing the Distributive Propert C. Binomial Multiplication The distributive propert is used when ou want to multipl a single term b an epression. E 1: 8(5 8 " 5 40! 9) + 8 " (! 9)! 7 II. Multipling Binomials the FOIL method When multipling two binomials (an epression with two terms), we use the FOIL method. The FOIL method uses the distributive propert twice! FOIL is the order in which ou will multipl our terms. First Outer Inner Last E. 1: ( + 6)( + 10) FIRST OUTER First " > ( + 6)( + 10) Outer Inner > > 6 INNER LAST Last > (After combining like terms) 9
9 Recall: 4 = 4 4 = E. ( + 5) ( + 5) = ( + 5)(+5) Now ou can use the FOIL method to get a simplified epression. PRACTICE SET 5 Multipl. Write our answer in simplest form. 1. ( + 10)( 9). ( + 7)( 1) 3. ( 10)( ) 4. ( 8)( + 81) 5. ( 1)(4 + 3) 6. (- + 10)(-9 + 5) 7. (-3 4)( + 4) 8. ( + 10) 9. (- + 5) 10. ( 3) 10
10 D. Factoring I. Using the Greatest Common Factor (GCF) to Factor. Alwas determine whether there is a greatest common factor (GCF) first. E ! In this eample the GCF is3. So when we factor, we have 3 (! ). Now we need to look at the polnomial remaining in the parentheses. Can this trinomial be factored into two binomials? In order to determine this make a list of all of the factors of Since = -11 and (-5)(-6) = 30 we should choose -5 and -6 in order to factor the epression. The epression factors into 3 (! 5)(! 6) Note: Not all epressions will have a GCF. If a trinomial epression does not have a GCF, proceed b tring to factor the trinomial into two binomials. II. Appling the difference of squares: a! b = ( a! b)( a + b) E. 4 3 "100 ( ) 4 " 5 ( )( + 5) 4 " 5 Since and 5 are perfect squares separated b a subtraction sign, ou can appl the difference of two squares formula. 11
11 PRACTICE SET 6 Factor each epression a b! 16ab + 8ab c 3.! 5 4. n + 8n g! 9g d + 3d! 8 7. z! 7z! m + 18m ! k + 30k! 135 1
12 E. Radicals To simplif a radical, we need to find the greatest perfect square factor of the number under the radical sign (the radicand) and then take the square root of that number. E. 1: 7 36 " 6 E. : " 9 " 10 4" 3" E. 3: OR E. 3 : " " 3 This is not simplified completel because 1 is divisible b 4 (another perfect square) 4 3 PRACTICE SET 7 Simplif each radical
13 F. Graphing Lines I. Finding the Slope of the Line that Contains each Pair of Points. Given two points with coordinates ( ) and ( ) 1, 1 the line containing the points is! m = 1.! 1, E. (, 5) and (4, 1) E. (-3, ) and (, 3) 1! 5! 4 3! 1 m = = =! m = = 4!! (! 3) 5 1 The slope is -. The slope is 5, the formula for the slope, m, of PRACTICE SET 8 1. (-1, 4) and (1, -). (3, 5) and (-3, 1) 3. (1, -3) and (-1, -) 4. (, -4) and (6, -4) 5. (, 1) and (-, -3) 6. (5, -) and (5, 7) 14
14 II. Using the Slope Intercept Form of the Equation of a Line. The slope-intercept form for the equation of a line with slope m and -intercept b is 3 E. = 3! 1 E. =! Slope: 3 -intercept: -1 Slope:! -intercept: 4 = m + b. Place a point on the -ais at -1. Place a point on the -ais at. Slope is 3 or 3/1, so travel up 3 on Slope is -3/4 so travel down 3 on the the -ais and over 1 to the right. -ais and over 4 to the right. Or travel up 3 on the -ais and over 4 to the left. PRACTICE SET = + 5. =! 3 Slope: -intercept: Slope: -intercept: 15
15 3. =! =! 3 Slope: Slope: -intercept: -intercept 5. =! + 6. = Slope: Slope: -intercept: -intercept 16
16 III. Using Standard Form to Graph a Line. An equation in standard form can be graphed using several different methods. Two methods are eplained below. a. Re-write the equation in = m + b form, identif the -intercept and slope, then graph as in Part II above. b. Solve for the - and - intercepts. To find the -intercept, let = 0 and solve for. To find the -intercept, let = 0 and solve for. Then plot these points on the appropriate aes and connect them with a line. E.! 3 = 10 a. Solve for. OR b. Find the intercepts:! 3 =! + 10 let = 0 : let = 0:! + 10 =! 3! 3(0) = 10 (0)! 3 = =! 3 3 = 10! 3 = 10 = 5 10 =! 3 So -intercept is (5, 0) & 10 # So -intercept is $ 0, '! % 3 " On the -ais place a point at On the -ais place a point at! 3 Connect the points with the line. =!
17 PRACTICE SET = = = ! 3 = 9 18
18 5.! + 6 = 1 6. =! 3 19
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