INTRODUCTION GOOD LUCK!

Transcription

1 INTRODUCTION The Summer Skills Assignment for has been developed to provide all learners of our St. Mar s Count Public Schools communit an opportunit to shore up their prerequisite mathematical skills before formall enrolling in in the fall. Educators from around our count have discussed the primar foundational mathematics skills with which students need to come to so that the can reach their full potential in that class. As a result, the Summer Skills Assignment for addresses eight different prerequisite skills referenced as Themes which have been designed to provide a thorough review of prerequisites that are essential for student success for all students. Completion of this packet over the summer before beginning is completel voluntar; however, it will be of great value to help students successfull meet the academic challenges awaiting them in and beond. These challenges include the following: The Preliminar Scholastic Aptitude Test (PSAT), Scholastic Aptitude Test (SAT) I and 2, and American College Test (ACT) College placement and advanced placement (AP) tests. Resources included in this packet are the following: A listing of the prerequisite objectives students need to have mastered to be successful in, Various sets of problems emphasizing important objectives that facilitate a student s foundational mathematical fluenc and accurac, Answers to those problems. Directions: Students are requested to work in pencil and show their work. The should check their answers using the ke provided and, if possible, correct the work for problems solved incorrectl. All answers should be rounded to the nearest tenth. Calculators ma be used as well as graphing calculators. If ou are still having questions about an of the above topics, please ask our teacher for assistance when school begins or acjaffurs@smcps.org. Families are encouraged to use the man resources available at the following websites that have links to a plethora of mathematics related websites containing activities, tutorials, games, puzzles, and lists of resources: Be persistent and resourceful until ou find a tutorial that is helpful, understandable, and provides good eamples with answers for ou to follow. Don t accept just getting an answer as it is important that ou understand how to successfull complete these tpes of review problems. If ou have difficult with an of these topics, review our notes from prior classes. GOOD LUCK!

2 Prerequisite Skills for Theme 0: The Comple Number Tree Theme 1: 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. Theme 2: 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. Theme 3: Rules of Eponents Objectives: The student will be able to: Simplif epressions using the laws of eponents. Evaluate powers that have zero or negative eponents. Theme 4: Binomial Multiplication Objectives: The student will be able to multipl two binomials. Theme 5: 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 2 + b + c b factoring and graphing. Theme 6: Radicals Objectives: The student will be able to simplif radical epressions. Theme 7: 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. Theme 8: Regression and Use of the Graphing Calculator Objectives: The student will be able to: Draw a scatterplot, find the line of best fit, and use it to make predictions. Graph and interpret real-world situations using linear models. 4

3 Theme 0: The Comple Number Tree Summer Skills Assignment June 2013 Page 5

4 Natural Numbers: These are numbers that are found in nature; 1, 2, 3, 4, 5, 6, 7, 8, 9. Zero: This number represents the absence of an amount: 0. Whole Numbers: These numbers are created b combining the Natural Numbers and Zero. For eample: 1, 10, 54, etc. Negative Numbers: These numbers are created b inverting the sign of the whole numbers. For eample: -1, -10, -54, etc. Integers: These numbers represent all positive and negative whole values. For eample: 1, -1, 10, -10, etc. Fractions: These numbers are created when one Integer is divided b another Integer. For eample: ½, 0.5, ¼. 0.25, ⅓, 0.333, etc. Rational Numbers: These numbers are created combining Integers and Fractions. For eample: 1, ½, -0.25, -10, etc. Irrational Numbers: These are numbers that cannot be epressed as Fractions. For eample: π, 3.14, e, 2.71, etc. Real Numbers: These numbers are created combining Rational and Irrational Numbers. For eample: π, -½, 0.5, etc. Imaginar Numbers: It is impossible to calculate the Square Root of a negative number; therefore imaginar numbers are those that are multiples of the Square Root of -1, also known as i. For eample: (-1), i, -2i, ½i, 0.5i, etc. Comple Numbers: These numbers are created b adding or subtracting an Imaginar Number from a Real Number. For eample: 1 + i, -5 2i, ½i, etc. Summer Skills Assignment June 2013 Page 6

5 Theme 1: 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. 2: -8h h 3-12h 2-15h 3-8h h 3-12h 2-15h 3-20h 2-5h 3 II. Appling the Distributive Propert - Ever term inside the parentheses is multiplied b the term outside of the parentheses. E. 1: E. 2: 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: E. 2: Summer Skills Assignment June 2013 Page 7

6 PRACTICE SET 1 Simplif n - (3 - n) (11b - 3) 5. 10q( ) 6. - (5-6) 7. 3(18z - 4w) + 2(10 z - 6w) 8. (8c + 3) + 12(4c - 10) (6 2) 3(9 3) 10. ( ) 6(5 7) Summer Skills Assignment June 2013 Page 8

7 I. Solving Two-Step Equations Theme 2: Solving Equations A couple of hints: 1. To solve an equation, UNDO the order of operations and work in the reverse order. 2. REMEMBER - Addition is "undone" b subtraction, and vice versa. Multiplication is "undone" b division, and vice versa. E.1: 4-2 = = = 8 E. 2: 87 = = = 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: = = = = 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) = = = = = 8 Summer Skills Assignment June 2013 Page 9

8 PRACTICE SET 2 Solve each equation. You must show all work = = (3-4) = = = 4(12-9) = = -5(3-8) = = -2(3-82) 10. (126) 12 6 Summer Skills Assignment June 2013 Page 10

9 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 E. 2 : 5a 10b 20, solve for a. 10b 10b 5a 2010b 5a 20 10b a 42b PRACTICE SET 3 Solve each equation for the specified variable. 1. Y + V = W, for V 2. 9wr = 81, for w 3. 2d 3f = 9, for f 4. d + t = 10, for 5. P = (g 9)180, for g h = 10 + u, for Summer Skills Assignment June 2013 Page 11

10 Theme 3: Rules of Eponents Multiplication: Recall ( m ) ( n ) = ( m + n ) E: (3 4 2 )(4 5 ) = (3-4)( 4-1 ) ( 2-5 ) = Division: Recall m n mn E: m j 42 m j m j 3 m j 2 m j Powers: Recall ( m ) n = ( m-n ) E: ( 2 abc ) ( 2) ( a) ( b )3 ( c ) 8abc Power of Zero: Recall 0 = 1, - 0 E: = (5)(1)( 4 ) = 5 4 PRACTICE SET 4 Simplif each epression. 1. (c 5 )(c)(c 2 ) 2. m m (k 4 ) 5 4. d 0 5. ( p 4 q 2 )( p 7 q 5 ) 6. 45z z 7. (-t 7 ) f 3 g 0 9. (4h 5 k 3 )(15k 2 h 3 ) a4 b (3m 2 n) (12 2 ) 0 36ab 2 c 13. (-5a 2 b)(2ab 2 c)(-3b) (2 2 ) (3 4 )(2 2 ) 3 Summer Skills Assignment June 2013 Page 12

11 Theme 4: Binomial Multiplication I. Reviewing the Distributive Propert The distributive propert is used when ou want to multipl a single term b an epression. E.: 8(5 2-9 ) (-9 ) 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.: ( + 6)( + 10) FIRST OUTER First > 2 ( + 6)( + 10) Outer Inner > > 6 INNER LAST Last > (After combining like terms) Summer Skills Assignment June 2013 Page 13

12 Recall: 4 2 = = E.: ( + 5) 2 ( + 5) 2 = ( + 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) 2. ( + 7)( 12) 3. ( 10)( 2) 4. ( 8)( + 81) 5. (2 1)(4 + 3) 6. ( )(-9 + 5) 7. (-3 4)(2 + 4) 8. ( + 10) 2 9. (- + 5) (2 3) 2 Summer Skills Assignment June 2013 Page 14

13 Theme 5: Factoring I. Using the Greatest Common Factor (GCF) to Factor Alwas determine whether there is a Greatest Common Factor (GCF) first. E.1: In this eample the GCF is 3 2. So when we factor, we have 3 2 ( ). 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 2 ( - 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: 2 2 a b ab a b ( )( ) E. 2: ( 2-25 ) 4 ( - 5)( + 5) Since 2 and 25 are perfect squares separated b a subtraction sign, ou can appl the difference of two squares formula. Summer Skills Assignment June 2013 Page 15

14 PRACTICE SET 6 Factor each epression a 2 b 2-16ab 3 + 8ab 2 c n 2 + 8n g 2-9g d 2 + 3d z 2-7 z m m k k Summer Skills Assignment June 2013 Page 16

15 Theme 6: 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: E. 2: E. 3: OR 4 3 PRACTICE SET 7 Simplif each radical. E. 4: This is not simplified completel because 12 is divisible b 4 (another perfect square) Summer Skills Assignment June 2013 Page 17

16 Theme 7: Graphing Lines I. Finding the Slope of the Line that Contains Each Pair of Points Given two points with coordinates ( 1, 1 ) and ( 2, 2 ), the formula for the slope, m, of the line containing the points is m = E.: (2, 5) and (4, 1) E.: (-3, 2) and (2, 3) m = = = -2 m = = (-3) 5 The slope is -2. The slope is 1 5 PRACTICE SET 8 1. (-1, 4) and (1, -2) 2. (3, 5) and (-3, 1) 3. (1, -3) and (-1, -2) 4. (2, -4) and (6, -4) 5. (2, 1) and (-2, -3) 6. (5, -2) and (5, 7) Summer Skills Assignment June 2013 Page 18

17 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 = m + b. E.: = 3-1 E.: = Slope: 3 -intercept: -1 Slope: intercept: 2 Place a point on the -ais at -1. Place a point on the -ais at 2. 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 9 1. = = Slope: -intercept: Slope: -intercept: Summer Skills Assignment June 2013 Page 19

18 3. = = -3 Slope: Slope: -intercept: -intercept 5. = = Slope: Slope: -intercept: -intercept Summer Skills Assignment June 2013

19 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. Rewrite the equation in in Part 2 above. = m + b form, identif the -intercept and slope, then graph as 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.: a. Solve for. OR b. Find the intercepts: - 3 = let = 0 : let = 0: (0) = 10 2(0) 3 10 = = 10-3 = = 5 = - 3 So -intercept is (5, 0) So -intercept is 10 0, 3 On the -ais place a point at 5. On the -ais place a point at - 10 = Connect the points with the line. Summer Skills Assignment June 2013

20 PRACTICE SET = = = Summer Skills Assignment June 2013

21 = = -3 Summer Skills Assignment June 2013

22 graph a function Press the Y= ke, Enter the function directl using the X, T,-, n ke to input. Press the GRAPH ke to view the function. Use the WINDOW ke to change the dimensions TI-83 Plus/TI-84 Graphing Calculator Tips How to and scale of the graph. Pressing TRACE lets ou move the cursor along the function with the arrow kes to displa eact coordinates. find the -value of an -value Once ou have graphed the function, press CALC 2nd TRACE and select 1:value. Enter the - value. The corresponding -value is displaed and the cursor find the maimum value of a function Once ou have graphed the function, press CALC 2nd TRACE and select 4:maimum. You can set the left and right boundaries of the area to be eamined and guess the maimum value either b entering values find the zero of a function Once ou have graphed the function, press CALC 2nd TRACE and select 2:zero. You can set the left and right boundaries of the root to be eamined and guess the value either b entering values find the intersection of two functions Once ou have graphed the function, press CALC 2nd TRACE and select 5:intersect. Use the up and down arrows to move among functions and press ENTER to select two. Net, enter lists of data Press the STAT ke and select 1:Edit. Store ordered pairs b entering the coordinates in L1 and the coordinates in L2. You can calculate new lists. To moves to that point on the function. directl or b moving the cursor along the function and pressing ENTER. The -value and -value of the point with the maimum -value are then displaed. directl or b moving the cursor along the function and pressing ENTER. The -value displaed is the root. enter a guess for the point of intersection or move the cursor to an estimated point and press ENTER. The -value and -value of the intersection are then displaed. create a list that is the sum of two previous lists, for eample, move the cursor onto the L3 heading. Then enter the formula L1 + L2 at the L3 prompt. Summer Skills Assignment June 2013

23 plot data Once ou have entered our data into lists, press STAT PLOT 2nd Y= and select Plot1. Select On and choose the tpe of graph ou want, e.g. scatterplot (points not connected) or connected dot for graph a linear regression of data Once ou have graphed our data, press STAT and move right to select the CALC menu. Select 4:LinReg(a + b). Tpe in the parameters L1, L2, Y1. To enter Y1, press VARS draw the inverse of a function Once ou have graphed our function, press DRAW 2nd PRGM and select 8:DrawInv. Then enter Y1 if our function is in Y1, or just enter the function itself. create a matri From the home screen, press 2nd -1 to select MATRX and move right to select the EDIT menu. Select 1:[A] and enter the number of rows and the number of columns. Then fill in the matri b entering a value in each element. solve a sstem of equations Once ou have entered the matri containing the coefficients of the variables and the constant terms for a particular sstem, press MATRX (2nd -1, move to MATH, and select B:rref. generate lists of random integers From the home screen, press MATH and move left to select the PRB menu. Select 5:RandInt and enter the lower integer bound, the upper integer bound, and the number of trials, separated b two variables, histogram for one variable. Press ZOOM and select 9:ZoomStat to resize the window to fit our data. Points on a connected dot graph or histogram are plotted in the listed order. and move right to select the Y-VARS menu. Select 1:Function and then 1:Y1. Press ENTER to displa the linear regression equation and Y= to displa the function. You ma move among elements with the arrow kes. When finished, press QUIT 2nd MODE to return to the home screen. To insert the matri into calculations on the home screen, press 2nd -1 to select MATRX and select NAMES and select 1:[A]. Then enter the name of the matri and press ENTER. The solution to the sstem of equations is found in the last column of the matri. commas, in that order. Press S TO and L1 to store the generated numbers in List 1. Repeat substituting L2 to store a second set of integers in List 2. Summer Skills Assignment June 2013

24 Summer Skills Packet Student Answer Ke Theme 1: Simplifing Polnomial Epressions PRACTICE SET n b q + 110q z - 24w 8. 56c Theme 2: Solving Equations PRACTICE SET 2 1. = 7 2. = = = = = = = = = 0 PRACTICE SET 3 1. V = W - Y 9 2. w = r 3. f = 9-2d = t 10 t d 4. = = d d d 5. g = P = P 9 + u + 5h = Summer Skills Assignment June 2013

25 Theme 3: Rules of Eponents PRACTICE SET 4 1. c 8 2. m k p 11 q z t f h 8 k a 3 b m 8 n c a 3 b 4 c Theme 4: Binomial Multiplication PRACTICE SET Theme 5: Factoring PRACTICE SET ( + 2) 2. 4ab 2 (a - 4b + 2c) 3. ( - 5)( + 5) 4. (n + 5)(n + 3) 5. ( g - 4)( g - 5) 6. (d + 7)(d - 4) 7. ( z - 10)( z + 3) 8. (m + 9) ( - 3)( + 3) 10. 5(k + 9)(k - 3) Summer Skills Assignment June 2013

26 Theme 6: Radicals PRACTICE SET Theme 7: Graphing Lines PRACTICE SET undefined PRACTICE SET 9 1. Slope: 2 -intercept: 5 2. Slope: 1 -intercept: -3 2 Summer Skills Assignment June 2013

27 3. = Slope: intercept: 4 4. = -3 Slope: -intercept = = Slope: -1 Slope: 1_ -intercept: 2 -intercept 0 Summer Skills Assignment June 2013

28 PRACTICE SET Summer Skills Assignment June 2013