FOR STUDENTS WHO HAVE COMPLETED ALGEBRA (Students entering Geometry) Dear Parent/Guardian and Student, Name: Date: Period: Attached you will find a review packet of skills which each student is expected to know upon the start of the year. Students will be given a test (no calculators) on this information during the second week of the school year. Teachers will go over the answers from the packet during the first week of school and minimal direct instruction will be given on these concepts, as they are a review from Algebra. Students may seek additional help during RECAP to ask questions. Topics from Algebra to be tested during the second week of school. Real Numbers and Operations Algebraic Expressions and Models Solving Linear Equations Rewriting Equations and Formulas Solving Linear Inequalities Solving Absolute Value Equations and Inequalities Functions and Their Graphs Slope and Rate of Change Quick Graphs of Linear Equations Writing Equations of Lines Correlation and Best Fitting Lines Linear Inequalities in Two Variables Solving Linear Systems by Graphing Solving Linear Systems Algebraically (Substitution and Linear Combinations) Graphing and Solving Systems of Linear Inequalities Adding, Subtracting, Multiplying Polynomials Factoring You may also access the following websites to assist your child. www.purplemath.com, www.math.com, www.khanacademy.com It is recommended that students who score between 80 and 00 continue with the current course. Students who score below an 80 may consider retaking Algebra, as it is imperative for future successes in math to have essential, baseline skills. Have a good summer. The PAMS Math Department
. Real Numbers and Operations Whole: 0,,, Integers:, -, -, 0,, Rational: Can be written as a fraction or a repeating or terminating decimal. Irrational: Cannot be written as a fraction or a repeating or terminating decimal. Properties of Addition and Multiplication. Commutative Property: a + b = b + a. Associative Property: (a + b) + c = a + (b + c). Identity Property: a + 0 = a for addition, a for multiplication. Inverse Property: a + (-a) = 0 for addition, a a 5. Distributive Property: a(b + c) = ab + bc Practice Set - Identify the following numbers as either W, In, R, or Irr. Remember, the numbers could fall into more than one category!..5. -7.. 5 5. 6 6. Plot the following values on a number line.,,.7,,. 7. a. What is the sum of -9 and 6? b. What property is represented here: + (+5) = (+) + 5 c. What property is represented here: 8 8 d. True or False: (a b) = (b a)
. Algebraic Expressions and Models Order of Operations Parentheses Exponents Multiplication Division Addition Subtraction From left to right! From left to right! Vocabulary Terms: Parts added together to make the expression. Coefficients: The number located in front of the variable. Constant: Numbers in an expression without a variable. Practice Set. Simplify the following: a. b. ( ) c. () ( 5). Simplify: a. b. x 5x 7, when x 5 c. 6 (8 5). Identify the following components from the following expression. 7 5x 8x 7 a. The number of terms b. Leading coefficient c. Constant Term. Simplify: a. x x 7 x b. ( x ) (x ) c. ( x ) ( 5 x )
. Solving Linear Equations Practice Set :. Solve the following equations. a. x 9 5 b. 5 7 9 7 n n c. 5 (x 5) 6 x 8 6x d. x x 6. Rewriting Equations and Formulas Practice Set :. Solve the following equation for y. 7x y 8. Solve the Fahrenheit equation for Celsius. 9 F C 5
5. Solving Linear Inequalities *Remember, when you multiply or divide both sides of an inequality by a negative, you must reverse the inequality symbol. *Closed dot represents and. This means our value is included in the solution. * Open dot represents and. This means our value is not included in the solution. *Compound Inequalities: Two simple inequalities joined by the words and or or. AND: x OR: x or x Example: - - 6x 6x 6x x 6 6 6 Practice Set 5. Graph the following solutions on the number line provided: a. x b. x 5 c. 7 x. Solve and graph the following inequalities. a. 5x 8 b. x 8 0 c. x or x 7
6. Solving Absolute Value Equations and Inequalities *The absolute value of a number x, is written x, is the distance the value is from zero. The absolute value of a number is always positive. x 5 or 5 Example : Solve: x 5 9 x 5 9 or x 5 9 x 7 or x *To solve an absolute value inequality, it is important to remember that a represents an and statement and a or represents an or statement. or Example : x 7 x 7 8 x 9 x Example : x 8 x 8 or x 8 x or 0 x Practice Set 6: Directions: Solve the following absolute value equations.. 6x 5. x 7 Directions: Solve and graph the following absolute value inequalities.. x 0 0. x 6
7. Finding Slope Finding Slope from Graph Finding Slope from Two Points m = y y x x = rise run = m = y y = rise x x run Example: Find the slope of the line that crosses (-, 7) and (, -) 7 8 ( ) 5 Practice Set 7: Directions: Find the slope of the following lines..... 5. (8, 0), (-7, ) 6. (-9, 6), (5, 6)
8. Graphing Linear Equations Slope-Intercept Form Standard Form Graph y x Graph x y *The y-intercept is -. *Use x and y intercepts. *The slope is. *x = 0 for y intercept and y = 0 for x intercept x 0 (0) y x y Practice Set 8: Directions: Graph the following equations.. y x. y x. x y 6 m = m = x int. b = b = y int.
9. Graphing Linear Inequalities. Graph the line the same way you would any other linear equation.. Remember, or represents a dashed line and or represents a solid line.. Chose a test point on the graph to see if it satisfies the inequality. If it does, shade to cover the test point as it is a solution. If it does not work, shade away from it. Example: Graph y x Test Point: (0, 0) 0 (0) 0 TRUE, so shade toward (0, 0) Practice Set 9: Directions: Graph the following inequalities.. y x. y x 6. y
0. Writing Linear Equations in Slope-Intercept Form Slope-intercept form: y = mx + b (-, 5), (, -9) ) Find the slope of the line. m = 9 5 = ( ) 5 ) Find the y-intercept. -9 = () + b 5-9 = + b 5-9 ( ) = 5-5 ( ) 5 5 = b Important! *. Two lines are parallel if they have the same slope. *. Two lines are perpendicular if they have opposite reciprocal slopes. ) Write an equation of the line. y = x + 5 5 Practice Set 0:. (, ), m Directions: Write the equation of the line in slope-intercept form.. (,), m. Write the equation of the line that is parallel to y x 5 and passes through (-, 0).. Write the equation of the line perpendicular to y x 7 and passes through (, -9). 5. Passes through (, ),( 9, ) 6. Passes through ( 7, 8),( 7, )
. Writing Linear Equations in Point-Slope Form Point-slope form: Example : Example : y y = m(x x ). Find the slope of the line. m, ( 5,) (,) and (5, ) m m = = 5 ( ) 9. Substitute m, x, and, y y ( x 5) y ( x ) OR y ( x 5) 9 9 Practice Set : Directions: Write the equation in point-slope form.. (, ), m. (,), m. (5,), (6, ). (, ), ( 9,)
. Writing Linear Equations in Standard Form Standard Form: Example : Example : Ax By C. Isolate the variable terms on the left and the constant term on the right.. Multiply each side by -5 to have integer coefficients and A to be positive. y = 5 x (, -), (, -) 5 x + y = m = ( ) = 5( 5 x + y) = 5( ) y = (x + ) x 5y = 5 y = x 8 x + y = 5 Practice Set : Directions: Write the equation in standard form with integer coefficients.. x y 7 0. 5 y x. 7 x 9 y. (,9), m 7
. Writing an Equation for Line of Best Fit Approximate the best fitting line for the data. ) Graph the table Hours Studying 6 5 7 % Increase - 5 7 6 ) Sketch a line that best fits the data. ) Estimate the coordinates of two points on the line, **not necessarily data points. (7, 6) and (, ) 6 m 7 y ( x ) y x y x * Positive correlation. Practice Set : Directions: Draw a scatter plot of the data, approximate a best-fit line, and state the correlation.. (0) () () () () (5) (6) (7) (8) Year, t 980 98 98 98 98 985 986 987 988 Pounds, b.6.8...7.9.5.6. ***Use t = 0 to represent 980
. Years Since 990 Home Runs 0 5 6 5 6 0 50 5
. Solving Linear Systems by Graphing Graphing: ) Graph both equations. (ie: slope-intercept form, x- and y-intercepts, table of values) ) Find the point of intersection. y = x + -x + y = -x + y = y = - ½ x -x + y = - x - y = -6 The solution is (-, ). The system has no solution. The system has infinitely many solutions. Practice Set : Directions: Solve the following systems of equations by graphing.. y = x + y = x. x + y = x y = 9 Solution: Solution:
. y = 5x y + = 0x. y = x + y + x = Solution: Solution:
5. Solving Linear Systems using Substitution Substitution: ) Solve one of the equations for one of its variables. x + y = x + y = x = y + ) Substitute the expression into the other equation and solve for the other variable. ( y + ) + y = 6y + 6 + y = y + 6 = y = 0 y = 5 ) Substitute the value into any equation from step to solve for the other variable. x = y + x = (5) + ) Write you answer as an ordered pair. x = 8 (-8, 5) Problem Set 5: Directions: Solve the system using substitution.. x 5y = 9 x + y = 7. 6x + y = x + y = 6. x y = 5x 0y =. 6x + y = 9 x + y =
6. Solving Linear Systems using Linear Combinations (Elimination) Linear Combinations: ) Arrange all equations with like terms in columns. x 6y x 7y 6 x 6y x 7y 6 ) Multiply one or both of the equations by a constant to obtain the same coefficients that differ in sign. ) Add the revised equations. ) Substitute the value into the equation from step. 5) Write your answer as an ordered pair. 6x 8y 6x y y 0 y 0 (,0) x 6y x 6(0) x x Problem Set 6: Directions: Solve the system using linear combinations.. x y = 6x + 6y =. x y = 5x 0y = 5. x + y = 8 x 5y =. x 9y = x + y =
7. Solving Linear Inequalities by Graphing Graphing: ) Graph both inequalities. (ie: slope-intercept form, x- and y-intercepts, etc) ) Use a test point to determine which way to shade (solutions to the inequality). ) Shade the areas that are satisfied by the inequalities and find the region shaded by both inequalities. x y < y < - 5 x + x > 0 Test Point for first inequality only. *Use the test point (0, 0) as line does not cross this point. x y (0) (0) 0 TRUE! (0, 0) creates a TRUE statement so I will shade to cover (0, 0) as it is a solution to the first inequality. Problem Set 7: Directions: Solve the system of linear inequalities by graphing.. x + y > x y <. y 5 x y
. x x y < x + 5. y > x y x + y x
8. Writing and Solving Real World Systems Writing a System of Equations ) Define your unknowns. ) Create a system to represent the problem. ) Solve the system using the method of your choice. Example: Rex wants to sign up with a new internet provider. Internet Provider Monthly Membership Hourly Use Fee Inter-Speedway $85 $0.85 Cyber-Zone $60 $.5 Write a system of equations to represent this situation. x = # hours y = total cost y.85x 85 y.5x 60 After how many hours will the cost of the different companies be the same? Use mathematics to explain your answer. Use words, symbols, or both to support your explanation. Use substitution to solve: y.85x 85 y.5x 60.85x 85.5x 60.5x 5 x 50 y.85x 85 y.85(50) 85 y.5 After 50 hours, the cost at each company will be $.50.
Problem Set 8: Directions: Create a system to represent each real world scenario. Then, solve using the method of your choice.. A piggy bank contains a total of 8 coins in nickels and quarters worth $.80. How many quarters and nickels are in the piggy bank?. The Williams family is going to the Johnstown Summer Carnival. They have two ticket options as shown in the table. Ticket Option Admission Price Price per Ride A $5 $0.0 B $ $0.80 How many rides will the Williams have to ride in order for the cost of ticket option A to be the same as the cost for ticket option B? What will the cost be at that time?
9. Polynomials. Multiply Polynomials Use the distributive property to multiply. Ex. (5x)(9x 8) Ex. (x )( 9x + ) ***You can FOIL!!! 5x 0x 8x + 6x + 6x 8x + x. Factoring Factor each polynomial. Ex. (6x 8x 5x) Ex. a b 8ab + ab 6x(x x 9) ab( a b + ). Factoring using ac method (when a = ). Ex. Factor x + 9x 6. Multiply the a and c terms. a c = ( 6) = 6. Determine what two values multiply ( ) = 6 to the ac term but add to the b term. + ( ) = 9. Rewrite your middle term with these x + 9x 6 two numbers, split the new polynomial x + x 9x 6 and factor. x + x 9x 6 x(x + ) 9(x + ) (x + )(x 9). Solving using ac method (when a ). Ex. Solve x 7x + 0 = 0. Multiply the a and c terms. a c = (0) = 0. Determine what two values multiply ( 5) = 0 to the ac term but add to the b term. + ( 5) = 7. Rewrite your middle term with these x 7x + 0 = 0 two numbers, split the new polynomial x x 5x + 0 = 0 and factor. x x 5x + 0 = 0 x(x ) 5(x ) = 0 (x 5)(x ) = 0. Solve for your solutions. x 5 = 0 x = 0 x = 5 x = x = 5,
Problem Set 9: Use the directions to complete each problem. Multiply.. y(6y y + 5). (x + )( 5x + 9). (x + 5)(5x x )) Factor.. x + 5x + 5. x x 6. x 0x + 56
Solve. 7. x + 9x 8 = 0 8. x x + 0 = 0 9. x + 7x = 5 (HINT: Set problem equal to zero before beginning!)
ANSWER KEY for Summer Math Packet Students Who Have Completed Algebra Practice Set. R. In, R. R. Irr 5. W, In, R 6.,.,,.7, 7. a. 7 b. Association of Addition c. Identity of Multiplication d. False Practice Set. a. 8 b. 8 c. -8. a. b. 5 c.. a. Three b. 5 c. 7. a. b. x c. x x 7 x Practice Set. a. x b. n 0 c. 7 x d. 5 x 8 Practice Set. y 7 8 x. C F 5 9 Practice Set 5. a. b. -5 c. 7. a. x b. x 6 c. x.5 or x 7-6.5 7
Practice Set 6. x,. x,. 5 7 x. x 8 or x 0-7.5.5-8 0 Practice Set 7. 5. m. m 6. 5 m. 5 m 7 0 m 0. Undefined Practice Set 8. m, b. 0 8 6 m, b. xint (,0) yint (0, ) 0 0 8 8 6 6-0 -8-6 - - 6 8 0 - - -6-8 -0-0 -8-6 - - 6 8 0 - - -6-8 -0-0 -8-6 - - 6 8 0 - - -6-8 -0 Practice Set 9. y x. y x. y 0 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0
Practice Set 0. y x 7. 5. 7 y x 5 5 6. 5 y x. y x. y x 6 y x 5 5 Practice Set. y ( x ). y ( x ). y ( x 5) OR y ( x 6). y ( x ) OR y ( x 9) 5 5 Practice Set. x y 7. x y 0. 6x 7y. 7x y Practice Set. The equation should be close to y.x.9. The equation should be close to y.x 5.9 Practice Set 0... 8 6 6 0 8 0 8 6-0 -8-6 - - 6 8 0-0 -8-6 - - 6 8 0-0 -8-6 - - 6 8 0 - - - - - - -6-6 -6-8 -8-8 -0-0 (, -) (-, ) Infinite Solutions -0
. 0 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0 No Solution Practice Set 5. (, -)., 6. No Solution. Infinite Solutions Practice Set 6.,. Infinite Solutions. (-, 0). No Solution Practice Set 7 0.. 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0 0 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0 0.. 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0 0 8 6-0 -8-6 - - 6 8 0 - - -6-8 -0
Practice Set 8. x y 8.05 x.5y.80 x quarters y nickels. y.x 5 y.8x x rides beforethey are equal y Thetotal will be$6.0 Practice Set 9. y 5 8y + 0y. 5x + 7x + 6. 0x + x x 5. (x + )(x + ) 5. (x 7)(x + 6) 6. (x 8)(x ) 7. x =, 6 8. x = 5, 9. x = 5,