Chapters 1, 2, 3, 4, 5, 6

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1 Chapters 1, 2,, 4,, 6 Name Period 2 Revised 201

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3 Graph the following on a number line. COMPARING RATIONAL NUMBERS 1) 0.01, 0.001, 0.1, and ) 2.2, 0.2, 0.248, and ) 0.8, 1., 0.47, and ) 0.006,.02, 0.0, ) 0.98, 0.89, 0.94, and 0.9 6) 0.201, 0.19, 1.2, and ) 0.46, 0.40, 0.47, and 4. 8) 0.1, 0.8, 0.60, and ) 0.04, 1.2, 0.16, and 2. 10) 0.76, 07, 0.076, and ) 1, 1, and 1 12), 2, and 2 8 1) 1 1 1,, and 14) 1 2,, and ) 11,, and 16) 2 11,, and ) 2 2, 1,and ) , 1, and ) ,, and 20) , 0.6, 0.1, ) 8., 9.62,.72, and ) 16 9, 7, 42 2) 2, 27,.6, ) 12,.7, 7, 6. 2) 24, 2 10, 8, 26) 4., 4.62, 4.72, and 21 27)., 6.,.27 and 0 28) , 2, 2,and ), 0.2,, 1, 0. 0) 1 2.,.6, and

4 Change the following repeating decimals to fraction: PACC Rational & Irrational Nos Approximate the following radicals to the nearest whole number without using a calculator: Use the approximation method to place the following radicals in the correct position on the number line below: , 6, 87, 111, 11, 10, 11, 70, 6, 7, 17, 6, ,, 9, 10, 140, 22, 12, 2,, 90,, 119,

5 Simplify the following. When necessary, write answer in simplest radical form. Simplify Radicals (4) x (88) xy (12) 2 (12) xy x x y x 2 y x 2 y x 4 y a ( 121 ) a 6. 4a b a 2 b b ( 26 ) x x a a 6 b (16) x 2 y a 8 b Holt McDougal, Larson Algebra 1, 2011

6 Simplify the following. When necessary, write answer in simplest radical form. Simplify Radicals a b ab a 4 b a 20 b ( 2) ( 2 21) ab 6 ab ab 2 14ab a 2 b ( a 2 ) ab 2 ( 10a b 4 ) x x 81. 6x 2x a 2 b 6 b a 9 b a x 2 x a 4 b 7 14a x 8x c 7 27c x 2 10x 8. 6m 4 18m

7 Closure Take two numbers from a set and perform an operation. If the answer is always in the set, then the set is closed. Properties Identity Property: For all real numbers a, a + 0 = a and 0 + a = a For all real numbers a, a 1 = a and 1 a = a Additive Inverse Property For every real number, there is exactly one real number a, such that a + (-a) = 0 and a + a = 0. Multiplicative Inverse Property For every non-zero real number a, there is exactly one number 1 a, such that a 1 1 and 1 = a = 1. The number 1 a is called the reciprocal or multiplicative inverse of a. a a Multiplicative Property of Zero The product of any real number and zero is zero. a 0= 0 Other Properties of Zero Zero divided by any nonzero real number is zero. 0 = 0 a Division by zero is undefined. (Division by zero cannot be simplified.) Commutative Property. For all real numbers and b, a + b = b + a and a b = b a Associative Property Distributive Property For all real numbers a, b and c, (a + b) + c = a + (b + c) and (a b) c = a (b c). For all real numbers a, b and c, a(b + c) = ab + ac and a(b c) = ab ac. Properties of Equality For all real numbers a, b and c: Reflexive Property a = a (A number equals itself.) Symmetric Property If a = b, then b = a. Transitive Property If a = b and b = c, then a = c. Substitution Property If a = b, then a can be replaced by b and b can be replaced by a. Additive Property of Equality: If a = b then a + c = b + c. Multiplicative Property of Equality: If a = b then a c = b c. Underlined Properties are on the Keystone Algebra 1 Exam. 7

8 Identify the properties displayed below: 1. + ( ) = ( + 7 ) = 2 2. If + = 8 and 8 = then + = is a natural number = x 7 + x 7 = ( 2 + ) = = ( x + y) = 11x + 11y 20. a ( x y ) = ( a x ) y = = is a rational number (16 + 1) = y = y 2. a 0 = x y = -y + 6x 24. 2y + x = 2y + 1x = = 14 so 14 = (7 4) y = y = = (a + ) = 4( + a) ( 2h + h ) = 6 + (2 + )h is an integer = (4 + 9)x = 1x = 11 so 11 = c + 2c = ( + 2)c 8

9 Name the property used in each step. Algebraic Proofs Show all the steps and name the properties used to simplify each expression below: ( ) / 2 + 6(8 4 2 ) (12 / 4 ) 10. 2( 6)( x) x(2.8)( ) 1. x + 2 = x = 10x

10 Solve the following equations showing all the work. Equations Review 1 1. x + 7 = x = = x = 8 + x. 19 = x x = x + 16 = = 0 x 6 9. x x = x = x = = 4 9 x 1. 12x = = 9x 1. 8x = = 2x 17. x = x = = x = x x = x = x + 2 = x = = 4x = 12x x = = 6c x = = 7x x + + 4x = x + 6 x = = 6x 2 8x = 4x (2x ) = (x + 4) = = (6x 11) = 9 + (x + 4) 9. 8(x 6) + 12 = (x 6) = (x 1) = x + 6 = x 4. 9x 11 = 12x 44. x = 8x x = x x + = 4x x 1 = 6x x = 10 x x 17 = 1 x x 19 = x x + 2 = x 4 2. x 11 = x x = 11 6x x = 1 7x. x x = 8x 2 6. x x = 7x x + 2 = 17 + x x = + 4x 9. 4x 16 = 18 4x x + x + x = x x 8 = x 6. 8(4x + 6) = x x 2 = 7(x ) 6. (2x 8) = (x + 6) 66. 7(x 1) = 11(x 4) 67. 8(4x + 2) = 12(2x + 4) 68. 6(2x ) = 8(9 8x) 69. 4(x +6) + x = 1x (2x + 8) = x (6x + 4) = 6 + 2(x + ) (x 1) = + 8(9x +2) 7. 8 (x + 2) = 9 6(10x + ) (4x 1) = 8 2(11x 2) 7. 4(x 6) 6(2x ) = (2x 1) 7(x + 6) = x x x + 4 = 7 = 8 + 2x 1 = x 7 = x x = 8.1x x 7.2 = 0.4x + 10

Functions A relation is a pairing between two sets of numbers. The first coordinate is called the domain. The second coordinate is called the range.

(x, y) (Number of units, Predicted Value) (Domain, Range) (Input, Output) (Input, Solution Set) (Independent variable, Dependent Variable) Solve y = 4x if the domain is {, 2, 0, 1, 2} Solve 8x + 4y =

11 Functions A relation is a pairing between two sets of numbers. The first coordinate is called the domain. The second coordinate is called the range. A function is a special type of relation that pairs each x (domain value) with exactly one y (range value). (x, y) (Number of units, Predicted Value) (Domain, Range) (Input, Output) (Input, Solution Set) (Independent variable, Dependent Variable) Solve y = 4x if the domain is {, 2, 0, 1, 2} Solve 8x + 4y = 24 if the domain is {-2, 0,, 8} Domain X 4x Range y (x, y) Domain X Range y (x, y) Use the graphing calculator to solve equations in two variables. Method 1 1. Store domain values in L1. 2. On the home screen, rewrite equation, replacing L1 for x. Method 2 1. Store domain values in L1. 2. On L2, rewrite equation, replacing L1 for x x y x y x y x y

12 Write an Equation for a Function Chapters 1 & 4 Find the first difference for x. Find the first difference for y. Find Changein y Changein x Δ y = Δ x Find the y-intercept: the y-coordinate in (0, ) Write equation in the form: y = Δ y Δ x x + original value Find the missing values in this table: x y Change in y Change in x Find the missing values in each of these tables and, if possible, find the equation for the table. 1. x y x y x y x y

13 Find the missing values in each of these tables and, if possible, find the equation for the table.. x y x y x y x y x y x y x y x y

14 Find the equation that corresponds to these tables x y x y x y x y x y x y x y x y x y x y x y x y

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Graphing Linear Equations 1. Rewrite the equation in slope-intercept form. 2. Identify the slope.. Identify the y-intercept. 4. Graph the y-intercept.

16 Graphing Linear Equations 1. Rewrite the equation in slope-intercept form. 2. Identify the slope.. Identify the y-intercept. 4. Graph the y-intercept.. Locate two other points on the line using the slope. 6. Draw a line through the points. 7. Label the line. 1. x + = y 2. y = x x y = 4. 4x + y = 6. 7x 2y = 6. y = and x = 16

17 Intercepts are points where the graph crosses the axes. x-intercept = (a, 0) or a graph crosses the x-axis. y-intercept = (0, b) or b graph crosses the y-axis. The STANDARD FORM of a linear equation is: Ax + By = C where A, B, C are integers and A 0 and A and B are both not equal to 0. A linear function is a linear equation were B 0 and whose domain is understood to be all real numbers. Graph using intercepts. 1. x-intercept is x + 7y = 42. x + 4y = 6 y-intercept is 2. SLOPE = m = Δy Δx = y 1 y 2 x 1 x 2 The SLOPE-INTERCEPT FORM of a linear equation is: y = mx + b where m = slope = Δy Δx and b = y-intercept or (0, b) The POINT-SLOPE FORM of a linear equation is: y y 1 = m(x x 1 ) given slope m and point (x 1, y 1 ). 17

Comparing Multiple Representations 1. Find and compare the slopes for the linear functions f and g. f(x) = ½ x 4 x -4 0 4 8 g(x) - - -1 0 Slope of f = Slope of g = Compare: 2.

They are climbing down a canyon wall. Connor starts from a cliff that is 200 feet above the canyon floor and climbs down at an average speed of 10 feet per minute.

18 Comparing Multiple Representations 1. Find and compare the slopes for the linear functions f and g. f(x) = ½ x 4 x g(x) Slope of f = Slope of g = Compare: 2. Find and compare the y-intercepts for the linear functions f and g. x f(x) y-intercept of f y-intercept of g Compare: Connor and Sheila are in a rock-climbing club. They are climbing down a canyon wall. Connor starts from a cliff that is 200 feet above the canyon floor and climbs down at an average speed of 10 feet per minute. Sheila climbs down the canyon wall as shown in the table. Time (min) Sheila s height (ft) Interpret the rates of change and initial values of the linear functions in terms of the situations they model. Connor Initial value Rates of change Sheila Initial value Rates of change Compare: Holt McDougal, Course, 1.4A 18

19 Scatterplots & Line of Best Fit The 1990 earnings per share and dividends per share for electric utility companies (in the central United States) are shown in the table. 1. Enter the data into L1 and L2. 2. Create a scatterplot.. 2 ND 0 CATALOG D DiagnosticON ENTER ENTER CLEAR 4. STAT CALC 4: LinReg(ax+b) ENTER. Write the line of best fit in space to right. 6. Write Correlation coefficient, r = Earnings Dividend Earnings Dividend Earnings Dividend Earnings Dividend The data in the table shows the age in years and the number of hours slept in a day by 28 infants who are less than one year old. 1. Enter the data into L1 and L2. 2. Create a scatterplot.. 2 ND 0 CATALOG D DiagnosticON ENTER ENTER CLEAR 4. STAT CALC 4: LinReg(ax+b) ENTER. Write the line of best fit in space to right. 6. Write Correlation coefficient, r = Sleep Sleep Sleep Sleep Age (yrs) Age (yrs) Age (yrs) Age (yrs) (hrs) (hrs) (hrs) (hrs)

20 ALGEBRA 1 FORMULA SHEET Formulas that you may need to solve questions on this exam are found below. You may use calculator π or the number.14 A=Lw Arithmetic Properties Additive Inverse: a + ( a) = 0 Multiplicative Inverse: a 1 a = 1 V=Lwh Commutative Property: a + b = b + a a b = b a Associative Property: (a + b) + c = a + (b + c) (a b) c = a (b c) Identity Property: a + 0 = a a 1 = a Linear Equations Distributive Property: a(b + c) = ab + ac Slope: m = y 2 y 1 x 2 x 1 Point-Slope Formula: (y y 1 ) = m(x x 1 ) Slope-Intercept Formula: y = mx + b Standard Equation of a Line: Ax + By = C Multiplicative Property of Zero: a 0 = 0 Additive Property of Equality: If a = b, then a + c = b + c Multiplicative Property of Equality: If a = b, then a c = b c 20

Exponential Properties Cone a m a n = a m+n (a m ) n = a mn a m a n = am n a 1 = 1 a V = 1 π r 2 h Algebraic Equations

21 PSSA MATHEMATICS GRADE 8 REFERENCE Formulas that you may need to work questions are found below. You may refer back to this age at any time during the mathematics test. You may use calculator π or the number.14. Exponential Properties Cone a m a n = a m+n (a m ) n = a mn a m a n = am n a 1 = 1 a V = 1 π r 2 h Algebraic Equations Cylinder Slope: m = y 2 y 1 x 2 x 1 Slope-intercept Form: y = mx + b Pythagorean Theorem V = π r 2 h a c Sphere b a 2 + b 2 = c 2 V = 4 π r 21

Addition add addend altogether bigger than greater than in all increased by larger than longer than more more than older than plus sum taller than tally together the sum of the tally of the total of

22 Addition add addend altogether bigger than greater than in all increased by larger than longer than more more than older than plus sum taller than tally together the sum of the tally of the total of total Subtraction amount of increase decreased by deduct deducted from difference diminished by fewer than how many fewer how many more how much greater less less than minus shorter than smaller than subtract subtracted from take away the difference between younger than Multiplication of use as many as as much as double use 2 factor multiply product times the product of thrice use triple use twice use 2 Division (any fraction) average divided by divided into dividend divisor out of quotient the average of the quotient of Variable number the number of... unknown h, x, y, n Exponent base power cubed (...) squared (...) 2 the square of (...) 2 square root of Equality = any verb equals is is the answer to is the same as Inequality does not exceed is at least is at most is between x is greater than is greater than or equal to is larger than is less than is less than or equal to is more than is no less than is no more than is not equal to 22 Natural Numbers Whole Numbers Integers {1, 2,, } {0, 1, 2,, } {0, 1, 2, } Rational Numbers Any number that can be written as where a and b are Integers and b 0. Includes all integers, fractions, terminating decimals, repeating decimals, percents and perfect roots. Irrational Numbers Any number that is not Rational. Includes non-terminating non-repeating decimals, and non-perfect roots. Real Numbers Includes all Rational and Irrational Numbers Additive Inverse: a + ( a) = 0 Multiplicative Inverse: a = 1 Commutative Property: a + b = b + a a b = b a Associative Property: (a + b) + c = a + (b + c) (a b) c = a (b c) Translation Reflection Rotation Dilation Transformation Type Identity Property: a + 0 = a AND a 1 = a Distributive Property: a (b + c) = a b + a c Multiplicative Prop. of Zero: a 0 = 0 Additive Prop of Equality: If a = b, then a + c = b + c. Multiplicative Prop of Equality: If a = b, then a c = b c Rule Move right or left a units Add a or a to each x-coor. Move up or down b units Add b or b to each y-coor. Across the y-axis Multiple each x-coor by -1 Across the x-axis Multiply each y-coor by Multiply both coor by clockwise Multiply each x-coor by -1, then switch x- & y-coor. 90 counterclockwise Multiply each y-coor by -1, then switch x- & y-coor. Scale Factor Multiply each coor. by scale factor.

TRIANGLE: Perimeter = side 1 + side 2 + side Area = base height 180 = Angle 1 + Angle 2 + Angle TRAPEZOID: Area = height (base 1 + base 2 ) Perimeter = side 1 + side 2 + side + side 4 PARALLELOGRAM:

Standard Form: Ax + By = C where A, B, C are integers and A Point-Slope Form: (y y 1 ) = m(x x 1 ) where m = slope and point is (x 1, y 1).

PROPORTIONS: OR SQUARE: Area = side 2 Perimeter = 4 side CIRCLE: Area = radius 2 Circumference = 2 radius CUBE: Volume = side Surface Area = 6 side 2 PRISM: Volume = length width height CYLINDER:

TEMPERATURE (choose one): FAHRENHEIT = Celsius + 2 OR CELSIUS = (Fahrenheit 2) Perimeter, Circumference, Distance, Base, Height, Length, Radius, Side, Width MEASUREMENT LABELS FOR WORD PROBLEMS mm,

23 TRIANGLE: Perimeter = side 1 + side 2 + side Area = base height 180 = Angle 1 + Angle 2 + Angle TRAPEZOID: Area = height (base 1 + base 2 ) Perimeter = side 1 + side 2 + side + side 4 PARALLELOGRAM: Area = base height Perimeter = side 1 + side 2 + side + side 4 FORMS OF LINEAR EQUATIONS: Slope Intercept Form: y = mx + b where m = slope and b = y-intercept. Standard Form: Ax + By = C where A, B, C are integers and A Point-Slope Form: (y y 1 ) = m(x x 1 ) where m = slope and point is (x 1, y 1). DISTANCE = rate time PREDICTED VALUE = Rate per Unit Number of Units PREDICTED VALUE = Original Value + Rate per Unit Number of Units RECTANGLE: Area = length width Perimeter = 2 length + 2 width PROPORTIONS: OR SQUARE: Area = side 2 Perimeter = 4 side CIRCLE: Area = radius 2 Circumference = 2 radius CUBE: Volume = side Surface Area = 6 side 2 PRISM: Volume = length width height CYLINDER: Volume = radius 2 height CONE: Volume = radius 2 height SPHERE: Volume = radius PYTHAGOREAN THEOREM: c 2 = a 2 + b 2 SLOPE = m = for points (x 1, y 1 ) and (x 2, y 2 ) SUM = first + second TEMPERATURE (choose one): FAHRENHEIT = Celsius + 2 OR CELSIUS = (Fahrenheit 2) Perimeter, Circumference, Distance, Base, Height, Length, Radius, Side, Width MEASUREMENT LABELS FOR WORD PROBLEMS mm, cm, m, km, inches, feet, yards, miles Area, Surface Area mm , cm, m, km, inches, feet, yards, miles Volume cm, m, km, inches, feet, yards, miles Rate rate per unit, mph, mpg, cost per unit, % (percent) EXPONENT RULES: a m a n = a m + n (a m ) n = a m n a 0 = 1 ANGLE MEASURE: Sum of angles = 180(n 2) 180 (n 2) = a 1 + a a n Angle = 180( n 2 ) / n 2

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Virginia Unit-Specific Learning Pathways. Grades 6-Algebra I: Standards of Learning

Virginia Unit-Specific Learning Pathways. Grades 6-Algebra I: Standards of Learning BUI L T F O VIR R G INIA 2014 2015 Virginia -Specific Learning Pathways Grades 6-Algebra I: Standards of Learning Table of Contents Grade 6...3 Grade 7...6 Grade 8...9 Algebra I... 11 Grade 6 Virginia