STANDARDS OF LEARNING CONTENT REVIEW NOTES ALGEBRA I Part I 1 st Nine Weeks, 2016-2017
OVERVIEW Algebra I Content Review Notes are designed by the High School Mathematics Steering Committee as a resource for students and parents. Each nine weeks Standards of Learning (SOLs) have been identified and a detailed explanation of the specific SOL is provided. Specific notes have also been included in this document to assist students in understanding the concepts. Sample problems allow the students to see step-by-step models for solving various types of problems. A section has also been developed to provide students with the opportunity to solve similar problems and check their answers. The answers to the at the end of the document. problems are found The document is a compilation of information found in the Virginia Department of Education (VDOE) Curriculum Framework, Enhanced Scope and Sequence, and Released Test items. In addition to VDOE information, Prentice Hall textbook series and resources have been used. Finally, information from various websites is included. The websites are listed with the information as it appears in the document. Supplemental online information can be accessed by scanning QR codes throughout the document. These will take students to video tutorials and online resources. In addition, a self-assessment is available at the end of the document to allow students to check their readiness for the nine-weeks test. To access the database of online resources scan this QR code, or visit http://spsmath.weebly.com The Algebra I Blueprint Summary Table is listed below as a snapshot of the reporting categories, the number of questions per reporting category, and the corresponding SOLs. Algebra I Blueprint Summary Table Reporting Categories No. of Items SOL Expressions & Operations 12 A.1 A.2a c A.3 Equations & Inequalities 18 A.4a f A.a d A.6a b Functions & Statistics 20 A.7a f A.8 A.9 A.10 A.11 Total Number of Operational Items 0 Field-Test Items* 10 Total Number of Items 60 * These field-test items will not be used to compute the students scores on the test. It is the Mathematics Instructors desire that students and parents will use this document as a tool toward the students success on the end-of-year assessment. 2
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Expressions and Order of Operations A.1 The student will represent verbal quantitative situations algebraically and evaluate these expressions for given replacement values of the variables. Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables. An expression can be represented algebraically Example 1: 6x + Example 2: a 9b or in written form. Example 1: The sum of a number and eleven Example 2: One half of a number squared minus four Some common words are used to indicate each operation. Many of these are shown in the table below, but there are others. Add Subtract Multiply Divide Equals Plus Sum More than Increased by Total All together Add to And Difference Minus Less than Decreased by Take away How many left Remaining Subtracted by Less Times Product Multiplied By Doubled (x2) Tripled (x3) By Squared (a a) Cubed (a a a) Part Quotient Divided by Each Half ( 2) Split ( 2) Is Are Is Equal To Is equivalent to Equals Expressions and Order of Operations Translate the written expressions to algebraic expressions, and algebraic expressions to written expressions. 1. the difference of eleven and x 2. three times the sum of a number and ten 3. four times the difference of n squared and five 4. 12g 4. a² - b⁴ 4
Expressions are simplified using the order of operations and the properties for operations with real numbers. The order of operations is as follows: First: Complete all operations within grouping symbols. If there are grouping symbols within other grouping symbols, do the innermost operation first. Grouping symbols include parentheses (a), brackets [a], radical symbols a, absolute value bars a, and the fraction bar a b. Second: Evaluate all exponents. Third: Multiply and/or divide from left to right. Fourth: Add and/or subtract from left of right. To evaluate an algebraic expression substitute in the replacement values of the variables and then evaluate using the order of operations. Example 1: 1 9 ( 3) 2 Step 1: 1 9 ( 3) 2 Scan this QR code to go to an order of operations video tutorial! Step 2: 1 9 9 Step 3: 1 1 Step 4: The answer is 14 Example 2: (p 3) 2 + 2p 4, p = 7 Step 1: ( 7 3) 2 + 2( 7) 4 Step 2: ( 10) 2 + 2( 7) 4 Step 3: 100 + 2( 7) 4 Step 4: 100 + ( 14) 4 Step : 86 4
Step 6: The answer is 82 Example 3: 20 3 10 14 + 7 4 Step 1: 20 30 14 + 7 20 Scan this QR code to go to a video for more complicated order of operations help. Step 2: 20 16 + 13 Step 3: 20 4 + 13 Step 4: The answer is 29 Expressions and Order of Operations Evaluate each expression. a=2, b=, x= - 4, and n=10. 6. [a + 8(b 2)] 2 4 7. (2x) 2 + an b 8. n 2 + 3(a + 4) 9. bx ax Evaluate each expression using the order of operations. 10. 6 3 10 2 + 2 22 3 11. 7 + (9 + 3) 2 3 12. (7 3 18) 3 63 14+2 1 13. 6 3 3 + 22 8 + 19 6
Properties of Real Numbers A.4 The student will solve multistep linear and quadratic equations in two variables, including b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; A. The student will solve multistep linear inequalities in two variables, including b) justifying steps used in solving inequalities, using axioms of inequality and properties Property Definition Examples Multiplicative Property of Zero Any number multiplied by zero always equals zero. a (0) = 0 0 ( 14) = 0 Additive Identity Any number plus zero is equal a + 0 = a to the original number. 126 + 0 = 126 Multiplicative Identity Any number times one is the a 1 = a original number. 1 78 = 78 Additive Inverse A number plus its opposite a + ( a) = 0 always equals zero. 21 + 21 = 0 Multiplicative Inverse Associative Property Commutative Property Distributive Property A number times its inverse (reciprocal) is always equal to one. When adding or multiplying numbers, the way that they are grouped does not affect the outcome. The order that you add or multiply numbers does not change the outcome. For any numbers a, b, and c: a(b + c) = ab + ac a 1 a = 1 2 2 = 1 (a + b) + c = a + (b + c) + (3 + 8) = ( + 3) + 8 (ab)c = a(bc) 6(3a) = (6 3) a a + b = b + a 14 + 6 = 6 + 14 ab = ba 8 3 = 3 8 ( 3 2) = 3 2 3 (a + b) = 3a + ( 3b) or 3 (a + b) = 3a 3b 7
Substitution property of equality Reflexive Property of Equality Transitive Property of Equality Symmetric Property of Equality If a = b, then b can replace a. A quantity may be substituted for its equal in any expression. Any quantity is equal to itself. If one quantity equals a second quantity and the second quantity equals a third, then the first equals the third. If one quantity equals a second quantity, then the second quantity equals the first. If + 2 = 7, then ( + 2) 4 = 7 4 If a =, then 11a = 11 a = a 3 = 3 If a = b, and b = c, then a = c. If 2 + 4 = 6, and 2(3) = 6, then 2 + 4 = 2(3) If a = b, then b = a If 2 = 13a 1, then 13a 1 = 2 Properties of Real Numbers Match the example on the left to the appropriate property on the right. 1. (x + 3) + y = x + (3 + y) 2. 1 = 1 2x 2x 3. 3x + 6 = 6 + 3x 4. ( x)2 = 10 2x. b + 0 = b 6. (x + 3) + y = y + (x + 3) 7. 0 17n = 0 8. x + ( x) = 0 9. xyz = xyz 10. If one dollar is the same as four quarters, and four quarters is the same as ten dimes, then ten dimes is the same as one dollar. A. Multiplicative Property of Zero B. Additive Identity C. Multiplicative Identity D. Additive Inverse E. Multiplicative Inverse F. Associative Property G. Commutative Property H. Distributive Property I. Substitution Property of Equality J. Reflexive Property of Equality K. Transitive Property of Equality L. Symmetric Property of Equality 8
Solving Equations A.4 The student will solve multistep linear and quadratic equations in two variables, including d) solving multistep linear equations algebraically and graphically; f) solving real-world problems involving equations and systems of equations. You will solve an equation to find all of the possible values for the variable. In order to solve an equation, you will need to isolate the variable by performing inverse operations (or undoing what is done to the variable). Any operation that you perform on one side of the equal sign MUST be performed on the other side as well. Drawing an arrow down from the equal sign may help remind you to do this. Example 1: m 9 = 3 +9 + 9 m = 6 Check your work by plugging your answer back in to the original problem. 6 9 = 3 Scan this QR code to go to a video tutorial on two-step equations. Example 2: x+4 = 12 x + 4 = 60 4 4 x = 64 Check your work by plugging your answer back in to the original problem. 64 +4 = 60 = 12 9
You may have to distribute a constant and combine like terms before solving an equation. Example 3: 4(g 7) + 2g = 10 4g + 28 + 2g = 10 2g + 28 = 10 2g 28 28 = 38 ( 2) ( 2) g = 19 Check your work by plugging your answer back in to the original problem. 4 (19 7) + 2(19) = 10 4(12) + 2(19) = 10 48 + 38 = 10 Scan this QR code to go to a video tutorial on multi-step equations. If there are variables on both sides of the equation, you will need to move them all to the same side in the same way that you move numbers. Example 4: 3p = 7(p 3) 3p = 7p 21 3p 3p = 4p 21 +21 + 21 16 = 4p 4 4 p = 4 Check your work by plugging your answer back in to the original problem. 3(4) = 7(4) 21 12 = 28 21 7 = 7 10
Example : x+10 x = 1 (x + 10) = 1(x) x + 0 = x +x 10x + 0 = 0 + x 0 0 10x = 0 10 10 x = You can begin this problem by cross multiplying! Check your work by plugging your answer back in to the original problem. +10 = 1 ( ) 2 = 1 1 = 1 Scan this QR code to go to a video tutorial on equations with variables on both sides. Solve each equation 1. k + 11 = 8 2. 9 3x = 4 3. 17 = y 6 2 Solving Equations 4. (2n + 6) + 8 = 33. 3 (4k + 2) = 1 6. g + 4 = 9g 10 7. 2( 4m 1) + 3m = 4m 8 + m 8. x 4 = 2(3x 3) 2 6 11
Justifying Steps using Properties A.4 The student will solve multistep linear and quadratic equations in two variables, including b) justifying steps used in simplifying expressions and solving equations, using field properties and axioms of equality that are valid for the set of real numbers and its subsets; You are using the properties of real numbers to solve equations and inequalities, and to simplify expressions. You will need to be able to identify the property that you are using in each step of the simplification or solution. When you solve an equation or inequality and perform the same operation on both sides of the equal sign this is a special property of equality. Property Equation Example Inequality Example Addition Property of Equality and Inequality If a = b, then a + c = b + c x 6 = 4 +6 + 6 x = 10 If a > b, then a + c > b + c If a < b, then a + c < b + c w 1 4 +1 + 1 w Subtraction Property of Equality and Inequality Multiplication Property of Equality and Inequality If a = b, then a c = b c d + 4 = 3 4 4 d = 7 If a = b, then ac = bc n = 1 n = If a > b, then a c > b c If a < b, then a c < b c g + 1 < 3 1 1 g < 2 If a > b, then ac > bc If a < b, then ac < bc *Don t forget to switch the sign if you multiply or divide by a negative! c 4 2 ( 2) ( 2) c 8 Division Property of Equality and Inequality If a = b, and c 0, then a c = b c 2h = 12 ( 2) ( 2) h = 6 If a > b, and c 0, then a c > b c If a < b, and c 0, then a c < b c *Don t forget to switch the sign if you multiply or divide by a negative! 9f < 81 9 9 f < 9 12
Example 1: 2x + 3 ( 6x) 4 2x + 1 18x 4 2x 18x + 1 4 16x + 1 4 16x + 11 Distributive Property Commutative Property of Addition Add(Substitution Property) Subtract (Substitution Property) Example 2: 7x + ( 3x) = 21 7x + ( 3x + ) = 21 Commutative Property of Addition (7x 3x) + = 21 Associative Property of Addition 4x + = 21 Add (Substitution Property) Subtraction Property of Equality 4x = 16 Subtract (Substitution Property 4 4 Division Property of Equality x = 4 Divide (Substitution Property) Example 3: Example 4: 12 < x + 6 4x 12 < x 4x + 6 Commutative Property of Addition 12 < 3x + 6 Subtract (Substitution Property) 6 6 Subtraction Property of Inequality 6 < 3x Subtract (Substitution Property) ( 3) ( 3) Division Property of Inequality 2 > x Divide (Substitution Property) x < 2 Symmetric Property of Inequality Solve 3 (2x y) = 12 for y 6x 3y = 12 Distributive Property 6x 6x Subtraction Property of Equality 3y = 6x + 12 Commutative Property of Addition ( 3) ( 3) Division Property of Equality y = 2x 4 Division (Substitution Property) 13
Justifying Steps using Properties List the properties used to justify each step in the problems below. 1. 8 + 4x 3(2 + x) 8 + 4x 6 3x 8 6 + 4x 3x 2 + 4x 3x 2 + x 2. Solve ( + 3x) + x 4y 8 for x + (3x + x) 4y 8 + 4x 4y 8 4x 4y 13 4 4 x 4y 13 4 14
Answers to the problems: Expressions and Order of Operations 1. 11 x 2. 3 (n + 10) 3. 4 (n² ) 4. The product of twelve and a number divided by four. a squared minus b to the fourth power 6. 169 7. 9 8. 118 9. 12 10. 209 11. 2 12. 3 13. 7 Properties of Real Numbers 1. F - Associative 2. E - Multiplicative Inverse 3. G - Commutative 4. H - Distributive. B - Additive Identity 6. G - Commutative 7. A - Multiplicative Property of Zero 8. D - Additive Inverse 9. J - Reflexive Property of Equality 10. K - Transitive Property of Equality Solving Equations 1. k = 19 2. x = 1 3. y = 28 4. n = 1 2. k = 4 6. g = 1 7. m = 3 8. x = 2 Justifying Steps using Properties 1. Distributive Commutative Subtraction (Substitution) Subtraction (Substitution) 2. Associative Addition (Substitution) Subtraction Property of Inequality Subtraction (Substitution) Division Property of Inequality Divide (Substitution) 1