Mathematics. Grade 8 Curriculum Guide. Curriculum Guide Revised 2016

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1 Mathematics Grade 8 Curriculum Guide Curriculum Guide Revised 2016

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3 Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction and assessment. It defines the content knowledge, skills, and understandings that are measured by the Standards of Learning assessment. It provides additional guidance to teachers as they develop an instructional program appropriate for their students. It also assists teachers in their lesson planning by identifying essential understandings, defining essential content knowledge, and describing the intellectual skills students need to use. This Guide delineates in greater specificity the content that all teachers should teach and all students should learn. The format of the Curriculum Guide facilitates teacher planning by identifying the key concepts, knowledge, and skills that should be the focus of instruction for each objective. The Curriculum Guide is divided into sections:, Essential Knowledge and Skills, Key Vocabulary,,, Resources, and Sample Instructional Strategies and Activities. The purpose of each section is explained below. : This section includes the objective and, focus or topic, and in some, not all, foundational objectives that are being built upon. Essential Knowledge and Skills: Each objective is expanded in this section. What each student should know and be able to do in each objective is outlined. This is not meant to be an exhaustive list nor a list that limits what is taught in the classroom. This section is helpful to teachers when planning classroom assessments as it is a guide to the knowledge and skills that define the objective. : This section includes vocabulary that is key to the objective and many times the first introduction for the student to new concepts and skills. : This section delineates the key concepts, ideas and mathematical relationships that all students should grasp to demonstrate an understanding of the objectives. : This section includes background information for the teacher. It contains content that is necessary for teaching this objective and may extend the teachers knowledge of the objective beyond the current grade level. It may also contain definitions of key vocabulary to help facilitate student learning. Resources: This section lists various resources that teachers may use when planning instruction. Teachers are not limited to only these resources. Sample Instructional Strategies and Activities: This section lists ideas and suggestions that teachers may use when planning instruction. 1

4 The following chart is the pacing guide for the Prince William County 8 th Grade Mathematics Curriculum. The chart outlines the order in which the objectives should be taught; provides the suggested number blocks to teach each unit; and organizes the objectives into Units of Study. The Prince William County cross-content vocabulary terms that are in this course are: analyze, compare and contrast, conclude, evaluate, explain, generalize, question/inquire, sequence, solve, summarize, and synthesize. Unit Objectives Approximate 1. Real Numbers 8.5, Blocks 2. Expressions 8.1, Blocks 3. Equations 8.15a, c 19 Blocks 4. Inequalities 8.15 b 10 Blocks 5. Problem Solving Blocks 6. Probability and Statistics 8.12, Blocks 7. Angles and Transformations 8.6, Blocks 8. 2-D Figures 8.10, Blocks 9. 3-D Figures 8.9, Blocks 10. Patterns, Functions, and Algebra 8.17, 8.14, Blocks GRADE 8 SOL TEST QUESTION BREAKDOWN (50 QUESTIONS TOTAL) (Based on 2009 SOL Objectives and Reporting Categories) Number, Number Sense, Computation and Estimation 14 Questions 28 % of the Test Measurement and Geometry 14 Questions 28 % of the Test Probability, Statistics, Patterns, Functions, and Algebra 22 Questions 44 % of the Test 2

5 Table of Contents Objective 8.1 Page Page Page Page Page Page Page Page Page Page Page Page Page Page Page Page Page 63 Page 3

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7 Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.5 a. determine whether a given number is a perfect square; and b. find the two consecutive whole numbers between which a square root lies. Unit 1: Real Numbers Time: 3 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Identify the perfect squares from 0 to 400. Identify the two consecutive whole numbers between which the square root of a given whole number from 0 to 400 lies (e.g., 57 lies between 7 and since 7 = 49 and 8 = 64). Estimate the square root of a nonperfect square to the nearest integer. Define a perfect square. Find the positive or positive and negative square roots of a given whole number from 0 to 400. (Use the symbol to ask for the positive root and when asking for the negative root.) consecutive irrational number negative root perfect square positive root rational number square root How does the area of a square relate to the square of a number? The area determines the perfect square number. If it is not a perfect square, the area provides a means for estimation. Why do numbers have both positive and negative roots? The square root of a number is any number which when multiplied by itself equals the number. A product, when multiplying two positive factors, is always the same as the product when multiplying their opposites (e.g., 7 7 = 49 and 7 7 = 49). The square root of a number is any number which when multiplied by itself equals the number. Whole numbers have both positive and negative roots. The square root of 36 is 6 and 6, where 6 is the positive root and 6 is the negative root. This can be expressed as 36 6 (±6 is read as plus or minus 6 ). The 36 (which is read as the positive square root of 36) is 6 and 36 (which is read as the negative square root of 36) is 6. A perfect square is a whole number whose square root is an integer (e.g., The square root of 25 is 5 and 5 ; thus, 25 is a perfect square.). The product of an integer and itself is a perfect square. Students can use grid paper and estimation to determine what is needed to build a perfect square. The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form a b where b 0 (e.g., 25, 1 4, 2.3, 75%, 4.59 ). Consecutive terms immediately follow each other in some order. For example 5 and 6 are consecutive whole numbers. Any whole number other than a perfect square has a square root that lies between two consecutive whole numbers. The square root of a whole number that is not a perfect square is an irrational number (e.g., 2 is an irrational number). An irrational number cannot be expressed exactly as a ratio (fraction). Estimation can be used to express a non-perfect square root to the nearest whole number (e.g., 11 is between 9 and 16. The 11 is a little more than 3 because 11 is closer to 9 than to 16. Therefore 11 estimated to the nearest whole number is 3. A number line can be used to illustrate this example

8 Resources Sample Instructional Strategies and Activities Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.5 Foundational Objectives 7.1d determine square roots. 6.5 investigate and describe concepts of positive exponents and perfect squares. 5.3 identify and describe the characteristics of prime and composite numbers; and even and odd numbers. Text: Virginia PreAlgebra 2012, Carter, Cuevas, Day, Malloy, McGraw-Hill School Education Group PWC Mathematics Website Virginia Department of Education Website thematics/index.shtml 6

9 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.2 describe orally and in writing the relationships between the subsets of the real number system. Unit 1: Real Numbers Time: 5 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Describe orally and in writing the relationships among the sets of natural or counting numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. Illustrate the relationships among the subsets of the real number system by using graphic organizers such as Venn diagrams. Subsets include rational numbers, irrational numbers, integers, whole numbers, and natural or counting numbers. Identify the subsets of the real number system to which a given number belongs. Determine whether a given number is a member of a particular subset of the real number system and explain why. Describe each subset of the set of real numbers and include examples and non-examples. Recognize that the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. How are the real numbers related? Some numbers can appear in more than one subset (e.g., 4 is an integer, a whole number, a counting or natural number, and a rational number.). The attributes (characteristics) of one subset can be contained in whole or in part in another subset. The set of real numbers includes the subsets (parts of sets) natural or counting numbers, whole numbers, integers, rational, and irrational numbers. The set of natural numbers is the set of counting numbers {1, 2, 3, 4, }. The set of whole numbers includes the set of all the natural numbers or counting numbers and zero {0, 1, 2, 3, }. The set of integers includes the set of whole numbers and their opposites {, 3, 2, 1, 0, 1, 2, 3, }. The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form a b where b 0 (e.g., 25, 1 4, 2.3, 75%, 4.59 ). Fractions such as 1 8, can be represented as terminating decimals (e.g., 1 = 0.125, which 8 has a finite number of decimal places) and fractions such as 2, can be represented as 9 repeating decimals (e.g., , whose decimal representation does not end but 9 continues to repeat). The repeating decimal can be written with ellipses (three dots) as in or denoted with a bar above the digit(s) that repeat as in 0.2. The set of irrational numbers is the set of all non-repeating, non-terminating decimals. An irrational number cannot be written in fraction form (e.g.,, 2, ). (continued) (continued) 7

10 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.2 describe orally and in writing the relationships between the subsets of the real number system. Unit 1: Real Numbers Time: 5 Blocks Essential Knowledge and Skills (continued) attributes integers irrational numbers natural/counting numbers non-repeating decimals non-terminating decimals rational numbers real numbers repeating decimals terminating decimals Venn diagram whole numbers (continued) A Venn diagram is a pictorial way of representing relationships among sets. Venn diagrams and number lines can be used to illustrate how different subsets relate to one another. Real Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Natural Numbers Examples: Which classifications of the real number system describe all the numbers in the each set? 6 2,, 1, , 2, 3, 7 5 All the numbers in this set are classified as: real, rational, integers, whole numbers, natural numbers All the numbers in this set are classified as: real, rational (continued) 8

11 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.2 describe orally and in writing the relationships between the subsets of the real number system. Unit 1: Real Numbers Time: 5 Blocks (continued) Graphic organizers help visualize relationships. Example: Given this diagram, determine which classifications of the real number system could set A and subset B represent. Multiple solutions to this problem exist. Possible solutions are set A could be rational and set B could be whole or set A could be real and subset B could be natural. 2 Example: Given this diagram, identify a number within subset B, such as. What could be the classifications represented by set A 3 and subset B. In this example set A could be real and set B could be rational. The sum of any two rational numbers is rational Example: = 13, , A B The product of two rational numbers is rational Example: 7(6) = 42, 4(1.6) 3, ( 4) The sum of a rational number and an irrational number is irrational. Example: , The product of a nonzero rational number and an irrational number is irrational. 2 Example: (3) 5 3 5, 2 (3 2)

12 Resources Sample Instructional Strategies and Activities Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.2 Foundational Objectives Text: Virginia PreAlgebra 2012, Carter, Cuevas, Day, Malloy, McGraw-Hill School Education Group PWC Mathematics Website Virginia Department of Education Website thematics/index.shtml Display the real number system as a Venn diagram. Give students problems that include examples and non-examples. For instance: Given the following numbers, which do not belong to the set of integers and why? 2, 0, 1 2, 3.4, 8, 1002, 75%, 47 10

13 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.1 a. simplify numerical expressions involving positive exponents, using rational numbers, order of operations and properties of operations with real numbers; and b. compare and order decimals, fractions, percents, and numbers written in scientific notation. Unit 2: Expressions Time: 14 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Simplify numerical expressions containing: 1. exponents (where the base is a rational number and the exponent is a positive whole number); 2. fractions, decimals, integers and square roots of perfect squares; and 3. grouping symbols (no more than 2 embedded grouping symbols). Order of operations and properties of operations with real numbers should be used. Compare and order no more than five fractions, decimals, percents, and numbers written in scientific notation using positive and negative exponents. Ordering may be in ascending or descending order. absolute value additive identity property (identity property of addition additive inverse property (inverse property of addition) associative property of addition associative property of multiplication base commutative property of addition commutative property of multiplication distributive property exponent grouping symbols identity elements inverses (continued) What is the role of the order of operations when simplifying numerical expressions? The order of operations prescribes the order to use to simplify a numerical expression. How does the different ways rational numbers can be represented help us compare and order rational numbers? Numbers can be represented as decimals, fractions, percents, and in scientific notation. It is often useful to convert numbers to be compared and/or ordered to one representation (e.g., fractions, decimals or percents). What is a rational number? A rational number is any number that can be written in fraction form. When are numbers written in scientific notation? Scientific notation is used to represent very large and very small numbers. The set of rational numbers includes the set of all numbers that can be expressed as fractions in the form a b where b 0 (e.g., 25, 1 4, 2.3, 75%, 4.59, 45 ). A rational number is any number that can be written in fraction form. A variable is a symbol ( a placeholder) used to represent an unspecified member of a set. Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables. Expressions do not contain equal or inequality signs. A numerical expression contains only numbers and the operations on those numbers (e.g., 7 + 4). A variable expression is an expression that contains a variable (e.g., 2x). A constant is a numerical expression that is part of an algebraic expression (e.g., In the expression 4x + 9, 9 is the constant.). Expressions are simplified using the order of operations and the properties for operations with real numbers (e.g., associative, commutative, distributive, and inverse properties). The commutative property of addition states that changing the order of the addends does not change the sum (e.g., = 4 + 5). The commutative property of multiplication states that changing the order of the factors does not change the product (e.g., 5 4 = 4 5). The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3]. The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 (4 3) = (5 4) 3]. Subtraction and division are neither commutative nor associative. (continued) 11

14 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.1 a. simplify numerical expressions involving positive exponents, using rational numbers, order of operations and properties of operations with real numbers; and b. compare and order decimals, fractions, percents, and numbers written in scientific notation. Unit 2: Expressions Time: 14 Blocks Essential Knowledge and Skills (continued) multiplicative identity property (identity property of multiplication multiplicative inverse property (inverse property of multiplication multiplicative property of zero numerical expression order of operations perfect square radical rational number simplify square root (continued) The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of the products of the number and each other number [e.g., 5 (3 + 7) = (5 3) + (5 7), or 5 (3 7) = (5 3) (5 7)]. Identity elements are numbers that combine with other numbers without changing the other numbers. Zero (0) is the identity element for addition and one (1) is the identity element for multiplication. There are no identity elements for subtraction and division. The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., = 5). The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 1 = 8). Inverses are numbers that combine with other numbers and result in identity elements. The additive inverse property states that the sum of a number and its additive inverse always equals zero (e.g., 5 ( 5) 0 ). The multiplicative inverse property states that the product of a number and its multiplicative inverse (reciprocal) always equals one (e.g., 1 5 1). Zero has no multiplicative inverse. 5 The multiplicative property of zero states that the product of zero and any real number is zero. Division by zero is not a possible arithmetic operation. A square root of a number is a number which, when multiplied by itself, produces the given number (e.g., 121 is 11 since = 121. is the radical sign and 121 is called a radical.). A whole number that can be named as a product of a number with itself is a perfect square (e.g., 81 = 9 9, where 81 is a perfect square). The absolute value of a number is the distance from 0 on the number line regardless of direction e.g.,,,, and (continued) 12

15 Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.1 a. simplify numerical expressions involving positive exponents, using rational numbers, order of operations and properties of operations with real numbers; and b. compare and order decimals, fractions, percents, and numbers written in scientific notation. Unit 2: Expressions Time: 14 Blocks (continued) 0 Any real number raised to the zero power is 1. The only exception to this rule is zero itself ( 0 1). Zero raised to the zero power is undefined. 4 A power of a number represents repeated multiplication of the number. For example, ( 5) means ( 5) ( 5) ( 5) ( 5). The base is the number that is multiplied, and the exponent represents the number of times the base is used as a factor. In this example, ( 5) is the base and 4 is the exponent. The product is 625. Notice that the base appears inside the grouping symbols. The meaning changes with the 4 4 removal of the grouping symbols. For example, 5 means negated which results in a product of 625. The expression (5) means to take the opposite of which is 625. Students should be exposed to all three representations. The order of operations, a mathematical convention, defines the order in which operations are performed to simplify an expression. To simplify an expression, regroup and combine like terms (integers and/or terms with the same variable) The order of operations is as follows: 1. Complete all operations within grouping symbols.* If there are grouping symbols within other grouping symbols, (embedded), do the innermost operation first. 2. Evaluate all exponential expressions. 3. Multiply and/or divide in order from left to right. 4. Add and/or subtract in order from left to right. *Parentheses ( ), brackets [ ], braces { }, absolute value, division/fraction bar, and the square root symbol, should be treated as grouping symbols. The overuse of the acronym PEMDAS tends to reinforce inaccurate use of the order of operations. Students frequently multiply before dividing and add before subtracting because they do not understand the correct order of operations. Example 1: Example 2: Example 3: 4 2(3 5) 4 2(8) 2(8) ( 3) ( 3 ) 5 ( 9) Scientific notation is used to represent very large or very small numbers. A number written in scientific notation is the product of two factors: a decimal greater than or equal to 1 but less than 10, multiplied by a power of 10 (e.g., = 310,000 and = ). All state approved scientific calculators use algebraic logic (follow the order of operations). 13

16 Resources Sample Instructional Strategies and Activities Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.1 Foundational Objectives 7.1 a. investigate and describe the concept of negative exponents for powers of ten; b. determine scientific notation for numbers greater than zero; c. compare and order fractions, decimals, percents and numbers written in scientific notation; d. determine square roots; and e. identify and describe absolute value for rational numbers a. write verbal expressions as algebraic expressions and sentences as equations and vice versa; b. evaluate algebraic expressions for given replacement values of the variables; 7.16 apply the following properties of operations with real numbers: a. the commutative and associative properties for addition and multiplication; b. the distributive property; c. the additive and multiplicative identity properties; Text: Virginia PreAlgebra 2012, Carter, Cuevas, Day, Malloy, McGraw-Hill School Education Group PWC Mathematics Website Virginia Department of Education Website thematics/index.shtml Using three single-digit numbers and two operations, chosen at random, determine how many different answers can be found by changing the operations. DO NOT change the order of the numbers. Using five single-digit numbers, four operations, and grouping symbols (e.g., parentheses ( ); brackets [ ]; braces { }; and fraction bars), determine how many different answers can be found by changing the grouping or operations. To understand the importance of having a common set of rules for operating on 12 numbers, have students work a problem similar to the following: 1 7( 3) 15 6 (answer 28). Discuss why there might be many different answers, and why mathematicians had to agree on a correct order of operations. Students wrestle with the classic problem: Use exactly four 4 s to write a numerical expression that equals each number from 1 to 25 (e.g., = 0). Students can play the Game of 24. (continued) 14

17 Sample Instructional Strategies and Activities Number, Number Sense, Computation and Estimation Relationships within the Real Number System Virginia SOL 8.1 Foundational Objectives (continued) d. the additive and multiplicative inverse properties; and e. the multiplicative property of zero. 6.2b, c, d b. identify a given fraction, decimal or percent from a representation; c. demonstrate equivalent relationships among fractions, decimals, and percents; d. compare and order fractions, decimals, and percents; 6.3 a. identify and represent integers; b. order and compare integers; and c. identify and describe absolute value of integers. 6.5 investigate and describe concepts of positive exponents and perfect squares investigate and recognize a. the identity properties for addition and multiplication; b. the multiplicative property of zero; and c. the inverse property for multiplication. (continued) Foundational Objectives (continued) 5.2 a. recognize and name fractions in decimal form and vice versa; and b. compare and order fractions and decimals in a given set from least to greatest and greatest to least. 5.3 identify and describe the characteristics of prime and composite numbers; and even and odd numbers investigate and recognize the distributive property of multiplication over addition. 4.16b investigate and describe the associative property for addition and multiplication a. investigate the identity and the commutative properties for addition and multiplication; and b. identify examples of the identity and commutative properties for addition and multiplication. 15

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19 Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.4 apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Unit 2: Expressions Time: 5 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Substitute rational numbers for variables in algebraic expressions and simplify the expressions by using the order of operations. Exponents are positive and limited to whole numbers less than 4. Square roots are limited to perfect squares. Apply the order of operations to evaluate formulas. Problems will be limited to positive exponents. Square roots may be included in the expressions but limited to perfect squares. absolute value algebraic expression coefficient evaluate exponent grouping symbols numerical expression order of operations perfect square radical simplify square root substitution variable What is the role of the order of operations when evaluating expressions? Using the order of operations assures only one correct answer for an expression. Expression is a word used to designate any symbolic mathematical phrase that may contain numbers and/or variables. Expressions do not contain equal or inequality signs and cannot be solved. A numerical expression contains only numbers and the operations on those numbers. An algebraic expression consists of one or more terms. Algebraic expressions use operations with algebraic symbols (variables) and numbers. A variable is a letter or other symbol that represents a number. Substitution is replacing one symbol with another. A coefficient is the numerical factor of a term (e.g., the numerical coefficient of 2x is 2, the numerical coefficient of 5y 2 is 5, and the numerical coefficient of n is 1). The order of operations, a mathematical convention, defines the order in which operations are performed to simplify an expression. To simplify an expression, regroup and combine like terms (integers and/or terms with the same variable) The order of operations is as follows: 1. Complete all operations within grouping symbols.* If there are grouping symbols within other grouping symbols, (embedded), do the innermost operation first. 2. Evaluate all exponential expressions. 3. Multiply and/or divide in order from left to right. 4. Add and/or subtract in order from left to right. *Parentheses ( ), brackets [ ], braces { }, absolute value, division/fraction bar, and the square root symbol, should be treated as grouping symbols. (continued) 17

20 Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.4 apply the order of operations to evaluate algebraic expressions for given replacement values of the variables. Unit 2: Expressions Time: 5 Blocks (continued) Algebraic expressions are evaluated by substituting rational numbers (fractions, decimals, positive integers and negative integers) for variables and applying the order of operations to simplify the resulting expression. Instruction should include a variety of problems such as the examples below. Example 1: 2 What is the value of 2x x when x = 1.5? Example 2: What is the value of y when y? x 2 Example 3: What is the value of 2 when x = 11? 12 3 Example 4: Which expression has a value of 27 when n = 2? 3 A n(4 8) 3 B n (4 8) 3 C ( n 4) 8 3 D ( n 4 ) 8 Example 5: 2 What is the value of 3( x 4 x) when x = 5? Example 6: What is the value of 5 3 x x when x = 4? The absolute value of a number is the distance from 0 on the number line regardless of direction e.g.,,,, and An exponent tells how many times the base is used as a factor. In the expression 3 2, 3 is the base and 2 is the exponent. A square root of a number is a number which, when multiplied by itself, produces the given number (e.g., 121 is 11 since = 121). A whole number that can be named as a product of a number with itself is a perfect square (e.g., 81 = 9 9, where 81 is a perfect square). A root of a number is a radical (e.g., 5 is called a radical). 18

21 Resources Sample Instructional Strategies and Activities Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.4 Foundational Objectives 7.13 a. write verbal expressions as algebraic expressions and sentences as equations and vice versa; and b. evaluate algebraic expressions for given replacement values of the variables. 6.8 evaluate whole number numerical expressions, using the order of operations. 5.7 evaluate whole number numerical expressions using the order of operations limited to parentheses, addition, subtraction, multiplication, and division. 5.18a, b a. investigate and describe the concept of variable; and b. write an open sentence to represent a given mathematical relationship using a variable. Text: Virginia PreAlgebra 2012, Carter, Cuevas, Day, Malloy, McGraw-Hill School Education Group PWC Mathematics Website Virginia Department of Education Website thematics/index.shtml Activity: Guess My Expression Each member of a group writes four algebraic expressions using the numbers 1 9 and a variable. The four expressions such as 3x 1; 5y 9; 2 y 7 ; 4z 8, are displayed for 6 the entire group to see. Players take turns spinning the spinner. Each player substitutes the spinner number into one of their expressions and announces the value to the other players. The other players must then guess which of the four expressions was used. The first player to get the correct expression scores one point. 19

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23 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Combine like terms to simplify expressions. Solve two- to four-step linear equations in one variable using concrete materials, pictorial representations and paper and pencil illustrating the steps performed. Solve two-step inequalities in one variable by showing steps and using algebraic sentences. Graph solutions to two-step linear inequalities on a number line. Identify properties of operations used to solve an equation from among: the commutative properties of addition and multiplication; the associative properties of addition and multiplication; the distributive property; the identity properties of addition and multiplication; the zero property of multiplication; the additive inverse property; and the multiplicative inverse property. Identify properties of equality used to solve an equation from among: - the addition property of equality; - the subtraction property of equality; - the multiplication property of equality; and - the division property of equality. How does the solution to an equation differ from the solution to an inequality? While a linear equation has only one replacement value for the variable that makes the equation true, an inequality can have more than one. A linear equation is an equation in which the variables are raised to the first power. A linear equation may have one variable or several variables. An equation in one variable can be of the form ax + b = 0 where x is the variable, a is the numerical coefficient, and b is the constant. A multi-step equation is an equation that requires more than one different mathematical operation to solve. Sometimes terms contain the same variable and must be combined. Combining like terms means to combine terms that have the same variable and the same exponent. Example: 8x x can be 5x +11 by combining the like terms of 8x and 3x. Note: In this example 8 and 3 are coefficients (numerical factors) of the terms. Variables can be on both sides of the equation (e.g., 3x 4 x 17) and can also be in the denominator of fractions (e.g., 4 2 x ) The process of solving an equation involves isolating the variable (with a coefficient of one) on one side of the equation. Each step in this process results in equivalent equations. Equivalent equations have the same solution. Because the equations y 5 10 and y 15 have the same solution, 15, they are equivalent equations. A linear inequality is an inequality in which the variables are raised to the first power. A linear inequality in one variable can be of the form ax + b > 0, or 0 < ax + b, or ax + b 0, or 0 ax + b where x is the variable, a is the numerical coefficient, and b is the constant. (continued) (continued) 21

24 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks Essential Knowledge and Skills addition property of equality additive identity property (identity property of addition) additive inverse property (inverse property of addition) associative property of addition associative property of multiplication coefficient commutative property of addition commutative property of multiplication distributive property division property of equality equivalent equations identity elements inverses like terms linear equation linear inequality multiplication property of equality multiplicative identity property (identity property of multiplication) multiplicative inverse property (inverse property of multiplication) substitution property of equality subtraction property of equality zero property of multiplication (multiplicative property of zero) (continued) Solving equations can be modeled using concrete materials and pictorial representations such as the following that can be done with Algebra Tiles or Algeblocks. Represent the equation 4x 1 2x = Remove two x-tiles and one 1-tile from each side = = Divide the x-tiles on the left into two groups and the 1-tiles on the right into two groups = Interpret the remaining tiles. The solution is 2. The commutative property of addition states that changing the order of the addends does not change the sum (e.g., = 4 + 5). The commutative property of multiplication states that changing the order of the factors does not change the product (e.g., 5 4 = 4 5). The associative property of addition states that regrouping the addends does not change the sum [e.g., 5 + (4 + 3) = (5 + 4) + 3]. The associative property of multiplication states that regrouping the factors does not change the product [e.g., 5 (4 3) = (5 4) 3]. Subtraction and division are neither commutative nor associative. (continued) 22

25 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks (continued) The distributive property states that the product of a number and the sum (or difference) of two other numbers equals the sum (or difference) of the products of the number and each other number [e.g., 5(3 + 7) = (5 3) + (5 7), or 5(3 7) = (5 3) (5 7)]. Identity elements are numbers that combine with other numbers without changing the other numbers. The additive identity is zero (0). The multiplicative identity is one (1). There are no identity elements for subtraction and division. The additive identity property states that the sum of any real number and zero is equal to the given real number (e.g., = 5). The multiplicative identity property states that the product of any real number and one is equal to the given real number (e.g., 8 1 = 8). Inverses are numbers that combine with other numbers and result in identity elements [e.g., 5 + ( 5) = 0; 1 5 1]. The additive inverse 5 property states that the sum of a number and its additive inverse always equals zero [e.g., 5 + ( 5) = 0]. The multiplicative inverse property 1 states that the product of a number and its multiplicative inverse (or reciprocal) always equals one (e.g., 4 1). Zero has no 4 multiplicative inverse. The zero property of multiplication states that the product of any real number and zero is zero. Division by zero is not a possible arithmetic operation. In addition to the properties of operations, properties of equality are also used when solving equations. These are true for all real numbers. The addition property of equality states that if the same number is added to each side of an equation, the two sides remain equal and the resulting equivalent equation is true. For any numbers a, b, and c, if a b, then a c b c. If x 2 4, then x The subtraction property of equality states that if the same number is subtracted from each side of an equation, the two sides remain equal and the resulting equivalent equation is true. For any numbers a, b, and c, if a b, then a c b c. If x 2 4, then x The multiplication property of equality states that if each side of an equation is multiplied by the same nonzero number, the two sides remain equal and the resulting equivalent equation is true. 1 1 For any numbers a, b, and c, c 0, if a b, then ac bc. If 2x 4, then 2x The division property of equality states that if each side of an equation is divided by the same nonzero number, the two sides remain equal and the resulting equivalent equation is true. a b 2x 4 For any numbers a, b, and c, c 0, if a b, then =. If 2x 4, then. c c 2 2 (continued) 23

26 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks (continued) The substitution property of equality (substitution) states that one name of a number can be substituted for another name of the same number in any expression. Substitution can be used if one statement is replaced with an equivalent one and no other property or definition works. For any numbers a and b, if a b, then either a or b may be substituted for the other in any equation. If 7 2 5, then 7 7. In addition to identifying equations that illustrate a given property, properties that justify the work between steps when solving an equation are also identified. In the following example the addition property of equality has been applied to the given equation. Step 1 3x 4 9 Step 2 3x 4 ( 4) 9 ( 4) In the following example the additive inverse property justifies the work between steps 2 and 3. Step x Step ( 5) 9 ( 5) 5 x Step ( 5) 5 x Properties of operations and properties of equality can be used to justify a step by step procedure for solving equations. Example 1: 19x 2x 3(2x 30) Step 1 Given equation 21x 3(2x 30) Step 2 Substitution 21x 6x 90 Step 3 Distributive property 21 x ( 6 x) 6 x ( 6 x) 90 Step 4 Addition property of equality 15 x 6 x ( 6 x) 90 Step 5 Substitution 15 x 0 90 Step 6 - Additive inverse 15 x 90 Step 7 Substitution x Step 8 Multiplication property of equality 1 x Step 9: Multiplicative inverse x 6 Step 10: Substitution (continued) 24

27 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks (continued) Example 2: x Step 1 Given equation x Step 2 Addition property of equality x Step 3 Additive inverse 2 3 x 22 6 Step 4 Additive identity property 2 3 x 28 Step 5 Substitution 3 2 x Step 6 Multiplication property of equality 3 1x 28 2 Step 6 Multiplicative inverse 3 x 28 2 Step 7 Multiplicative identity x 42 Step 8 Substitution (continued) 25

28 Probability, Statistics, Patterns, Functions, and Algebra (continued) Example 3: (Using the LCM) Linear Relationships Virginia SOL 8.15 d. solve multi-step linear equations in one variable on one and two sides of the equation; e. solve two-step linear inequalities and graph the results on a number line; and f. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks 2 3 x Step 1 Given equation x Step 2 Multiplicative property of equality (Multiply each side by LCM) x 20(2) Step 3 Distributive property 8 x Step 4 Substitution 8x Step 5 Subtraction property of equality 8 x Step 6 Additive inverse 8 x Step 7 Additive identity property 8x 25 Step 8 Substitution 1 1 (8 x) ( 25) 8 8 Step 8 Multiplication property of equality 1 1 x ( 25) 8 Step 9 Multiplicative inverse 1 x ( 25) 8 Step 10 Multiplicative identity 25 x 8 Step 11 Substitution (continued) 26

29 Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships (continued) Graphing can be used to demonstrate that both x < 5 and 5 > x, represent the same solution set. To illustrate why an inequality is reversed when multiplying or dividing with a negative number, use the inequality x < 0, or the opposite of a number is less than zero. For this to be true, the original number must be greater than zero. Because the graph of 3 is to the right of the graph of 2, 3 > 2. Multiplying both numbers by 1 gives 3 and 2. Because the graph of 3 is to the left of the graph of 2, 3 < 2, that is, the inequality is reversed. Virginia SOL 8.15 a. solve multi-step linear equations in one variable on one and two sides of the equation; b. solve two-step linear inequalities and graph the results on a number line; and c. identify properties of operations used to solve an equation. Unit 3: Equations (Objective 8.15a, 8.15c) Time: 19 Blocks Unit 4: Inequalities (Objective 8.15b) Time: 10 Blocks A two-step inequality is defined as an inequality that requires the use of two different operations to solve [e.g., 3x 4 9 (add 4 then multiply by 1 3 ), or 2(x 5) < 12 (multiply by 1 2 then add 5), or x 4 10 (multiply by 3 then add 4)]. 3 In an equation, the equal sign indicates that the value on the left is the same as the value on the right. To maintain equality, an operation that is performed on one side of an equation must be performed on the other side. The same procedures that work for equations work for inequalities, except when both expressions of an inequality are multiplied or divided by a negative number then the inequality sign reverses. Solutions for inequalities may be graphed on a number line using appropriate graphing symbols. For example: 3x + 4 < 19 ( 4) ( 4) 3x + 0 < (3 x) (15) 3 3 x <

30 Resources Sample Instructional Strategies and Activities Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 Foundational Objectives 7.14 a. solve one- and two-step linear equations in one variable; and b. solve practical problems requiring the solution of one- and two-step linear equations a. solve one-step inequalities in one variable and b. graph solutions to inequalities on the number line apply the following properties of operations with real numbers: a. the commutative and associative properties for addition and multiplication; b. the distributive property; c. the additive and multiplicative identity properties; d. the additive and multiplicative inverse properties; and e. the multiplicative property of zero. Text: Virginia PreAlgebra 2012, Carter, Cuevas, Day, Malloy, McGraw-Hill School Education Group PWC Mathematics Website Virginia Department of Education Website thematics/index.shtml Students will use algebra tiles to solve linear equations. Students will use Algeblocks to model solving equations. Use "Hands-on-Equations" to model solving equations. If you do not have this set, you may use cups to represent variables and beans to represent the numbers or rectangles to represent variables and squares to represent the numbers. Use a graphic organizer to compare and contrast equations and inequalities. Discover the rationale for reversing the inequality symbols when multiplying or dividing by negatives by creating a pattern using several examples with numbers such as: 5 > 3 and 5(2) > 3(2) but 5( 2 ) < 3( 2 ). Write the steps out for solving an equation or inequality on 3 by 5 cards, one step per card. Students put them in the correct order and identify the property used for each step. (continued) 28

31 Sample Instructional Strategies and Activities Probability, Statistics, Patterns, Functions, and Algebra Linear Relationships Virginia SOL 8.15 Foundational Objectives (continued) 6.18 solve one-step linear equations in one variable involving whole number coefficients and positive rational solutions investigate and recognize a. the identity properties for addition and multiplication; b. the multiplicative property of zero; and c. the inverse property for multiplication graph inequalities on a number line a. investigate and describe the concept of variable; b. write an open sentence to represent a given mathematical relationship using a variable; and c. model one-step linear equations in one variable using addition and subtraction. Foundational Objectives (continued) 5.19 investigate and recognize the distributive property of multiplication over addition. 4.16b investigate and describe the associative property for addition and multiplication a. investigate the identity and the commutative properties for addition and multiplication; and b. identify examples of the identity and commutative properties for addition and multiplication. (continued) 29

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33 Number, Number Sense, Computation and Estimation Practical Applications of Operations with Real Numbers Virginia SOL 8.3 a. solve practical problems involving rational numbers, percents, ratios, and proportions; and b. determine the percent increase or decrease for a given situation. Unit 5: Problem Solving Time: 16 Blocks Essential Knowledge and Skills use problem solving, mathematical communication, mathematical reasoning, connections and representations to: Solve practical problems by using computation procedures for whole numbers, integers, fractions, percents, ratios, and proportions. Some problems may require the application of a formula. Maintain a checkbook and check registry for five or fewer transactions. Compute a discount or markup and the resulting sale price for one discount or markup. Write a proportion given the relationship of equality between two ratios. Compute the percent increase or decrease for a one-step equation found in a real life situation. Compute the sales tax and/or tip and resulting total. Substitute values for variables in given formulas. For example, use the simple interest formula I=prt to determine the value of any missing variable when given specific information. Compute the simple interest and new balance earned in an investment or on a loan for a given number of years. What is a percent? A percent is a special ratio with a denominator of 100. What is the difference between percent increase and percent decrease? Percent increase and percent decrease are both percents of change measuring the percent a quantity increases or decreases. Percent increase shows a growing change in the quantity while percent decrease shows a lessening change. Practical problems may include, but not be limited to, those related to economics, sports, science, social sciences, transportation, and health. Some examples include problems involving the amount of a paycheck per month, balancing a checkbook, the discount price on a product, temperature, simple interest, sales tax, and installment buying. Multi-step practical problems are the focus of instruction at this level. Practical problems also include interpreting information represented as percentages in a graph. A rate is a ratio that compares two quantities measured in different units. A unit rate is a rate with a denominator of 1. Examples of unit rates include miles/hour and revolutions/minute. A discount rate is the percent off an item (e.g., If an item is reduced in price by 20%, 20% is the discount rate.) The amount of discount (discount) is how much is subtracted from the original amount. The sale price (discount price) is the result of subtracting the discount from the original price. A sales tax rate is the percent of tax (e.g., Virginia has a 5% tax rate on most items purchased.) Sales tax is the amount added to the price of an item based on the tax rate. A tip is a small sum of money given as acknowledgment of services rendered, (a gratuity). It is often times computed as a percent of the bill or service. A percent is a special ratio with a denominator of 100. amount of discount discount price (sale price) discount rate formula interest (continued) A markup is a price increase. It is the difference between a cost of an item and its selling price. Interest is an amount of money paid for the use of money. The percent of the invested or borrowed amount on which the interest is based is called the interest rate. Simple interest for a number of years is determined by multiplying the principal (loan amount) by the rate of interest by the number of years of the loan or investment (I=prt). (continued) 31