West Windsor-Plainsboro Regional School District Math A&E Grade 6 Page 1 of 20
Unit 1: Integers and Expressions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale Students extend understandings of addition, subtraction, multiplication, and division together with their properties, to all rational numbers, including negative integers. 20 days Standard 4.7.NS The Number System Recommended Pacing State Standards 1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing real world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. Instructional Focus Page 2 of 20
Unit Enduring Understandings In order to model the world around us we need a set of negative numbers in addition to positive numbers and we must be able to work with these numbers Computation involves taking apart and combining numbers using a variety of approaches Unit Essential Questions What meaning do numbers convey? Why are negative numbers necessary? How is mathematics used to quantify and compare situations, events, and phenomena? Objectives Students will know: The definitions of ordered pairs, axes, quadrants, origin and how to plot and identify each. The conventions of the number line. The definition of absolute value. The rules for adding, subtracting, multiplying, and dividing with integers The procedure for simplifying and solving one and two step equations with integers Students will be able to: Represent integers, opposites, and absolute value Add integers using models, patterns, and rules Subtract integers using models, patterns, and rules Solve problems by looking for a pattern Multiply integers using repeated addition, patterns, and rules Divide integers Apply integer operations to problems Assign variables and write variable expressions. Study order of operations Evaluate variable expressions Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 3 of 20
Unit 2: Solving Equations Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale Students will use the language of algebra to represent and analyze relationships among variable quantities and solve problems involving algebraic concepts and processes involving positive and negative values. 20 days Standard 4.7.EE Expressions and Equations Recommended Pacing State Standards 1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. 2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 3 Solve multi step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 4 Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Standard 4.8.EE Expressions and Equations 7 Solve linear equations in one variable. b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. Standard 4.7.NS The Number System 1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, Page 4 of 20
in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q = p/( q). Interpret quotients of rational numbers by describing real world contexts. c. Apply properties of operations as strategies to multiply and divide rational numbers. Instructional Focus Unit Enduring Understandings One representation may sometimes be more helpful than another; and, used together, multiple representations give a fuller understanding of a problem Mathematical rules like order of operations are important and necessary to evaluate and solve expressions, equations and inequalities In order to model the world around us we need a set of negative numbers in addition to positive numbers and we must be able to work with these numbers Computation involves taking apart and combining numbers using a variety of approaches Unit Essential Questions How can real world phenomena be expressed algebraically and can accurate expressions help us understand and explore these situations? What rules and conventions do we need to know in order to simplify expressions and solve equations or inequalities? What meaning do numbers convey? Why are negative numbers necessary? Objectives Students will know: What an exponent represents The definition of square root and the perfect square numbers up to 144 What a variable represents The correlation between verbal terms and algebraic symbols Order of operations The definitions of commutative, associative, and identity properties of multiplication and addition; and the distributive property Page 5 of 20
The procedure for simplifying and solving one and two step equations The procedure for simplifying and solving one and two step inequalities The conventions for graphing a simple inequality on the number line Students will be able to: Determine if an equations is true or false Determine if a given number is a solution of an open equation Recognize and use the commutative, associative, and identity properties of addition and multiplication Recognize and use the distributive property of multiplication over addition/subtraction Use the properties of addition and multiplication to simplify expressions Solve one step equations involving addition and subtraction Solve one step equations involving multiplication and division Write the correct equation for a word problem or model Use guess and test as a solution to a problem Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 6 of 20
Unit 3: Decimals and Equations Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale Students will develop number sense and will perform standard numerical operations, write and solve equations and use estimations with decimal numbers including positive and negative values. 20 days Standard 4.7.NS The Number System Recommended Pacing State Standards 1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. d. Apply properties of operations as strategies to add and subtract rational numbers. 2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. c. Apply properties of operations as strategies to multiply and divide rational numbers. 3 Solve real world and mathematical problems involving the four operations with rational numbers. Standard 4.7.EE Expressions and Equations CPI # Cumulative Progress Indicator (CPI) 2 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. 3 Solve multi step real life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 4 Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. Instructional Focus Unit Enduring Understandings Page 7 of 20
To compare, accurately round, and order decimals To estimate sums, differences, products, and quotients of decimal numbers To evaluate and simplify expressions involving decimals To solve decimal equations using addition and subtraction To solve decimal equations using multiplication and division To understand and solve density problems using metric units To accurately use formulae To solve problems using a simpler problem Unit Essential Questions The need for different number systems is a response to fundamentally different types of phenomena that need quantifying Computational fluency includes understanding the meaning and the appropriate use of numerical operations Objectives Students will know: Place value names Conventions of naming decimal values Rules for rounding decimals Strategies to compare positive and negative decimal values Techniques for estimating decimal operations The algorithms for addition/subtraction/multiplication and division of decimal numbers The procedure for solving one and two step equations and equations with simplifying The procedures for writing equations That multiplying by a value between 0 and 1 will result in a product less than the largest factor That dividing by a value between 0 and 1 will result in a quotient larger than the dividend Students will be able to: Compare, accurately round, and order decimals Estimate sums, differences, products, and quotients of decimal numbers Evaluate and simplify expressions involving decimals Solve decimal equations using addition and subtraction Solve decimal equations using multiplication and division Understand and solve density problems using metric units Accurately use formulae Solve problems using a simpler problem Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 8 of 20
Unit 4: Number Theory Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale All students will develop number sense and will perform standard numerical operations and estimations on fractions. Students will also learn how the concepts of multiplication and division can be applied to algebraic expressions with exponents. 20 days Standard 4.7.NS The Number System Recommended Pacing State Standards 1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. d. Apply properties of operations as strategies to add and subtract rational numbers. 2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 3 Solve real world and mathematical problems involving the four operations with rational numbers. Standard 4.8 EE Expressions and Equations 1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 3 5 = 3 3 = 1/33 = 1/27. 3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. 4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Instructional Focus Unit Enduring Understandings The need for different number systems is a response to fundamentally different types of phenomena that need quantifying Page 9 of 20
Computational fluency includes understanding the meaning and the appropriate use of numerical operations Fractions and decimals can be used to represent equivalent forms of the same number Multiplication and division can be performed on algebraic exponent expressions as well as numerical expressions Unit Essential Questions How can fractional values be used to describe or quantify fundamental realities? How do operations affect numbers and algebraic expressions? Objectives Students will know: The definitions of greatest common factor and least common multiple The process of repeated division to find greatest common factor and least common multiple Common fraction decimal equivalent pairs (halves, thirds, fourths, fifths, eighths, and tenths) The procedures for converting fractions and decimals The algorithms of addition/subtraction/multiplication and division of fractions (positive and negative values) The procedure for solving fraction equations with traditional algebra steps and with clearing The procedure for writing and solving fraction equations The meaning of negative exponents The value of n 0 and n 1 The rules for multiplying and dividing exponent expressions with like bases The rules for raising exponent expressions to powers What scientific notation is and why it is used Students will be able to: Discover the rules governing exponential numbers Evaluate and write exponential number Use exponential rules to multiply expression with powers Use exponential numbers to write and compute with numbers in scientific notation Solve problems using scientific notation Identify whole number factor and multiples of a number Understand and use divisibility rules in factoring Determine whether a number is prime or composite Determine the prime factorization (PF) of a number Determine the GCF or LCM of a number by: the listing method using prime factorization Use the problem solving technique of accounting for all possibilities (combinations) Resources Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Page 10 of 20
Unit 5: Rational Numbers and Expressions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale All students will develop number sense and will perform standard numerical operations and estimations on fractions 20 days Standard 4.7.NS The Number System Recommended Pacing State Standards 1 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. d. Apply properties of operations as strategies to add and subtract rational numbers. 2 Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. 3 Solve real world and mathematical problems involving the four operations with rational numbers. Instructional Focus Unit Enduring Understandings The need for different number systems is a response to fundamentally different types of phenomena that need quantifying Computational fluency includes understanding the meaning and the appropriate use of numerical operations Fractions and decimals can be used to represent equivalent forms of the same number Unit Essential Questions How can fractional values be used to describe or quantify fundamental realities? Objectives Students will know: The definitions of greatest common factor and least common multiple The process of repeated division to find greatest common factor and least common multiple Common fraction decimal equivalent pairs (halves, thirds, fourths, fifths, eighths, and tenths) The procedures for converting fractions and decimals The algorithms of addition/subtraction/multiplication and division of fractions The procedure for solving one step equations Students will be able to: Write equivalent fractions and fractions in lowest terms using GCF and/or PF Page 11 of 20
Write mixed numbers as improper fractions and decimal numbers Understand the meaning of the term rational number Compare and order fractions and decimal numbers Add/subtract fractions and mixed numbers Use the technique of working backwards to solve problems Understand and use the algorithms for multiplication and division of fractions and mixed numbers Understand the origin of negative exponents Simplify expressions with negative exponents Solve rational number equations using addition and subtraction Solve rational number equations using multiplication Resources Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Page 12 of 20
Unit 6: Ratios, Proportions, and Percent Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale Relationships can be recorded and represented verbally, graphically, algebraically, and numerically to suit different goals. There is a need for different types of numbers and number systems to quantify or describe fundamentally different phenomena. 20 days Standard 4.7.RP Ratios and Proportional Relationships Recommended Pacing State Standards 2 Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. 3 Use proportional relationships to solve multistep ratio and percent problems. Standard 4.7.G Geometry CPI # Cumulative Progress Indicator (CPI) 1 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Instructional Focus Unit Enduring Understandings Ratios, rates, and proportions are ideas that help us compare and communicate about the relationships between quantities Proportional reasoning is an idea that can be used to solve problems where a relationship exists between quantities Fractions, decimals and percents can be used to represent equivalent forms of the same value There are situations, contexts and problems where each of these is the most efficient way to communicate about the quantity Unit Essential Questions What kinds of questions can be answered using proportional reasoning? How is proportional reasoning of geometric figures used to solve problems? How are percent s used to describe or quantify real world situations? When is it most appropriate to use fractions, decimals, or percent s? How are fractions, decimals, and percent s related? Objectives Students will know: Page 13 of 20
Definitions of: ratio, rate, unit price, proportion, similarity A ratio shows a comparison of two quantities and a rate is a ratio between two quantities that have different units Rates can be simplified to unit rates The distance formula Scale describes how a figure is enlarged or reduced A scale drawing is a proportional representation of an object Similar figures are two figures in which corresponding angles are congruent and corresponding sides are proportional Common fraction decimal percent equivalencies (halves, thirds, fourths, fifths, eighths, tenths) Strategies for solving mental math percent problems including tips The proportion and equation techniques for solving percent problems Definitions and uses of discount, sales tax, commission, paycheck taxes, simple interest and percent of change Students will be able to: Define and write ratios and proportions Use proportions in problem solving Apply proportional reasoning to scaling problems Express ratios and rates as percent s Evaluate percent s using proportions Explore the relationships between decimals, fractions, and percent s Solve percent equations using: a) triangle diagram, b) sentence conversion, c) proportions Determine percent increase or decrease Use a diagram as a method of problem solving Resources Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Page 14 of 20
Unit 7: Equations and Inequalities Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale Students will use the language of algebra to represent and analyze relationships among variable quantities and solve problems involving algebraic equations and inequalities. All students will represent and analyze relationships among variable quantities and solve problems involving patterns and functions. 20 days Standard 4.7.EE Expressions and Equations Recommended Pacing State Standards 4 Use variables to represent quantities in a real world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Standard 4.8.F Functions 1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Instructional Focus Unit Enduring Understandings Common mathematical processes are important and necessary to solve complex equations and inequalities While some problems have only one solution, other problems have multiple solutions An inequality represents a solution set for a real world problem Functional relationships can be expressed verbally, graphically, numerically, and symbolically Unit Essential Questions How can real world phenomena be expressed algebraically and can accurate expressions help us understand and explore these situations? What rules and conventions do we need to know in order to solve complex equations and inequalities? How can real world phenomena be modeled in patterns? What do graphs convey about the real world situation they model? Objectives Page 15 of 20
Students will know: The procedure for solving equations with variables on both sides. The procedure for writing equations with two expressions using one variable. An inequality represents a range of solutions. The meaning of and differences between less than, greater than, less than or equal to, and greater than or equal to symbols. The conventions of a number line The meaning of an open and closed circle on a number line The procedure for solving one step inequalities with positive values That graphs can model real world situations That graphs can help to make predictions There is a relationship between input and output and that output is dependent on the input That input output data can be graphed The meaning of function notation Students will be able to: Solve elementary two step equations Solve two step equations with like terms Solve problems by writing equations Solve equations with an unknown on both sides Write and graph simple inequalities with one variable Discover the properties of inequalities Solve one step inequalities using properties of inequalities Solve simple two step inequalities Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 16 of 20
Unit 8: Graphing in the Coordinate Plane Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale All students will represent and analyze relationships among variable quantities and solve problems involving patterns and functions. 20 days Standard 4.8.F Functions Recommended Pacing State Standards 3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. 5 Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Instructional Focus Unit Enduring Understandings Functional relationships can be expressed verbally, graphically, numerically, and symbolically Unit Essential Questions How can real world phenomena be modeled in patterns, graphs and equations? What do graphs convey about the real world situations they model? Objectives Students will know: That graphs can model real world situations That graphs can help to make predictions There is a relationship between input and output and that output is dependent on the input The input output data form ordered pairs that can be graphed on the coordinate plane The definition of slope and y intercept What positive and negative slope values look like in a graph The technique to transform an equation to slope intercept form Students will be able to: Locate and graph a coordinate pair Solve linear equations in two variables Graph a linear equation and determine the x and y intercepts Explore slope, including positive, negative, zero, and no slope Determine the slope and y intercept for the graph of a linear equation Solve problems by graphing Solve two equations in two variables by graphing Solve linear inequalities in two variables Graph a linear inequality in two variables Graph systems of inequalities Solve problems of direct and inverse variation Page 17 of 20
Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 18 of 20
Unit 9: Algebra in Geometry and Measurement Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 6 Summary and Rationale All students will develop spatial sense and the ability to use geometric concepts, relationships, and measurement to model, describe and analyze real world situations. 20 days Standard 4.7.G Geometry Recommended Pacing State Standards 4 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. 5 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi step problem to write and solve simple equations for an unknown angle in a figure. Standard 4.8.G Geometry 5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle angle criterion for similarity of triangles. Instructional Focus Unit Enduring Understandings Geometric concepts, relationships, and measurement are an attempt to model, describe, classify, and analyze real world objects Unit Essential Questions How well do geometric concepts and relationships model real world phenomena? What attributes of objects are important to recognize? How can recognizing these attributes be useful? Objectives Students will know: Symbols for line, ray, segment, angle, plane, parallel and perpendicular Definitions of vertical angles, alternate interior angles, alternate exterior angles, corresponding angles. (Review: parallel and perpendicular lines, complementary and supplementary angles, polygon classifications, congruent figures, translations, reflections, and rotations.) Classifications of 2 D figures (triangles, quadrilaterals, and regular and irregular: pentagons, hexagons, heptagons, octagons, nonagons, decagons, hendecagons, and dodecagons and circles) Properties of polygons and be able to apply algebraic equations to find missing measures Definition of corresponding parts of congruent figures and will be able to name congruent figures accordingly Definitions of translations, reflections and rotations Students will be able to: Understand and review geometric concepts, symbols, and vocabulary Explore angle measures and their relationships Page 19 of 20
Review properties of polygons and quadrilaterals Review properties of triangles and circles Explore the concepts of congruence and symmetry Determine correspondence in similar figures Find missing measures in similar figures Determine the perimeter or circumference of a figure Review the concept of tessellations Core Text: Prentice Hall Pre Algebra, Davison Suggested Resources: Resources Page 20 of 20