West Windsor-Plainsboro Regional School District Math A&E Grade 7
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1 West Windsor-Plainsboro Regional School District Math A&E Grade 7 Page 1 of 24
2 Unit 1: Introduction to Algebra Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of real numbers and the language of algebra. Using the language and mechanics of algebra to analyze, compare and communicate is foundational to representing and solving real world situations. 15 days Standard 4.8.NS The Number System Recommended Pacing State Standards 1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions. Standard 4.A SSE Seeing Structure in Expressions 1 Interpret expressions that represent a quantity in terms of its context. 2 Use the structure of an expression to identify ways to rewrite it. 3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Instructional Focus Unit Enduring Understandings Equivalent expressions can be represented in a variety of forms Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Many real world problems can be solved through the use of algebraic representation and reasoning Unit Essential Questions What are the different ways that numbers can be represented and classified? How can numerical operations and properties be extended to algebraic expressions? What makes an algebraic algorithm both effective and efficient? How can I represent real life situations algebraically? Objectives Students will know: The order of operations The concept of solution space as it applies to equations and inequalities Algebraic properties, including commutative, associative, distributive, additive and multiplicative identities & inverses, zero, and negative one property The different classifications of real numbers and how they relate to one another The definition of like terms The definition of absolute value Students will be able to: Simplify and evaluate numerical and variable expressions Page 2 of 24
3 Simplify expressions with and without grouping symbols Find solution sets of equations Represent numerical relationships stated in words by mathematical expressions Translate word sentences describing the equality of members into equations Identify and use the five steps in solving word problems Graph real numbers on a number line Identify the opposite and absolute values of a real number Compare and order real numbers using their positions on the number line Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 3 of 24
4 Unit 2: Working with Real Numbers Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of real numbers and the language of algebra. Using the language and mechanics of algebra to analyze, compare and communicate is foundational to representing and solving real world situations. 15 days Standard 4.8.NS The Number System Recommended Pacing State Standards 1 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. 2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions. Standard 4.A SSE Seeing Structure in Expressions 1 Interpret expressions that represent a quantity in terms of its context. 2 Use the structure of an expression to identify ways to rewrite it. 3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. Instructional Focus Unit Enduring Understandings Equivalent expressions can be represented in a variety of forms Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole Many real world problems can be solved through the use of algebraic representation and reasoning Unit Essential Questions What are the different ways that numbers can be represented and classified? How can numerical operations and properties be extended to algebraic expressions? What makes an algebraic algorithm both effective and efficient? How can I represent real life situations algebraically? Objectives Students will know: The order of operations The concept of solution space as it applies to equations and inequalities Algebraic properties, including commutative, associative, distributive, additive and multiplicative identities & inverses, zero, and negative one property The different classifications of real numbers and how they relate to one another The definition of like terms The definition of absolute value Students will be able to: Use number properties (axioms) to simplify expressions Page 4 of 24
5 Add real numbers using established addition rules Subtract real numbers and to recognize subtraction as addition of the opposite Simplify expressions using sums and differences Use the distributive axiom to simplify expressions Multiply real numbers Write equations to represent relationships between and among integers Use reciprocals and to recognize their properties Divide real numbers Simplify expressions involving products and quotients Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 5 of 24
6 Unit 3: Solving Equations and Problems Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of single variable equations and inequalities. Modeling inequalities, absolute value equations and inequalities using words, tables, number lines, and symbols will enable students to apply their algebraic thinking to real world contexts. 15 days Standard 4.A CED Creating Equations Recommended Pacing State Standards 1 Create equations and inequalities in one variable and use them to solve problems. Standard 4.A REI Reasoning with Equations and Inequalities 1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Instructional Focus Unit Enduring Understandings Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns The value of a particular representation depends on its purpose Rules of arithmetic and algebra can be used together with (the concept of) equivalence to transform equations and inequalities so solutions can be found to solve problems Proportionality involves a relationship in which the ratio of two quantities remains constant as the corresponding values of the quantities change Unit Essential Questions How can patterns and equations be used as tools to best describe and help explain real life situations? What makes an algebraic algorithm both effective and efficient? How can arithmetic operations be extended to solve algebraic equations and inequalities? When is it appropriate to use proportions to model relationships in the real world? Objectives Students will know: Procedures for simplifying, solving and graphing one, two and multi step single variable equations and inequalities Procedures for simplifying, solving and graphing single variable compound inequalities Procedures for simplifying and solving single variable absolute value equations and inequalities Single variable equations and inequalities may have infinitely many (identity), no real number solutions (empty set) or a unique solution Students will be able to: Solve equations using addition and/or subtraction transformations Page 6 of 24
7 Solve equations using multiplication and/or division transformations Solve equations using a series of transformations Use the five step plan to solve word problems Solve equations having variables on both sides Use charts to arrange the facts of a word problem Solve problems using equations in one variable (consecutive integer, age, area, and cost value) Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 7 of 24
8 Unit 4: Polynomials Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves knowledge and skills relative to polynomials, the basic building blocks of algebraic expressions. Using previous knowledge of the properties of numbers students will be able to transform polynomial expressions into equivalent forms. The concepts covered in this unit are foundational in finding solutions to quadratic equations and functions. 15 days Standard 4.A SSE Seeing Structure in Expressions Recommended Pacing State Standards 2 Use the structure of an expression to identify ways to rewrite it. Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 2 Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a)=0 if and only if (x a) is a factor of p(x). Instructional Focus Unit Enduring Understandings Rules of algebra can be used together with equivalents to transform expressions Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole as it applies to real life situations Unit Essential Questions How do the operations of addition, subtraction and multiplication apply to polynomial expressions? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Procedures for adding, subtracting, multiplying polynomial expressions Procedures for the application of polynomial expressions in real life situations The correct terminology for identifying polynomial expressions by degree and number of terms Students will be able to: Write and simplify exponential expressions Add and subtract polynomials Multiply monomials using exponent rules Find powers of monomials using exponent rules Multiply polynomial by monomial both horizontally and vertically Multiply two polynomials in vertical form Transform formulae Use charts to solve problems involving uniform motion Page 8 of 24
9 Solve area problems Recognize problems without solution Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 9 of 24
10 Unit 5: Factoring Polynomials Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale 15 days Standard 4.A SSE Seeing Structure in Expressions Recommended Pacing State Standards 2 Use the structure of an expression to identify ways to rewrite it. Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 2 Know and apply the Remainder Theorem. For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a)=0 if and only if (x a) is a factor of p(x). Instructional Focus Unit Enduring Understandings Rules of algebra can be used together with equivalents to transform expressions Algebraic and numeric procedures are interconnected and build on one another to produce a coherent whole as it applies to real life situations Unit Essential Questions How do the operations of addition, subtraction and multiplication apply to polynomial expressions? What makes an algebraic algorithm both effective and efficient? Objectives Students will know: Procedures for adding, subtracting, multiplying and factoring polynomial expressions Procedures for the application of polynomial expressions in real life situations The correct terminology for identifying polynomial expressions by degree and number of terms Students will be able to: Use prime factorization to factor integers and to calculate GCF Divide and factor monomials Divide a polynomial by a monomial Find the monomial factor(s) of a polynomial Find the product of two binomials mentally using the FOIL method Recognize and factor the difference of two squares Recognize and factor the sums and differences of two cubes Find the squares of binomials Recognize and factor trinomial squares Factor quadratic trinomials of the form x2 + bx + c, c 0 Page 10 of 24
11 Factor quadratic trinomials of the form x2 + bx + c, c 0 Factor quadratic trinomials of the form ax2 + bx + c, a 1, c 0 Factor by grouping using the distributive axiom Factor polynomials to their prime factors using a combination of methods Solve polynomial equations by factoring Solve problems involving quadratic equations Apply algebraic notations to the study of number theory Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 11 of 24
12 Unit 6: Fractions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale All students will develop number sense and will perform standard numerical operations and estimations on fractions 15 days Recommended Pacing State Standards Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Standard 4.A REI Reasoning with Equations and Inequalities 2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may rise. Instructional Focus Unit Enduring Understandings Equivalent expressions can be represented in a variety of forms There are algebraic models that represent the way of a number varies A variety of families of functions can be used to model and solve real world situations Unit Essential Questions When is a rational expression simplified? How are patterns of change related to the type or behavior of functions? What real life phenomena can be represented with rational expressions? Objectives Students will know: Operations with fractions extend to simplifying rational expressions Students will be able to: Simplify algebraic fractions using factorization Multiply algebraic fractions Divide algebraic fractions Express two fractions using their least common denominator Add and subtract algebraic fractions by finding their LCD Write mixed expressions as fractions in simplest form Divide polynomials by polynomials with two or more terms using long division Express complex fractions as a simple fraction Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 12 of 24
13 Unit 7: Applying Fractions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale 15 days Recommended Pacing State Standards Standard 4.A APR Arithmetic with Polynomials and Rational Expressions Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Standard 4.A REI Reasoning with Equations and Inequalities Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may rise. 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 Unit Essential Questions What kinds of questions can be answered using proportional reasoning? How is proportional reasoning of geometric figures used to solve problems? Objectives Students will know: 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 Students will be able to: Solve problems involving ratios Solve problems involving proportions Solve equations with fractional coefficients Solve fractional equations Define and work with the concept of percent Solve problems involving percent and percent of change Page 13 of 24
14 Use tables and/or charts to solve mixture problems Use tables to solve work problems Write, use, and understand the concept of scientific notation Use and explain the nature of negative exponents Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 14 of 24
15 Unit 8: Linear Equations and Systems Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of systems of linear equations and inequalities. By learning to solve the systems both algebraically and graphically and to appropriately interpret their solutions students will be able to apply systems to model real world situations. 15 days Recommended Pacing State Standards Standard 4.A REI Reasoning with Equations and Inequalities 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6 Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. 10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12 Graph the solutions of a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables, as the intersection of the corresponding half planes. Instructional Focus Unit Enduring Understandings A system of equations or inequalities can be used to model and solve world situations The value of the graphical representation of the solution to a system of linear equations or inequalities depends on the real world situation the system models Unit Essential Questions How can you solve systems of equations or inequalities? How can systems of equations and inequalities be used to solve real life situations? Objectives Students will know: Procedures for solving systems of linear equations and inequalities Procedures for writing systems of linear equations and inequalities for real life situations Students will be able to: Solve equations in two variables over given domains of the variables Graph ordered pairs and linear equations in two variables Solve a system of linear equations by graphing Solve a system of linear equations by substitution Use systems of equations to solve problems in two variables Solve systems of equations in two variables using addition and subtraction Use multiplication with the addition/subtraction method to solve systems of linear equations Use systems of equations to solve wind and water current problems Use systems of equations to solve digit, age, and fraction problems Page 15 of 24
16 Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 16 of 24
17 Unit 9: Introduction to Functions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves solving and graphing the solution sets of linear equations and inequalities. Words, tables, graphs, and symbols are used to represent, analyze and model linear functions. In contextual problems graphing and interpretation of results in terms of the context will be explored. 15 days Standard 4.A CED Creating Equations Recommended Pacing State Standards 2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non viable options in a modeling context. Standard 4.A REI Reasoning with Equations 3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Standard 4.F IF Interpreting Functions 1 Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. 2 Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. 5 Relate the domain of a function to its graph and where applicable, to the quantitative relationship it describes. 6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. Instructional Focus Unit Enduring Understandings Functions are a special type of relationship or rule that uniquely associates members of one set with members of another set Graphs and equations are alternative (and often equivalent) ways for depicting and analyzing patterns of change The value of a particular representation depends on its purpose Functional relationships can be expressed in real contexts, graphs, algebraic equations, tables, and words; each representation of a given function is simply a different way of expressing the same idea Linear and absolute value graphs and equations can be used to model and describe physical relationships Page 17 of 24
18 Unit Essential Questions How can change be best represented mathematically? How can we use mathematical language to describe linear change? How can we use mathematical models to describe change or change over time? How are patterns of change related to the behavior of functions? How are functions and their graphs related? Objectives Students will know: Terminology and notation for functions Slope as a rate of change The procedures for writing the equation of line in slope intercept, standard and point slope form The value of the correlation coefficient indicates the strength of the linear relationship between the elements of a set of data Students will be able to: Define and find the slope of a given line, including situations of positive, negative, zero, and no slopes. Write and use the slope intercept form of a linear equation Use the slope intercept form to recognize situations of one, no, and infinite solutions. Find the equations of a line given: a) the slope and a point and b) two points, using either slope intercept or point slope methods. Define a function in terms of domain and range and to determine its values. Graph linear and quadratic functions, determining min/max and line of symmetry for quadratics. Understand and use the algebraic definition of direct variation. Understand and use the geometric definition (graph) of direct variation. Understand and use the algebraic definition of inverse variation. Understand and use the geometric definition (graph) of inverse variation. Understand and use direct and inverse variations raised to the second power. Understand and solve problems using joint and combined variations. Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 18 of 24
19 Unit 10: Inequalities Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of systems of linear inequalities. By learning to solve the systems both algebraically and graphically and to appropriately interpret their solutions students will be able to apply systems to model real world situations. 15 days Recommended Pacing State Standards Standard 4.A REI Reasoning with Equations and Inequalities 5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 6 Solve systems of linear equations exactly and approximately (e.g. with graphs), focusing on pairs of linear equations in two variables. 10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 12 Graph the solutions of a linear inequality in two variables as a half plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables, as the intersection of the corresponding half planes. Instructional Focus Unit Enduring Understandings A system of inequalities can be used to model and solve world situations The value of the graphical representation of the solution to a system of linear inequalities depends on the real world situation the system models Unit Essential Questions How can you solve systems of inequalities? How can systems of inequalities be used to solve real life situations? Objectives Students will know: Procedures for simplifying, solving and graphing one, two and multi step single variable inequalities Procedures for simplifying, solving and graphing single variable compound inequalities Procedures for simplifying and solving single variable absolute value inequalities Single variable inequalities may have infinitely many (identity), no real number solutions (empty set) or a unique solution Students will be able to: Do order and graph inequalities on a number line Transform inequalities for the purpose of solution Solve inequality word problems Find solution sets of combined inequalities Solve equations and inequalities containing absolute values Understand and use the concepts of conjunction and disjunction Page 19 of 24
20 Graph linear inequalities in two variables Find the solution set of a systems of linear inequalities in two variables, using graphing (linear programming) Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 20 of 24
21 Unit 11: Rational and Irrational Numbers Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit involves the study of rational functions and the interconnectedness of arithmetic and algebraic skills. Rational functions allow students to explore real world situations combining several algebraic concepts into one mathematical model. 15 days Recommended Pacing State Standards Standard 4.A APR Arithmetic with Polynomials and Rational Expressions 7 Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. Standard 4.A REI Reasoning with Equations and Inequalities 2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may rise. Instructional Focus Unit Enduring Understandings Equivalent expressions can be represented in a variety of forms There are algebraic models that represent the way of a number varies A variety of families of functions can be used to model and solve real world situations Unit Essential Questions When is a rational expression simplified? How are patterns of change related to the type or behavior of functions? What real life phenomena can be represented with rational expressions? Objectives Students will know: Equations for direct and inverse variation Operations with fractions extend to simplifying rational expressions Students will be able to: Apply the properties of real numbers Express rational numbers as decimals or fractions and to recognize patterns Find the rational square roots of numbers that have them, using properties of square roots Simplify radicals and approximate decimal values to irrational square roots Find and use square roots of variable expressions in problem/equation solving Derive the Pythagorean Theorem Use the Pythagorean Theorem and its converse to solve geometric problems Simplify products and quotients of radicals Simplify sums and differences of radicals Multiply binomials containing square root radicals Page 21 of 24
22 Solve simple radical equations Use and understand rational exponent notation Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 22 of 24
23 Unit 12: Quadratic Functions Content Area: Mathematics Course & Grade Level: A&E Mathematics, Grade 7 Summary and Rationale This unit will be an extension of skills that will model real world situations with quadratic functions and equations. In addition to the algebraic methods for solving quadratic equations, the unit will explore graphical representations of quadratic equations and analyze and extrapolate pertinent information from these graphs. 15 days Standard 4.A SSE Seeing Structure in Expressions Recommended Pacing State Standards 4 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. Standard 4.A REI Reasoning with Equations and Inequalities 4 Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p) squared = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a 3 bi for real numbers a and b. Instructional Focus Unit Enduring Understandings Real world situations involving quadratic relationships can be modeled and solved using multiple representations Quadratic equations and functions can be solved using a variety of methods Unit Essential Questions How can we use mathematical language to describe non linear change? What are the characteristics of a quadratic function? How can you solve a quadratic equation? Objectives Students will know: Procedures for solving quadratic equations and functions A parabola is the graphical representation of a quadratic function When an estimate is an appropriate solution in a real life situation Students will be able to: Solve quadratic equations of perfect squares by factoring Page 23 of 24
24 Solve quadratic equations by completing the square Solve quadratic equations by using the quadratic formula Use the discriminant to determine the number and location of roots of a quadratic equation Choose and use the best method of solution for a quadratic equation, in a specific problem Resources Core Text: Algebra, Structure & Method, Houghton Mifflin Suggested Resources: Page 24 of 24
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