Norwalk La Mirada Unified School District. Algebra Scope and Sequence of Instruction

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1 1 Algebra Scope and Sequence of Instruction Instructional Suggestions: Instructional strategies at this level should include connections back to prior learning activities from K-7. Students must demonstrate understanding of algebraic concepts using multiple representations (verbal, numeric, pictorial, algebraic). Students build understanding of these concepts from concrete to abstract through use of manipulatives, pictures and their connections to verbal, numeric and algebraic representations. Focus on the connections between graphic and algebraic representations to create meaning. The development of inductive and deductive reasoning in Algebra 1 is key for success in future mathematics courses. Key Algebra Accomplishments: Graph, analyze, solve, and apply linear equations and inequalities. Graph, analyze, solve, and apply systems of linear equations and inequalities. Graph, analyze, solve, and apply quadratic equations. Fluency with laws of exponents and polynomial operations. Benchmark Windows: Benchmark 1: Nov (Units 1-2) - Discuss results Benchmark 2: Jan. 30-Feb. 2 (Cumulative including Units 1-3) - Discuss results Benchmark 3: April 2-5 (Cumulative including Units 1-6) - Discuss results

2 2 Algebra Scope and Sequence of Instruction UNIT 1: Prerequisite Skills for Algebra - 17 days (15 instructional/2flex) Unit Overview: This unit focuses on selected skills necessary for success in Algebra I. Some concepts might require a more complete conceptual development (integer operations, combine like terms) than others. Use the allotted time as you see fit to meet the needs of your students. Textbook Sections: Holt Algebra 1 Textbook Lesson 1-1 to 1-7 Glencoe 1-1 to 1-8, 2-1 Essential Questions: When translating verbal expressions to algebraic expressions, what are words that indicate addition? Subtraction? Multiplication? Division? What is a zero pair? How can integer tiles be used to enhance understanding of integer addition/subtraction? How can algebra tiles enhance your understanding of combining like terms? + More (below)

3 3 Unit 1 Objectives Essential Questions Resources Key Vocabulary Translate Verbal Expressions Into Algebraic Expressions When translating verbal expressions to algebraic expressions, what are words that indicate addition? Subtraction? Multiplication? Division? Holt - Lesson 1.1: Create Poster(s) of Words that Mean +,, x, Glencoe 2-1: Create Poster(s) of Words that Mean +,, x, PICTORIAL RESOURCES: Algebraic Expressions Algebraic Expression Evaluate Variable Constant Sum-Difference Product-Quotient More Than/Less Than Consecutive Perform Integer Operations What is a zero pair? How can integer tiles be used to enhance understanding of integer addition/subtraction? Include Integer Tile Activities to build Visual/Concrete Understanding Glencoe Warning!!! No review with operations on integers. You will need to supplement. Use integer tile activities to build visual/concrete understanding Integers Absolute Value PICTORIAL RESOURCES: Integer Addition Simplify Powers & Roots Use Area Model for explaining and teaching perfect squares. Base Exponent Power Perfect Square Square Root Cube Root

4 4 Combine Like Terms Apply Properties of Real Numbers How can algebra tiles enhance your understanding of combining like terms? What are similarities and differences between the associative, commutative and distributive property. Unit 1 Include Algebra Tile Activities to build Visual/Concrete Understanding Glencoe 1-5 Example 3 Combine like terms is only shown here. Intro with algebra tiles and more practice is suggested. See Glencoe 7-3 & 7-4 Algebra Labs - (pgs. 375, ) PICTORIAL RESOURCES: Combine Like Terms , Build understanding of distributive property using Area Model Glencoe 1-5 Algebra Lab (Pg. 27) Algebra Tiles for Distributive Property Glencoe Reading Math (Pg. 38) Do this in preparation for solving equations and justifying steps in unit 2. Coefficient Variable Like and Unlike Terms Commutative Property Associative Property Distributive Property Frayer Model Reciprocal Multiplicative Inverse Opposite Additive Inverse Counterexample Simplify using Order of Operations How do you know which operation to perform first when evaluating numeric expressions? What is the mathematic impact of performing the wrong operation first? Activity Math Tai Chi (Students act out each operation tai-chi style.) Order of Operations

5 5 Algebra Scope and Sequence of Instruction UNIT 2: Equations and Inequalities: Solving - 30 days (27 instructional/3 flex) Unit Overview: This unit focuses on solving equations and inequalities. Some teachers choose to solve equations and inequalities simultaneously, while others separate this unit into two parts (equations, then inequalities). Require students to justify steps while solving. Use the allotted time as you see fit to meet the needs of your students. Textbook Sections: Holt Algebra 1 Chapter 2 Lessons 2-1 to 2-7 Holt Algebra 1 Deductive Reasoning and Equations (Pg 99) Holt Algebra 1 Chapter 3 Lessons 3-1 to 3-7 Holt Algebra 1 Deductive Reasoning and Inequalities (Pg 169) Glencoe 2-2 to 2-9 Glencoe 6-1 to 6-6 (See Algebra Academy Resource File - PATTERNS (Page 12), SOLVING EQUATIONS TEMPLATES (Page 27) Connections: Have students graph the solution to linear and absolute value equations as a point(s) on a number line in preparation for graphing solutions of inequalities. Justify each step in solving equations. (Prep for Geometric Proof) Use concrete representations (algebra tiles, stories, money, pictures, etc.) to build understanding of equations. Essential Questions: How can equations be represented in the following ways: verbally, numerically, pictorially, algebraically? How do you know which property to apply first when solving equations? What is the difference between an equation and an inequality? How do you verify a solution to an equation or an inequality? + More (below

6 6 Applications Warning: Glencoe addresses weighted averages, mixtures, and rate in section 2-9; Holt addresses these as applications of systems of equations in section 6-5. High school students should do Holt section 6-5 as single linear functions at this point. Students are NOT expected to use systems on benchmarks 1 and 2. Unit 2 Objectives Essential Questions Resources Key Vocabulary Use Deductive Reasoning to Justify Solutions How do you verify a solution to an equation or an inequality? Glencoe Reading Math (Pg. 38) Refer back to this in preparation for solving equations and justifying steps in solving equations. Equation Frayer Model Solution Properties of Equality Deductive Reasoning Solution Set No Solution Infinite Number of Solutions Solve 1-Step & 2- Step Equations What is the inverse of addition? Subtraction? Multiplication? Division? How can equations be represented in the following ways: verbally, numerically, pictorially, algebraically? Glencoe 2-2 Algebra Lab Solve using Algebra Tiles (pg. 77) (See Algebra Academy Resource File - SOLVING EQUATIONS TEMPLATES Page 27) Solve Multi-Step Equations How do you know which property to apply first when solving equations? Glencoe 2-4 Algebra Lab Solve Multi- Step Equations using Algebra Tiles (pg. 91)

7 7 Solve with Variables on Opposite Sides Which side should you move the variable when solving? How can equations be represented in the following ways: verbally, numerically, pictorially, algebraically? Unit 2 Solve Proportions Solve Equations with Absolute Values What types of problems can proportions be used to solve? How can proportions be represented in the following ways: verbally, numerically, pictorially, algebraically? Ratio Proportion Cross Product Scale Drawing Scale Model Formula Percent Unit Rate Percent of Change Absolute Value Solve Inequalities & Graph on a Number Line What is the difference between an equation and an inequality? Glencoe 6-2 Algebra Lab Solve Inequalities (pg. 300) Inequality Compound Inequality Union Intersection and/or

8 8 Algebra Scope and Sequence of Instruction UNIT 3: Linear Functions - 36 days (31 instructional/5 flex) Unit Overview: This unit focuses on solving and writing linear equations. Although students have experience with slope and linear functions from previous grade levels, conceptual development is necessary using real applications and pictorial representations. Use the allotted time as you see fit to meet the needs of your students. Textbook Sections: Holt Algebra 1 Chapter 4 Lessons 4-1 to 4-3, 4-5 Holt Algebra 1 - Inductive Reasoning (Pg 233) Holt Algebra 1 Chapter 5 Lessons 5-1 to 5-3; 5-5 to 5-7 Glencoe 1-9 Glencoe 3-1 to 3-5 Glencoe 4-1 to 4-7 (See Algebra Academy Resource File Pages 3-59) Connections: Use concrete models (algebra tiles, stories, money, pictures, etc.) to build understanding of equations. Represent these equations by graphing in Cartesian plane. Use multiple representations (algebraic, verbal, pictorial, and numeric) build connections. Essential Questions: What does the rate of change describe? How can a linear function be represented in the following ways: verbally, numerically, pictorially, algebraically? How does the rate of change relate to a linear function? Which form of a linear equation is most useful? (Standard Form, Slope-Intercept Form, or Point-Slope Form) Explain. + More (below)

9 9 Unit 3 Objectives Essential Questions Resources Key Vocabulary Intro to Functions: Use related vocabulary accurately Plot Points Recognize and Describe Patterns How can a linear function be represented in the following ways: verbally, numerically, pictorially, algebraically? How does the rate of change relate to a linear function? Holt - Inductive Reasoning (Pg 233) & Holt Algebra 1 One-Stop Planner-Exploration 4-5 (Arithmetic Sequences) Have student make predictions about arithmetic sequences. It is suggested NOT to use nth term formula listed here. A stronger connection can be made to slopeintercept form by identifying the initial term as the 0 th term (same as y-intercept) and using the common difference (same as the slope) to extend the pattern. (See Algebra Academy Resource File - PATTERNS Page 12) Glencoe 1-9 Intro to Functions & Graphs * Holt - Lesson 4-4 Scatterplots and Trend Lines is not an Algebra 1 standard but does appear on the CAHSEE Relation X-Axis Y-Axis Ordered Pair Inductive Reasoning Function Function Notation Domain-Input Range-Output Arithmetic Sequence * Positive Correlation *Negative Correlation *No Correlation *Scatterplot Graph Lines using a Table How does the rate of change relate to a table? Which variable in used for the input? Output? *Glencoe Lesson 4.6 Statistics: Scatter Plots and Lines of Fit is not an Algebra 1 standard but does appear on the CAHSEE Algebra Walk This in an introductory lesson where students are human points and predictions are made about linear equations. See attached lesson resources. (See Algebra Academy Resource File - Algebra Walk Page 4) Independent Variable Dependent Variable Linear Equation Linear Function

10 10 Find x- and y- intercepts Find the Slope of a Line Convert Equations from Standard Form to Slope-Intercept Form Where do you substitute zero to calculate the x- intercept? Y-intercept? What does the rate of change describe? How does the slope help you to predict what the graph of a linear function looks like? Do you prefer to find the slope graphically or algebraically? Explain. Why is it helpful to convert from standard form into slope-intercept form? Unit 3 (See Algebra Academy Resource File - SLOPE Page 42) X-Intercept Y-Intercept Rise/Run Slope Frayer Model Rate of Change Vertical Line Undefined Slope Horizontal Line Parallel Lines Perpendicular Lines Standard Form Slope-Intercept Form Graph Equations using Slope- Intercept Form Write Linear Equations with & without a graph Write Linear Equations using Point-Slope Form Where do you start? What does the slope tell you? What two things do you need to write a linear equation? Which form of a linear equation is most useful? (Standard Form, Slope- Intercept Form, or Point- Slope Form) Explain. Glencoe 4-3 Graphing Calculator Lab Investigation PICTORIAL RESOURCES (from Pictorial Mathematics by Mendieta) Start writing equations by graphing and transition to non-graphing approaches after a need for algebraic methods is developed. Point-Slope Form

11 11 Algebra Scope and Sequence of Instruction UNIT 4- Systems of Linear Equations & Inequalities: Solving & Graphing - 17 days (15 instructional/2flex) Unit Overview: This unit focuses on solving systems of equations. Students solve applications numerically, algebraically and by graphing. Use the allotted time as you see fit to meet the needs of your students. Textbook Sections: Holt Algebra 1 Chapter 6 Lesson 6-1 to 6-7 Glencoe 5-1 to 5-5 Glencoe 6-7 to 6-8 (Systems of Linear Inequalities) (See Algebra Academy Resource File SYSTEMS OF LINEAR EQUATIONS - Pages 60-85) Connections: Help students make meaning by exploring with real applications (word problems) involving systems. Transition from graphing to non-graphing approaches after a need for algebraic method is developed. Use Multiple Representation Templates to build conceptual understanding. If you require students represent the same problem in multiple ways, they are able to better understand the connection between solutions. Essential Questions: What is a system of equations? How can a system of equations be represented in the following ways: verbally, numerically, pictorially, algebraically? What are the methods that can be used to solve a system of equations? When should each method be used? What does a system of equations look like that has: 1 solution, no solutions, infinitely-many solutions? What is the difference between a system of equations and a system of inequalities?

12 12 Unit 4 Objectives Essential Questions Resources Key Vocabulary Apply Systems of Equations What is a system of equations? (See Algebra Academy Resource File VERBAL & ALGEBRAIC PRACTICE, pg 80) Solve by Graphing How do you find the solution to a system by graphing? Model with Graphing Calculator (See Algebra Academy Resource File SYSTEMS WALK, pg 61) System of Linear Equations Solution of a System Solve by Substitution Why do you need to know algebraic methods for solving systems of equations? Where does the substitution method get its name? Holt - LAB Model Systems of Equations with Algebra Tiles (Pg. 335) Glencoe 5-2 Algebra Lab Substitution with Algebra Tiles (pg. 260) Substitution Solve by Elimination When should you use the elimination method instead of the substitution method? Elimination Apply Systems of Equations How can a system of equations be represented in the following ways: verbally, numerically, pictorially, algebraically? Have students work in pairs to create their own applications of systems. (See Algebra Academy Resource File VERBAL & ALGEBRAIC PRACTICE, pg 80, MATCHING MULTIPLE REPRESENTATIONS (pg. 86 & Separate Attachment)) Identify Systems with no solutions/many solutions What does a system of equations look like that has: 1 solution, no solutions, infinitelymany solutions? Holt - Lesson 6.4 Work concepts in this section into 6-1to 6-3 Parallel Lines No solution Infinitely many solutions Graph Systems of Linear Inequalities What is the difference between a system of equations and a system of inequalities? Linear Inequality System of Linear Inequalities

13 13 Algebra Scope and Sequence of Instruction UNIT 5 - Quadratics: Solving & Graphing - 40 days (36 instructional/4flex) Unit Overview: This unit focuses on factoring, solving and graphing quadratic functions. In addition to solving by factoring, students use square roots, the quadratic formula, and completing the square. Identifying the roots from the solutions of quadratics is using to sketch graphs of parabolas. Textbook Sections: Exponent Rules: Holt Algebra 1 Chapter 7 Lesson 7-3, 7-4 Operations with Polynomials: Holt Algebra 1 Chapter 7 Lesson 7-6 to 7-9 Factoring: Holt Algebra 1 Chapter 8 Lesson 8-1 to 8-6 Solving: Holt Algebra 1 Chapter 9 Lesson 9-1 to 9-6 Holt Algebra 1 Chapter 11 Lesson 11-2 (Simplify Square Roots) Solving: Holt Algebra 1 Chapter 9 Lesson 9-7 to 9-9 Exponent Rules: Glencoe 7-1, 7-2 Operations with Polynomials: Glencoe Factoring & Solving 8-1 to 8-6 Graphing Quadratics 9-1 to 9.3 Simplify Radicals 10-1 (example 1 only) Quadratic Formula 9.4 (See Algebra Academy Resource File QUADRATIC FUNCTIONS - Pages )

14 14 Connections: Make strong connections between numeric solutions and x-intercepts of parabolas. Provide algebra tiles for students to use while factoring and solving. Physical Applications of Quadratics (Projectile Motion) Essential Questions: How can a quadratic equation be represented in the following ways: verbally, numerically, pictorially, algebraically? What does factoring mean? How are the solutions to a quadratic equation related to the graph of a quadratic function? How can algebra tiles be used to multiply binomials and to factor quadratics? What is the zero product property and how is it used in solving quadratics? What methods can be used to mind solutions to quadratic equations? When is it preferable to use each method? How is completing the square related to the quadratic formula? + More (below)

15 15 Unit 5 Objectives Essential Questions Resources Key Vocabulary Know and Apply Exponent Rules (products and quotients of powers) Holt - Lesson 7-3 Example #1 (simple applications with variables) Holt - Lesson 7-4 Example #1a (simple applications with variables) Negative Exponent Frayer Model Perform Operations on Polynomials (Add, Subtract, Multiply) How can algebra tiles be used to multiply binomials? Holt - LAB 7-7: Use Algebra Tiles for Polynomial Addition and Subtraction (Pg ) Holt - LAB 7-8: Use Algebra Tiles for Polynomial Multiplication (Pg ) Holt - Lessons 7-8 to 7-9: Use area model Glencoe 7-6 Use area model in addition to FOIL for connection to algebra tile work and preview of factoring. Glencoe 7-3 Algebra Lab Polynomials with Algebra Tiles (pg. 375) Glencoe 7-4 Algebra Lab Combine Like Terms with Algebra Tiles (pg ) Glencoe 7-6 Algebra Lab Multiply Polynomials (pg ) Polynomial Standard Form (of a Polynomial) Degree Constant Monomial Binomial Trinomial Quadratic Area Model F.O.I.L. PICTORIAL RESOURCES: Multiplication/Area Model 256,257, 260, 262, 263 Holt - Section Includes work on simplifying radicals. Simplify Radicals How are perfect squares helpful in simplifying radicals? Radical Expression Square Root Perfect Square

16 16 Unit 5 Factor using GCF What does factoring mean? Factoring out a GCF is the opposite of applying which property? Holt - LAB 8-2: Use Algebra Tiles for Factoring GCF (Page 486) Glencoe 8-2 Algebra Lab Factoring (GCF) with Algebra Tiles (pg. 425) Factor (noun and verb) Prime Factorization Greatest Common Factor Factored Form Parabola Factored Form X-Intercepts Roots Zeros Vertex Axis of Symmetry Graph Quadratics and Know that their roots are the x- intercepts. How are the solutions to a quadratic equation related to the graph of a quadratic function? Holt - LAB 9-4: Explore and make predictions about x-intercepts using the graphing calculator (quadratics with integer roots are suggested) Glencoe 9-1 Graphing Calculator Lab Quadratic Functions (pg ) (See Algebra Academy Resource File EXPLORE X-INTERCEPTS OF PARABOLAS, pg.106, PARABOLA MATCHING, pg. 120) Factor Quadratics in the form x 2 +bx+c How can algebra tiles be used to factor quadratics? What does factoring mean? How can you determine if a quadratic is unfactorable? Holt - LAB 8-3: Use Algebra Tiles for Factoring: x 2 +bx + c (Page 494) Glencoe 8-3 Algebra Lab Factoring x 2 +bx + c with Algebra Tiles (pg ) Glencoe 8-5 Algebra Lab Factoring Difference of Squares (pg. 447) PICTORIAL RESOURCES: Factoring Other Resources: Factor with Algebra Tiles (CPM AP 2 resource page) (See Algebra Academy Resource File BUILD GEOMETRIC MODELS WITH ALGEBRA TILES, pg. 94, WRITE AREAS AS SUMS & PRODUCTS, pg. 99) Factored Form Difference of Squares

17 17 Unit 5 Factor Quadratics in the form ax 2 +bx+c How is factoring ax 2 +bx+c different from x 2 +bx+c? How can an area model be used to factor quadratics? Holt - LAB 8-4: Use Algebra Tiles for Factoring: ax 2 +bx + c (Page 504) Glencoe 8-4 When factoring ax 2 +bx + c repeat Algebra Lab 8-3 (pg ) with new problems. (See Algebra Academy Resource File MATCH ALGEBRAIC & GEOMETRIC REPRESENTATIONS, pg. 101) Factored Form Factor by Grouping Why can using an area model be helpful when factoring by grouping? Do you prefer the traditional method of factoring by grouping or the area model method? Suggestion: Use Area Model Solve Quadratics using Multiple Methods (Square Root, Factor, Complete the Square, Quadratic Formula) What is the zero product property and how is it used in solving quadratics? When should you use each method of solving? How do you use each method? Glencoe 9-3 Use algebra tiles to complete the square Square Root Zero Product Property Quadratic Formula Roots Zeros Apply Quadratic Functions What types of problems require the use of quadratic equations? Glencoe 9-4 Algebra Lab Applying Quadratics (See Algebra Academy Resource File MULTIPLE REPRESENTATIONS TEMPLATES, pg. 126)

18 18 Algebra Scope and Sequence of Instruction UNIT 6: Rational Expressions - 18 days (16 instructional/2 flex) Unit Overview: This unit focuses on simplification of rational expression (algebraic fractions) and operations on rational expressions. Students should also master rules of exponents and their use in simplification. Textbook Sections: Holt Algebra 1 Chapter 7 Lessons 7-1 to 7-5 Holt Algebra 1 Chapter 10 Lessons 10-3 to 10-5, 10-6*, 10.7 Glencoe 11-2 to 11-4, 11-6 to 11-7 Connections: Operations on algebraic fractions found in this unit should reference operations used in numeric fractions from previous grade levels. The concept of the BIG 1 (fractions equivalent to one) should be familiar to students. Essential Questions: Why must you factor before simplifying algebraic fractions? What is the BIG 1? How are operations with algebraic fractions (rational expressions) similar to operations with numeric fractions? How do you know if a solution is extraneous?

19 19 Unit 6 Objectives Essential Questions Resources Key Vocabulary Know and Apply Exponent Rules *Holt - Selected problems from Lesson 10-6 may be included with Lesson 10-3 Simplify Rational Expressions (Algebraic Fractions) Why must you factor before simplifying algebraic fractions? What is the BIG 1? Review with students how to reduce fractions by factoring. Numerator Denominator Rational Expression (Algebraic Fraction) Excluded Value BIG 1 Perform Operations with Rational Expressions (Add, Subtract, Multiply, Divide) How are operations with algebraic fractions (rational expressions) similar to operations with numeric fractions? (Note: Polynomial Long Division and is not an Algebra 1 standards) Solve Equations with Rational Expressions How do you know if a solution is extraneous? Extraneous Solution

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