Algebra Discovering Algebra An Investigative Approach California: Algebra 1 Grades Eight Through Twelve Mathematics Content Standards 1.0 Students identify and use the arithmetic properties of subsets of integers and rational, irrational, and real numbers, including closure properties for the four basic arithmetic operations where applicable: Content Standards 1.1 Students use properties of numbers to demonstrate whether assertions are true or false. Chapter 0: Fractions and Fractals 0.4: Going Somewhere? 2.7: Evaluating Expressions Chapter 6 Review Content Standards 2.0 Students understand and use such operations as taking the opposite, finding the reciprocal, taking a root, and raising to a fractional power. They understand and use the rules of exponents. 2.1: Proportions 2.7: Evaluating Expressions 2.8: Undoing Operations 3.6: Solving Equations Using the Balancing Method 6.2: Exponential Equations 6.3: Multiplication and Exponents 6.4: Scientific Notation for Large Numbers 6.5: Looking Back with Exponents 6.6: Zero and Negative 6.7: Fitting Exponential Models to Data 6.8: Activity Day: Decreasing Exponential Models and Half-Life Chapter 6 Review 7.4: Function Notation CA correlation of 1
9.1: Solving Quadratic Equations Chapter 10: Probability 10.3: Random Outcomes 11.3: Squares, Right Triangles, and Areas 11.4: The Pythagorean Theorem 11.5: Operations with Roots 11.6: A Distance Formula 3.6, 6.3, 6.4, 6.5, 6.6, and 11.5 3.0 Students solve equations and inequalities involving absolute values. 4.0 Students simplify expressions before solving linear equations and inequalities in one variable, such as 3(2x 5) + 4(x 2) = 12. 7.5: Defining the Absolute-Value Function 7.6: Squares, Squaring, and Parabolas Chapter 10: Probability 10.3: Random Outcomes 7.5 and 7.6 3.6: Solving Equations Using the Balancing Method Chapter 3 Review 4.7: Applications of Modeling Chapter 4 Review 5.5: Inequalities in One Variable 7.3: Graphs of Real-World Situations 8.1: Translating Points 3.6, 4.4, and 5.5 CA correlation of 2
5.0 Students solve multi-step problems, including word problems, involving linear equations and linear inequalities in one variable and provide justification for each step. 6.0 Students graph a linear equation and compute the x- and y- intercepts (e.g., graph 2x + 6y = 4). They are also able to sketch the region defined by linear inequality (e.g., they sketch the region defined by 2x + 6y < 4). 7.0 Students verify that a point lies on a line, given an equation of the line. Students are able to derive linear equations by using the point-slope formula. 3.4: Linear Equations and the Intercept Form 3.5: Linear Equations and Rate of Change 3.6: Solving Equations Using the Balancing Method Chapter 3 Review 4.1: A Formula for Slope 4.5: Writing Point-Slope Equations to Fit Data 4.6: More on Modeling 4.7: Applications of Modeling Chapter 4 Review 5.5: Inequalities in One Variable 4.4, 4.7, and 5.5 2.4: Direct Variation 3.2: Linear Plots 3.4: Linear Equations and the Intercept Form 3.5: Linear Equations and Rate of Change 4.2: Writing a Linear Equation to Fit Data 4.5: Writing Point-Slope Equations to Fit Data 4.6: More on Modeling Project: State of the States 5.6: Graphing Inequalities in Two Variables 5.7: Systems of Equations 4.5, 5.6, and 5.7 4.1: A Formula for Slope 4.3: Point-Slope Form of a Linear Equation 4.5: Writing Point-Slope Equations to Fit Data 4.7: Applications of Modeling Chapter 4 Review 5.1: Solving Systems of Equations 5.2: Solving Systems of Equations Using Substitution CA correlation of 3
6.1: Recursive Routines 7.5: Defining the Absolute-Value Function Chapter 10: Probability 10.5: Multiple-Stage Experiments 4.5, 5.1, and 5.2 8.0 Students understand the concepts of parallel lines and perpendicular lines and how those slopes are related. Students are able to find the equation of a line perpendicular to a given line that passes through a given point. 9.0 Students solve a system of two linear equations in two variables algebraically and are able to interpret the answer graphically. Students are able to solve a system of two linear inequalities in two variables and to sketch the solution sets. 10.0 Students add, subtract, multiply, and divide monomials and polynomials. Students solve multistep problems, including word problems, by using these techniques. 11.2: Finding the midpoint 11.3: Squares, Right Triangles, and Areas 11.1 and 11.2 5.1: Solving Systems of Equations 5.2: Solving Systems of Equations Using Substitution 5.3; Solving Systems of Equations Using Elimination 5.4: Solving Systems of Equations Using Matrices 5.7: Systems of Inequalities 7.2: Functions and Graphs 7.5: Defining the Absolute-Value Function 5.2, 5.3, and 5.7 3.6: Solving Equations Using the Balancing Method 5.1: Solving Systems of Equations 5.2: Solving Systems of Equations Using Substitution 5.3; Solving Systems of Equations Using Elimination 6.3: Multiplication and Exponents 6.4: Scientific Notation for Large Numbers 6.5: Looking Back with Exponents 6.6: Zero and Negative 6.7: Fitting Exponential Models to Data Chapter 6 Review CA correlation of 4
7.1: Secret Codes 7.6: Squares, Squaring, and Parabolas Chapter 8 Review Chapter 10: Probability 10.4: Counting Techniques 3.6, 6.3, 6.4, 6.5, and 9.3 11.0 Students apply basic factoring techniques to second-and simple third-degree polynomials. These techniques include finding a common factor for all terms in a polynomial, recognizing the difference of two squares, and recognizing perfect squares of binomials. 12.0 Students simplify fractions with polynomials in the numerator and denominator by factoring both and reducing them to the lowest terms. 4.2: Writing a Linear Equation to fit Data 9.4 and 9.8 Chapter 8 Review 9.4 and 9.8 CA correlation of 5
13.0 Students add, subtract, multiply, and divide rational expressions and functions. Students solve both computationally and conceptually challenging problems by using these techniques. 14.0 Students solve a quadratic equation by factoring or completing the square. 15.0 Students apply algebraic techniques to solve rate problems, work problems, and percent mixture problems. 8.4: Stretching and Shrinking Graphs Chapter 8 Review 8.6 and 9.8 9.4 and 9.6 2.2: Capture-Recapture 3.2: Linear Plots 3.4: Linear Equations and the Intercept Form 3.5: Linear Equations and Rate of Change 4.5: Writing Point-Slope Equations to Fit Data 4.7: Applications of Modeling 5.2: Solving Systems of Equations Using Substitution 5.7: Systems of Inequalities See also the following lesson in the ancillary booklet More Practice 5.2 CA correlation of 6
16.0 Students understand the concepts of a relation and a function, determine whether a given relation defines a function, and give pertinent information about given relations and functions. 17.0 Students determine the domain of independent variables and the range of dependent variables defined by a graph, a set of ordered pairs, or a symbolic expression. 18.0 Students determine whether a relation defined by a graph, a set of ordered pairs, or a symbolic expression is a function and justify the conclusion. 19.0 Students know the quadratic formula and are familiar with its proof by completing the square. 7.1: Secret Codes 7.2: Functions and Graphs 7.3: Graphs of Real-World Situations 7.4: Function Notation 7.1, 7.2, and 7.4 7.1: Secret Codes 7.2: Functions and Graphs 7.3: Graphs of Real-World Situations 7.4: Function Notation See also the following lesson in the ancillary booklet More Practice 7.4 7.1: Secret Codes 7.2: Functions and Graphs 7.3: Graphs of Real-World Situations 7.1 and 7.2 See also the following lesson in the ancillary booklet More Practice 9.7 CA correlation of 7
20.0 Students use the quadratic formula to find the roots of a second-degree polynomial and to solve quadratic equations. 21.0 Students graph quadratic functions and know that their roots are the x- intercepts. 22.0 Students use the quadratic formula or factoring techniques or both to determine whether the graph of a quadratic function will intersect the x-axis in zero, one, or two points. 23.0 Students apply quadratic equations to physical problems, such as the motion of an object under the force of gravity. 11.6: A Distance Formula See also the following lesson in the ancillary booklet More Practice 9.7 9.1: Solving Quadratic Equations 9.2: Finding the Roots and Vertex 9.2 and 9.4 9.1: Solving Quadratic Equations 11.5: Operations with Roots 9.4 and 9.7 9.1: Solving Quadratic Equations 9.2: Finding the Roots and Vertex 9.5: Activity Day: Projectile Motion See also the following lesson in the ancillary booklet More Practice 9.1 CA correlation of 8
24.0 Students use and know simple aspects of a logical argument: Content Standards 24.1 Students explain the difference between inductive and deductive reasoning and identify and provide examples of each. 24.2 Students identify the hypothesis and conclusion in logical deduction. 24.3 Students use counterexamples to show that an assertion is false and recognize that a single counterexample is sufficient to refute an assertion. 11.2: Finding the Midpoint 11.4: The Pythagorean Theorem Project: Show Me Proof This topic is addressed extensively in Key Curriculum s Discovering Geometry. This topic is addressed extensively in Key Curriculum s Discovering Geometry. 7.2: Functions and Graphs This topic is addressed extensively in Key Curriculum s Discovering Geometry. CA correlation of 9
25.0 Students use properties of the number system to judge the validity of results, to justify each step of a procedure, and to prove or disprove statements: Content Standards 25.1 Students use properties of numbers to construct simple, valid arguments (direct and indirect) for, or formulate counterexamples to, claimed assertions. 25.2 Students judge the validity of an argument according to whether the properties of the real number system and the order of operations have been applied correctly at each step. 25.3 Given a specific algebraic statement involving linear, quadratic, or absolute value expressions or equations or inequalities, students determine whether the statement is true sometimes, always, or never. 2.7: Evaluating Expressions 6.6: Zero and Negative 11.5: Operations with Roots Chapter 0: Fractions and Fractals 0.4: Going Somewhere? 2.2: Capture-Recapture 2.7: Evaluating Expressions 2.8: Undoing Operations 6.3: Multiplication and Exponents 5.1: Solving Systems of Equations 5.5: Inequalities in One Variable This type of problem is not precisely addressed. CA correlation of 10