Topic: Equations and inequalities |
Mathematics |
| Standard 8. Understands and applies basic and advanced properties of functions and algebra |
| | Level II (Grade 3-5) |
| | | Benchmark 4. | Understands the basic concept of an equality relationship (i.e., an equation is a number sentence that shows two quantities that are equal) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of an equation |
| | | Benchmark 5. | Solves simple open sentences involving operations on whole numbers (e.g., ? + 17 = 23) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of an open sentence |
| | Level III (Grade 6-8) |
| | | Benchmark 1. | Knows that an expression is a mathematical statement using numbers and symbols to represent relationships and real-world situations (e.g., equations and inequalities with or without variables) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of a mathematical expression |
| | | | | 2. | Understands the role of numbers in math expressions |
| | | | | 3. | Understands the role of variables in math expressions |
| | | | | 4. | Understands the concept of an inequality |
| | | | | 5. | Understands the role of symbols in math expressions |
| | | Benchmark 5. | Solves linear equations using concrete, informal, and formal methods (e.g., using properties, graphing ordered pairs, using slope-intercept form) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of a linear equation |
| | | | | 2. | Solves linear equations using knowledge of properties |
| | | | | 3. | Solves linear equations by graphing ordered pairs |
| | | | | 4. | Solves linear equations using slope-intercept form |
| | | | | 5. | Solves linear equations using concrete methods |
| | | | | 6. | Solves linear equations using informal methods |
| | | | | 7. | Solves linear equations using formal methods |
| | | Benchmark 6. | Solves simple inequalities and non-linear equations with rational number solutions, using concrete and informal methods |
| | | | Knowledge/skill statements |
| | | | | 1. | Solves simple inequalities using informal methods |
| | | | | 2. | Understands the concept of a non-linear equation |
| | | | | 3. | Solves simple non-linear equations using informal methods |
| | | | | 4. | Understands the concept of an inequality |
| | | | | 5. | Solves inequalities using concrete methods |
| | | | | 6. | Solves non-linear equations using concrete methods |
| | | Benchmark 10. | Solves simple systems of equations graphically |
| | | | Knowledge/skill statements |
| | | | | 1. | Solves simple systems of equations graphically |
| | | | | 2. | Understands the concept of systems of equations |
| | Level IV (Grade 9-12) |
| | | Benchmark 2. | Uses expressions, equations, inequalities, and matrices to represent situations that involve variable quantities and translates among these representations |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the relationship between inequalities and equations |
| | | | | 2. | Transforms inequalities to equations |
| | | | | 3. | Understands the relationship between equations and matrices |
| | | | | 4. | Represents equations in matrix form |
| | | | | 5. | Understands the relationship between expressions and equations |
| | | | | 6. | Understands the relationship between expressions and inequalities |
| | | | | 7. | Understands the relationship between expressions and matrices |
| | | | | 8. | Understands the relationship between inequalities and matrices |
| | | | | 9. | Understands the concept of an algebraic expressions |
| | | Benchmark 5. | Solves polynomial equations (e.g., find the real or complex roots of a quadratic equation) using a variety of methods (e.g., symbolically, graphically, numerically). |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of polynomials |
| | | | | 2. | Understands the concept of roots |
| | | | | 3. | Solves polynomial equations symbolically. |
| | | | | 4. | Models real world problems using polynomial equations without the use of technology. |
| | | | | 5. | Solves real world problems using polynomial equations with the use of technology. |
| | | | | 6. | Solves polynomial equations by finding the real roots of a quadratic equation. |
| | | | | 7. | Solves polynomial equations by finding the complex roots of a quadratic equation. |
| | | Benchmark 7. | Uses a variety of models (e.g., written statement, algebraic formula, table of input-output values, graph) to represent functions, patterns, and relationships |
| | | | Knowledge/skill statements |
| | | | | 1. | Represents functions using graphs |
| | | | | 2. | Represents functions using table of input/output values |
| | | | | 3. | Represents functions using algebraic formula |
| | | | | 4. | Represents functions using written statements |
| | | | | 5. | Represents functions using models |
| | | | | 6. | Represents patterns using graphs |
| | | | | 7. | Represents patterns using table of input/output values |
| | | | | 8. | Represents patterns using algebraic formula |
| | | | | 9. | Represents patterns using written statements |
| | | | | 10. | Represents patterns using models |
| | | | | 11. | Represents relationships using graphs |
| | | | | 12. | Represents relationships using table of input/output values |
| | | | | 13. | Represents relationships using algebraic formula |
| | | | | 14. | Represents relationships using written statements |
| | | | | 15. | Represents relationships using models |
| | | Benchmark 11. | Uses a variety of methods (e.g., with graphs, algebraic methods, and matrices) to solve systems of equations and inequalities |
| | | | Knowledge/skill statements |
| | | | | 1. | Solves systems of equations using graphs |
| | | | | 2. | Solves systems of equations using matrices |
| | | | | 3. | Solves systems of equations using algebraic methods |
| | | | | 4. | Understands the concept of systems of inequalities |
| | | | | 5. | Solves systems of inequalities using algebraic methods |
| | | | | 6. | Solves systems of inequalities using graphs |
| | | | | 7. | Solves systems of inequalities using matrices |
| | | Benchmark 13. | Adds, subtracts, multiplies, and divides polynomials |
| | | | Knowledge/skill statements |
| | | | | 1. | Adds polynomials |
| | | | | 2. | Subtracts polynomials |
| | | | | 3. | Multiplys polynomials |
| | | | | 4. | Divides polynomials |
| | | Benchmark 14. | Understands the meaning of and writes equivalent forms of expressions, equations, inequalities, and relations, including all of their parts(e.g., terms, factors, coefficients). |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the meaning of expressions. |
| | | | | 2. | Understands the meaning of equations. |
| | | | | 3. | Understands the meaning of inequalities. |
| | | | | 4. | Understands the meaning of relations. |
| | | | | 5. | Writes expressions in equivalent forms. |
| | | | | 6. | Writes equations in equivalent forms. |
| | | | | 7. | Writes inequalities in equivalent forms. |
| | | | | 8. | Writes relations in equivalent forms. |
| | | | | 9. | Understands the meanings of terms of an equation. |
| | | | | 10. | Understands the meanings of terms of an expression. |
| | | | | 11. | Understands the meanings of terms of an inequality. |
| | | | | 12. | Understands the meanings of terms of a relation. |
| | | | | 13. | Understands the meanings of factors of an equation. |
| | | | | 14. | Understands the meanings of factors of an expression. |
| | | | | 15. | Understands the meanings of factors of an inequality. |
| | | | | 16. | Understands the meanings of factors of a relation. |
| | | | | 17. | Understands the meanings of coefficient of an equation. |
| | | | | 18. | Understands the meanings of coefficient of an expression. |
| | | | | 19. | Understands the meanings of coefficient of an inequality. |
| | | | | 20. | Understands the meanings of coefficient of a relation. |
| | | Benchmark 15. | Models and solves real world problems using polynomial equations, with and without the use of technology. |
| | | | Knowledge/skill statements |
| | | | | 1. | Models real world problems using polynomial equations with the use of technology. |
| | | | | 2. | Models real world problems using polynomial equations without the use of technology. |
| | | | | 3. | Solves real world problems using polynomial equations with the use of technology. |
| | | | | 4. | Solves real world problems using polynomial equations without the use of technology. |
| | | Benchmark 16. | Solves quadratic equations in one variable(e.g. using quadratic formula, by completing the square) |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of quadratic equations |
| | | | | 2. | Knows the quadratic formula |
| | | | | 3. | Knows how completing the square can be used to solve a quadratic equation |
| | Level V (Grade (College Readiness)) |
| | | Benchmark 1. | Solves equations and inequalities involving absolute value |
| | | | Knowledge/skill statements |
| | | | | 1. | Solves equations involving absolute value |
| | | | | 2. | Solves inequalities involving absolute value |
| | | Benchmark 2. | Understands graphs of solution sets of quadratic inequalities |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of solution sets |
| | | | | 2. | Identifies solution sets of quadratic inequalities |
| | | | | 3. | Interprets graphs of solution sets of quadratic inequalities |
| | | Benchmark 3. | Factors polynomials using a variety of methods (e.g., difference of squares, perfect square trinomials) |
| | | | Knowledge/skill statements |
| | | | | 1. | Factors polynomials using the difference of squares |
| | | | | 2. | Factors polynomials using perfect square trinomials |
| | | Benchmark 5. | Solves problem situations that require linear, quadratic, or exponential modeling |
| | | | Knowledge/skill statements |
| | | | | 1. | Understands the concept of modeling a situation using equations |
| | | | | 2. | Solves problem situations that require linear modeling |
| | | | | 3. | Solves problem situations that require quadratic modeling |
| | | | | 4. | Solves problem situations that require exponential modeling |