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- Seeing Structure in Expressions
- Interpret the structure of expressions
- 1. Interpret expressions that represent a quantity in terms of its context.
- Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 y2).
- Find all factors of Polynomial
- Factorising with Blocks
- Factorizing with Identities
- Factorizing Quadratic Equation
- Represent a Quadratic Polynomial and Find factors
- Factorization using Common Factors
- Factorization with identities - Difference of two Squares
- Expansion of Algebraic Identities
- Algebraic Identities - Missing term
- Application of Algebraic Identities
- Factorization with Cubic Identities
- Write expressions in equivalent forms to solve problems
- 3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression
- 4. Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments
- Interpret the structure of expressions
- Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials
- Understand the relationship between zeros and factors of 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).
- 3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.
- Use polynomial identities to solve problems
- 4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 y2)2 = (x2 – y2)2 (2xy)2 can be used to generate Pythagorean triples.
- 5. ( ) Know and apply the Binomial Theorem for the expansion of (x y)n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal’s Triangle.
- Rewrite rational expressions
- 6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.
- 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
- Creating Equations
- Create equations that describe numbers or relationships
- 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.
- 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 nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.
- 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R
- Create equations that describe numbers or relationships
- Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning
- 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.
- 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
- Solve equations and inequalities in one variable
- Solve systems of equations
- 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.
- 7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x2 y2 = 3.
- 8. ( ) Represent a system of linear equations as a single matrix equation in a vector variable.
- 9. ( ) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).
- 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
- Represent and solve equations and inequalities graphically
- 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).
- 11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.
- 12. Graph the solutions to a linear inequality in two variables as a halfplane (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.
- Understand solving equations as a process of reasoning and explain the reasoning