Nebraska (NE)
9th Grade · High school · Math
Official Nebraska (NE) math standards for 9th Grade. Click a standard for the full text and study materials. Study units below reuse the same Kindergarten unit library.
Standards
Click a standard for the full text and study materials.
Number and Algebraic Relationships
- 9.N.1.a — 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.
- 9.N.2.a — 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).
- 9.N.2.b — 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.
- 9.N.3.a — 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.
- 9.N.3.b — (+) 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.
- 9.N.4.a — 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.
- 9.N.4.b — (+) 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.
- 9.N.1.b — 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.
- 9.N.1.c — Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
- 9.N.1.d — 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.
- 9.N.1.e — 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.
- 9.N.1.f — 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.
- 9.N.1.g — Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.
- 9.N.2.c — Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.
- 9.N.2.d — Solve quadratic equations in one variable.
- 9.N.3.c — 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.
- 9.N.3.d — Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.
- 9.N.3.e — 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.
- 9.N.3.f — (+) Represent a system of linear equations as a single matrix equation in a vector variable.
- 9.N.3.g — (+) 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).
- 9.N.4.c — 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).
- 9.N.4.d — 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.*
- 9.N.4.e — Graph the solutions to 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.
- 9.N.1.h — Interpret expressions that represent a quantity in terms of its context.*
- 9.N.1.i — 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).
- 9.N.2.e — Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.*
- 9.N.2.f — 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.*
Number
- 9.N.1.j — Know there is a complex number i such that i² = -1, and every complex number has the form a + bi with a and b real.
- 9.N.1.k — Use the relation i² = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
- 9.N.1.l — (+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.
- 9.N.2.g — (+) Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.
- 9.N.2.h — (+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.
- 9.N.2.i — (+) Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.
- 9.N.3.h — Solve quadratic equations with real coefficients that have complex solutions.
- 9.N.3.i — (+) Extend polynomial identities to the complex numbers. For example, rewrite x² + 4 as (x + 2i)(x - 2i).
- 9.N.3.j — (+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.
- 9.N.1.m — Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
- 9.N.1.n — Define appropriate quantities for the purpose of descriptive modeling.
- 9.N.1.o — Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
- 9.N.1.p — Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.
- 9.N.1.q — Rewrite expressions involving radicals and rational exponents using the properties of exponents.
- 9.N.2.j — Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.
- 9.N.1.r — (+) Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v, |v|, ||v||, v).
- 9.N.1.s — (+) Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.
- 9.N.1.t — (+) Solve problems involving velocity and other quantities that can be represented by vectors.
- 9.N.2.k — (+) Add and subtract vectors.
- 9.N.2.l — (+) Multiply a vector by a scalar.
- 9.N.3.k — (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- 9.N.3.l — (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- 9.N.3.m — (+) Add, subtract, and multiply matrices of appropriate dimensions.
- 9.N.3.n — (+) Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- 9.N.3.o — (+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
- 9.N.3.p — (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
- 9.N.3.q — (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.
View official Nebraska (NE) framework
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