United States · Common Core State Standards
9th Grade Mathematics
This course bridges middle school arithmetic and advanced high school mathematics by focusing on the structure of expressions and the logic of functions. Students develop a deep understanding of linear and non-linear relationships through rigorous problem solving, statistical analysis, and geometric reasoning.
01The Language of Algebra
Students investigate the structure of expressions and the properties of real numbers to simplify complex mathematical statements. This unit emphasizes the transition from arithmetic operations to generalized algebraic rules.
Analyzing the component parts of algebraic expressions to interpret their meaning in real-world contexts.
Deepening understanding of commutative, associative, and distributive properties as the foundation of algebra.
Viewing equation solving as a logical process of maintaining equality rather than a series of memorized steps.
Using units as a guide to set up and solve multi-step problems involving various scales and measurements.
Rearranging formulas to highlight a quantity of interest using the same reasoning as solving equations.
Solving equations involving absolute value by considering the distance from zero on a number line.
Understanding the concept of inequalities and their basic properties, including graphing on a number line.
Applying algebraic properties to solve inequalities with multiple steps, similar to solving equations.
Solving and graphing compound inequalities (AND/OR) to represent complex constraints.
Solving inequalities involving absolute value by considering distance from zero and applying compound inequality rules.
Defining functions, identifying domain and range, and distinguishing functions from relations.
Learning the formal language of functions to describe inputs, outputs, and relationships.
Interpreting key features of function graphs, including intercepts, intervals of increase/decrease, and end behavior.
02Linear Relationships and Modeling
An exploration of constant rates of change and their representations in tables, graphs, and equations. Students learn to model real-world phenomena using linear functions.
Understanding slope not just as a formula, but as a constant ratio that defines linear growth.
Exploring slope-intercept, point-slope, and standard forms of linear equations and their applications.
Developing linear equations from tables, graphs, and verbal descriptions of real-world situations.
Modeling constraints and possibilities using inequalities to find viable solutions in complex scenarios.
Representing linear inequalities on the coordinate plane, including shading and boundary lines.
Finding the intersection of multiple constraints to identify unique solutions or regions of feasibility using substitution and elimination.
Visualizing solutions to systems of equations and inequalities on the coordinate plane.
Using systems of equations to determine when revenue equals costs in a small business model.
Connecting the concept of constant difference in sequences to linear functions.
Using census data to create linear models that predict population shifts and trends.
Investigating the relationship between slopes of parallel and perpendicular lines.
Defining and graphing functions that are composed of multiple sub-functions, each defined over a certain interval.
Exploring specific types of piecewise functions like step functions and their applications in pricing or taxation.
03Statistical Reasoning and Data
Students analyze data sets to identify trends, measures of center, and variability. The unit focuses on making informed decisions based on statistical evidence.
Evaluating mean, median, and mode to determine the most representative value of a data set.
Visualizing data distribution and variability using five-number summaries and box plots.
Quantifying how much data values deviate from the mean to understand consistency.
Identifying normal, skewed, and bimodal distributions and their implications.
Analyzing categorical data to identify associations and conditional probabilities between variables.
Creating and interpreting scatter plots to visualize relationships between two quantitative variables.
Using scatter plots and residuals to determine the strength and direction of linear correlations.
Examining the difference between observed and predicted values to validate linear models.
Distinguishing between situations where correlation implies causation and where it does not.
Exploring different methods of collecting data and understanding the importance of random sampling.
Understanding the principles of experimental design, including control groups, randomization, and blinding.
Critically analyzing statistical claims and identifying potential misrepresentations or biases.
Introducing fundamental concepts of probability, including events, outcomes, and calculating probabilities.
04Geometric Transformations and Logic
Students explore the properties of shapes through rigid motions and dilations. This unit builds the foundation for congruence and similarity proofs.
Investigating translations as rigid motions and representing them using vectors.
Exploring reflections across lines and their role in creating symmetrical figures.
Understanding rotations about a point and identifying rotational symmetry in figures.
Investigating the effects of combining multiple rigid transformations.
Investigating translations, reflections, and rotations to understand how shapes remain congruent under movement.
Exploring how scaling factors change the size of a figure while maintaining its proportional shape.
Developing logical arguments and formal proofs based on definitions, axioms, and theorems.
Exploring the angle relationships formed when a line intersects two parallel lines.
Proving triangles are congruent using SSS, SAS, ASA, and AAS criteria.
Using the distance formula and midpoint formula to analyze geometric figures on a coordinate plane.
Calculating perimeter and area of polygons on the coordinate plane.
Identifying line and rotational symmetry in cultural artifacts and biological organisms.
Performing basic geometric constructions using a compass and straightedge.
05Exponent Laws and Polynomials
Extending the properties of exponents to rational numbers and performing operations on polynomial expressions.
Reviewing and applying the laws of exponents for integer powers.
Connecting radical notation to fractional exponents and applying exponent laws to simplify expressions.
Applying exponent laws to very large and very small numbers in the context of astronomy and biology.
Defining polynomials, identifying their degree, leading coefficient, and classifying them by terms.
Performing addition and subtraction of polynomials by combining like terms.
Multiplying polynomials using the distributive property and various methods (FOIL, box method).
Breaking down complex polynomials into their irreducible factors using various algebraic techniques, starting with GCF.
Factoring quadratic trinomials where the leading coefficient is 1.
Factoring quadratic trinomials where the leading coefficient is greater than 1.
Identifying and factoring differences of squares and perfect square trinomials.
Using long division to divide polynomials and understand the relationship between divisor, dividend, quotient, and remainder.
Using synthetic division as an efficient method for polynomial division and exploring the Remainder Theorem.
Connecting the roots of a polynomial to its factors using the Factor Theorem.
06Quadratic Functions and Equations
Students shift from linear to non-linear thinking by exploring the parabolic nature of quadratic functions and their applications.
Identifying key attributes of quadratic graphs including the vertex, axis of symmetry, and intercepts from standard form.
Understanding how shifts and stretches affect the graph and equation of a quadratic.
Using factoring to find the zeros of quadratic functions and solve quadratic equations.
Developing methods to solve quadratic equations when the expression is not easily factorable.
Deriving and applying the quadratic formula to find solutions for any quadratic equation.
Using quadratic functions to model the path of objects in flight under the influence of gravity.
Comparing the rates of change between linear and quadratic functions in various contexts.
Finding the intersection of a line and a parabola algebraically and graphically.
Using the vertex of a quadratic function to find maximum area or minimum cost.
Representing quadratic inequalities on the coordinate plane, including shading and boundary curves.
Introducing the concept of imaginary numbers and complex numbers as solutions to quadratic equations.
Performing addition, subtraction, and multiplication of complex numbers.
Using the quadratic formula to find complex solutions when the discriminant is negative.
07Exponential Functions and Finance
An introduction to non-linear growth through exponential functions, with a focus on interest and decay.
Identifying the constant percent rate of change in exponential relationships.
Analyzing the behavior of exponential graphs, including asymptotes and y-intercepts.
Modeling patterns that grow by a constant ratio as exponential functions.
Applying exponential functions to model savings, loans, and investment growth over time.
Synthesizing the three major function types to choose the best model for a given data set.
Examining Malthusian growth models and their implications for resource management.
Introducing logarithms as the inverse of exponential functions and their basic properties.
08Advanced Geometry and Trigonometry
Applying similarity and right triangle relationships to solve complex spatial problems.
Using side lengths to identify right triangles and solve for missing distances.
Exploring the altitude-on-hypotenuse theorem and geometric means.
Defining Sine, Cosine, and Tangent as ratios of side lengths in right triangles.
Using trig ratios and inverse trig functions to find all missing sides and angles.
Identifying the unique ratios in 45-45-90 and 30-60-90 triangles.
Calculating the area of various polygons, including triangles, quadrilaterals, and regular polygons.
Calculating the circumference and area of circles and sectors.
Calculating the volume of prisms and cylinders.
Calculating the volume of pyramids and cones.
Calculating the volume and surface area of spheres.
Calculating the surface area of prisms, cylinders, pyramids, and cones.