Grade 9 math

What makes a number irrational, why every rational number has a terminating or repeating decimal, and how to place an irrational number on a number line.

Solve absolute value equations by splitting into two cases, why two answers appear, when there is no solution, and how absolute value inequalities differ.

Solve absolute value inequalities by thinking in distances: less than gives one stretch of the number line, greater than gives two rays pointing away.

Learn completing the square: why the method works geometrically, how to handle a leading coefficient, and how it produces vertex form and the quadratic formula.

Solve compound inequalities joined by AND or OR, graph the solution on a number line, and tell which combinations give no solution or every number.

What domain and range mean, how to find them from a graph, a table or an equation, and why some inputs have to be excluded.

Understand f(x) as a name and an input rather than multiplication, how to evaluate it, and how to read statements like f(3) = 7 in context.

Graph a line three ways — from slope-intercept form, from intercepts, or from a table — and understand why every point on the line is a solution.

Four reliable tests for deciding whether a relationship is linear — from its equation, its table, its graph, or its rate of change.

Rearrange a formula to solve for any variable using the same inverse operations you use on numbers, with worked examples from geometry, physics, and finance.

Solve multi-step equations by distributing, combining like terms, then undoing operations in reverse order, with worked examples and practice problems.

Solve multi-step inequalities by distributing, combining like terms, and undoing operations in reverse order, including when to flip the inequality sign.

Parallel lines have equal slopes and perpendicular lines have negative reciprocal slopes — why both rules are true, and how to write equations using them.

Learn point-slope form, why it is the slope formula rearranged, and how to use it to write a line's equation from a single point and a slope.

Learn the quadratic formula, where it comes from, and how to use it to solve any quadratic equation, with worked examples and practice problems.

What average rate of change means, how it relates to slope, how to compute it from a table or a function, and how to read its units and sign.

Learn what slope measures, how to find it with rise over run and the slope formula, and why similar triangles guarantee it is the same everywhere on a line.

Understand y = mx + b, what m and b each control, how to graph a line from it in two moves, and how to rewrite any linear equation into this form.

Solve quadratics by factoring using the zero product property, why the equation must equal zero first, and how to factor the common trinomial patterns.

What standard form is, when it beats slope-intercept form, how to convert between the two, and how to find the slope without rearranging.

Solve a system by graphing, substitution or elimination, why the solution is the intersection point, and how to spot systems with none or infinitely many.

Solve equations with variables on both sides by collecting variable terms on one side, plus how to recognize no solution and infinitely many solutions.

Find a line's equation from any two points: compute the slope, substitute into point-slope form, then simplify — with worked examples and full solutions.

Find the x- and y-intercepts of any equation by setting the other variable to zero, why that works, and what intercepts mean in a real situation.

What a polynomial is, how to read its degree and leading coefficient, why only like terms combine, and why subtracting means distributing the minus sign to every term.

How to tell an arithmetic sequence from a geometric one, how to write each recursively and explicitly, and why the explicit rules are linear and exponential functions.

How to compare two functions given in different forms, build a linear function from a table or a description, and read the rate of change from each one.

How to tell an exponential situation from a linear one, what the start value and growth factor mean, and why exponential growth eventually passes any straight line.

How to build an exponential model from a description, a table or two points, what each parameter means in context, and how to read a growth curve's key features.

How to factor a quadratic by finding a pair that multiplies to c and adds to b, when to pull out a common factor first, and how to spot the two special patterns.

What each change to f(x) does to its graph, why a change inside the parentheses moves the graph the opposite way, and how to read the constants off a picture.

How a linear inequality splits the plane into a half-plane of solutions, when the boundary line is solid and when it is dashed, and how to shade a system of inequalities.

How to find the vertex and axis of symmetry of a parabola, why the axis sits at -b over 2a, and what each of the three forms of a quadratic reveals at a glance.

How to find an inverse by swapping x and y, why the graph reflects across the line y = x, and how the horizontal line test decides whether an inverse is a function.

How to solve a system of one linear and one quadratic equation by substitution and by graphing, and why such a system has two, one or no solutions.

How to multiply polynomials with the distributive property, why the box method never misses a term, where FOIL comes from, and the two special products to know.

How to read, evaluate and graph a function built from several rules, why endpoints need open and closed dots, and how absolute value and step functions fit the pattern.

Why a power of one half has to mean a square root, how to read the top and bottom of a fractional exponent, and how to switch between radical and exponent form.

Why the x-coordinates where y = f(x) and y = g(x) cross are the solutions of f(x) = g(x), and how to approximate them with a graph, a table and successive narrowing.

How units guide a multi-step problem and catch errors, what the units of each part of a formula mean, how to define a quantity, and how many digits an answer deserves.

How to turn a situation into an equation or inequality and solve it, why some solutions are not viable, and how a system of inequalities shows every option that works.

Read a scatter plot, describe its association, draw and use a line of best fit, and understand why correlation does not prove causation.

What a normal distribution is, how the 68-95-99.7 rule estimates percentages from the mean and standard deviation, how z-scores compare values, and when it fails.

How to calculate a residual, what a residual plot reveals about whether a line fits, and how to read the correlation coefficient r for direction and strength.

What standard deviation measures, how to calculate it step by step, why the deviations are squared, and when median and IQR describe the data better.

How to build and read a two-way frequency table, how relative frequencies reveal an association between two categories, and why raw counts can mislead.

Step-by-step constructions for copying, bisecting, perpendiculars, parallels and inscribed polygons, with the congruence argument that proves each one works.

Proofs that a parallelogram's opposite sides and angles are congruent, that its diagonals bisect each other, and that equal diagonals make it a rectangle.

Transformations as functions on points, which ones preserve distance and angle, precise definitions of each rigid motion, and congruence defined by motion.

Why two equal angles are enough to prove triangles similar, how corresponding sides stay in a fixed ratio, and how to use that ratio to measure things you cannot reach.

Why three matching parts are enough to prove two triangles congruent, how each criterion follows from rigid motions, why SSA and AAA fail, and how CPCTC finishes a proof.

Complete proofs that a triangle's angles sum to 180°, that isosceles base angles are equal, that a midsegment is half the third side, and that the three medians meet.

How a geometry proof is built from definitions, postulates and earlier theorems, with complete proofs about vertical angles, parallel lines and perpendicular bisectors.