Grade 10 math

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.

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

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

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 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 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.

Add the terms of an arithmetic sequence with Sₙ = n(a₁ + aₙ)/2, write sums compactly in sigma notation, and use the formula for counting and money problems.

How to build a model by adding, subtracting or dividing functions you know, why adding functions adds their graphs, and how composition feeds one function into another.

Why mathematicians introduced i with i² = −1, how to add, subtract and multiply complex numbers, how the powers of i cycle, and how the conjugate makes division possible.

Interest compounded n times a year, what happens as n grows without bound, the number e ≈ 2.718, and continuous growth with A = Pe^(rt).

Model quantities that vary together: direct variation y = kx, inverse variation y = k/x, joint and combined variation, and finding the constant of variation.

How polynomial long division mirrors division of numbers, how to write a quotient as q(x) + r(x)/b(x), and why synthetic division is a shortcut for dividing by x − a.

Test whether a function is even, odd or neither by computing f(−x), and see the answer in its graph: a mirror image in the y-axis or a half turn about the origin.

Model decay, growth and cooling with exponential functions, solve for the time with logarithms, and compare values on pH, decibel and earthquake scales.

Factor polynomials beyond quadratics: sums and differences of cubes, factoring by grouping, and polynomials in quadratic form, then use the factors to solve equations.

A polynomial of degree n has exactly n complex zeros, counted with multiplicity. Use that count and conjugate pairs to find every zero and build polynomials.

How to derive the sum of a finite geometric series by multiplying by the ratio and subtracting, and how the same sum sets a savings total and a monthly loan payment.

How to sketch a polynomial from its factored form: where it crosses the x-axis, why a repeated factor makes it bounce, and how the leading term decides both ends.

How factoring a rational function reveals its zeros, holes and vertical asymptotes, why the graph shoots off near an asymptote, and how degrees decide the end behavior.

How unrolling the unit circle produces a wave, what amplitude, period, midline and phase shift each control, and how to fit a sine model to a repeating situation.

Graph y = tan x with its period π and vertical asymptotes, and the reciprocal functions secant, cosecant and cotangent built from sine and cosine.

Graph y = bˣ and y = log_b x: growth and decay, asymptotes, domain and range, the mirror image in y = x, and shifts and stretches of both.

Add all the terms of a geometric sequence: when −1 < r < 1 the sum is a/(1 − r). Use it to turn repeating decimals into fractions and total a bouncing ball's path.

Find the best value of a linear objective over a region cut out by linear inequalities, and see why the maximum and minimum always sit at a corner.

What a logarithm is, why it is the inverse of an exponential, how to use logs to solve for an unknown exponent, and how to evaluate any logarithm with a calculator.

Solve polynomial and rational inequalities with a sign chart: find every critical number, test each interval, and handle zeros of the denominator correctly.

What makes an equation an identity, how to prove one, how identities generate Pythagorean triples and mental-math shortcuts, and how Pascal's triangle expands (x + y)ⁿ.

The three logarithm rules for products, quotients and powers, why they follow from the exponent rules, expanding and condensing logs, and change of base.

Solve inequalities like x² − x − 6 > 0 by finding where the parabola crosses the x-axis, testing the intervals between, and writing the answer as intervals.

How the quadratic formula gives complex solutions when the discriminant is negative, why they come in conjugate pairs, and what the Fundamental Theorem of Algebra says.

Why every solving step is a claim that follows from the one before, why squaring or clearing denominators can add false answers, and how to solve and check both kinds.

How rational expressions follow the rules of fractions, why only common factors cancel, how to multiply and divide them, and how to add them over a common denominator.

List the possible rational zeros of a polynomial, test them with synthetic division, and use each zero you find to factor and solve the polynomial completely.

Why dividing p(x) by x − a leaves the remainder p(a), why p(a) = 0 means x − a is a factor, and how the two theorems turn one known root into a full factorization.

Simplify square roots and cube roots, multiply and divide radicals, add like radicals, and rationalize denominators, including with conjugates.

Solve equations with the unknown in an exponent or inside a logarithm: matching bases, taking logs of both sides, condensing logs, and rejecting extraneous solutions.

Solve equations like 2 sin x = 1: find one angle, use the unit circle to find every solution in one cycle, and add multiples of the period to get them all.

The shapes of y = √x and y = ∛x, why one starts at a point and the other runs both ways, and how shifts and stretches move the starting point, center, domain and range.

Solve linear systems in x, y and z by elimination: reduce to two equations in two unknowns, solve, substitute back, and see why a solution is where three planes meet.

Why the Pythagorean theorem on the unit circle gives sin²θ + cos²θ = 1, what makes it an identity, and how to find sine, cosine or tangent from one value and a quadrant.

Why the right-triangle definition stops at 90°, how the unit circle replaces it, the exact values at π/6, π/4 and π/3, and why sine and cosine repeat.

Find the probability of exactly k successes in n independent trials with the binomial formula, see why it works, and answer at least and at most questions.

How reshuffling an experiment's results shows what chance alone can produce, how to judge whether a gap between two groups is significant, and how to read a data report.

What 'given that' does to a sample space, finding a conditional probability from a table or the formula, why P(A|B) and P(B|A) differ, and the multiplication rule.

What it means for two events to be independent, the two equivalent tests for it, how to decide from a two-way table, and why independent is not mutually exclusive.

Why two honest samples give different answers, how simulating a thousand samples reveals the spread, where the square-root rule comes from, and how to read a poll.

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.

The counting principle, factorials, when order matters and when it does not, why a combination divides a permutation by r!, and how to use counts to find probabilities.

What each kind of study can and cannot show, where randomization enters, why random selection and random assignment do different jobs, and how confounding works.

How to describe events with and, or and not, how a Venn diagram shows them, why the addition rule subtracts the overlap, and what makes two events mutually exclusive.

Why all circles are similar, why an arc's length is proportional to its radius, how that ratio defines the radian, and the formulas for arc length and sector area.

Why the area of any triangle is one half ab sin C, derived by dropping a height from a vertex, why the formula still works when the angle is obtuse, and how to use it.

Why an inscribed angle is half the central angle on the same arc, why an angle in a semicircle is a right angle, and why a tangent meets its radius at 90°.

What each plane cut through a cylinder, cone, sphere, cube and pyramid exposes, why every slice of a sphere is a circle, and which flat shapes sweep out which solids.

How the Pythagorean theorem gives the equation of a circle, how to read the center and radius from it, and how completing the square recovers them from expanded form.

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

Constructing the circles through and inside a triangle, why opposite angles of an inscribed quadrilateral add to 180°, and how to draw a tangent from an outside point.

Why c² = a² + b² − 2ab cos C is the Pythagorean theorem with a correction, proved with coordinates, and how to find a side from SAS or an angle from SSS.

Why each side of a triangle over the sine of its opposite angle gives the same number, how to solve triangles with it, and why SSA can give two triangles or none.

How to replace a real object with a shape that answers the question, how density works in two and three dimensions, and how to solve a design problem under constraints.

Why a parabola is every point equally far from a point and a line, how to derive its equation from that definition with the distance formula, and what the focus does.

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.

How to find the point that divides a directed segment in a ratio such as 1:2, why the ratio becomes a fraction of the step from start to end, and why direction matters.

How to dilate from any center, why a dilation sends a line to a parallel line and leaves lines through the center in place, and why every length scales by k.

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

How to find a missing angle with inverse sine, cosine and tangent, when the Pythagorean theorem is faster, and how to set up an elevation or depression problem.

Where the side ratios of the two special right triangles come from, how to use them without a calculator, and the exact sine, cosine and tangent values they give.

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.

Why a side ratio in a right triangle depends only on the angle, what SOH CAH TOA names, why sine equals the cosine of the complement, and how to find a missing side.

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

Informal arguments for circumference, the area of a circle, and the volumes of a cylinder, pyramid, cone and sphere, including Cavalieri's principle.