Skill Guides: Step-by-Step Math Methods

One guide for each problem type in the Math Racer curriculum, from adding whole numbers to multivariable calculus. Every guide explains the idea, lists the steps in order, and works through an example.

Elementary (Grades K-5)

Addition — Addition means putting numbers together to find the total. The numbers you add are called addends, and the answer is called the sum.
Subtraction — Subtraction means taking one number away from another to find the difference. The number you start with is on top, and the number you take away is on the…
Multiplication — Multiplication is a fast way to add the same number again and again. For example, 4 x 3 means 4 + 4 + 4.
Division — Division means splitting a number into equal groups. The number being split is the dividend, the number of groups (or the size of each group) is the…
Order of Operations — When a problem has more than one operation, the order you work in matters. PEMDAS is a way to remember the order: Parentheses, Exponents, Multiplication…
Perimeter — The perimeter of a shape is the distance all the way around it. To find it, add up the lengths of all the sides. Some shapes have shortcuts.
Area — Area tells you how much flat space is inside a shape. We measure it in square units, like square centimeters.
Polygon Properties — A polygon is a closed shape made of straight sides. Polygons have properties you can figure out, like how many sides they have and what their angles add…
Classifying Shapes — We sort shapes into groups by their properties: how many sides they have, whether the sides are equal, what kinds of angles they have, and whether sides…

Middle School (Grades 6-8)

Exponents — An exponent tells you how many times to multiply a number by itself. In 2^3, the 2 is the base and the 3 is the exponent, so 2^3 = 2 x 2 x 2 = 8.
Evaluating Expressions — Evaluating an expression means replacing each letter (variable) with a number you are given and then working out the answer.
Simplifying Linear Expressions — Simplifying an expression means rewriting it in a shorter form that has the same value.
Triangle Area — The area of a triangle is half of the base times the height. The height is the straight-up distance from the base to the opposite corner.
Missing Angles — In many shapes, the angles add up to a number you can count on. The angles in a triangle add up to 180 degrees.
Circumference — The circumference of a circle is the distance around it. It is equal to pi times the diameter, or 2 times pi times the radius.
Circle Area — The area of a circle is pi times the radius times the radius. The radius is the distance from the center to the edge.
Pythagorean Theorem — In a right triangle (a triangle with a 90 degree corner), the longest side is called the hypotenuse.

Algebra 1 (Grade 9)

Solving Linear Equations — A linear equation like 3x + 7 = 22 has one unknown number, x. Solving means getting x all by itself on one side of the equals sign.
Solving Multi-Step Linear Equations — Some equations need more than two steps. They might have parentheses, fractions, or x on both sides.
Solving for a Variable (Symbolic Isolation) — Sometimes you need to rearrange a formula to get a different letter by itself. For example, you might change d = rt into t = d/r.
Solving Linear Inequalities — An inequality compares two things with a symbol like <, >, <=, or >= instead of an equals sign. You solve it almost the same way as an equation.
Evaluating Functions — A function is a rule that takes an input and gives back one output. We write f(x) to name a function, and it is read 'f of x'.
Direct Variation — Two quantities have direct variation when one is always the same number times the other: y = kx. The number k is called the constant of variation.
Inverse Variation — Two quantities have inverse variation when their product stays the same: xy = k, which can also be written y = k/x.
Find the Slope of a Line — The slope of a line tells you how steep it is. It is the rise (how far the line goes up or down) divided by the run (how far it goes across).
Write the Equation of a Line — The equation of a line tells you every point that sits on that line. Two common forms are slope-intercept form, y = mx + b, and point-slope form, y - y1…
Solving Systems of Linear Equations — A system of linear equations is a set of two (or more) equations that use the same variables.
Simplifying Polynomials — A polynomial is an expression made of terms like 3x^2, 5x, and 7, added or subtracted together.
Factoring Polynomials — Factoring means rewriting an expression as a product of simpler expressions, like writing 12 as 3 x 4.
Factoring Trinomials — A trinomial like ax^2 + bx + c has three terms. Many trinomials can be written as a product of two binomials, like (x + 2)(x + 3).
Solve Quadratic Equation — A quadratic equation has an x^2 term and looks like ax^2 + bx + c = 0. It can have up to two solutions.
Solve Quadratic by Completing the Square — Completing the square is a way to rewrite a quadratic equation so it looks like (x + h)^2 = k.
Solve Quadratic Using the Quadratic Formula — The quadratic formula solves any quadratic equation of the form ax^2 + bx + c = 0. It comes from completing the square on the general equation.

Geometry (Grade 10)

Surface Area — Surface area is the total area of all the outside faces of a 3D shape. Think of it as the amount of wrapping paper you would need to cover it.
Volume — Volume is the amount of space inside a 3D shape. It is measured in cubic units.
Triangle Inequality — Not every set of three lengths can make a triangle. The triangle inequality says that any two sides added together must be longer than the third side.
Find the Midpoint of a Segment — The midpoint of a line segment is the point exactly halfway between its two endpoints.
Find the Distance Between Two Points — The distance between two points is the length of the straight line that connects them. You can find it by making a right triangle.
Finding Missing Sides — In a right triangle, if you know one acute angle and one side, you can find the other sides using SOH-CAH-TOA.
Arcs and Sectors — An arc is a piece of the edge of a circle. A sector is a slice of the circle, like a slice of pizza, with two radii and an arc as its edges.
Circle Equation in Standard Form — A circle with its center at (h, k) and a radius of r has the equation (x - h)^2 + (y - k)^2 = r^2.
Reflections — A reflection flips a shape over a line, like a mirror. The line is called the line of reflection.
Rotations — A rotation turns a shape around a fixed point (the center) by a certain angle. When the center is the origin, there are simple rules.
Translations — A translation slides a shape without turning it, flipping it, or changing its size. Every point moves the same distance in the same direction.
Dilations — A dilation makes a shape bigger or smaller by multiplying all distances from a center point by a scale factor k.
Triangle Congruence — Two triangles are congruent if they are exactly the same size and shape, so all their matching sides and angles are equal.
Triangle Similarity — Two triangles are similar if they have the same shape but can be different sizes.
Similar Triangle Sides — When two triangles are similar, their matching sides are in proportion. That means the ratio of one pair of matching sides equals the ratio of every…
Angle Relationships — When lines cross, or when a line crosses a pair of parallel lines, the angles it makes follow patterns. Complementary angles add up to 90 degrees.
Geometric Constructions — A geometric construction is a drawing made with only a compass (for drawing circles and arcs) and a straightedge (a ruler with no numbers).

Algebra 2 (Grade 11)

Simplifying Symbolic Expressions — Some expressions have letters, exponents, multiplication, and division all mixed together.
Function Composition — Function composition means using the output of one function as the input of another. (f o g)(x) is the same as f(g(x)).
Finding Inverse Functions — The inverse of a function f, written f^-1, undoes what f does. If f(a) = b, then f^-1(b) = a.
Piecewise Functions — A piecewise function uses different rules for different inputs. To find an output, first look at which range of numbers your x falls in.
Solving Compound Inequalities — A compound inequality puts two inequalities together with AND or OR. With AND, both parts must be true at the same time.
Solving Absolute Value Inequalities — Absolute value tells you how far a number is from 0. An absolute value inequality compares that distance to a number.
Solving Nonlinear Systems — A nonlinear system is a set of equations where at least one equation has a term that is not a straight line, like x^2, xy, or 1/x.
Simplifying Rational Expressions — A rational expression is a fraction with polynomials on the top and bottom. To simplify one, factor both the top and the bottom completely and cancel the…
Rational Expression Operations — You add, subtract, multiply, and divide rational expressions (fractions with polynomials) the same way you do with number fractions, but factoring is a…
Solve Rational Equation — A rational equation has fractions with x in the bottom. The usual plan is to multiply every term by the least common denominator (LCD) to clear the…
Simplifying Radicals — A radical (like a square root) is in simplest form when nothing under the root sign can be pulled out, there are no fractions under the root, and there…
Radical Operations — Radicals (square roots, cube roots, and so on) follow rules a lot like exponents do.
Solve Radical Equation — A radical equation has a variable under a square root (or another root). To solve it, get the root by itself on one side and then square both sides to…
Rationalizing Denominators — Rationalizing a denominator means rewriting a fraction so there is no square root (or other root) in the bottom.
Nested Radicals — A nested radical has a square root inside another square root, like the square root of (a + b root c).
Logarithms — A logarithm answers the question 'what power do I need?' log_b(x) = y means the same thing as b^y = x. For example, log_2(8) = 3 because 2^3 = 8.
Solve Exponential Equation — An exponential equation has the variable in the exponent, like 3^x = 27. If you can write both sides with the same base, then the exponents must be…
Solve Logarithmic Equation — A logarithmic equation has the variable inside a log, like log_2(x) = 3. To solve it, use the log rules to combine everything into a single log, then…
Complex Numbers — A complex number looks like a + bi. The number a is the real part, b is the imaginary part, and i is a special number where i^2 = -1.
Complex Modulus — The modulus (or absolute value) of a complex number z = a + bi is its distance from the origin on the complex plane. It is written |z|.
Polar-Complex Conversion — A complex number can be written in rectangular form, a + bi, or in polar form, r(cos(theta) + i sin(theta)).
Unit Circle — The unit circle is a circle with radius 1 centered at the origin. For any angle theta, measured counterclockwise from the positive x-axis, the point on…

Trigonometry (Grades 11-12)

Cosine Function — On the unit circle (a circle with radius 1), the cosine of an angle is the x-coordinate of the point where the angle's side meets the circle.
Tangent Function — Tangent is defined as tan(theta) = sin(theta) / cos(theta). In a right triangle, it is the opposite side divided by the adjacent side.
Cosecant Function — Cosecant is the reciprocal of sine: csc(theta) = 1 / sin(theta). It is undefined wherever sin(theta) = 0 (at k pi). It repeats every 2 pi.
Finding Missing Angles — In a right triangle, if you know any two sides, you can find an unknown acute angle.
Secant Function — Secant is the reciprocal of cosine: sec(theta) = 1 / cos(theta). It is undefined wherever cos(theta) = 0 (at pi/2 + k pi). It repeats every 2 pi.
Cotangent Function — Cotangent is the reciprocal of tangent: cot(theta) = cos(theta) / sin(theta) = 1 / tan(theta).
Sine Function — On the unit circle (a circle with radius 1), the sine of an angle is the y-coordinate of the point where the angle's side meets the circle.
Simplifying Trig Expressions — Simplifying a trig expression means rewriting it in a shorter form that has the same value.
Pythagorean Identities — The Pythagorean identities connect the squares of trig functions: sin^2(x) + cos^2(x) = 1, 1 + tan^2(x) = sec^2(x), and 1 + cot^2(x) = csc^2(x).
Double Angle Identities — Double angle identities let you find the trig functions of 2x using the trig functions of x. sin(2x) = 2 sin(x) cos(x).
Sum and Difference Identities — Sum and difference identities let you find the trig functions of a + b or a - b from the trig functions of a and b.
Half Angle Identities — Half angle identities let you find the trig functions of x/2 using cos(x). sin(x/2) = plus or minus the square root of ((1 - cos x) / 2).
Cofunction Identities — Cofunction identities connect the trig functions of two angles that add up to 90 degrees (pi/2).
Law of Sines — The Law of Sines connects the sides of a triangle to the sines of the angles across from them: a / sin(A) = b / sin(B) = c / sin(C).
Law of Cosines — The Law of Cosines is like the Pythagorean theorem, but it works for any triangle: c^2 = a^2 + b^2 - 2ab cos(C).
Oblique Triangles — An oblique triangle is any triangle that does not have a right angle. To solve one, you find all three sides and all three angles.
Inverse Sine — Arcsine, written arcsin(x) or sin^-1(x), is the inverse of sine. It answers the question 'what angle has this sine?' Because many angles have the same…
Inverse Tangent — Arctangent, written arctan(x) or tan^-1(x), is the inverse of tangent. It answers 'what angle has this tangent?' We only allow answers strictly between…
Inverse Reciprocal Trig — The inverse reciprocal trig functions arcsec(x), arccsc(x), and arccot(x) undo secant, cosecant, and cotangent.
Inverse Cosine — Arccosine, written arccos(x) or cos^-1(x), is the inverse of cosine. It answers 'what angle has this cosine?' Because many angles have the same cosine,…
Basic Trig Equations — A basic trig equation looks like sin(x) = k, cos(x) = k, or tan(x) = k. To solve one, get the trig function by itself, use the inverse function to find…
Multiple Angle Equations — Some equations have a multiple of x inside the trig function, like sin(2x) = k or cos(3x + b) = k.
Quadratic Trig Equations — Some trig equations look like a quadratic: a x f(x)^2 + b x f(x) + c = 0, where f(x) is one trig function.
Polar Coordinates — Polar coordinates describe a point using its distance r from the origin and its angle theta from the positive x-axis.

Calculus 1 (AP Calc AB / College)

Direct Limits — Direct substitution evaluates a limit by plugging the target value of x straight into the function.
Limits by Factoring — When direct substitution gives the indeterminate form 0/0, factoring often resolves the limit.
Limits by Rationalization — When a limit gives 0/0 and the expression contains a square root, multiplying by the conjugate clears the radical from one part of the fraction and…
Limits at Infinity — A limit at infinity describes the end behavior of a function as x grows without bound (positive or negative).
L'Hopital's Rule — L'Hopital's Rule states that for indeterminate forms 0/0 or infinity/infinity, the limit of f(x)/g(x) equals the limit of f'(x)/g'(x), provided the…
Vertical Asymptote Limits — A vertical asymptote at x = a occurs when f(x) tends to plus or minus infinity as x approaches a.
Power Rule for Derivatives — The power rule states that d/dx[x^n] = n*x^(n-1) for any real exponent n. Combined with the constant multiple, sum, and difference rules, it lets you…
Product Rule — The product rule differentiates a product of two functions: if y = u(x)*v(x), then y' = u'v + u v'.
Quotient Rule — The quotient rule differentiates a ratio of two functions: if y = u(x)/v(x), then y' = (u'v - u v')/v^2.
Chain Rule — The chain rule differentiates a composition of functions: if y = f(g(x)), then dy/dx = f'(g(x)) * g'(x).
Trig Derivatives — The six trig functions have memorable derivatives: d/dx sin x = cos x, d/dx cos x = -sin x, d/dx tan x = sec^2 x, d/dx cot x = -csc^2 x, d/dx sec x = sec…
Exponential and Log Derivatives — The natural exponential e^x is its own derivative, and the natural log has derivative 1/x.
Implicit Differentiation — Implicit differentiation finds dy/dx for an equation in which y is not explicitly solved for in terms of x (such as x^2 + y^2 = 25).
Higher Order Derivatives — A higher-order derivative is the derivative of a derivative. The second derivative f''(x) is d/dx of f'(x), the third derivative f'''(x) is d/dx of…
Logarithmic Differentiation — Logarithmic differentiation simplifies derivatives of complex products, quotients, and especially functions with variable bases AND variable exponents…
Critical Points — A critical point of f is a value x = c in the domain of f where either f'(c) = 0 or f'(c) does not exist.
Optimization — Optimization problems use calculus to find the maximum or minimum value of a quantity subject to a constraint.
Related Rates — Related rates problems link the rates of change of two or more quantities through an equation.
Continuity — A function f is continuous at x = a when three conditions all hold: f(a) is defined, the limit of f(x) as x approaches a exists, and that limit equals…
Mean Value Theorem — The Mean Value Theorem (MVT) says that if f is continuous on [a, b] and differentiable on (a, b), then there exists at least one c in (a, b) such that…
Rolle's Theorem — Rolle's Theorem states that if f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists at least one c in (a, b) such that…
Curve Sketching — Curve sketching uses calculus to draw an accurate graph of a function: domain, intercepts, asymptotes, intervals of increase/decrease (from f'),…

Calculus 2 (AP Calc BC / College)

Power Rule for Integration — The power rule for integration is the inverse of the power rule for differentiation: to integrate x^n (with n not equal to -1), add 1 to the exponent and…
Integration by Substitution — u-substitution is the integral analog of the chain rule: if an integrand contains a function and a multiple of its derivative, we set u equal to the…
Trig Integrals — Standard trig integrals include: integral of sin x dx = -cos x + C, integral of cos x dx = sin x + C, integral of sec^2 x dx = tan x + C, integral of…
Integrals of Exponential and Logarithmic Functions — The exponential function e^x is its own antiderivative, and 1/x integrates to ln|x|.
Definite Integral — A definite integral computes the net signed area between a function and the x-axis over a closed interval [a, b].
Integration by Parts — Integration by parts is the integral version of the product rule. It says integral of u dv equals u*v minus integral of v du, and it converts a difficult…
Area Under a Curve — The area under a curve y = f(x) between x = a and x = b is given by the definite integral of f(x) from a to b, provided f(x) >= 0 on that interval.
Volume of Revolution — When a region in the plane is rotated around an axis, the resulting 3D solid has a volume that can be computed by integration.
Series Convergence Tests — A series sum of a_n converges if its partial sums approach a finite limit; otherwise it diverges.
Taylor Series — A Taylor series represents a smooth function as an infinite polynomial built from its derivatives at a point a.
Power Series — A power series centered at c has the form sum c_n (x - c)^n. It converges for x in an interval centered at c, with a radius of convergence R found by the…

Calculus 3 (College)

Partial Derivatives — A partial derivative measures how a multivariable function changes as one variable varies while all others are held constant.
Gradient — The gradient of a scalar function f is the vector of its partial derivatives. It points in the direction of greatest increase of f at each point, and its…
Directional Derivative — The directional derivative D_u f(P) measures the rate of change of f at a point P in the direction of a unit vector u.
Double Integral — A double integral computes the signed volume under a surface z = f(x, y) over a 2D region R.
Triple Integral — A triple integral integrates a function f(x, y, z) over a 3D region E and is evaluated as three nested single integrals.
Line Integral — A line integral integrates a function or vector field along a curve C. The scalar form integrates f ds (against arc length);
Vector Fields — A vector field assigns a vector to every point in a region of space. Common examples include velocity fields, force fields, and gradient fields.
Divergence and Curl — For a vector field F = (P, Q, R), the divergence div F is a scalar that measures net outflow per unit volume, and the curl curl F is a vector that…