Mathematics - StudyReady

Foundations

Algebra formula sheet

Concept	Formula
Quadratic formula	`x = (-b ± √(b² - 4ac)) / 2a`, for `ax² + bx + c = 0`
Slope of a line	`m = (y² - y¹) / (x² - x¹)`
Slope-intercept form	`y = mx + b`
Point-slope form	`y - y¹ = m(x - x¹)`
Difference of squares	`a² - b² = (a + b)(a - b)`
Perfect square trinomial	`(a ± b)² = a² ± 2ab + b²`
Exponent rules	`a^m·aⁿ = a^m+n`, `(a^m)ⁿ = a^mn`, `a⁰ = 1`
Logarithm identity	`log_b(x) = y` means `b^y = x`

Worked example: solve 2x² + 3x - 2 = 0. Here a=2, b=3, c=-2. Plug into the quadratic formula: x = (-3 ± √(9 + 16)) / 4 = (-3 ± 5) / 4, giving x = 0.5 or x = -2.

Shapes & Space

Geometry formula sheet

Shape	Area	Perimeter / Circumference
Rectangle	`A = lw`	`P = 2(l + w)`
Triangle	`A = ½bh`	`P = a + b + c`
Circle	`A = πr²`	`C = 2πr`
Trapezoid	`A = ½(b¹ + b²)h`	-

Solid	Volume	Surface Area
Rectangular prism	`V = lwh`	`SA = 2(lw + lh + wh)`
Cylinder	`V = πr²h`	`SA = 2πrh + 2πr²`
Sphere	`V = &frac43;πr³`	`SA = 4πr²`
Cone	`V = ⅓πr²h`	-

Pythagorean theorem: for a right triangle with legs a, b and hypotenuse c, a² + b² = c².

Angles

Trigonometry reference

Ratio	Definition (right triangle)
Sine	`sin(θ) = opposite / hypotenuse`
Cosine	`cos(θ) = adjacent / hypotenuse`
Tangent	`tan(θ) = opposite / adjacent`
Pythagorean identity	`sin²(θ) + cos²(θ) = 1`
Law of sines	`a/sin(A) = b/sin(B) = c/sin(C)`
Law of cosines	`c² = a² + b² - 2ab·cos(C)`

Memory trick: SOH-CAH-TOA - Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.

Change & Accumulation

Calculus quick reference

Rule	Formula
Power rule (derivative)	`d/dx[xⁿ] = nx^n-1`
Product rule	`d/dx[fg] = f'g + fg'`
Quotient rule	`d/dx[f/g] = (f'g - fg') / g²`
Chain rule	`d/dx[f(g(x))] = f'(g(x))·g'(x)`
Power rule (integral)	`∫xⁿdx = xⁿ⁺¹/(n+1) + C`
Derivative of sin/cos	`d/dx[sin x] = cos x`, `d/dx[cos x] = -sin x`

Strategy

How to study math effectively

Do problems, don't just read them

Watching a worked example feels like understanding, but math is a skill - you only build it by solving problems yourself, including ones you get wrong.

Redo your mistakes from scratch

When you miss a problem, don't just read the correct answer. Close the solution and try the problem again a day later.

Build a formula sheet as you go

Use the Flashcard Builder to turn each new formula into a card the moment you learn it, rather than cramming them all before a test.

Common Mistakes

Where students lose marks in math

Patterns we see again and again in student work. Each one is fixable once you know to look for it.

Sign errors when distributing a negative: -(x - 3) becomes -x - 3 instead of -x + 3. This single error accounts for a surprising share of wrong answers on algebra tests. Slow down at every minus sign in front of parentheses and distribute it to every term inside.
Cancelling terms instead of factors: You can cancel a common factor from the top and bottom of a fraction, not a common term. (x + 2)/2 is not x + 1. Before cancelling anything, ask whether it multiplies the whole numerator and the whole denominator.
Using the formula without checking what it needs: The quadratic formula requires the equation to be in the form ax² + bx + c = 0. Students plug coefficients from 2x² + 3x = 2 without moving the 2 across first. Any time you reach for a formula, check the form it expects.
Degrees and radians: Calculators default to one or the other. A trig answer that is wildly wrong is very often a mode problem. Check the mode before the test, and get used to which unit each course expects.
Forgetting the constant of integration: Every indefinite integral ends in + C. It is a small thing that graders check every time, and it matters when you are solving for a specific solution using an initial condition.