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Elementary and Middle school learning

Full Math Cheat Sheet Grades 1–10

Full Math Cheat Sheet — Grades 1–10

One‑page Summary — Quick Reference (Grades 1–10)

Essential formulas — print single sheet for quick lookups

Basic Arithmetic & Number Facts

Add/Subtract/Multiply/Divide properties: commutative, associative, distributive: a(b+c)=ab+ac

Sum of first n naturals: \(\displaystyle \sum_{k=1}^n k=\frac{n(n+1)}{2}\)

Sum of first n odd numbers: \(\displaystyle 1+3+\dots+(2n-1)=n^2\)

Sum of first n even numbers: \(\displaystyle 2+4+\dots+2n=n(n+1)\)

Fractions · Decimals · Percents

Convert: fraction → decimal: divide; decimal → percent: ×100; percent → fraction: ÷100 → simplify

Common: \(\tfrac12=0.5=50\%\)

\(\tfrac13\approx0.333=33.33\%\)

\(\tfrac14=0.25=25\%\)

Common fractions, decimals, and percentages:

  • \(\tfrac{1}{2} = 0.5 = 50\%\)
  • \(\tfrac{1}{3} \approx 0.333 = 33.33\%\)
  • \(\tfrac{1}{4} = 0.25 = 25\%\)
  • \(\tfrac{1}{5} = 0.2 = 20\%\)
  • \(\tfrac{1}{8} = 0.125 = 12.5\%\)

Add/Subtract: common denominator; Multiply: multiply numerators/denominators; Divide: multiply by reciprocal.

Algebra & Polynomials

Expand: \((a+b)^2=a^2+2ab+b^2\), \((a-b)^2=a^2-2ab+b^2\)

Factor: \(a^2-b^2=(a-b)(a+b)\), \(ax+ay=a(x+y)\)

Quadratic: \(ax^2+bx+c=0\). Roots: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\). Discriminant \(\Delta=b^2-4ac\)

Binomial theorem: \((x+y)^n=\sum_{k=0}^n {n\choose k}x^{n-k}y^k\)

Exponents, Roots & Logarithms

Exponent rules: \(a^m a^n=a^{m+n}\), \(\dfrac{a^m}{a^n}=a^{m-n}\), \((a^m)^n=a^{mn}\)

Negative/power: \(a^{-n}=\dfrac{1}{a^n}\), \(\sqrt[n]{a}=a^{1/n}\)

Log rules: \(\log_b(xy)=\log_b x+\log_b y\), \(\log_b(x^k)=k\log_b x\), change of base \(\log_b x=\dfrac{\log x}{\log b}\)

Sequences & Series

Arithmetic sequence: \(a_n=a_1+(n-1)d\). Sum: \(S_n=\dfrac{n}{2}(a_1+a_n)\)

Geometric sequence: \(a_n=a_1 r^{n-1}\). Sum (r≠1): \(S_n=a_1\dfrac{1-r^n}{1-r}\)

Perimeter, Area & Angles

Rectangle: \(P=2(l+w),\ A=lw\). Square: \(P=4a,\ A=a^2\).

Triangle: \(P=a+b+c,\ A=\tfrac12 bh\). Trapezium: \(A=\tfrac12(a+b)h\).

Circle: \(C=2\pi r,\ A=\pi r^2\). Sector: \(A=\tfrac12 r^2\theta\) (θ in radians). Arc length: \(s=r\theta\).

Polygon interior sum: \((n-2)\times180^\circ\). Interior angle (regular n-gon): \(\dfrac{(n-2)180^\circ}{n}\).

Surface Area & Volume

Cube: \(V=a^3,\ SA=6a^2\). Cuboid: \(V=lwh,\ SA=2(lb+bh+hl)\).

Cylinder: \(V=\pi r^2h,\ SA=2\pi r(h+r)\). Sphere: \(V=\tfrac{4}{3}\pi r^3,\ SA=4\pi r^2\).

Cone: \(V=\tfrac13\pi r^2 h,\ SA=\pi r(l+r)\) where l is slant height.

Coordinate Geometry & Conics

Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\). Midpoint: \(M=(\tfrac{x_1+x_2}{2},\tfrac{y_1+y_2}{2})\).

Line: \(y=mx+c\), point-slope: \(y-y_1=m(x-x_1)\). Perpendicular slopes: \(m_1 m_2=-1\).

Circle: \((x-h)^2+(y-k)^2=r^2\). Parabola: standard \(y=ax^2+bx+c\), vertex \(y=a(x-h)^2+k\).

Trigonometry — Identities & Rules

Definitions: \(\sin\theta=\tfrac{opp}{hyp},\ \cos\theta=\tfrac{adj}{hyp},\ \tan\theta=\tfrac{opp}{adj}\).

Reciprocals: \(\csc\theta=\tfrac{1}{\sin\theta},\ \sec\theta=\tfrac{1}{\cos\theta},\ \cot\theta=\tfrac{1}{\tan\theta}\).

Pythagorean: \(\sin^2\theta+\cos^2\theta=1\). Double-angle: \(\sin2\theta=2\sin\theta\cos\theta,\ \cos2\theta=\cos^2\theta-\sin^2\theta\).

Sum/difference: \(\sin(a\pm b)=\sin a\cos b\pm\cos a\sin b\), \(\cos(a\pm b)=\cos a\cos b\mp\sin a\sin b\).

Sine rule: \(\dfrac{a}{\sin A}=\dfrac{b}{\sin B}=\dfrac{c}{\sin C}\). Cosine rule: \(c^2=a^2+b^2-2ab\cos C\).

Basic solved forms: \(\sin^2\theta=\tfrac{1-\cos2\theta}{2},\ \cos^2\theta=\tfrac{1+\cos2\theta}{2}\).

Permutations · Combinations · Probability

Factorial: \(n!=n\times(n-1)\times\cdots\times1\).

Permutations: \(P(n,r)=\dfrac{n!}{(n-r)!}\). Combinations: \(C(n,r)=\dfrac{n!}{r!(n-r)!}\).

Probability: \(P(E)=\dfrac{\text{favourable}}{\text{total}}\). Conditional: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\).

Independent: \(P(A\cap B)=P(A)P(B)\). Complement: \(P(\overline{E})=1-P(E)\).

Statistics & Data

Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\). Median: middle value. Mode: most frequent.

Variance (population): \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\). Std dev: \(\sigma=\sqrt{\sigma^2}\).

Grouped data: estimated mean using midpoints, cumulative frequency for median quartiles.

Finance & Growth

Simple interest: \(I=Prt\). Amount: \(A=P(1+rt)\).

Compound (annual): \(A=P(1+r)^t\). Compound (n times yearly): \(A=P\left(1+\dfrac{r}{n}\right)^{nt}\).

Continuous growth/decay: \(A=Pe^{kt}\).

Useful Identities & Constants

Euler: \(e^{i\pi}+1=0\). \(\pi\approx3.14159,\ e\approx2.71828\).

Factor & expansion shortcuts: \((x+y)^3=x^3+3x^2y+3xy^2+y^3\).

Angle conversion: \(1^\circ=\dfrac{\pi}{180}\) rad. Unit tips: 1 m = 100 cm = 1000 mm.

This summary covers common formulas used through Grade 10. For printable clarity, use A4 landscape and increase print-scale if needed.

Grades 1–2 — Foundations

Basic arithmetic, number facts, shapes

Counting & Place Value

Digits: 0–9. Place values: units, tens, hundreds.

Even / Odd: even ends 0,2,4,6,8. odd ends 1,3,5,7,9.

Addition & Subtraction

Commutative: a+b=b+a

Associative: (a+b)+c = a+(b+c)

Column addition/subtraction → carry / borrow rules.

Multiplication basics

Repeated addition: 3×4 = 3+3+3+3

Tables: learn ×1–×12 (practice)

Division basics

Sharing and grouping. Division fact: a ÷ b = c ⇔ b×c = a

Shapes

Square, rectangle, triangle, circle — name corners/edges.

Perimeter (simple): add sides.

Time & Money

Clock: 60 sec = 1 min, 60 min = 1 hr. Coins & notes: practice sums.

Tip: use flash cards for tables and number bonds (e.g. 7 = 3+4).

Grades 3–4 — Elementary

fractions, decimals, area, perimeter, factors/multiples

Fractions

Proper: numerator < denominator. Improper: numerator ≥ denominator.

Equivalent: \(\frac{a}{b}=\frac{ka}{kb}\)

Add/Subtract (common denom): \(\dfrac{a}{b}+\dfrac{c}{b}=\dfrac{a+c}{b}\)

Multiply: \(\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}\)

Divide: \(\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}\)

Decimals & Percent

Decimals: place values tenths, hundredths, thousandths.

Convert: \(0.25=25\%= \tfrac{25}{100}=\tfrac{1}{4}\)

Percent of: \(p\%\text{ of }N = \dfrac{p}{100}\times N\)

Factors & Multiples

Factors: integers dividing a number. Multiples: product with integers.

GCF/LCM basics: list method or prime factors.

Perimeter & Area

Rectangle: \(P=2(l+w)\), \(A=l\times w\)

Square: \(P=4a\), \(A=a^2\)

Triangle: \(A=\tfrac{1}{2}bh\)

Circle: \(C=2\pi r\), \(A=\pi r^2\)

Angles & Polygons

Sum angles triangle = 180°

Regular polygon interior angle = \(\dfrac{(n-2)\times180^\circ}{n}\)

Basic Data

Mean (average): \(\bar{x}=\dfrac{\sum x_i}{N}\)

Mode: most frequent. Median: middle when ordered.

Tip: practise converting between fractions, decimals and percentages.

Grades 5–6 — Upper Elementary

ratio, proportion, algebra basics, volume, exponents

Ratio & Proportion

Ratio: 3:2 means \(\dfrac{3}{2}\). Proportion: \(\dfrac{a}{b}=\dfrac{c}{d}\)

Cross-multiply: \(ad=bc\)

Exponents & Roots

\(a^m \times a^n = a^{m+n}\)

\(\dfrac{a^m}{a^n}=a^{m-n}\), \((a^m)^n=a^{mn}\)

Square root: \(\sqrt{a}\) such that \(\sqrt{a}^2=a\)

Algebra basics

Solve linear: ax+b=c → \(x=\dfrac{c-b}{a}\)

Simple identities: \((a+b)^2=a^2+2ab+b^2\)

Volume

Cube: \(V=a^3\)

Cuboid: \(V=lbh\)

Cylinder: \(V=\pi r^2 h\)

Prism: \(V=\text{base area}\times height\)

Percent & Interest (intro)

Percent change: \(\dfrac{\text{change}}{\text{original}}\times100\%\)

Simple interest: \(I=Prt\) (P principal, r rate, t time)

Statistics basics

Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\)

Range = max − min

Frequency tables → mode, median estimation.

Tip: use algebra tiles or simple balancing to visualise equations.

Grades 7–8 — Middle School

advanced arithmetic, linear algebra, coordinate geometry, basic trigonometry

Linear equations

One variable: \(ax+b=0\Rightarrow x=-\dfrac{b}{a}\)

Two variables (slope form): \(y=mx+c\)

Slope between points: \(m=\dfrac{y_2-y_1}{x_2-x_1}\)

Coordinate Geometry

Distance: \(d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}\)

Midpoint: \(M\!=\!\Big(\dfrac{x_1+x_2}{2},\dfrac{y_1+y_2}{2}\Big)\)

Equation of line (point-slope): \(y-y_1=m(x-x_1)\)

Quadratic basics

Standard: \(ax^2+bx+c=0\)

Quadratic formula: \(x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}\)

Discriminant \(\Delta=b^2-4ac\) (real roots if \(\Delta\ge0\))

Trigonometry (basic)

Right triangle definitions: \(\sin\theta=\dfrac{\text{opp}}{\text{hyp}},\ \cos\theta=\dfrac{\text{adj}}{\text{hyp}},\ \tan\theta=\dfrac{\text{opp}}{\text{adj}}\)

Identity: \(\sin^2\theta+\cos^2\theta=1\)

Polynomials & Factorisation

Factor: \(a^2-b^2=(a-b)(a+b)\)

Common factor: \(ax+ay=a(x+y)\)

Probability

Simple probability: \(P(E)=\dfrac{\text{favourable}}{\text{total}}\)

Complement: \(P(\overline{E})=1-P(E)\)

Surface Area & Volume (refresher)

Cube: \(SA=6a^2, V=a^3\)

Cuboid: \(SA=2(lb+bh+hl), V=lbh\)

Sphere: \(SA=4\pi r^2, V=\tfrac{4}{3}\pi r^3\)

Cone: \(V=\tfrac13\pi r^2h\)

Sequences & Series (intro)

Arithmetic sequence: \(a_n=a_1+(n-1)d\)

Sum of n terms: \(S_n=\dfrac{n}{2}(a_1+a_n)\)

Geometric sum (r≠1): \(S_n=a_1\dfrac{1-r^n}{1-r}\)

Statistics (middle)

Mean: \(\bar{x}=\dfrac{\sum x_i}{N}\)

Median & Mode definitions. Variance (intro): \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\)

Tip: print each sheet separately or use the page numbers to focus on the grade you need.

Grade 8 — Key Formulas

Advanced arithmetic, linear algebra, basic trig & geometry

Linear Algebra & Manipulation

Linear eqns: \(ax+b=0\Rightarrow x=-\dfrac{b}{a}\)

Simultaneous (2×2) solve (substitution/elimination).

Functions & Graphs

Function notation: \(f(x)=mx+c\). Evaluate: \(f(2)=2m+c\).

Trigonometry (right triangles)

\(\sin\theta=\dfrac{opp}{hyp}\), \(\cos\theta=\dfrac{adj}{hyp}\), \(\tan\theta=\dfrac{opp}{adj}\)

Use SOHCAHTOA to find missing sides/angles.

Geometry & Transformations

Scale factor area → \(k^2\), volume → \(k^3\)

Angle facts: parallel lines, alternate interior angles equal.

Data & Probability

Mean, median, mode, range. Simple probability and complements.

Number Theory

Prime, composite, HCF/GCF by prime factors, LCM from prime factors.

Practice solving linear pairs and applying trig to right triangles.

Grade 9 — Key Formulas

Polynomials, quadratic, coordinate geometry, trigonometry

Polynomials & Factoring

Factorisation techniques: common factor, grouping, quadratic trinomials.

Remainder theorem: f(a) remainder when dividing by (x-a).

Quadratics

Vertex: \(x_v=-\dfrac{b}{2a}\). Axis of symmetry \(x=x_v\).

Roots via quadratic formula and discriminant tests.

Coordinate Geometry

Equation line: \(y=mx+c\). Perpendicular slopes: \(m_1m_2=-1\).

Circle eqn and distance/midpoint refresher.

Trigonometry (advanced basics)

Compound formulas, double-angle: \(\sin2\theta=2\sin\theta\cos\theta\).

Sine and cosine rule for any triangle (see summary).

Statistics

Grouped data: mean estimate, cumulative frequencies, median interpolation (concept).

Finance basics

Simple interest: \(I=Prt\). Compound annually: \(A=P(1+r)^t\)

Grade 9 prepares for algebraic manipulation and coordinate proofs.

Grade 10 — Key Formulas

Quadratics, advanced trig identities, log rules, permutations/combinations

Advanced Algebra

Quadratic forms, completing the square: \(ax^2+bx+c=a(x+\tfrac{b}{2a})^2+\dots\)

Binomial theorem and Pascal’s triangle applications.

Trigonometric Identities

Product-to-sum and sum-to-product (reference): \(\sin A\cos B=\tfrac12[\sin(A+B)+\sin(A-B)]\)

Triple-angle basics and solving trig equations (intro).

Logarithms & Exponentials

Exponential growth/decay models: \(A=Pe^{kt}\) (continuous).

Log rules and solving simple log equations (use change of base).

Probability & Combinatorics

Permutations & combinations (see summary). Conditional probability: \(P(A|B)=\dfrac{P(A\cap B)}{P(B)}\)

Independent events: \(P(A\cap B)=P(A)P(B)\)

Coordinate Geometry & Conics

Circle: \((x-h)^2+(y-k)^2=r^2\). Parabola: \(y=ax^2+bx+c\).

Focus/directrix form and basic locus concepts (intro).

Statistics (intro to distributions)

Mean, variance: \(\sigma^2=\dfrac{\sum (x_i-\bar{x})^2}{N}\). Std dev: \(\sigma=\sqrt{\sigma^2}\)

Normal approximation (concept) and uses of z-scores (intro).

Grade 10 consolidates algebra/trig and introduces exponential/logarithmic modelling.
Created for quick reference — adapt font-size for printed copies if needed.