Addition
9 + 2 = ?
carry when a column reaches ten · 4 concept cases
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Speed test
10 questions against the clock. Type the answer, no Enter needed. Stars come from your average pace, and any miss voids them.
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Addition
Subtraction
Multiplication
Division
Everything
Every problem, built backwards from a small answer.
6 courses, 112 skills, 367 named concept cases. Each quiz deals its cases from a shuffled deck so every variant shows up, missed concepts come back as redemption questions, and your per concept mastery is tracked between sessions.
Practice today to start a streak.
Your map
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Basic Math
Mental Math Tricks
Algebra 1
Algebra 2
Geometry
Precalculus
Basic Math
The four operations
Addition, subtraction, multiplication, and division, split by the regrouping and sign traps that cost points.
Long addition with carryingExplainer
457 + 368
Watch a random sum added column by column, with every carry moving into place.
Subtraction
8 − 4 = ?
borrow when the top digit is smaller · 4 concept cases
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Long subtraction with borrowingExplainer
503 − 278
Watch a random subtraction worked column by column, borrowing ten whenever a top digit is too small.
Adding and subtracting negatives on a number lineExplainer
−7 + 15
Watch a random sum with a negative worked out by hops on a number line, split at 0, including subtracting a negative.
Multiplication
6 × 6 = ?
split into tens and ones, then add the parts · 4 concept cases
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Long multiplicationExplainer
47 × 26
Watch a random multiplication built row by row: one row per digit, carries on top, then the rows added.
Division
36 ÷ 9 = ?
how many groups fit, and what is left over · 4 concept cases
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Long divisionExplainer
285 ÷ 6
Watch a random division worked digit by digit: divide, multiply, subtract, bring down.
Mental Math Tricks
Add and subtract like a mathemagician
Mental arithmetic runs left to right, the order you say the answer in. Round awkward numbers, then correct.
Adding left to right
63 + 47 = ?
big parts first, the way you say the answer; pair numbers that make tens · 4 concept cases
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Subtracting by complements
63 − 48 = ?
overshoot to a round number and add back, or count up like making change · 5 concept cases
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Multiplication shortcuts
Most products hide an easier one: a round number nearby, a factor of ten, or a split into single digits.
Special multipliers
36 × 11 = ?
times 4, 5, 8, 9, 11, 12, 15, 25, 50, 99, 101 and 125 without a times table · 12 concept cases
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Multiplying left to right
46 × 9 = ?
big part first, then correct or factor · 6 concept cases
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Squares and near squares
One identity, (a + d)(a minus d) = a squared minus d squared, turns squares and close pairs into one easy product and a correction.
Squaring in your head
75² = ?
up and down to a round number, add the distance squared · 4 concept cases
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Close together products
96 × 99 = ?
work from the round number beside or between them · 5 concept cases
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Division and percent shortcuts
Division by 5, 25, 4 and 8 is really multiplication or halving, and every percent is built from 10 percent.
Dividing in your head
410 ÷ 5 = ?
double, quarter or halve instead of long division · 4 concept cases
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Percent shortcuts
20% of 20 = ?
build from 10 percent, or swap the two numbers · 3 concept cases
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Check your answer
Reduce every number to one digit and the arithmetic still works on the digits, so a wrong answer shows itself.
Casting out nines and divisibility
digital root of 57090 = ?
prove an answer wrong without redoing it · 5 concept cases
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Estimation
Most answers can be judged before they are worked out: round to friendly numbers for the size, and check the last digit for the rest. One option is near the estimate; the others are far off or end wrong.
Estimate the size
7 094 × 101
round to friendly numbers, then pick the only option near the estimate · 10 concept cases
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Check the last digit
4 405 × 7 845
the last digit, odd or even, or the zeros decide it without the full sum · 4 concept cases
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Calendar calculation
The weekday of any date from three small codes added together, with every seven thrown away as you go.
Day of the week for any date
year code for 2087 = ?
month code plus date plus year code, mod 7 · 3 concept cases
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Algebra 1
Solving linear equations
Undo what was done to x, one inverse operation at a time.
One-step: add / subtract
x − 2 = −6
undo + and −
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One-step: multiply / divide
12x = −7
undo ×, fraction answers
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Two-step equations
7x + 1 = −62
undo +/− first, then ÷
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Solving an equation, step by stepExplainer
5x + 1 = −44
Watch a random equation solved by keeping it balanced: undo each operation on both sides, then check the answer.
Variables on both sides
9x + 8 = 2x + 36
collect x on one side
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Distributive property
6(x − 3) = −6
a(x + b) = c
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Absolute value equations
|x + 4| = 4
split into two cases, or spot the impossible one · 3 concept cases
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Rearranging formulas
d = ?
solve for the letter, undo operations in reverse · 3 concept cases
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Proportions
x4 = 93
cross-multiply
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Solving a proportion by cross multiplyingExplainer
x/3 = 8/12
Watch a random proportion solved: multiply across both diagonals, then get x on its own and check.
Expressions
Rewrite without solving: distribute, combine, substitute.
Combine like terms
x + 6 + 8x + 6
add coefficients
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Distribute & simplify
2(x + 4) − 7x
multiply through, then combine · 3 concept cases
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Distributing a negativeExplainer
−3(x − 4) + 5x
Watch a random number distributed over parentheses in a box: it multiplies every term inside, signs and all, then like terms combine.
Evaluate expressions
−5x + 11
substitute and compute
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Lines & slope
Rise over run, and y = mx + b.
Write y = mx + boften tested
y = mx + b
solve for b from a point
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Linear models in contextoften tested
C = 10x + 45
the slope is the rate, the intercept is the start · 4 concept cases
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Forms of a lineoften tested
2x + 6y = 14
standard, point slope, parallel and perpendicular · 4 concept cases
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Slope from two points
(4, −5) and (12, −15)
Δy over Δx, reduced
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Slope from two pointsExplainer
(1, −2) and (5, 4)
Watch a random slope worked out: the rise over the run, both subtracted in the same order, then reduced.
Inequalities
Solve like equations, but flip when multiplying or dividing by a negative.
One- and two-step inequalitiesoften tested
−6x + 3 > 27
remember the flip
0/1
Solving an inequality, and when to flip the signExplainer
−3x + 4 ≥ −11
Watch a random inequality solved like an equation, the sign flipped when dividing by a negative, and the answers drawn on a number line.
Absolute value inequalities
|x − 1| < 6
less than is AND, greater than is OR · 3 concept cases
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Systems of equations
Two equations, two unknowns, eliminate one variable by adding or subtracting.
Systems of equations
x + y = −3
x − y = 1
x − y = 1
eliminate, and know when lines never meet · 3 concept cases
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Polynomials
Pure times-tables in disguise: FOIL multiplies, factoring runs it backwards.
Multiply binomials (FOIL)
(x + 6)(x − 3)
sum and product of a, b
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Multiplying two binomials with a boxExplainer
(x − 4)(x + 3)
Watch two random binomials multiplied in a box: one box for every pair of terms, then the like terms joined.
Factor x² + bx + c
x² − 25
two numbers: multiply to c, add to b
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Factoring x² + bx + cExplainer
x² − x − 12
Watch a random trinomial factored: find the two numbers that multiply to the last term and add up to the middle one.
Numbers & data
Percent problems the way tests ask them, and the statistics questions everyone meets.
Two way tablesoften tested
| Bus | Walk | Total | |
|---|---|---|---|
| Freshmen | 20 | 20 | 40 |
| Seniors | 25 | 35 | 60 |
| Total | 45 | 55 | 100 |
joint uses the grand total, conditional shrinks to one row · 3 concept cases
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Scatterplots & lines of best fitoften tested
y = ?
read the model, then interpret the slope · 3 concept cases
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Percent problems
50% of 26 = ?
percents are fractions; change compares to the ORIGINAL · 4 concept cases
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Percents as fractionsExplainer
40 → 50
Watch a random percent worked out as a fraction: a percent of a number, or a percent change compared with the original amount.
Ratios & rates
$? per pound
per-unit thinking: divide by the count, split by the parts · 4 concept cases
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Probability
P = ?
favorable over total; complements add to 1; "and" multiplies · 3 concept cases
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Scientific notation
5,000 = 5 × 10?
the exponent counts decimal moves; multiplying adds exponents · 3 concept cases
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Scientific notation: hopping the decimal pointExplainer
45000 = 4.5 × 10⁴
Watch a random number written in scientific notation: the decimal point hops one digit at a time, and the hops are the exponent.
Spread, outliers & weighted averages
34, 46, 54, 31
range, deviation, and what an outlier moves · 4 concept cases
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Unit conversion
? ounces
multiply going down, divide going up · 3 concept cases
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Mean & median
5, 8, 2, 1
mean uses the total; median needs a sort · 3 concept cases
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Algebra 2
Quadratics
Solve, locate the vertex, and read the discriminant, every root pattern covered.
Completing the squareoften tested
x² − 6x + ?
halve the middle coefficient, then square it · 2 concept cases
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Completing the square, as a squareExplainer
x² + 6x + 11
Watch a random quadratic turned into a square: half of the middle term along each side, and the missing corner that completes it.
Solve by factoring
x² − 6x + 8 = 0
zero product, including x = 0 and double roots · 4 concept cases
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Nonlinear systems
y = x²
y = −x + 2
y = −x + 2
set the two expressions equal and factor
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Quadratic formula
x² − 1 = 0
clean discriminants, fraction roots · 2 concept cases
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The quadratic formula, one piece at a timeExplainer
2x² − x − 6 = 0
Watch a random quadratic solved by the formula: read a, b and c, work out the part under the root first, then the two roots.
Vertex & axis of symmetry
y = x² + 4x + 8
−b/2a, and the vertex-form sign trap · 3 concept cases
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Counting solutions
x² − 2x − 15 = 0
b² − 4ac decides everything · 3 concept cases
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Functions
Composition, inverses, and reading transformations off the formula.
Composition f(g(x))
f(x) = −3x − 1
g(x) = −2x + 1
g(x) = −2x + 1
inside-out, both orders · 3 concept cases
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Inverse functions
f(x) = x − 6
undo in reverse order · 3 concept cases
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Transformations
g(x) = f(x) + 1
inside vs outside; x − h moves RIGHT · 4 concept cases
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Exponents, radicals & logs
The rules as pure add/subtract/multiply moves, negatives and fractions included.
Rational exponentsoften tested
43/2
the denominator is the root, the numerator is the power · 2 concept cases
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Exponent rules
x6 · x5 = x?
add, subtract, multiply, never memorize backwards · 5 concept cases
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Evaluate logarithms
log2 16 = ?
"base to what power?", including 0, negatives, fractions · 4 concept cases
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Radical equations
√(x + 3) = 5
square both sides, then check every answer · 2 concept cases
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Polynomial division
p(x) = x² − 6x − 9
synthetic division ends at the remainder
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Simplify radicals
√27 = a√b
pull out the biggest perfect square · 3 concept cases
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Exponential growth & decay
P(t) = ?
the base is 1 ± rate, what remains, not what changes · 3 concept cases
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Reading graphs
Name the family from its shape, match a moved graph to its equation, and read features off the grid.
Identify the function family
f(x) = ?
corner vs curve, endpoint vs both ways, steady vs accelerating · 6 concept cases
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Match the moved graph to its equation
y = ?
outside acts as written; inside flips the sign · 4 concept cases
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Read features off the graph
m = ?
count squares: rise/run, crossing points, the turning point · 4 concept cases
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Rational expressions
Factor, cancel, and spot opposite factors, the classic exam trap.
Rational expression arithmeticoften tested
x − 3x − 5 · x − 5x + 4
multiply across, flip to divide, find an LCD to add · 5 concept cases
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Adding fractions with different denominatorsExplainer
1/4 + 1/6
Watch two random fractions added: find the least common denominator, rewrite both fractions, add the tops, then reduce.
Rational equationsoften tested
4x = 7
clear the denominator, then check for extraneous roots · 2 concept cases
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Simplify rational expressions
x² − x − 12x − 4
factor first, then cancel · 4 concept cases
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Sequences
Spot the pattern, extend it, and use the formula.
Arithmetic & geometric sequences
1, −1, −3, −5, …
differences vs ratios · 5 concept cases
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Complex numbers
i² = −1 and everything that follows, small numbers, big concept.
Powers of i
i23 = ?
the 4-cycle
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Add & multiply complex numbers
(2 − 3i) + (2 − 3i)
track real and imaginary parts · 3 concept cases
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Geometry
Angles & lines
Angle pairs and parallel-line relationships, equal or supplementary, and which is which.
Angle pair relationships
m∠A = 58°. Find m∠B.
90 vs 180, and solving for x · 4 concept cases
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Parallel lines & a transversal
x = ?
name the pair, then equal or 180 − x · 4 concept cases
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Polygon angles
? °
(n − 2)·180 inside; always 360 outside · 3 concept cases
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Triangles
The 180° budget, the exterior-angle shortcut, and right triangles that never need a calculator.
Triangle angle relationships
x = ?
angle sum, exterior angle, isosceles · 4 concept cases
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Pythagorean theorem
x = ?
triples only, add for the hypotenuse, subtract for a leg · 3 concept cases
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The Pythagorean theoremExplainer
6² + 8² = x²
Watch a random right triangle solved: find the hypotenuse first, then add the squares, or subtract them for a leg.
Triangle congruence & proof reasons
three pairs of equal sides
name the shortcut, and know why SSA and AAA fail · 5 concept cases
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Special right triangles
hyp = ?√2
1:1:√2 and 1:√3:2, know which side is which · 4 concept cases
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Similarity
One scale factor controls every length.
Similar triangles
x = ?
find k, multiply, fractions welcome · 5 concept cases
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Area & volume
Formulas with the traps built in: the ½, the ⅓, the 4/3, and π stays exact.
Area & circumference
A = ?
which formula, and where the ½ goes · 6 concept cases
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Volume
V = ?
prisms fill, cones are ⅓, spheres are 4/3 · 5 concept cases
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Coordinate geometry
Midpoints, distances, and the slopes of parallel and perpendicular lines.
Midpoint & distance
(−6, −7) and (0, 1)
average vs Pythagorean, and when to just subtract · 4 concept cases
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Parallel & perpendicular slopes
m = ?
same vs negative reciprocal · 4 concept cases
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Transformations
Six rules for moving a point, each one a single sign or swap.
Image of a point
(7, −1) → (?, ?)
which coordinate changes, and how · 6 concept cases
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Circles
Central, inscribed, and the arc between them.
Angles in circles
x = ?
inscribed is half, arcs match the center · 4 concept cases
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Precalculus
Trigonometry
The unit circle as sign rules + three memorized values, no calculator, ever.
Trig graphsoften tested
amplitude = ?
amplitude outside, period from the inside coefficient · 3 concept cases
0/3
Right-triangle trigoften tested
sin θ = ?
SOH-CAH-TOA on Pythagorean triples · 4 concept cases
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Unit circle exact values
tan 60° = ?
ASTC signs on three memorized values · 4 concept cases
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Degrees ↔ radians
330° = ?π
multiply by π/180 and reduce · 3 concept cases
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Reference angles
132°
distance to the x-axis, per quadrant · 3 concept cases
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Trig identities
cos θ = ?
sin squared plus cos squared is always one · 2 concept cases
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Trig equations & inverses
sin θ = √3/2
read the unit circle backwards · 2 concept cases
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Law of Sines & Cosines
b = ?
pair each side with the angle across from it · 2 concept cases
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Arc length & sector area
s = ?π
s = r theta, area = half r squared theta · 2 concept cases
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Functions & graphs
Where functions break, which branch owns the boundary, and how graphs end.
Difference quotientoften tested
f(x+h) − f(x)h = ?
expand, cancel, divide by h: the bridge to calculus · 2 concept cases
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Asymptotes of rational functionsoften tested
x + 4x − 5
compare the degrees · 4 concept cases
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Domain
2x − 1
denominators ≠ 0, even roots ≥ 0 · 3 concept cases
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Piecewise evaluation
f(x) = x + 4 if x < 0
f(x) = −2x if x ≥ 0
f(x) = −2x if x ≥ 0
pick the branch, the boundary has an owner · 3 concept cases
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Even, odd, or neither
f(x) = 2x² + 3
test f(−x) by power parity · 3 concept cases
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Slant asymptotes
degree of top = degree of bottom + 1
top degree one higher means a slanted line · 2 concept cases
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Partial fractions
1(x − 1)(x − 3)
cover up a factor and substitute its root
0/1
Exponential & log equations
Everything reduces to matching exponents on a common base.
Log properties
log2 (82) = ?
combine first, evaluate second · 3 concept cases
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Exponential equations
3x = 27
same base, match exponents · 3 concept cases
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Log equations
log2 x = 4
rewrite in exponential form · 3 concept cases
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Polynomial behavior
Read the graph off the formula: leading term, multiplicities, and p(a).
End behavior
f(x) = 2x2 − 1x + 5
degree parity × leading sign · 4 concept cases
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Zeros & multiplicity
f(x) = (x + 1)2(x + 1)
even touches, odd crosses · 3 concept cases
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Remainder & factor theorems
p(x) = x² − 2x − 6
evaluate p(a), skip the division · 2 concept cases
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Conics & series
Identify the shape from its squares and signs; sum the pattern, not the terms.
Identify the conic
x² + y² = 36
which variables are squared, and with what signs · 4 concept cases
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Circle equations
(x − 2)² + (y − 5)² = 25
center flips the signs; the right side is r² · 2 concept cases
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Sums & sigma notation
Σ (3k + 2) for k = 1 to 4
expand small sums, formula for big ones · 3 concept cases
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Vectors, matrices & polar
The precalculus toolkit beyond functions: components and magnitude, matrix size rules, and a second way to name a point.
Vectors
v = <6, 8>
components add separately, magnitude is Pythagoras · 4 concept cases
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Matrices
A is 2 × 3, B is 2 × 3
rows by columns, and inner dimensions must match · 3 concept cases
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Polar coordinates
(r, θ) = (4, 0°)
x = r cos, y = r sin, and back again · 3 concept cases
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Mental math tricks
Shortcuts that make arithmetic fast enough to do in your head. Many come from Arthur Benjamin, the Harvey Mudd College mathematician whose books Secrets of Mental Math and Mathemagics teach calculating from left to right; the rest are classic and Vedic shortcuts. Each trick gives its rule, why it works, and a worked example you can step through. Practice it as a quiz, or race it as a timed sprint.
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