MathBeasts

Align ones under ones, tens under tens (and decimal points under decimal points) before adding or subtracting.

✗ 305 + 42 = 725 (42 slid under the 30)

✓ 305 + 42 = 347

Prerequisites: counting, place-value

Keep full precision through the working; round once, at the end, to the requested place.

✗ 2.46 × 3.46 → 2.5 × 3.5 = 8.75 ≈ 8.8

✓ 2.46 × 3.46 = 8.5116 ≈ 8.5

Prerequisites: place-value, decimals

a − (−b) = a + b; same signs multiply or divide to a positive, different signs to a negative.

✗ −7 − (−3) = −10

✓ −7 − (−3) = −7 + 3 = −4

Prerequisites: number-line, integer-addition

Rewrite over a common denominator, then add numerators only.

✗ 1/3 + 1/4 = 2/7

✓ 4/12 + 3/12 = 7/12

Prerequisites: equivalent-fractions, multiples

A product has as many decimal places as the factors combined; estimate to confirm.

✗ 0.3 × 0.2 = 0.6

✓ 0.3 × 0.2 = 0.06

Prerequisites: place-value, multiplication

Keep the order: boys : girls = 3 : 2 means boys/girls = 3/2.

✗ Boys : girls = 3 : 2, 30 girls → 20 boys

✓ boys = (3/2) × 30 = 45

Prerequisites: fractions, multiplication

Percent means per hundred: 5% = 5/100 = 0.05.

✗ 5% of 80 = 0.5 × 80 = 40

✓ 5% of 80 = 0.05 × 80 = 4

Prerequisites: decimals, fractions

Grouping, exponents, multiplication/division (left to right), then addition/subtraction (left to right).

✗ 8 + 3 × 4 = 44

✓ 8 + 3 × 4 = 8 + 12 = 20

Prerequisites: four-operations

Only terms with the same variable raised to the same power combine; add their coefficients.

✗ 3x + 2x² = 5x³

✓ 3x + 2x² is already simplified; 3x + 2x = 5x

Prerequisites: variables, exponents-basic

a(b + c) = ab + ac: the outside factor multiplies every term inside, sign included.

✗ 3(x + 4) = 3x + 4

✓ 3(x + 4) = 3x + 12

Prerequisites: signed-numbers, like-terms

Whatever you do to one side, do to the other, so the equation stays true.

✗ 2x + 5 = 13 → 2x = 13 → x = 6.5

✓ 2x = 13 − 5 = 8 → x = 4

Prerequisites: inverse-operations, like-terms

Multiplying or dividing both sides by a negative number reverses the inequality.

✗ −3x < 12 → x < −4

✓ −3x < 12 → x > −4

Prerequisites: signed-numbers, linear-equations

(xᵃ)ᵇ = xᵃᵇ — multiply the exponents; add them only when multiplying same-base powers.

✗ (x³)² = x⁵

✓ (x³)² = x³ · x³ = x⁶

Prerequisites: multiplication, exponents-basic

(a + b)² = a² + 2ab + b²; multiply every term by every term.

✗ (x + 3)² = x² + 9

✓ (x + 3)² = x² + 6x + 9

Prerequisites: distribution, like-terms

Only common factors of the whole numerator and denominator cancel.

✗ (x + 6)/6 = x + 1

✓ (x + 6)/6 = x/6 + 1

Prerequisites: fractions, factoring

√(ab) = √a·√b for nonnegative a, b, but √(a + b) is not √a + √b.

✗ √(9 + 16) = 3 + 4 = 7

✓ √(9 + 16) = √25 = 5

Prerequisites: exponents-basic, order-of-operations

The first coordinate is horizontal (x), the second vertical (y).

✗ (2, 5): 5 right, 2 up

✓ (2, 5): 2 right, 5 up

Prerequisites: number-line, signed-numbers

Slope = change in y ÷ change in x = (y₂ − y₁)/(x₂ − x₁).

✗ (1, 2), (3, 8): slope = 2/6 = 1/3

✓ slope = (8 − 2)/(3 − 1) = 3

Prerequisites: coordinates, fractions

x-intercept: set y = 0. y-intercept: set x = 0.

✗ y = 2x − 6: x-intercept (0, −6)

✓ 0 = 2x − 6 → x = 3; x-intercept (3, 0)

Prerequisites: linear-equations, coordinates

f(x + 2) means substitute x + 2 for every x in the rule.

✗ f(x) = x²: f(x + 2) = x² + 2

✓ f(x + 2) = (x + 2)² = x² + 4x + 4

Prerequisites: substitution, distribution

A solution to a system in two variables is an ordered pair that satisfies every equation.

✗ x + y = 10, x − y = 4 → x = 7. Done.

✓ x = 7, y = 3: solution (7, 3)

Prerequisites: linear-equations, substitution

x² = k (k > 0) has two solutions: x = ±√k.

✗ x² = 9 → x = 3

✓ x² = 9 → x = 3 or x = −3

Prerequisites: radicals, signed-numbers

Growth of r% per period multiplies by (1 + r) each period: A = P(1 + r)ᵗ.

✗ 1,000 growing 10%/yr for 3 yrs: 1,000 + 3 × 100 = 1,300

✓ 1,000 × 1.1³ = 1,331

Prerequisites: percentages, exponents-basic

log(ab) = log a + log b, but log(a + b) has no such split.

✗ log(10 + 100) = 1 + 2 = 3

✓ log(110) ≈ 2.04; log(10 × 100) = 3

Prerequisites: exponents-basic, functions

Convert to a common unit first, and carry units through every step.

✗ 2 m + 50 cm = 52

✓ 2 m + 0.5 m = 2.5 m

Prerequisites: decimals, multiplication

Average speed = total distance ÷ total time.

✗ 60 mi at 30 mph, 60 mi back at 60 mph: (30 + 60)/2 = 45 mph

✓ 120 mi ÷ (2 h + 1 h) = 40 mph

Prerequisites: units, fractions

Divide the annual rate by the number of compounding periods per year.

✗ $1,000 at 6%/yr compounded monthly, 1 month: $60 interest

✓ 1,000 × 0.06/12 = $5 interest

Prerequisites: percentages, decimals

Successive percent changes multiply, and each applies to a different base.

✗ $100 +20% then −20% → $100

✓ 100 × 1.2 × 0.8 = $96

Prerequisites: percentages

Scaling lengths by k scales area by k² and volume by k³.

✗ Square side 3 → 6: area 9 → 18

✓ area 9 → 36 (× 2² = × 4)

Prerequisites: area, exponents-basic

A model is trustworthy only near the range of its data; check whether the trend can continue.

✗ Grew 6 cm/yr, 85 → 121 cm from age 2 to 8, so 121 + 32 × 6 = 313 cm at 40

✓ The linear model fits ages 2–8 only; growth stops, so don't extrapolate to 40

Prerequisites: slope, rates

Estimate first and compare: does the size of the answer make sense in context?

✗ 15% tip on $40 = $600

✓ 0.15 × 40 = $6 — a tip can't exceed the bill

Prerequisites: estimation, percentages

Compare the values themselves; check where the axis starts.

✗ Bars 52 vs 48, axis from 46: bar 3× taller → 'three times as many'

✓ 52 ÷ 48 ≈ 1.08: about 8% more

Prerequisites: ratios, percentages

Sort the data first; the median is the middle of the ordered list.

✗ 7, 2, 9, 4, 5 → median 9

✓ 2, 4, 5, 7, 9 → median 5

Prerequisites: ordering-numbers

Spread measures how far values sit from each other (range, IQR, SD), not how large they are.

✗ {98, 100, 102} is more spread than {1, 10, 19}

✓ Range 4 vs range 18: {1, 10, 19} is more spread

Prerequisites: center, signed-numbers

For independent trials, past outcomes don't change the next probability.

✗ Fair coin, 5 heads in a row → tails now more likely than 1/2

✓ P(tails) is still 1/2

Prerequisites: fractions, probability-basic

Conditional probabilities reverse only through the base rate (Bayes' rule).

✗ Test catches 90% of cases, so a positive means 90% chance of the condition

✓ With 1% prevalence and 9% false positives: 0.009 ÷ (0.009 + 0.0891) ≈ 9.2%

Prerequisites: percentages, probability-basic

Only a random (probability) sample supports generalizing to the population.

✗ 80% of 2,000 website poll respondents oppose → 80% of the town opposes

✓ Self-selected respondents aren't representative; a random sample is needed

Prerequisites: percentages

Correlation alone doesn't show cause; look for lurking variables or a randomized experiment.

✗ Ice-cream sales and drownings correlate (r = 0.8), so ice cream causes drowning

✓ Summer heat drives both; the data show association, not cause

Prerequisites: interpreting-graphs

A p-value is the probability of data at least this extreme, assuming the null is true.

✗ p = 0.03 → 3% chance the null is true

✓ If the null were true, results this extreme would occur about 3% of the time

Prerequisites: probability-basic, sampling