MTTC Mathematics (Elementary) (089) cheat sheet

Place value: each position is 10 times the value of the position to its right; 0.1 = 1/10, 0.01 = 1/100, so 3.47 = 3 + 4/10 + 7/100.

Divisibility rules: by 3 if digit sum is divisible by 3; by 4 if the last two digits form a multiple of 4; by 9 if digit sum is divisible by 9; by 6 if divisible by both 2 and 3.

Order of operations: parentheses, exponents, multiplication/division left to right, addition/subtraction left to right.

Fraction division: a/b ÷ c/d = a/b × d/c; dividing by a fraction less than 1 yields a quotient larger than the dividend.

Integer operations: subtracting a negative adds its absolute value; a negative times a negative is positive; a negative times a positive is negative.

Percent change = (new − old)/old × 100; a 20% increase followed by a 20% decrease returns to 96% of the original, not 100%.

Proportion setup: cross-multiplication of a/b = c/d gives ad = bc; unit rate = quantity/1 unit, found by dividing both terms by the second term.

Rounding: locate the target place, then round up if the digit to its right is 5 or more; 4.567 rounded to hundredths is 4.57.

GCF is the product of shared prime factors; LCM is the product of the highest power of each prime appearing in either number.

Distributive property: a(b + c) = ab + ac; factoring reverses it, so 6x + 9 = 3(2x + 3).

Solving linear equations: perform the same operation on both sides; isolate the variable by undoing addition/subtraction first, then multiplication/division.

Slope = (y₂ − y₁)/(x₂ − x₁); slope-intercept form y = mx + b, where m is the slope and b is the y-intercept.

Arithmetic sequence: nth term = a₁ + (n − 1)d; geometric sequence: nth term = a₁ · r^(n − 1).

Inequality rule: multiplying or dividing both sides by a negative number reverses the inequality symbol.

Function test: a relation is a function if each input (x-value) maps to exactly one output; the vertical line test detects failures on a graph.

Equivalent expressions: substituting a test value into both expressions is a valid check; matching values for all inputs is the definition of equivalence.

Exponent rules: x^a · x^b = x^(a+b); (x^a)^b = x^(ab); x^0 = 1 for x ≠ 0; x^(−a) = 1/x^a.

Area formulas: rectangle A = lw; triangle A = ½bh; parallelogram A = bh; trapezoid A = ½(b₁ + b₂)h; circle A = πr².

Circumference C = πd = 2πr; π ≈ 3.14 or 22/7 for estimation.

Volume: rectangular prism V = lwh; cylinder V = πr²h; cone V = ⅓πr²h; sphere V = 4/3πr³.

Triangle angle sum = 180°; quadrilateral sum = 360°; exterior angle equals the sum of the two remote interior angles.

Pythagorean theorem a² + b² = c² applies only to right triangles, with c the hypotenuse; converse tests for a right angle.

Similar figures: corresponding angles are equal and side lengths are proportional; scale factor k multiplies lengths, k² multiplies areas, k³ multiplies volumes.

Complementary angles sum to 90°; supplementary angles sum to 180°; vertical angles are equal.

Unit conversion: multiply by a conversion fraction equal to 1 (e.g., 12 in/1 ft) arranged so unwanted units cancel.

Perimeter is a linear measure (units), area is square units, and volume is cubic units; doubling all dimensions of a solid multiplies volume by 8.

Mean = sum of values ÷ number of values; median = middle value of ordered data (average the two middle values if n is even); mode = most frequent value.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1, with Q1 and Q3 the medians of the lower and upper halves.

Outliers pull the mean toward them but leave the median nearly unchanged; the median is more resistant to skew.

Probability of an event = favorable outcomes ÷ total equally likely outcomes, always between 0 and 1.

Complement rule: P(not A) = 1 − P(A); for independent events P(A and B) = P(A) · P(B).

A box plot displays minimum, Q1, median, Q3, and maximum; a histogram shows frequency within intervals and reveals distribution shape.

Correlation describes the strength and direction of a linear association, not causation; a line of best fit predicts y from x.

Counting: fundamental counting principle multiplies choices across stages; permutations order items (nPr), combinations do not (nCr).