MTEL Math Elementary (68) cheat sheet

Place value: each position is 10 times the value of the position to its right; 4,706 = 4 thousands + 7 hundreds + 0 tens + 6 ones.

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: evaluate grouping symbols, then exponents, then multiplication/division left to right, then 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.

Percent change = (new − old) ÷ old × 100; a 20% increase followed by a 20% decrease does not return to the original value (result is 96%).

LCM × GCF of two positive integers equals their product; use prime factorization to find either.

Rounding: identify the target place, then round up if the digit to its right is 5 or more, keeping all digits to the left unchanged.

Integer operations: subtracting a negative is adding a positive; the product or quotient of two negatives is positive, of one negative and one positive is negative.

Arithmetic sequence: common difference d; nth term aₙ = a₁ + (n − 1)d.

Geometric sequence: common ratio r; nth term aₙ = a₁ · r^(n−1).

Linear function slope-intercept form y = mx + b, where m is the constant rate of change and b is the y-intercept (initial value).

Slope m = (y₂ − y₁)/(x₂ − x₁); a horizontal line has slope 0, a vertical line has undefined slope.

Distributive property: a(b + c) = ab + ac; combine like terms before solving multi-step equations.

Solving an inequality: multiplying or dividing both sides by a negative number reverses the inequality symbol.

Direct variation y = kx passes through the origin; inverse variation y = k/x has a constant product xy = k.

Function definition: each input in the domain is paired with exactly one output; the vertical line test detects a function on a graph.

Triangle angle sum is 180°; the exterior angle equals the sum of the two nonadjacent interior angles.

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

Circumference C = 2πr = πd; circle area uses the radius, not the diameter, so halve a given diameter first.

Volume: prism/cylinder V = Bh (B = base area); pyramid/cone V = ⅓Bh; sphere V = 4/3 πr³.

Surface area of a rectangular prism = 2(lw + lh + wh); of a cylinder = 2πr² + 2πrh.

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

Unit conversions: 1 km = 1,000 m, 1 m = 100 cm, 1 kg = 1,000 g, 1 L = 1,000 mL, 1 hour = 60 minutes; convert by multiplying by unit ratios.

Similar figures have equal corresponding angles and proportional sides; scale factor k multiplies lengths, k² multiplies areas, k³ multiplies volumes.

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

Outliers pull the mean toward them but leave the median largely unchanged, so the median better describes skewed data.

Range = maximum − minimum; interquartile range IQR = Q3 − Q1 and marks the middle 50% of ordered data.

Theoretical probability P(event) = favorable outcomes ÷ total equally likely outcomes; experimental probability uses observed frequencies.

Probability of a complement: P(not A) = 1 − P(A); for independent events P(A and B) = P(A) · P(B).

Mutually exclusive events satisfy P(A or B) = P(A) + P(B); otherwise subtract P(A and B) to avoid double counting.

A random sample must represent the population; biased sampling (convenience, voluntary response) undermines valid inference.

Correlation measures the strength and direction of a linear association and does not by itself establish causation.

Convert fractions to decimals by dividing numerator by denominator; a fraction in lowest terms has a terminating decimal only if the denominator's prime factors are 2 and/or 5.

Scientific notation: a × 10ⁿ with 1 ≤ |a| < 10; a positive exponent shifts the decimal right, a negative exponent shifts it left.

Absolute value |x| is the distance from 0 on the number line; |−7| = 7 and |x| = k has solutions x = k and x = −k for k > 0.

Estimating products: round each factor to a convenient place value, then adjust; 498 × 21 ≈ 500 × 20 = 10,000.

Benchmark fractions for comparison: 1/4 = 0.25, 1/3 ≈ 0.333, 1/2 = 0.5, 2/3 ≈ 0.667, 3/4 = 0.75.

Multiplying by a power of 10 moves the decimal point one place per zero; dividing moves it in the opposite direction.

Ratio reasoning: set up a proportion a/b = c/d and cross-multiply (ad = bc) to solve for an unknown quantity.

Slope formula applied to a table: divide the change in output by the change in input between any two rows; a constant result signals a linear pattern.

Point-slope form y − y₁ = m(x − x₁) builds an equation from one point and the slope.

Exponent rules: xᵃ · xᵇ = x^(a+b); xᵃ ÷ xᵇ = x^(a−b); (xᵃ)ᵇ = x^(ab); x⁰ = 1 for x ≠ 0.

Systems of two linear equations: substitution isolates one variable; elimination adds or subtracts equations to cancel a variable.

Factoring a quadratic x² + bx + c requires two numbers whose product is c and whose sum is b.

Translate verbal phrases: '5 less than n' is n − 5, while '5 is less than n' is 5 < n; order matters in subtraction and division.

A recursive rule states each term from the previous term (e.g., aₙ = aₙ₋₁ + 3), while an explicit rule gives aₙ directly from n.

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

Parallel lines cut by a transversal create congruent alternate interior and corresponding angles, and same-side interior angles that are supplementary.

Sum of interior angles of an n-gon = (n − 2) × 180°; each interior angle of a regular n-gon is that sum divided by n.

Perimeter is a linear measure (units), area is square units, and volume is cubic units; label answers with the correct power.

Distance and midpoint in the coordinate plane: d = √((x₂−x₁)² + (y₂−y₁)²); midpoint = ((x₁+x₂)/2, (y₁+y₂)/2).

Transformations: translations and rotations preserve size and orientation relationships; reflections reverse orientation; dilations change size but preserve shape.

Time and elapsed-time problems: convert to a single unit (minutes) before adding or subtracting intervals.

Reading a bar or line graph requires checking axis scales and labels; a truncated y-axis exaggerates apparent differences.

In a box plot, the box spans Q1 to Q3 with the median line inside; whiskers extend to the minimum and maximum (or to non-outlier extremes).

Weighted mean = Σ(value × weight) ÷ Σ(weights), used when data values occur with different frequencies.

Counting principle: if one choice has m options and a second has n options, the pair has m × n outcomes.

Permutations count ordered arrangements: nPr = n!/(n − r)!; combinations count unordered selections: nCr = n!/(r!(n − r)!).

Sample space for two independent events is the set of all ordered pairs; list it systematically to compute probabilities.

Mean absolute deviation measures typical distance from the mean; a larger value indicates greater spread.

Multi-step word problems: identify the unknown, translate to an equation, solve, then check the answer against the original context and units.

Fractions with unlike denominators: rewrite using the least common denominator before adding or subtracting.

Mixed numbers: convert to improper fractions before multiplying or dividing; convert back afterward.

Estimation check: if an exact answer differs greatly from a reasonable estimate, recheck the operation or decimal placement.

Percent of a number: convert the percent to a decimal and multiply; finding the whole given a part uses part ÷ percent.

Greatest common factor simplifies fractions; least common multiple finds common denominators.

Ordering rational numbers: convert all to decimals or a common denominator before comparing on the number line.

A graph's x-intercept is where y = 0; the y-intercept is where x = 0; intercepts help sketch a line quickly.

Rate of change from a graph equals rise ÷ run between two labeled points.

A function table with a constant second difference indicates a quadratic relationship, not a linear one.

Domain is the set of allowable inputs; range is the set of resulting outputs.

Solving one-variable equations: perform the same inverse operation on both sides to isolate the variable.

Checking a solution: substitute the value back into the original equation; both sides must be equal.

Proportional relationships graph as straight lines through the origin with constant unit rate.

Angle classification: acute < 90°, right = 90°, obtuse between 90° and 180°, straight = 180°.

A protractor measures angles from 0° along one ray; check which scale matches the ray's direction.

Congruent figures have equal corresponding side lengths and angle measures; similar figures preserve angle measures only.

Composite figures: decompose into rectangles, triangles, or semicircles, compute each area, then add or subtract.

Volume of a rectangular prism = length × width × height; doubling one dimension doubles volume, doubling all three multiplies it by 8.

Nets of a solid show all faces unfolded; a cube net has six congruent squares.

Units of area convert by squaring the linear conversion factor: 1 m = 100 cm, so 1 m² = 10,000 cm².

Histograms group continuous data into intervals; bar graphs compare categorical data with separated bars.

Stem-and-leaf plots preserve individual data values and show shape; read the key to interpret stems and leaves.

Line graphs suit change over time; scatter plots suit paired numerical variables.

Mean is affected by every value; adding a constant to every data value shifts the mean by that constant.

Probability ranges from 0 (impossible) to 1 (certain); a probability near 0.5 indicates an even chance.

With replacement, probabilities stay constant across draws; without replacement, the denominator and numerator change after each draw.

Simulation estimates probability by repeating a random trial many times; more trials generally produce estimates closer to theoretical values.