MTEL 65 cheat sheet

Divisibility tests: by 3 or 9, test the digit sum; by 4, test the last two digits; by 5, test the last digit 0 or 5; by 11, test whether the alternating digit sum is divisible by 11.

For positive integers a and b, gcd(a,b) × lcm(a,b) = a × b; using prime factorizations, use the lowest shared prime powers for GCD and the highest prime powers for LCM.

Exponent laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n for a ≠ 0, and a^0 = 1 for a ≠ 0.

A rational number can be written as p/q where p and q are integers and q ≠ 0; its decimal form terminates or eventually repeats, while an irrational decimal is nonterminating and nonrepeating.

Order of operations: evaluate grouping symbols first, then exponents and radicals, then multiplication and division from left to right, then addition and subtraction from left to right.

Percent change = ((new value − old value) ÷ old value) × 100%; successive percent changes multiply by factors such as 1.20 and 0.85 rather than adding percentages.

Fraction division: a/b ÷ c/d = a/b × d/c for b, c, d ≠ 0; division by a fraction is multiplication by its reciprocal.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiply by multiplying coefficients and adding exponents, and divide by dividing coefficients and subtracting exponents.

Direct variation has the form y = kx, so y/x is constant; inverse variation has the form y = k/x, so xy is constant for nonzero x and y.

Slope between distinct points (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1); point-slope form is y−y1 = m(x−x1), and slope-intercept form is y = mx+b.

Parallel nonvertical lines have equal slopes; perpendicular nonvertical lines have slopes whose product is −1, so their slopes are negative reciprocals.

For ax^2+bx+c=0 with a ≠ 0, x = (−b ± √(b^2−4ac))/(2a); the discriminant b^2−4ac is positive for two real roots, zero for one repeated real root, and negative for no real roots.

The parabola y = ax^2+bx+c has vertex x-coordinate x = −b/(2a); substituting this x gives the maximum or minimum y-value.

Exponential growth or decay is y = a(1±r)^t; compound interest is A = P(1+r/n)^(nt), and continuous compounding is A = Pe^(rt).

A relation is a function when each input has exactly one output and passes the vertical line test; exclude denominator zeros and require even-root radicands to be nonnegative and logarithm arguments to be positive.

A linear system has one solution when slopes differ, no solution when slopes are equal and intercepts differ, and infinitely many solutions when the equations represent the same line.

Absolute value |x−a| = b means x = a+b or x = a−b when b ≥ 0, and it has no solution when b < 0; |x| is distance from 0 on a number line.

Logarithm definition: log_b x = y exactly when b^y = x for x > 0, b > 0, and b ≠ 1; log_b(xy) = log_b x + log_b y and log_b(x/y) = log_b x − log_b y.

Circle formulas: circumference C = 2πr = πd and area A = πr^2; arc length = (θ/360)(2πr) and sector area = (θ/360)(πr^2) when θ is in degrees.

Triangle area = (1/2)bh; the Pythagorean theorem states a^2+b^2 = c^2 for a right triangle with hypotenuse c; 45-45-90 sides are in ratio 1:1:√2, and 30-60-90 sides are in ratio 1:√3:2.

A polygon with n sides has interior angles summing to (n−2)180°; a regular polygon has each interior angle equal to (n−2)180°/n, and any convex polygon has exterior angles summing to 360°.

Volume formulas: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr^3; sphere surface area = 4πr^2.

For similar figures with scale factor k, corresponding lengths have ratio k, areas have ratio k^2, and volumes have ratio k^3.

Coordinate geometry: distance = √((x2−x1)^2+(y2−y1)^2), midpoint = ((x1+x2)/2,(y1+y2)/2), and a circle with center (h,k) and radius r has equation (x−h)^2+(y−k)^2 = r^2.

Angle pairs: complementary angles sum to 90° and supplementary angles sum to 180°; vertical angles are equal, and alternate interior angles are equal when lines are parallel.

Unit conversion by dimensional analysis multiplies by ratios equal to 1, such as 1 in = 2.54 cm, 1 L = 1000 cm^3, and 1 ft^3 = 1728 in^3.

Mean = sum of values ÷ number of values; median is the middle ordered value or average of the two middle values; mode is the most frequent value; range = maximum − minimum.

Interquartile range IQR = Q3−Q1; a common outlier rule flags values below Q1−1.5(IQR) or above Q3+1.5(IQR).

Standard deviation measures typical distance from the mean and equals the square root of variance; adding a constant c to every value adds c to the mean and median but leaves standard deviation and IQR unchanged.

Multiplying every data value by c multiplies the mean and median by c and multiplies standard deviation and IQR by |c|; the median and IQR resist outliers more than the mean and standard deviation.

Probability satisfies 0 ≤ P(E) ≤ 1; P(not E) = 1−P(E); P(A or B) = P(A)+P(B)−P(A and B), subtracting the overlap to avoid double counting.

Conditional probability P(A|B) = P(A and B)/P(B) for P(B) > 0; events A and B are independent exactly when P(A and B) = P(A)P(B).

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

In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3; z = (x−μ)/σ standardizes a value.

Correlation coefficient r lies between −1 and 1 and measures strength and direction of linear association; r near 0 indicates weak linear association, and correlation does not imply causation.

Random sampling reduces selection bias; increasing sample size reduces random variability and margin of error, but it does not correct bias caused by a flawed sampling method.

Bar charts display categorical data, histograms display quantitative data in bins, and skewness pulls the mean toward the longer tail while the median remains more resistant.

Divisibility tests: by 3 or 9, test the digit sum; by 4, test the last two digits; by 5, test the last digit 0 or 5; by 11, test whether the alternating digit sum is divisible by 11.

For positive integers a and b, gcd(a,b) × lcm(a,b) = a × b; using prime factorizations, use the lowest shared prime powers for GCD and the highest prime powers for LCM.

Exponent laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n for a ≠ 0, and a^0 = 1 for a ≠ 0.

A rational number can be written as p/q where p and q are integers and q ≠ 0; its decimal form terminates or eventually repeats, while an irrational decimal is nonterminating and nonrepeating.

Order of operations: evaluate grouping symbols first, then exponents and radicals, then multiplication and division from left to right, then addition and subtraction from left to right.

Percent change = ((new value − old value) ÷ old value) × 100%; successive percent changes multiply by factors such as 1.20 and 0.85 rather than adding percentages.

Fraction division: a/b ÷ c/d = a/b × d/c for b, c, d ≠ 0; division by a fraction is multiplication by its reciprocal.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiply by multiplying coefficients and adding exponents, and divide by dividing coefficients and subtracting exponents.

Direct variation has the form y = kx, so y/x is constant; inverse variation has the form y = k/x, so xy is constant for nonzero x and y.

Slope between distinct points (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1); point-slope form is y−y1 = m(x−x1), and slope-intercept form is y = mx+b.

Parallel nonvertical lines have equal slopes; perpendicular nonvertical lines have slopes whose product is −1, so their slopes are negative reciprocals.

For ax^2+bx+c=0 with a ≠ 0, x = (−b ± √(b^2−4ac))/(2a); the discriminant b^2−4ac is positive for two real roots, zero for one repeated real root, and negative for no real roots.

The parabola y = ax^2+bx+c has vertex x-coordinate x = −b/(2a); substituting this x gives the maximum or minimum y-value.

Exponential growth or decay is y = a(1±r)^t; compound interest is A = P(1+r/n)^(nt), and continuous compounding is A = Pe^(rt).

A relation is a function when each input has exactly one output and passes the vertical line test; exclude denominator zeros and require even-root radicands to be nonnegative and logarithm arguments to be positive.

A linear system has one solution when slopes differ, no solution when slopes are equal and intercepts differ, and infinitely many solutions when the equations represent the same line.

Absolute value |x−a| = b means x = a+b or x = a−b when b ≥ 0, and it has no solution when b < 0; |x| is distance from 0 on a number line.

Logarithm definition: log_b x = y exactly when b^y = x for x > 0, b > 0, and b ≠ 1; log_b(xy) = log_b x + log_b y and log_b(x/y) = log_b x − log_b y.

Circle formulas: circumference C = 2πr = πd and area A = πr^2; arc length = (θ/360)(2πr) and sector area = (θ/360)(πr^2) when θ is in degrees.

Triangle area = (1/2)bh; the Pythagorean theorem states a^2+b^2 = c^2 for a right triangle with hypotenuse c; 45-45-90 sides are in ratio 1:1:√2, and 30-60-90 sides are in ratio 1:√3:2.

A polygon with n sides has interior angles summing to (n−2)180°; a regular polygon has each interior angle equal to (n−2)180°/n, and any convex polygon has exterior angles summing to 360°.

Volume formulas: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr^3; sphere surface area = 4πr^2.

For similar figures with scale factor k, corresponding lengths have ratio k, areas have ratio k^2, and volumes have ratio k^3.

Coordinate geometry: distance = √((x2−x1)^2+(y2−y1)^2), midpoint = ((x1+x2)/2,(y1+y2)/2), and a circle with center (h,k) and radius r has equation (x−h)^2+(y−k)^2 = r^2.

Angle pairs: complementary angles sum to 90° and supplementary angles sum to 180°; vertical angles are equal, and alternate interior angles are equal when lines are parallel.

Unit conversion by dimensional analysis multiplies by ratios equal to 1, such as 1 in = 2.54 cm, 1 L = 1000 cm^3, and 1 ft^3 = 1728 in^3.

Mean = sum of values ÷ number of values; median is the middle ordered value or average of the two middle values; mode is the most frequent value; range = maximum − minimum.

Interquartile range IQR = Q3−Q1; a common outlier rule flags values below Q1−1.5(IQR) or above Q3+1.5(IQR).

Standard deviation measures typical distance from the mean and equals the square root of variance; adding a constant c to every value adds c to the mean and median but leaves standard deviation and IQR unchanged.

Multiplying every data value by c multiplies the mean and median by c and multiplies standard deviation and IQR by |c|; the median and IQR resist outliers more than the mean and standard deviation.

Probability satisfies 0 ≤ P(E) ≤ 1; P(not E) = 1−P(E); P(A or B) = P(A)+P(B)−P(A and B), subtracting the overlap to avoid double counting.

Conditional probability P(A|B) = P(A and B)/P(B) for P(B) > 0; events A and B are independent exactly when P(A and B) = P(A)P(B).

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

In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3; z = (x−μ)/σ standardizes a value.

Correlation coefficient r lies between −1 and 1 and measures strength and direction of linear association; r near 0 indicates weak linear association, and correlation does not imply causation.

Random sampling reduces selection bias; increasing sample size reduces random variability and margin of error, but it does not correct bias caused by a flawed sampling method.

Bar charts display categorical data, histograms display quantitative data in bins, and skewness pulls the mean toward the longer tail while the median remains more resistant.

Divisibility tests: by 3 or 9, test the digit sum; by 4, test the last two digits; by 5, test the last digit 0 or 5; by 11, test whether the alternating digit sum is divisible by 11.

For positive integers a and b, gcd(a,b) × lcm(a,b) = a × b; using prime factorizations, use the lowest shared prime powers for GCD and the highest prime powers for LCM.

Exponent laws: a^m × a^n = a^(m+n), a^m ÷ a^n = a^(m−n), (a^m)^n = a^(mn), a^(−n) = 1/a^n for a ≠ 0, and a^0 = 1 for a ≠ 0.

A rational number can be written as p/q where p and q are integers and q ≠ 0; its decimal form terminates or eventually repeats, while an irrational decimal is nonterminating and nonrepeating.

Order of operations: evaluate grouping symbols first, then exponents and radicals, then multiplication and division from left to right, then addition and subtraction from left to right.

Percent change = ((new value − old value) ÷ old value) × 100%; successive percent changes multiply by factors such as 1.20 and 0.85 rather than adding percentages.

Fraction division: a/b ÷ c/d = a/b × d/c for b, c, d ≠ 0; division by a fraction is multiplication by its reciprocal.

Scientific notation writes a number as a × 10^n with 1 ≤ |a| < 10; multiply by multiplying coefficients and adding exponents, and divide by dividing coefficients and subtracting exponents.

Direct variation has the form y = kx, so y/x is constant; inverse variation has the form y = k/x, so xy is constant for nonzero x and y.

Slope between distinct points (x1,y1) and (x2,y2) is m = (y2−y1)/(x2−x1); point-slope form is y−y1 = m(x−x1), and slope-intercept form is y = mx+b.

Parallel nonvertical lines have equal slopes; perpendicular nonvertical lines have slopes whose product is −1, so their slopes are negative reciprocals.

For ax^2+bx+c=0 with a ≠ 0, x = (−b ± √(b^2−4ac))/(2a); the discriminant b^2−4ac is positive for two real roots, zero for one repeated real root, and negative for no real roots.

The parabola y = ax^2+bx+c has vertex x-coordinate x = −b/(2a); substituting this x gives the maximum or minimum y-value.

Exponential growth or decay is y = a(1±r)^t; compound interest is A = P(1+r/n)^(nt), and continuous compounding is A = Pe^(rt).

A relation is a function when each input has exactly one output and passes the vertical line test; exclude denominator zeros and require even-root radicands to be nonnegative and logarithm arguments to be positive.

A linear system has one solution when slopes differ, no solution when slopes are equal and intercepts differ, and infinitely many solutions when the equations represent the same line.

Absolute value |x−a| = b means x = a+b or x = a−b when b ≥ 0, and it has no solution when b < 0; |x| is distance from 0 on a number line.

Logarithm definition: log_b x = y exactly when b^y = x for x > 0, b > 0, and b ≠ 1; log_b(xy) = log_b x + log_b y and log_b(x/y) = log_b x − log_b y.

Circle formulas: circumference C = 2πr = πd and area A = πr^2; arc length = (θ/360)(2πr) and sector area = (θ/360)(πr^2) when θ is in degrees.

Triangle area = (1/2)bh; the Pythagorean theorem states a^2+b^2 = c^2 for a right triangle with hypotenuse c; 45-45-90 sides are in ratio 1:1:√2, and 30-60-90 sides are in ratio 1:√3:2.

A polygon with n sides has interior angles summing to (n−2)180°; a regular polygon has each interior angle equal to (n−2)180°/n, and any convex polygon has exterior angles summing to 360°.

Volume formulas: prism or cylinder V = Bh; pyramid or cone V = (1/3)Bh; sphere V = (4/3)πr^3; sphere surface area = 4πr^2.

For similar figures with scale factor k, corresponding lengths have ratio k, areas have ratio k^2, and volumes have ratio k^3.

Coordinate geometry: distance = √((x2−x1)^2+(y2−y1)^2), midpoint = ((x1+x2)/2,(y1+y2)/2), and a circle with center (h,k) and radius r has equation (x−h)^2+(y−k)^2 = r^2.

Angle pairs: complementary angles sum to 90° and supplementary angles sum to 180°; vertical angles are equal, and alternate interior angles are equal when lines are parallel.

Unit conversion by dimensional analysis multiplies by ratios equal to 1, such as 1 in = 2.54 cm, 1 L = 1000 cm^3, and 1 ft^3 = 1728 in^3.

Mean = sum of values ÷ number of values; median is the middle ordered value or average of the two middle values; mode is the most frequent value; range = maximum − minimum.

Interquartile range IQR = Q3−Q1; a common outlier rule flags values below Q1−1.5(IQR) or above Q3+1.5(IQR).

Standard deviation measures typical distance from the mean and equals the square root of variance; adding a constant c to every value adds c to the mean and median but leaves standard deviation and IQR unchanged.

Multiplying every data value by c multiplies the mean and median by c and multiplies standard deviation and IQR by |c|; the median and IQR resist outliers more than the mean and standard deviation.

Probability satisfies 0 ≤ P(E) ≤ 1; P(not E) = 1−P(E); P(A or B) = P(A)+P(B)−P(A and B), subtracting the overlap to avoid double counting.

Conditional probability P(A|B) = P(A and B)/P(B) for P(B) > 0; events A and B are independent exactly when P(A and B) = P(A)P(B).

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

In a normal distribution, about 68% of values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3; z = (x−μ)/σ standardizes a value.

Correlation coefficient r lies between −1 and 1 and measures strength and direction of linear association; r near 0 indicates weak linear association, and correlation does not imply causation.

Random sampling reduces selection bias; increasing sample size reduces random variability and margin of error, but it does not correct bias caused by a flawed sampling method.

Bar charts display categorical data, histograms display quantitative data in bins, and skewness pulls the mean toward the longer tail while the median remains more resistant.