FAST Grade 7 Math cheat sheet

Adding integers with the same sign: keep the sign and add absolute values. With different signs: subtract the smaller absolute value from the larger and keep the sign of the number with the larger absolute value.

Subtracting integers: a − b = a + (−b). Change the subtraction to addition of the opposite, then apply integer addition rules.

Multiplying or dividing two numbers with the same sign gives a positive result; with different signs the result is negative.

Order of operations: parentheses and grouping symbols, then exponents, then multiplication and division left to right, then addition and subtraction left to right.

Product rule: a^m · a^n = a^(m+n). Quotient rule: a^m ÷ a^n = a^(m−n) for a ≠ 0. Power rule: (a^m)^n = a^(mn).

Any nonzero base to the zero power equals 1: a^0 = 1. A negative exponent gives a reciprocal: a^(−n) = 1/a^n.

A rational number can be written as a fraction p/q where p and q are integers and q ≠ 0. Terminating and repeating decimals are rational.

To add or subtract fractions, rewrite with a common denominator, combine numerators, and simplify. To divide by a fraction, multiply by its reciprocal.

Absolute value |x| is the distance from x to 0 on the number line, so |x| ≥ 0 and |−x| = |x|. On a number line, −5 < −2 because −5 lies farther left.

A ratio compares two quantities; a rate is a ratio comparing quantities with different units, and a unit rate has a denominator of 1.

To find a unit rate, divide the first quantity by the second. Example: 150 miles in 3 hours gives 50 miles per hour.

A proportion states two ratios are equal: a/b = c/d. Cross-multiply to solve: a·d = b·c.

A relationship is proportional if it can be written y = kx with constant k ≠ 0; its graph is a straight line through the origin (0, 0).

The constant of proportionality k equals y/x for any corresponding pair, and the unit rate is that same constant.

Percent means per 100: p% = p/100. To find p% of n, compute (p/100)·n.

Percent change = (new − original)/original × 100%. A positive result is an increase; a negative result is a decrease.

Percent proportion: part/whole = percent/100. Use it to find a missing part, whole, or percent.

To convert between forms: decimal to percent multiply by 100; percent to decimal divide by 100; fraction to percent divide numerator by denominator then multiply by 100.

Combine like terms by adding coefficients of terms with identical variable parts; 3x + 5x = 8x, but 3x + 5 cannot be combined.

Distributive property: a(b + c) = ab + ac. Apply it before combining terms, and watch signs: −2(x − 4) = −2x + 8.

Solving a two-step equation reverses the order of operations: undo addition or subtraction first, then undo multiplication or division.

Whatever operation is applied to one side of an equation must be applied to the other side to keep it balanced.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol: −2x > 6 gives x < −3.

Graphing an inequality on a number line: open circle for < or >, closed circle for ≤ or ≥, with the arrow pointing in the direction of the solution values.

A word problem can be modeled as px + q = r, where p is the rate per unit, q a fixed amount, and r the total.

Check a solution by substituting it back into the original equation or inequality; a true statement confirms the answer.

Terms with the same variable but different exponents, such as x and x², are not like terms and cannot be combined.

Circumference of a circle: C = 2πr or C = πd. Area of a circle: A = πr². Use π ≈ 3.14 or 22/7 unless told otherwise.

The diameter is twice the radius (d = 2r), and the radius is half the diameter.

Area of a triangle: A = ½bh, where b is the base and h is the perpendicular height. Area of a parallelogram: A = bh.

A scale drawing compares drawing length to actual length, such as 1 cm : 5 m. Set up a proportion to find an unknown actual or drawing length.

Surface area of a rectangular prism: SA = 2lw + 2lh + 2wh. Surface area of a cube with edge s: SA = 6s².

Volume of a rectangular prism: V = lwh. Volume of a triangular prism: V = (area of triangular base) × height of prism.

Vertical angles are congruent. Complementary angles sum to 90°, supplementary angles sum to 180°, and angles on a straight line sum to 180°.

When two parallel lines are cut by a transversal, alternate interior angles and corresponding angles are congruent, and same-side interior angles are supplementary.

The sum of the interior angles of any triangle is 180°, and the sum of the interior angles of a quadrilateral is 360°.

A random sample gives every member of the population an equal chance of being selected, which supports valid inferences about the whole population.

A representative sample reflects the population's characteristics. A biased sample, such as surveying only one group, tends to over- or under-represent the population.

Measures of center: mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent value.

Measures of spread: range = maximum − minimum; the mean absolute deviation (MAD) is the average distance of data values from the mean.

The mean is affected by outliers, while the median is resistant to them. Compare mean and median to judge skew.

Probability of an event = number of favorable outcomes ÷ total number of equally likely outcomes, giving a value from 0 to 1.

The probabilities of all outcomes in a sample space sum to 1, so P(not A) = 1 − P(A).

Theoretical probability comes from analyzing equally likely outcomes; experimental probability comes from actual trial results and may differ, especially with few trials.

A uniform probability model assigns equal probability to each outcome; a non-uniform model assigns different probabilities, and the model's probabilities must still sum to 1.

A ratio compares two quantities; a rate is a ratio comparing quantities with different units, and a unit rate has a denominator of 1.

To find a unit rate, divide the first quantity by the second. Example: 150 miles in 3 hours gives 50 miles per hour.

A proportion states two ratios are equal: a/b = c/d. Cross-multiply to solve: a·d = b·c.

A relationship is proportional if it can be written y = kx with constant k ≠ 0; its graph is a straight line through the origin (0, 0).

The constant of proportionality k equals y/x for any corresponding pair, and the unit rate is that same constant.

Percent means per 100: p% = p/100. To find p% of n, compute (p/100)·n.

Percent change = (new − original)/original × 100%. A positive result is an increase; a negative result is a decrease.

Percent proportion: part/whole = percent/100. Use it to find a missing part, whole, or percent.

To convert between forms: decimal to percent multiply by 100; percent to decimal divide by 100; fraction to percent divide numerator by denominator then multiply by 100.

Combine like terms by adding coefficients of terms with identical variable parts; 3x + 5x = 8x, but 3x + 5 cannot be combined.

Distributive property: a(b + c) = ab + ac. Apply it before combining terms, and watch signs: −2(x − 4) = −2x + 8.

Solving a two-step equation reverses the order of operations: undo addition or subtraction first, then undo multiplication or division.

Whatever operation is applied to one side of an equation must be applied to the other side to keep it balanced.

Multiplying or dividing both sides of an inequality by a negative number reverses the inequality symbol: −2x > 6 gives x < −3.

Graphing an inequality on a number line: open circle for < or >, closed circle for ≤ or ≥, with the arrow pointing in the direction of the solution values.

A word problem can be modeled as px + q = r, where p is the rate per unit, q a fixed amount, and r the total.

Check a solution by substituting it back into the original equation or inequality; a true statement confirms the answer.

Terms with the same variable but different exponents, such as x and x², are not like terms and cannot be combined.

Circumference of a circle: C = 2πr or C = πd. Area of a circle: A = πr². Use π ≈ 3.14 or 22/7 unless told otherwise.

The diameter is twice the radius (d = 2r), and the radius is half the diameter.

Area of a triangle: A = ½bh, where b is the base and h is the perpendicular height. Area of a parallelogram: A = bh.

A scale drawing compares drawing length to actual length, such as 1 cm : 5 m. Set up a proportion to find an unknown actual or drawing length.

Surface area of a rectangular prism: SA = 2lw + 2lh + 2wh. Surface area of a cube with edge s: SA = 6s².

Volume of a rectangular prism: V = lwh. Volume of a triangular prism: V = (area of triangular base) × height of prism.

Vertical angles are congruent. Complementary angles sum to 90°, supplementary angles sum to 180°, and angles on a straight line sum to 180°.

When two parallel lines are cut by a transversal, alternate interior angles and corresponding angles are congruent, and same-side interior angles are supplementary.

The sum of the interior angles of any triangle is 180°, and the sum of the interior angles of a quadrilateral is 360°.

A random sample gives every member of the population an equal chance of being selected, which supports valid inferences about the whole population.

A representative sample reflects the population's characteristics. A biased sample, such as surveying only one group, tends to over- or under-represent the population.

Measures of center: mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent value.

Measures of spread: range = maximum − minimum; the mean absolute deviation (MAD) is the average distance of data values from the mean.

The mean is affected by outliers, while the median is resistant to them. Compare mean and median to judge skew.

Probability of an event = number of favorable outcomes ÷ total number of equally likely outcomes, giving a value from 0 to 1.

The probabilities of all outcomes in a sample space sum to 1, so P(not A) = 1 − P(A).

Theoretical probability comes from analyzing equally likely outcomes; experimental probability comes from actual trial results and may differ, especially with few trials.

A uniform probability model assigns equal probability to each outcome; a non-uniform model assigns different probabilities, and the model's probabilities must still sum to 1.