8th Grade Math Formula Sheet
Every formula and rule, each with an example
Numbers and exponents
Rational and irrational numbers
WorksheetRational: can be written as a fraction of integers (denominator not 0). The decimal ends or repeats.
Irrational: the decimal never ends and never repeats, so it can’t be written as a fraction, like √2 and π.
Repeating decimal to fraction: x = 0.333…, so 10x = 3.333…, 9x = 3 and x = 13
Example: 0.25 and 0.333… are rational; √5 is irrational
Exponent rules
Worksheeta to the power m × a to the power n = a to the power m + n
a to the power m ÷ a to the power n = a to the power m − n (a ≠ 0)
(a to the power m) to the power n = a to the power mn
(ab) to the power n = a to the power nb to the power n
a to the power 0 = 1 and a to the power −n = 1a to the power n (a ≠ 0)
Example: 2 to the power 3 × 2 to the power 4 = 2 to the power 7 = 128. 5 to the power −2 = 125
Square and cube roots
Worksheet√49 = 7 because 7 to the power 2 = 49
∛64 = 4 because 4 to the power 3 = 64
If x to the power 2 = p (p ≥ 0), x = ±√p. √p means the positive root. If x to the power 3 = p, x = ∛p.
Example: x to the power 2 = 81, so x = 9 or x = −9
Perfect squares and cubes
WorksheetSquares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144
Cubes: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000
Estimating roots
WorksheetFind the perfect squares just below and above the number.
Example: √50 is between √49 = 7 and √64 = 8, close to 7.1
Numbers and exponents (continued)
Scientific notation
Worksheeta × 10 to the power n, where 1 ≤ a < 10 and n is an integer
(a × 10 to the power m) × (b × 10 to the power n) = ab × 10 to the power m + n. If ab ≥ 10, rewrite it: 20 × 10 to the power 5 = 2 × 10 to the power 6.
Example: 45,000 = 4.5 × 10 to the power 4. 0.0032 = 3.2 × 10 to the power −3
Equations
Variables on both sides
WorksheetGet the variable terms on one side and the numbers on the other.
Example: 5x − 4 = 2x + 11, so 3x = 15 and x = 5
How many solutions?
WorksheetOne: you end with x = a number.
None: you end with something false, like 3 = 7.
Infinitely many: you end with something always true, like 4 = 4.
Systems of equations
WorksheetThe solution is the point where the two lines cross: it makes both equations true.
Parallel lines: no solution. The same line: infinitely many.
Example: y = 2x and y = x + 3: 2x = x + 3, so x = 3 and y = 6
Functions and lines
Functions
WorksheetA function gives exactly one output for each input.
Example: (1, 4), (2, 5), (2, 6) is not a function: the input 2 has two outputs
Slope
Worksheetm = riserun = y sub 2 − y sub 1x sub 2 − x sub 1
Up to the right: positive. Down to the right: negative. Flat: 0. Vertical: undefined.
Example: (1, 2) and (3, 8): m = 8 − 23 − 1 = 62 = 3
Functions and lines (continued)
Slope-intercept form
Worksheety = mx + b
m is the slope (rate of change); b is the y-intercept (initial value).
From two points: find m, then put one point into y = mx + b to find b.
Proportional: y = mx (b = 0), and the slope is the unit rate.
Example: y = 2x + 5 starts at 5 and goes up 2 for every 1 across. Through (1, 2) and (3, 8): m = 3, 2 = 3 × 1 + b, so b = −1 and y = 3x − 1
Linear or nonlinear
WorksheetLinear: a constant rate of change and a straight-line graph.
Nonlinear: the rate of change varies, as in y = x to the power 2.
Geometry
Pythagorean theorem
WorksheetIn a right triangle with legs a and b: a to the power 2 + b to the power 2 = c to the power 2 (c is the hypotenuse, across from the right angle)
If a to the power 2 + b to the power 2 = c to the power 2 and c is the longest side, the triangle is a right triangle.
Example: Legs 6 and 8: c = √(36 + 64) = √100 = 10
Distance between two points
Worksheetd = √((x sub 2 − x sub 1) to the power 2 + (y sub 2 − y sub 1) to the power 2)
Example: (1, 2) to (4, 6): d = √(9 + 16) = √25 = 5
Triangle angles
WorksheetThe three angles add up to 180°.
An exterior angle equals the sum of the two interior angles that aren’t next to it.
Example: Angles of 40° and 75°: the third is 65°, and the exterior angle next to it is 115°
Parallel lines cut by a transversal
WorksheetCorresponding angles are equal.
Alternate interior angles are equal, and so are alternate exterior angles.
Same-side interior angles add up to 180°.
Geometry (continued)
Transformations
WorksheetTranslation: (x, y) → (x + a, y + b)
Reflection over the x-axis: (x, y) → (x, −y). Over the y-axis: (x, y) → (−x, y).
Rotation about (0, 0): 90° counterclockwise (x, y) → (−y, x), 180° (x, y) → (−x, −y), 90° clockwise (x, y) → (y, −x)
Dilation by scale factor k from (0, 0): (x, y) → (kx, ky)
Congruent and similar
WorksheetCongruent: same shape and size (after translations, reflections and rotations).
Similar: same shape, maybe a different size (dilations too). Matching angles are equal and matching sides are proportional.
Two triangles with two pairs of equal angles are similar (AA).
Volume
WorksheetCylinder: V = πr to the power 2h
Cone: V = 13πr to the power 2h
Sphere: V = 43πr to the power 3
Example: Cylinder with r = 3 and h = 10: V = π × 9 × 10 = 90π ≈ 282.7 cubic units
Statistics
Scatter plots
WorksheetPositive association: y tends to rise as x rises. Negative: y tends to fall. No association: no pattern.
Look for clusters and outliers (points far from the rest), and whether the pattern is a line (linear) or a curve.
A trend line runs through the middle of the points, with about as many above it as below.
Two-way tables
Worksheetrelative frequency = countrow or column total
Example: 12 of the 30 seventh graders have a pet: 1230 = 40%