6th Grade Math Formula Sheet
Every formula and rule, each with an example
Ratios, rates and percents
Ratio
Worksheeta to b = a : b = ab (b ≠ 0)
Example: 3 cats to 5 dogs is 3 : 5. The fraction of the animals that are cats is 38.
Equivalent ratios
WorksheetMultiply or divide both numbers by the same nonzero number.
Example: 2 : 3 = 4 : 6 = 10 : 15
Unit rate
Worksheetunit rate = total ÷ number of units
Example: $6 for 4 notebooks: 6 ÷ 4 = $1.50 per notebook
Percent
WorksheetPercent means per 100: 35% = 35100 = 0.35
part = percent (as a decimal) × whole
whole = part ÷ percent (as a decimal)
percent = partwhole × 100
Example: 25% of 80 = 0.25 × 80 = 20. 15 is 30% of 15 ÷ 0.3 = 50
Converting units
1 ft = 12 in, 1 yd = 3 ft, 1 mi = 5,280 ft
1 lb = 16 oz, 1 gal = 4 qt, 1 qt = 2 pt, 1 pt = 2 cups
1 km = 1,000 m, 1 m = 100 cm, 1 kg = 1,000 g, 1 L = 1,000 mL
Example: 4 yd = 4 × 3 = 12 ft
Numbers
Dividing fractions
Worksheetab ÷ cd = ab × dc
Multiply by the reciprocal (flip the second fraction).
Example: 34 ÷ 12 = 34 × 21 = 64 = 1 12
Numbers (continued)
Decimals
WorksheetAdd or subtract: line up the decimal points.
Multiply: multiply as whole numbers. The answer has as many decimal places as both numbers together.
Example: 0.6 × 0.04 = 0.024 (6 × 4 = 24, with 1 + 2 = 3 decimal places)
GCF and LCM
WorksheetGCF: the greatest factor both numbers share.
LCM: the smallest positive multiple both numbers share.
Factor out the GCF: 36 + 8 = 4(9 + 2)
Example: 12 and 18: GCF = 6, LCM = 36
Absolute value
Worksheet|a| = the distance from a to 0, so it is never negative.
Example: |−7| = 7 and |7| = 7
Negative numbers
WorksheetOn a number line, numbers get bigger to the right.
The opposite of a is −a, and −(−a) = a.
Example: −8 < −3, and the opposite of −5 is 5
Coordinate plane
Worksheet(x, y): x is across, y is up or down.
Quadrants: I (+, +), II (−, +), III (−, −), IV (+, −)
Reflect over the x-axis: (x, −y). Over the y-axis: (−x, y).
Distance on a grid line
WorksheetSame x (or same y): if the other coordinates have the same sign, subtract their absolute values; if the signs differ, add them.
Example: (2, 5) to (2, −3): 5 + 3 = 8 units
Expressions and equations
Exponents
Worksheeta to the power n = a × a × … × a (n times)
Example: 2 to the power 3 = 2 × 2 × 2 = 8
Expressions and equations (continued)
Order of operations
WorksheetParentheses, then exponents, then × and ÷ (left to right), then + and − (left to right)
Example: 3 + 4 × 2 to the power 2 = 3 + 4 × 4 = 3 + 16 = 19
Writing expressions
Worksheetsum +, difference −, product ×, quotient ÷
“5 more than n” is n + 5; “5 less than n” is n − 5; “3 times x” is 3x
Parts of an expression
WorksheetIn 4x + 7: 4x is a term, 4 is the coefficient, x is the variable and 7 is the constant.
To evaluate, put the number in for the variable, then follow the order of operations.
Example: 4x + 7 when x = 3: 4 × 3 + 7 = 19
One-step equations
WorksheetDo the opposite operation to both sides.
Example: x + 7 = 12, so x = 12 − 7 = 5. 4x = 28, so x = 28 ÷ 4 = 7
Inequalities
Worksheet< less than, > greater than, ≤ less than or equal to (at most), ≥ greater than or equal to (at least)
Number line: open circle for < and >, closed circle for ≤ and ≥.
Example: x ≥ 3: closed circle at 3, shade to the right
Dependent and independent variables
WorksheetThe dependent variable (y) depends on the independent variable (x).
Example: Tickets cost $5 each: the total cost y = 5x depends on the number of tickets x
Geometry
Area
WorksheetRectangle: A = lw
Parallelogram: A = bh
Triangle: A = 12bh
Trapezoid: A = 12(b sub 1 + b sub 2)h
The height h meets the base at a right angle, not along a slanted side.
Example: Triangle with b = 10 and h = 6: A = 12 × 10 × 6 = 30 square units
Geometry (continued)
Volume of a rectangular prism
WorksheetV = lwh = Bh (B is the area of the base). It works for fraction edge lengths too.
Example: l = 4, w = 3, h = 2: V = 4 × 3 × 2 = 24 cubic units
Surface area
WorksheetAdd the areas of all the faces (a net shows them).
Rectangular prism: SA = 2lw + 2lh + 2wh
Example: 4 by 3 by 2: SA = 24 + 16 + 12 = 52 square units
Statistics
Mean, median and mode
Worksheetmean = sum of the valuesnumber of values
Median: the middle value in order (with two middle values, their mean).
Mode: the value that appears most often.
Example: 2, 4, 4, 10: mean 20 ÷ 4 = 5, median 4, mode 4
Range and MAD
Worksheetrange = greatest − least
MAD (mean absolute deviation): the mean of the distances from each value to the mean.
Example: 2, 4, 4, 10 (mean 5): range 8, MAD = (3 + 1 + 1 + 5) ÷ 4 = 2.5
Box plots
WorksheetFive-number summary: minimum, Q1, median, Q3, maximum
Q1 is the median of the lower half of the data; Q3 is the median of the upper half.
IQR (interquartile range) = Q3 − Q1
Example: Q1 = 4 and Q3 = 10: IQR = 10 − 4 = 6
Statistical questions
WorksheetA statistical question expects answers that vary.
Example: “How tall are the students in my class?” is one. “How tall am I?” is not.