FAST Math Grade 5 cheat sheet

Place value: each position to the left is 10 times the value to its right, so in 4,582,307 the 5 is worth 500,000 and the 3 is worth 300.

Rounding rule: locate the target place, then look at the single digit to its right; 5 or more rounds up, 4 or less keeps the digit and zeros the rest.

Multi-digit multiplication uses partial products: 34 × 26 = 34 × 20 + 34 × 6 = 680 + 204 = 884.

Long division steps repeat in order: divide, multiply, subtract, bring down, then record the remainder as a fraction of the divisor.

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

Decimal place values to the right of the point are tenths, hundredths, thousandths; 0.7 = 0.70, so annexing zeros does not change a decimal's value.

To compare decimals, align the decimal points and compare digit by digit from the left; 0.45 < 0.5 because 4 tenths < 5 tenths.

Multiplying by a power of 10 shifts the decimal point right once per zero; dividing shifts it left, so 3.6 ÷ 100 = 0.036.

A fraction a/b means a parts of size 1/b; the same whole can be named by equivalent fractions because multiplying numerator and denominator by the same nonzero number does not change value.

To add or subtract fractions, rewrite with a common denominator, combine numerators, and keep the denominator: 2/3 + 1/4 = 8/12 + 3/12 = 11/12.

Multiplying fractions multiplies numerators and denominators: 2/3 × 3/5 = 6/15 = 2/5; the product of two fractions less than 1 is smaller than either factor.

Dividing by a fraction equals multiplying by its reciprocal: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8.

A fraction times its reciprocal equals 1, so 5/8 × 8/5 = 1; this is why the invert-and-multiply rule works.

A mixed number converts to an improper fraction by multiplying the whole number by the denominator, adding the numerator, and keeping the denominator.

Scaling intuition: multiplying by a factor greater than 1 enlarges, by a factor between 0 and 1 shrinks, and by 1 leaves the size unchanged.

Fraction–decimal links: 1/2 = 0.5, 1/4 = 0.25, 1/5 = 0.2, 1/10 = 0.1; divide numerator by denominator to convert any fraction to a decimal.

An expression has no equals sign; an equation states two expressions are equal, so only equations can be solved for a value.

A variable is a letter standing for an unknown or changing number; 3n means 3 × n, and 3 + n means three more than n.

Evaluate by substituting the value for the variable and following order of operations: if x = 4, then 5x − 7 = 20 − 7 = 13.

Words to symbols: sum, more than, increased by mean addition; difference, less than, fewer mean subtraction; product, times mean multiplication; quotient, per mean division.

Order matters in subtraction and division: '5 less than n' is n − 5, not 5 − n; 'n divided by 4' is n ÷ 4.

The distributive property: a(b + c) = ab + ac, so 6(20 + 3) = 120 + 18 = 138.

A pattern rule describes how one term relates to its position; in 3, 7, 11, 15 the rule is start at 3 and add 4 each time, giving term n = 4n − 1.

Solve one-step equations by applying the inverse operation to both sides: x + 9 = 14 gives x = 5; 4y = 32 gives y = 8.

Volume of a rectangular prism = length × width × height; a prism 4 by 3 by 5 has volume 60 cubic units.

Volume is additive: a solid made of two prisms has volume equal to the sum of the parts' volumes.

Area of a rectangle = length × width; area of a triangle = 1/2 × base × height, where height is perpendicular to the base.

Perimeter is the distance around a figure; for a rectangle it is 2(length + width).

A unit cube has edge 1 unit and volume 1 cubic unit; filling a prism with unit cubes counts the volume.

Customary conversions: 1 foot = 12 inches, 1 yard = 3 feet, 1 gallon = 4 quarts, 1 pound = 16 ounces.

Metric conversions: 1 kilometer = 1,000 meters, 1 meter = 100 centimeters, 1 kilogram = 1,000 grams, 1 liter = 1,000 milliliters.

The coordinate plane has an x-axis (horizontal) and y-axis (vertical) meeting at the origin (0,0); an ordered pair (3,5) means 3 right and 5 up.

Classify two-dimensional figures by properties: a square is a rectangle and a rhombus, but a rectangle is not always a square.

Mean = sum of values ÷ number of values; median = middle value when ordered; mode = most frequent value; range = greatest minus least.

The median of an even number of values is the mean of the two middle values.

A line plot shows each data value as an X above its position on a number line, so clusters and gaps are visible.

Probability of an event = favorable outcomes ÷ total equally likely outcomes; a probability of 0 is impossible and 1 is certain.

Probabilities of all outcomes in a sample space add to 1, so P(not A) = 1 − P(A).

A fair coin has probability 1/2 for heads on every toss; past results do not change the next toss.

Bar graphs compare categories, line graphs show change over time, and the scale on an axis must be read before comparing heights.

Outliers are values far from the rest; they pull the mean toward them more than they affect the median.

Make sense of a problem by identifying what is given, what is asked, and what units the answer needs.

Estimate before computing: round to friendly numbers to check whether an exact answer is reasonable.

Use a model - tape diagram, number line, area model, or table - to represent the relationships before writing an equation.

Check by substituting the answer back into the original situation or by using inverse operations.

Explain reasoning with a claim, evidence from the computation, and a statement of why the method works.

When a problem has multiple steps, record intermediate results with labels so units stay consistent.

Look for structure: repeated addition is multiplication, and equal groups across many problems signal a rate or unit rate.

Attend to precision: label units, keep decimal points aligned, and state remainders in the form the question requires.