NY Grade 5 Math cheat sheet

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

A number like 3(4 + 5) means 3 × (4 + 5); a letter or parenthesis directly beside a number signals multiplication.

Words that signal operations: sum, more than, increased by → addition; difference, fewer than, less than → subtraction; product, times, twice → multiplication; quotient, shared equally, per → division.

Numerical expression = numbers and operations only (no equals sign); equation = two expressions joined by =; inequality uses <, >, ≤, or ≥.

Evaluate an expression by substituting a value for the variable and simplifying, e.g. 4x + 7 when x = 5 gives 4(5) + 7 = 27.

Two rules generate a pattern: a recursive rule tells how to get the next term (add 6); an explicit rule gives any term from its position (term n = 6n − 1).

In a table of paired values, check the change per step (rate) and the starting value; constant rate means equal differences between consecutive outputs.

A rule like y = 3x + 2 can be tested by substituting each x from a table and confirming the matching y.

Place values to the right of the decimal point: tenths, hundredths, thousandths; each place is 1/10 of the place to its left.

A digit in one place represents 10 times what it would represent in the place to its right, e.g. 0.7 = 7 tenths = 70 hundredths.

Powers of 10: multiplying by 10^n moves the decimal point n places right; dividing by 10^n moves it n places left.

0.4 × 10^3 = 400 and 0.4 ÷ 10^2 = 0.004; exponents on 10 count zeros in the whole-number form.

Read and write decimals in three forms: standard (3.047), word (three and forty-seven thousandths), expanded (3 + 0.04 + 0.007).

Compare decimals by aligning place values: compare whole parts first, then tenths, hundredths, and so on; 0.6 > 0.58 because 6 tenths > 5 tenths.

Round decimals by finding the target place, then looking at the digit to its right: 5 or more rounds up, 4 or less keeps the digit.

Multi-digit whole-number multiplication: partial products aligned by place value; long division follows divide, multiply, subtract, bring down, and record the remainder.

A fraction a/b means a parts of size 1/b; equivalent fractions come from multiplying or dividing numerator and denominator by the same nonzero number.

Add or subtract fractions with unlike denominators by rewriting with the least common denominator, then combining numerators only.

Multiplying a fraction by a whole number: (a/b) × n = (a × n)/b; a whole number n equals n/1.

Multiplying fraction by fraction: multiply numerators and multiply denominators; the product of two fractions less than 1 is smaller than either factor.

Dividing by a unit fraction 1/b is the same as multiplying by b: 4 ÷ 1/5 = 4 × 5 = 20.

Dividing a unit fraction by a whole number makes a smaller piece: 1/3 ÷ 4 = 1/12.

Scaling: multiplying by a factor greater than 1 enlarges, by a factor between 0 and 1 shrinks, and by exactly 1 keeps the size.

A fraction word problem's answer should be checked for reasonableness: dividing by a fraction less than 1 gives a quotient larger than the dividend.

Mixed numbers: convert to improper fractions before multiplying or dividing, e.g. 2 1/3 = 7/3.

Customary length conversions: 1 foot = 12 inches, 1 yard = 3 feet, 1 mile = 5,280 feet.

Customary capacity and weight: 1 cup = 8 fluid ounces, 1 pint = 2 cups, 1 quart = 2 pints, 1 gallon = 4 quarts; 1 pound = 16 ounces.

Metric conversions are powers of 10: 1 km = 1,000 m, 1 m = 100 cm, 1 m = 1,000 mm, 1 kg = 1,000 g, 1 L = 1,000 mL.

To convert to a smaller unit, multiply; to convert to a larger unit, divide.

Line plot with fractional data: each X marks one data value; find the total by adding all values, and the fair share by dividing the total by the number of X's.

Volume of a rectangular prism: V = length × width × height, or V = area of the base × height; units are cubic (cm³, in³).

Volume is additive: a solid made of two non-overlapping prisms has volume equal to the sum of the two prism volumes.

Coordinate plane: the x-axis is horizontal, the y-axis is vertical; an ordered pair (x, y) gives horizontal distance first, then vertical.

Two-dimensional figures form a hierarchy: every square is a rectangle and a rhombus, but not every rectangle is a square.

Quadrilateral categories: trapezoid has at least one pair of parallel sides; parallelogram has two pairs; rectangle has four right angles; rhombus has four equal sides.

Triangles classify by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse); angles sum to 180°.

Quadrilateral angles sum to 360°.

Coordinate plane rules: a point reflected over the x-axis changes (x, y) to (x, −y); over the y-axis it becomes (−x, y).

A translation slides a figure without turning or flipping it; the shape and size stay the same, so side lengths and angles are preserved.

A prism's faces are polygons, its edges are where faces meet, and its vertices are corners; a rectangular prism has 6 faces, 12 edges, 8 vertices.

A cube is a rectangular prism with all edges equal; its volume is s × s × s.

Order of operations: grouping symbols first, then multiplication/division left to right, then addition/subtraction left to right.

A letter stands for an unknown number; evaluate by substitution, e.g. 3n + 2 at n = 4 equals 14.

Words to symbols: '5 less than x' is x − 5, not 5 − x.

Pattern rules can be recursive (next term) or explicit (any term from position n).

Each place value is 10 times the place to its right, including decimals.

Multiplying by 10 moves the decimal one place right; dividing by 10 moves it one place left.

To compare decimals, compare digits place by place from left to right.

Expanded form of 4.206 is 4 + 0.2 + 0.006.

To add fractions with unlike denominators, use a common denominator first.

Multiplying fractions: numerator × numerator over denominator × denominator.

Dividing by 1/b is the same as multiplying by b.

Multiplying by a fraction less than 1 makes the product smaller than the starting number.

Volume of a rectangular prism = length × width × height.

12 inches = 1 foot; 3 feet = 1 yard; 1,000 millimeters = 1 meter.

On a line plot, each X represents one data value.

In an ordered pair (x, y), x is the horizontal distance and y is the vertical distance.

The interior angles of any triangle add to 180°.

The interior angles of any quadrilateral add to 360°.

A square is a special rectangle and a special rhombus.

Reflecting over the x-axis changes the sign of the y-coordinate.