OST Grade 5 Math cheat sheet

Place value: in a number, each place is 10 times the place to its right; 0.1 is one-tenth of 1, and 0.01 is one-hundredth of 0.1. Reading 4.307 gives 4 ones, 3 tenths, 0 ones in the hundredths place, and 7 thousandths.

Powers of 10: multiplying by 10, 100, or 1,000 shifts each digit left by 1, 2, or 3 places; dividing shifts digits right. So 3.6 × 100 = 360 and 47 ÷ 1,000 = 0.047.

Comparing decimals: compare whole-number parts first, then tenths, hundredths, and so on, place by place. 0.6 > 0.58 because 6 tenths beats 5 tenths, even though 58 has more digits.

Rounding decimals: find the target place, look at the single digit to its right; 5 or more rounds up, 4 or less keeps the digit. 7.286 rounded to the hundredths place is 7.29.

Multi-digit multiplication: 3-digit × 2-digit products can be estimated first (e.g., 412 × 38 ≈ 400 × 40 = 16,000) to check the exact answer for reasonableness.

Long division: divide, multiply, subtract, bring down, repeat. A remainder must be less than the divisor; 4,375 ÷ 6 gives 729 remainder 1, and the quotient rounds to about 729.17.

Order of operations: evaluate parentheses or brackets first, then multiplication and division left to right, then addition and subtraction left to right. 5 + 2 × (9 − 4) = 5 + 2 × 5 = 15.

Numerical expressions: a letter or symbol can stand for an unknown number; writing 3 × n + 4 for "4 more than triple a number" lets you evaluate it once n is known.

Equivalent fractions: multiplying or dividing numerator and denominator by the same nonzero number keeps the value; 2/3 = 8/12. Simplest form uses the greatest common factor.

Adding and subtracting fractions: rewrite with a common denominator, then combine numerators and keep the denominator. 3/4 + 5/6 = 9/12 + 10/12 = 19/12 = 1 7/12.

Multiplying fractions: multiply numerators and multiply denominators; "of" signals multiplication. 2/3 × 3/5 = 6/15 = 2/5, which is less than either factor because both are below 1.

Dividing with unit fractions: dividing by 1/n is the same as multiplying by n. 4 ÷ 1/3 = 12 because 12 groups of 1/3 fit into 4; a whole number divided by a fraction less than 1 gives a larger quotient.

Fraction scaling: multiplying by a fraction greater than 1 enlarges the number, by a fraction less than 1 shrinks it, and by 1 leaves it unchanged - a quick reasonableness check.

Customary units: 12 inches = 1 foot, 3 feet = 1 yard, 5,280 feet = 1 mile, 16 ounces = 1 pound, 4 cups = 1 quart, 4 quarts = 1 gallon. Convert by multiplying or dividing by the unit factor.

Metric units: kilo- means 1,000, centi- means 1/100, milli- means 1/1,000; 1 km = 1,000 m, 1 m = 100 cm, 1 kg = 1,000 g, 1 L = 1,000 mL.

Volume: a right rectangular prism has volume V = length × width × height, also written V = base area × height. A 4 × 3 × 2 prism has volume 24 cubic units.

Coordinate plane: the x-coordinate gives horizontal distance from the origin and the y-coordinate gives vertical distance; point (3, 5) lies 3 right and 5 up. The axes cross at (0, 0).

Line plots: each X above a value on a number line counts one data item; the mean is the sum of all values divided by the count, and it can be a fraction such as 3 1/2.