NYS Math Grade 3 cheat sheet

Multiplication a × b means a equal groups of b objects; the total is the product.

Division has two meanings: sharing into equal groups (how many in each group) and measuring (how many groups).

Multiplication and division are inverse operations: if 6 × 4 = 24, then 24 ÷ 6 = 4 and 24 ÷ 4 = 6.

The commutative property of multiplication: a × b = b × a, so 3 × 8 = 8 × 3.

The distributive property: a × (b + c) = (a × b) + (a × c), e.g., 6 × 7 = 6 × 5 + 6 × 2 = 30 + 12 = 42.

An unknown factor is found by dividing: in 4 × n = 32, n = 32 ÷ 4 = 8.

Two-step word problems are solved in order, using parentheses or a sketch to show which operation comes first.

A pattern rule describes how terms change: in 5, 10, 15, 20 the rule is add 5, and the next term is 25.

Rounding to estimate a product or quotient gives a check: 4 × 19 is about 4 × 20 = 80, so the exact answer is near 80.

Place value in a 4-digit number: each place is 10 times the place to its right, so 1 thousand = 10 hundreds = 100 tens = 1,000 ones.

Expanded form: 3,472 = 3,000 + 400 + 70 + 2; word form: three thousand, four hundred seventy-two.

Rounding to the nearest 10: look at the ones digit; 0-4 round down, 5-9 round up (247 rounds to 250).

Rounding to the nearest 100: look at the tens digit; 0-4 round down, 5-9 round up (1,362 rounds to 1,400).

Add and subtract within 1,000 using place value: line up ones under ones, tens under tens, hundreds under hundreds.

Regrouping: 10 ones make 1 ten and 10 tens make 1 hundred; when subtracting, borrow 1 from the next place.

Compatible numbers help estimate sums and differences, e.g., 198 + 403 is about 200 + 400 = 600.

Multiplication by multiples of 10: 3 × 40 = 3 × 4 tens = 12 tens = 120.

A fraction a/b names a parts of a whole cut into b equal parts; the denominator b is the number of equal parts.

On a number line, 1/b is one part when the interval from 0 to 1 is split into b equal parts.

Two fractions are equivalent if they name the same point on a number line: 1/2 = 2/4 = 4/8.

Whole numbers as fractions: 3 = 3/1 = 6/2, and 4/4 = 1.

Compare fractions with the same denominator by numerators: 3/8 < 5/8 because 3 < 5.

Compare fractions with the same numerator by denominators: 1/3 > 1/8 because eighths are smaller pieces.

Fractions that equal one whole have equal numerator and denominator; fractions greater than 1 have a numerator greater than the denominator.

Fraction models must use equal parts; a whole cut into unequal pieces does not show fractions correctly.

Telling time to the nearest minute: the minute hand moves 1 minute per small mark and 5 minutes per number; the hour hand is between numbers.

Elapsed time is found by counting forward or back in minutes: from 2:40 to 3:15 is 20 minutes to 3:00 plus 15 minutes, or 35 minutes.

Liquid volume: 1 liter = 1,000 milliliters; 1 gallon = 4 quarts = 8 pints = 16 cups.

Mass: 1 kilogram = 1,000 grams; 1 gram is about the mass of a paper clip.

A scaled picture graph uses a key, e.g., each picture = 4 books, so 3 pictures show 12 books.

A scaled bar graph: read the height against the scale, then subtract bars to find how many more or fewer.

Area of a rectangle = length × width; area is measured in square units.

Perimeter of a rectangle = 2 × (length + width); perimeter is the distance around the edge in linear units.

A square with side s has area s × s and perimeter 4 × s.

A quadrilateral has 4 sides and 4 angles; a rectangle has 4 right angles, a square has 4 right angles and 4 equal sides.

A rhombus has 4 equal sides; a parallelogram has two pairs of parallel sides; a trapezoid has exactly one pair of parallel sides.

A square is also a rectangle, a rhombus, and a parallelogram, because it satisfies each definition.

Equal shares of the same whole have equal area; a whole can be partitioned into halves, thirds, fourths, sixths, eighths.

A unit fraction 1/b is one equal share when a whole is split into b shares; the larger b, the smaller the share.

Shapes in different categories can share attributes, so a shape can belong to more than one group.

Attributes to compare include number of sides, number of angles, side lengths, and parallel or perpendicular sides.

Multiplication a × b means a equal groups of b objects; the total is the product.

Division has two meanings: sharing into equal groups (how many in each group) and measuring (how many groups).

Multiplication and division are inverse operations: if 6 × 4 = 24, then 24 ÷ 6 = 4 and 24 ÷ 4 = 6.

The commutative property of multiplication: a × b = b × a, so 3 × 8 = 8 × 3.

The distributive property: a × (b + c) = (a × b) + (a × c), e.g., 6 × 7 = 6 × 5 + 6 × 2 = 30 + 12 = 42.

An unknown factor is found by dividing: in 4 × n = 32, n = 32 ÷ 4 = 8.

Two-step word problems are solved in order, using parentheses or a sketch to show which operation comes first.

A pattern rule describes how terms change: in 5, 10, 15, 20 the rule is add 5, and the next term is 25.

Rounding to estimate a product or quotient gives a check: 4 × 19 is about 4 × 20 = 80, so the exact answer is near 80.

Place value in a 4-digit number: each place is 10 times the place to its right, so 1 thousand = 10 hundreds = 100 tens = 1,000 ones.

Expanded form: 3,472 = 3,000 + 400 + 70 + 2; word form: three thousand, four hundred seventy-two.

Rounding to the nearest 10: look at the ones digit; 0-4 round down, 5-9 round up (247 rounds to 250).

Rounding to the nearest 100: look at the tens digit; 0-4 round down, 5-9 round up (1,362 rounds to 1,400).

Add and subtract within 1,000 using place value: line up ones under ones, tens under tens, hundreds under hundreds.

Regrouping: 10 ones make 1 ten and 10 tens make 1 hundred; when subtracting, borrow 1 from the next place.

Compatible numbers help estimate sums and differences, e.g., 198 + 403 is about 200 + 400 = 600.

Multiplication by multiples of 10: 3 × 40 = 3 × 4 tens = 12 tens = 120.

A fraction a/b names a parts of a whole cut into b equal parts; the denominator b is the number of equal parts.

On a number line, 1/b is one part when the interval from 0 to 1 is split into b equal parts.

Two fractions are equivalent if they name the same point on a number line: 1/2 = 2/4 = 4/8.

Whole numbers as fractions: 3 = 3/1 = 6/2, and 4/4 = 1.

Compare fractions with the same denominator by numerators: 3/8 < 5/8 because 3 < 5.

Compare fractions with the same numerator by denominators: 1/3 > 1/8 because eighths are smaller pieces.

Fractions that equal one whole have equal numerator and denominator; fractions greater than 1 have a numerator greater than the denominator.

Fraction models must use equal parts; a whole cut into unequal pieces does not show fractions correctly.

Telling time to the nearest minute: the minute hand moves 1 minute per small mark and 5 minutes per number; the hour hand is between numbers.

Elapsed time is found by counting forward or back in minutes: from 2:40 to 3:15 is 20 minutes to 3:00 plus 15 minutes, or 35 minutes.

Liquid volume: 1 liter = 1,000 milliliters; 1 gallon = 4 quarts = 8 pints = 16 cups.

Mass: 1 kilogram = 1,000 grams; 1 gram is about the mass of a paper clip.

A scaled picture graph uses a key, e.g., each picture = 4 books, so 3 pictures show 12 books.

A scaled bar graph: read the height against the scale, then subtract bars to find how many more or fewer.

Area of a rectangle = length × width; area is measured in square units.

Perimeter of a rectangle = 2 × (length + width); perimeter is the distance around the edge in linear units.

A square with side s has area s × s and perimeter 4 × s.

A quadrilateral has 4 sides and 4 angles; a rectangle has 4 right angles, a square has 4 right angles and 4 equal sides.

A rhombus has 4 equal sides; a parallelogram has two pairs of parallel sides; a trapezoid has exactly one pair of parallel sides.

A square is also a rectangle, a rhombus, and a parallelogram, because it satisfies each definition.

Equal shares of the same whole have equal area; a whole can be partitioned into halves, thirds, fourths, sixths, eighths.

A unit fraction 1/b is one equal share when a whole is split into b shares; the larger b, the smaller the share.

Shapes in different categories can share attributes, so a shape can belong to more than one group.

Attributes to compare include number of sides, number of angles, side lengths, and parallel or perpendicular sides.