STAAR Grade 6 Math

Whole numbers are the set of numbers beginning with zero and continuing infinitely: 0, 1, 2, 3, 4, and so on. These numbers have no fractional or decimal parts. Whole numbers form the foundation of all mathematical operations and appear in virtually every area of mathematics.

Place value determines the value of each digit based on its position within a number. In our base-ten system, each position represents a power of ten. Moving from right to left, the positions are ones, tens, hundreds, thousands, ten thousands, hundred thousands, millions, and beyond. For example, in the number 5,432,187, the digit 5 represents 5,000,000, the digit 4 represents 400,000, the digit 3 represents 30,000, the digit 2 represents 2,000, the digit 1 represents 100, the digit 8 represents 80, and the digit 7 represents 7.

Understanding place value enables students to read, write, compare, and round numbers accurately. When comparing whole numbers with the same number of digits, compare the digits from left to right. The first position where the digits differ determines which number is greater. When numbers have different numbers of digits, the number with more digits is greater (assuming no leading zeros).

Addition combines two or more numbers to find their total or sum. The numbers being added are called addends, and the result is called the sum. Addition follows two important properties: the commutative property states that changing the order of addends does not change the sum (7 + 3 = 3 + 7), and the associative property states that changing the grouping of addends does not change the sum ((2 + 5) + 8 = 2 + (5 + 8)).

When adding multi-digit numbers, align the digits by place value and add column by column from right to left. When the sum of a column exceeds 9, carry the tens digit to the next column. For example, adding 2,847 and 1,396: start with 7 + 6 = 13, write 3 and carry 1; then 4 + 9 + 1 = 14, write 4 and carry 1; then 8 + 3 + 1 = 12, write 2 and carry 1; finally 2 + 1 + 1 = 4. The sum is 4,243.

The identity property of addition states that any number plus zero equals that number. Zero is called the additive identity. This property is useful when solving equations and simplifying expressions.

Subtraction finds the difference between two numbers. The number being subtracted from is called the minuend, the number being subtracted is called the subtrahend, and the result is called the difference. Unlike addition, subtraction is neither commutative nor associative.

When subtracting multi-digit numbers, align digits by place value and subtract column by column from right to left. When a digit in the minuend is smaller than the corresponding digit in the subtrahend, regroup (borrow) from the next higher place value. For example, in 5,023 − 2,758: starting from the ones place, 3 cannot subtract 8, so borrow from tens making it 13 − 8 = 5; in tens, now 1 cannot subtract 5, so borrow from hundreds making it 11 − 5 = 6; in hundreds, now 9 cannot subtract 7, so borrow from thousands making it 19 − 7 = 12, but we write 2 and carry; wait—let me recalculate properly. In 5,023 − 2,758: ones place, borrow to get 13 − 8 = 5; tens place becomes 1 (after borrowing), need to borrow again to get 11 − 5 = 6; hundreds place becomes 9 (after borrowing), 9 − 7 = 2; thousands place becomes 4, 4 − 2 = 2. The difference is 2,265.

Subtraction and addition are inverse operations. This relationship allows checking subtraction by adding the difference to the subtrahend to verify the result equals the minuend.

Multiplication is repeated addition. Multiplying 4 × 6 means adding 6 four times: 6 + 6 + 6 + 6 = 24. The numbers being multiplied are called factors, and the result is called the product.

Multiplication has several important properties. The commutative property states that changing the order of factors does not change the product (5 × 9 = 9 × 5). The associative property states that changing the grouping of factors does not change the product ((3 × 4) × 2 = 3 × (4 × 2)). The identity property states that any number multiplied by 1 equals that number; 1 is the multiplicative identity. The zero property states that any number multiplied by zero equals zero. The distributive property connects multiplication and addition: a × (b + c) = (a × b) + (a × c). This property is especially useful for mental math and algebraic manipulation.

Multi-digit multiplication requires multiplying each digit of one factor by each digit of the other factor, considering place values. For 234 × 56: first multiply 234 × 6 = 1,404; then multiply 234 × 50 = 11,700; finally add 1,404 + 11,700 = 13,104.

Division splits a number into equal groups or finds how many times one number fits into another. The number being divided is the dividend, the number dividing is the divisor, and the result is the quotient. Any amount remaining after division is the remainder.

Division is the inverse operation of multiplication. This relationship helps verify division: quotient × divisor + remainder = dividend. For example, 47 ÷ 5 = 9 remainder 2 because 9 × 5 + 2 = 47.

Long division breaks complex division into manageable steps. For 7,284 ÷ 12: determine how many times 12 goes into 72 (6 times, since 12 × 6 = 72); bring down 8 to get 08; 12 goes into 8 zero times with remainder 8; bring down 4 to get 84; 12 goes into 84 exactly 7 times (12 × 7 = 84). The quotient is 607.

When expressions contain multiple operations, follow the standard order of operations to evaluate correctly. The acronym PEMDAS helps remember the order: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right).

Parentheses and other grouping symbols (brackets, braces, fraction bars) take highest priority. Evaluate expressions inside grouping symbols first, working from innermost to outermost. Exponents come next, representing repeated multiplication.

Multiplication and division have equal priority and are performed left to right as they appear in the expression. Similarly, addition and subtraction have equal priority and are performed left to right.

For example, evaluate 3 + 4 × 2² − (6 − 1): first handle parentheses, 6 − 1 = 5; then exponents, 2² = 4; then multiplication, 4 × 4 = 16; finally left to right addition and subtraction, 3 + 16 − 5 = 14.