GACE Middle Grades Math (4–8) (704)

Same signs: Add absolute values and keep the common sign. 5 + 3 = 8 (-5) + (-3) = -8

Different signs: Subtract absolute values and take the sign of the number with the larger absolute value. 5 + (-3) = 2 (-5) + 3 = -2

Subtraction is equivalent to adding the opposite. 5 - 3 = 5 + (-3) = 2 5 - (-3) = 5 + 3 = 8 (-5) - 3 = (-5) + (-3) = -8 (-5) - (-3) = (-5) + 3 = -2

Same signs: Product is positive. 4 × 3 = 12 (-4) × (-3) = 12

Different signs: Product is negative. 4 × (-3) = -12 (-4) × 3 = -12

Same signs: Quotient is positive. 12 ÷ 3 = 4 (-12) ÷ (-3) = 4

Different signs: Quotient is negative. 12 ÷ (-3) = -4 (-12) ÷ 3 = -4

Fractions must have common denominators before adding or subtracting.

Finding the Least Common Denominator (LCD):

List multiples of each denominator

Identify the smallest common multiple

Or find the least common multiple using prime factorization

LCD of 3 and 4 is 12 2/3 = 8/12 and 3/4 = 9/12 8/12 + 9/12 = 17/12 = 1 5/12

Multiply numerators together and denominators together. Simplify the result. 2/3 × 4/5 = 8/15 3/4 × 2/9 = 6/36 = 1/6

Cross-canceling before multiplying can simplify calculations:

Divide by multiplying by the reciprocal of the divisor. 2/3 ÷ 4/5 = 2/3 × 5/4 = 10/12 = 5/6 3/4 ÷ 2 = 3/4 × 1/2 = 3/8

Align decimal points vertically and add or subtract as with whole numbers. Place the decimal point in the answer directly below the aligned decimal points.

Multiply as with whole numbers, ignoring decimal points. Count the total number of decimal places in both factors. Place the decimal point in the product so it has that many decimal places.

When dividing by a decimal, move the decimal point in the divisor to make it a whole number. Move the decimal point in the dividend the same number of places. Place the decimal point in the quotient directly above its new position in the dividend.

The standard order of operations (PEMDAS/GEMDAS):

Parentheses (or other grouping symbols including brackets, braces, fraction bars, and radicals)

Exponents (and roots)

Multiplication and Division (left to right)

Addition and Subtraction (left to right)

Order does not affect the result for addition or multiplication.

a + b = b + a

a × b = b × a

Grouping does not affect the result for addition or multiplication.

(a + b) + c = a + (b + c)

(a × b) × c = a × (b × c)

Multiplication distributes over addition and subtraction.

a(b + c) = ab + ac

a(b - c) = ab - ac

This property works in reverse for factoring:

ab + ac = a(b + c)

Additive identity: a + 0 = a

Multiplicative identity: a × 1 = a

Additive inverse: a + (-a) = 0

Multiplicative inverse: a × (1/a) = 1 (when a ≠ 0)

a × 0 = 0

An exponent indicates repeated multiplication. In the expression bⁿ, b is the base and n is the exponent. 2⁴ = 2 × 2 × 2 × 2 = 16 5³ = 5 × 5 × 5 = 125

Product of powers: aᵐ × aⁿ = aᵐ⁺ⁿ 2³ × 2⁴ = 2⁷ = 128

Quotient of powers: aᵐ ÷ aⁿ = aᵐ⁻ⁿ (when a ≠ 0) 5⁶ ÷ 5² = 5⁴ = 625

Power of a power: (aᵐ)ⁿ = aᵐⁿ (2³)² = 2⁶ = 64

Power of a product: (ab)ⁿ = aⁿbⁿ (3 × 4)² = 3² × 4² = 9 × 16 = 144

Power of a quotient: (a/b)ⁿ = aⁿ/bⁿ (when b ≠ 0) (2/3)³ = 2³/3³ = 8/27

Zero exponent: a⁰ = 1 (when a ≠ 0)

7⁰ = 1

(-3)⁰ = 1

Negative exponent: a⁻ⁿ = 1/aⁿ (when a ≠ 0) 2⁻³ = 1/2³ = 1/8 (1/3)⁻² = 3² = 9

The square root of a number n is a value that, when multiplied by itself, equals n. √25 = 5 because 5² = 25 √49 = 7 because 7² = 49

Perfect squares are numbers whose square roots are integers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, and so on.

Factor the radicand to identify perfect square factors. √72 = √(36 × 2) = √36 × √2 = 6√2 √50 = √(25 × 2) = √25 × √2 = 5√2

The cube root of a number n is a value that, when cubed, equals n. ∛8 = 2 because 2³ = 8 ∛27 = 3 because 3³ = 27 ∛-8 = -2 because (-2)³ = -8

Perfect cubes include: 1, 8, 27, 64, 125, 216, and so on.

Addition/Subtraction: Combine like radicals (same radicand). 3√5 + 2√5 = 5√5 4√3 - √3 = 3√3

Multiplication: Multiply coefficients and radicands separately. 2√3 × 4√5 = 8√15 √6 × √2 = √12 = 2√3

To round a number to a specific place value:

Look at the digit to the right of the target place

If 5 or greater, round up; if less than 5, round down

Front-end estimation: Use leading digits and adjust. Compatible numbers: Round to numbers that work well together. Clustering: When numbers cluster around a value, use that value.

Breaking apart numbers: 23 × 4 = 20 × 4 + 3 × 4 = 80 + 12 = 92 Compensation: 49 × 6 = 50 × 6 - 6 = 300 - 6 = 294 Using known facts: 8 × 7 = 8 × 7 = 56

Factors of a number divide it evenly. To find all factors, check divisibility by each number from 1 to the square root of the number, pairing each factor with its complement.

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Prime numbers have exactly two factors: 1 and themselves. Examples: 2, 3, 5, 7, 11, 13, 17, 19, 23.

Composite numbers have more than two factors. Examples: 4, 6, 8, 9, 10, 12, 14, 15.

The number 1 is neither prime nor composite.

Every composite number can be expressed as a product of prime factors. 60 = 2² × 3 × 5 84 = 2² × 3 × 7

The GCF is the largest number that divides two or more numbers evenly.

Method 1: List factors and find the largest common factor. Method 2: Use prime factorization—multiply the common prime factors with the smallest exponents.

GCF of 48 and 72: 48 = 2⁴ × 3 72 = 2³ × 3² GCF = 2³ × 3 = 24

The LCM is the smallest number that is a multiple of two or more numbers.

Method 1: List multiples and find the smallest common multiple. Method 2: Use prime factorization—multiply all prime factors with the largest exponents.

LCM of 12 and 18: 12 = 2² × 3 18 = 2 × 3² LCM = 2² × 3² = 36