GACE Middle Grades Math (4–8) (704)

The real number system forms the foundation of middle grades mathematics. Understanding how numbers are classified, represented, and related to one another is essential for teaching mathematical concepts effectively.

Natural numbers (counting numbers) include 1, 2, 3, 4, and so on. Whole numbers extend this set by including zero: 0, 1, 2, 3, 4, and continuing infinitely. These numbers represent quantities we encounter in everyday counting situations.

Integers expand the whole numbers to include negative values. The set of integers includes all positive whole numbers, zero, and their negative counterparts. On a number line, integers extend infinitely in both directions from zero. Understanding integers is crucial for representing real-world situations involving loss, debt, temperature below zero, or elevation below sea level.

Rational numbers include any number that can be expressed as a fraction p/q where p and q are integers and q is not zero. This definition encompasses integers (since any integer n can be written as n/1), fractions, terminating decimals, and repeating decimals.

Key characteristics of rational numbers:

Every fraction can be converted to a decimal by dividing the numerator by the denominator

Terminating decimals have a finite number of digits after the decimal point

Repeating decimals have a pattern of digits that repeats infinitely

Every repeating or terminating decimal can be converted back to a fraction

To convert a repeating decimal to a fraction, use algebraic manipulation. For example, to convert 0.333... to a fraction: Let x = 0.333...

Multiply both sides by 10: 10x = 3.333...

Subtract the original equation: 10x - x = 3.333... - 0.333...

Simplify: 9x = 3

Solve: x = 3/9 = 1/3

For decimals with longer repeating patterns, multiply by the appropriate power of 10 to shift the decimal point past one complete cycle of the repeating portion.

Irrational numbers cannot be expressed as a ratio of two integers. Their decimal representations are non-terminating and non-repeating. Common examples include:

The square root of any non-perfect-square positive integer is irrational. This can be proven using proof by contradiction.

The real number system combines rational and irrational numbers. Every point on the number line corresponds to exactly one real number, and every real number corresponds to exactly one point on the number line. This one-to-one correspondence is called the completeness property of real numbers.

Our decimal (base-10) number system uses place value, where the position of each digit determines its value. Moving left from the decimal point, each position represents a power of 10: ones (10⁰), tens (10¹), hundreds (10²), thousands (10³), and so on. Moving right from the decimal point, each position represents a negative power of 10: tenths (10⁻¹), hundredths (10⁻²), thousandths (10⁻³), and continuing.

Any number can be written in expanded form to show the value of each digit. For example: 3,472 = 3(1000) + 4(100) + 7(10) + 2(1) 3,472 = 3(10³) + 4(10²) + 7(10¹) + 2(10⁰)

Decimals follow the same pattern:

Scientific notation expresses numbers as a product of a coefficient (between 1 and 10) and a power of 10. This notation is useful for representing very large or very small numbers.

To convert a number to scientific notation:

Move the decimal point until there is exactly one non-zero digit to its left

Count the number of places moved

If you moved left, the exponent is positive; if you moved right, the exponent is negative

0.00032 = 3.2 × 10⁻⁴ (moved decimal 4 places right)

Multiplication: Multiply the coefficients and add the exponents. (3 × 10⁴) × (2 × 10³) = 6 × 10⁷

Division: Divide the coefficients and subtract the exponents. (8 × 10⁶) ÷ (2 × 10²) = 4 × 10⁴

Addition and subtraction require the same power of 10 before combining coefficients.

Proper fractions have numerators smaller than denominators (value less than 1). Improper fractions have numerators greater than or equal to denominators (value greater than or equal to 1). Mixed numbers combine a whole number with a proper fraction.

Improper fraction to mixed number: Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the numerator over the original denominator. 17/5 = 3 remainder 2 = 3 2/5

Mixed number to improper fraction: Multiply the whole number by the denominator, add the numerator, and place over the original denominator. 3 2/5 = (3 × 5 + 2)/5 = 17/5

Equivalent fractions represent the same value. Create equivalent fractions by multiplying or dividing both numerator and denominator by the same non-zero number.

To simplify a fraction to lowest terms, divide both numerator and denominator by their greatest common factor (GCF). 24/36: GCF of 24 and 36 is 12, so 24/36 = 2/3

To compare fractions:

Convert to common denominators and compare numerators

Convert to decimals and compare

Use cross multiplication: for a/b and c/d, compare a×d to b×c

A ratio compares two quantities. Ratios can be written as a:b, a to b, or a/b. Ratios should typically be simplified to lowest terms.

A rate compares quantities with different units (miles per hour, cost per item). A unit rate has a denominator of 1, making comparisons easier.

To find a unit rate, divide the first quantity by the second.

150 miles in 3 hours = 150/3 = 50 miles per hour

A proportion states that two ratios are equal. If a/b = c/d, then ad = bc (cross products are equal).

Use cross multiplication to solve for unknown values.

Two quantities are proportionally related if their ratio remains constant. In a proportional relationship:

The graph passes through the origin

The equation has the form y = kx, where k is the constant of proportionality

The constant k represents the unit rate

Percent means "per hundred." A percent represents a ratio with a denominator of 100. 45% = 45/100 = 0.45

Percent to decimal: Move decimal point two places left (divide by 100). 72% = 0.72 8% = 0.08 125% = 1.25

Decimal to percent: Move decimal point two places right (multiply by 100). 0.35 = 35% 0.075 = 7.5% 2.4 = 240%

Fraction to percent: Divide numerator by denominator, then convert the decimal to percent. 3/8 = 0.375 = 37.5%

Percent to fraction: Write percent over 100 and simplify. 60% = 60/100 = 3/5

Three basic percent problems involve finding:

Percent change = (New Value - Original Value) / Original Value × 100

Percent increase occurs when the new value exceeds the original. Percent decrease occurs when the new value is less than the original.

When applying multiple percent changes, multiply the resulting factors.

20% increase followed by 10% decrease: 1.20 × 0.90 = 1.08 (net 8% increase)

The absolute value of a number is its distance from zero on the number line. Absolute value is always non-negative.

|5| = 5

|-5| = 5

|0| = 0

|a| ≥ 0 for all real numbers a

|-a| = |a|

|ab| = |a| × |b|

|a/b| = |a|/|b| when b ≠ 0

|a + b| ≤ |a| + |b| (triangle inequality)

On a horizontal number line, numbers increase from left to right. Any number to the left of another is smaller.

The distance between two points a and b on a number line equals |a - b| or |b - a|.

The midpoint between two numbers a and b is (a + b)/2. Midpoint of -4 and 10: (-4 + 10)/2 = 6/2 = 3