Session 1: Number Sense & Basic Operations

Explanatory SVG

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Session Overview

90 minutes | Week 1 | No calculator for Part 1 · Calculator allowed for Part 2

These concepts appear in approximately 20% of GED® Mathematical Reasoning questions and are tested in both exam parts. Master them here before moving on.

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Section 1: Learning Objectives

By the end of this session you will be able to:

• Identify and classify integers, whole numbers, rational and irrational numbers
• Perform operations with fractions (add, subtract, multiply, divide) accurately
• Convert between fractions, decimals, and percentages
• Apply the order of operations (PEMDAS) to solve multi-step expressions
• Calculate with absolute values and understand number lines
• Solve GED®-style word problems involving basic number operations

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Section 2: GED® Test Connection

Why this session matters on the GED®:

• Approximately 20% of GED® Mathematical Reasoning questions draw on number sense
• Tested in Part 1 (no calculator) AND Part 2 (calculator allowed)
• Question types: Multiple choice and fill-in-the-blank

Common traps to avoid:

• Forgetting order of operations (PEMDAS)
• Sign errors when working with negative numbers
• Fraction vs. decimal confusion in word problems

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Section 3.1 — The Number System

Understanding the types of numbers is the foundation of all GED® math. Numbers are organised into sets that build on each other.

• Natural Numbers — Counting numbers starting from 1. Example: 1, 2, 3, 4, 5 …
• Whole Numbers — Natural numbers plus zero. Example: 0, 1, 2, 3, 4 …
• Integers — All whole numbers and their negatives. Example: … −3, −2, −1, 0, 1, 2, 3 …
• Rational Numbers — Any number expressible as a fraction p/q where q ≠ 0. Examples: 3/4, 0.5, −2, 0.333…
• Irrational Numbers — Cannot be expressed as a fraction; decimals are non-terminating and non-repeating. Examples: π ≈ 3.14159…, √2 ≈ 1.41421…

Number Line:

Negative numbers lie to the LEFT of zero; positive numbers lie to the RIGHT.

← −3 · · · −2 · · · −1 · · · 0 · · · 1 · · · 2 · · · 3 →

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Section 3.2 — Absolute Value

Absolute value is the distance a number is from zero on the number line. Distance is always positive (or zero).

Notation: \(|x|\) means "the absolute value of x"

Examples:

• \(|-5| = 5\)
• \(|3| = 3\)
• \(|0| = 0\)

GED® Tip: Absolute value is often combined with other operations. Always resolve the \(| \; |\) bars first before doing any addition, subtraction, or multiplication outside them.

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Section 3.3 — Fractions

A fraction represents a part of a whole. The top number is the numerator (how many parts you have); the bottom number is the denominator (how many equal parts in the whole).

Example: \(\dfrac{3}{4}\) → Numerator = 3 (parts you have), Denominator = 4 (total equal parts)

Equivalent Fractions

Fractions that have the same value but different numerators and denominators:

\[\frac{1}{2} = \frac{2}{4} = \frac{4}{8} = \frac{8}{16}\]

Simplifying Fractions

Divide both the numerator and denominator by their Greatest Common Factor (GCF).

Example: Simplify \(\dfrac{12}{18}\)

• Step 1: Find GCF of 12 and 18 → GCF = 6
• Step 2: Divide both: \(12 ÷ 6 = 2\) and \(18 ÷ 6 = 3\) → Answer: \(\dfrac{2}{3}\)

Adding and Subtracting Fractions

Fractions must have the same denominator (LCD = Least Common Denominator) before you can add or subtract.

Example: \(\dfrac{1}{3} + \dfrac{1}{4}\)

• Step 1: LCD of 3 and 4 = 12
• Step 2: Convert: \(\dfrac{1}{3} = \dfrac{4}{12}\) and \(\dfrac{1}{4} = \dfrac{3}{12}\)
• Step 3: Add: \(\dfrac{4}{12} + \dfrac{3}{12} = \dfrac{7}{12}\)

Multiplying Fractions

Multiply numerator × numerator and denominator × denominator. Simplify if possible.

Example: \(\dfrac{2}{3} \times \dfrac{3}{5} = \dfrac{2 \times 3}{3 \times 5} = \dfrac{6}{15} = \dfrac{2}{5}\)

Dividing Fractions — KCF Method (Keep, Change, Flip)

Keep the first fraction · Change ÷ to × · Flip the second fraction (take its reciprocal) · then multiply.

Example: \(\dfrac{2}{3} \div \dfrac{4}{5}\)

• Step 1: Keep \(\dfrac{2}{3}\) | Change ÷ to × | Flip \(\dfrac{4}{5}\) to \(\dfrac{5}{4}\)
• Step 2: \(\dfrac{2}{3} \times \dfrac{5}{4} = \dfrac{10}{12} = \dfrac{5}{6}\)

Mixed Numbers

Always convert mixed numbers to improper fractions before performing operations.

Example: \(2\frac{1}{2} \rightarrow \dfrac{(2 \times 2 + 1)}{2} = \dfrac{5}{2}\)

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Section 3.4 — Decimals

Decimals are another way to write fractions. Each position after the decimal point has a specific place value:

• 0.1 = tenths
• 0.01 = hundredths
• 0.001 = thousandths

Converting Fractions to Decimals: Divide the numerator by the denominator.

Example: \(\dfrac{3}{4} = 3 \div 4 = 0.75\)

Converting Decimals to Fractions: Read the decimal using its place value name, then simplify.

Example: Convert 0.35 to a fraction

• Step 1: 0.35 = 35 hundredths = \(\dfrac{35}{100}\)
• Step 2: Simplify — GCF of 35 and 100 is 5 → \(\dfrac{35}{100} = \dfrac{7}{20}\)

Operations with Decimals:

• Adding / Subtracting: Always align decimal points before computing. → 3.25 + 1.8 = 5.05
• Multiplying: Multiply as whole numbers, then count total decimal places and insert the point. → 1.2 × 0.3 = 0.36 (2 decimal places total)
• Dividing: Move the decimal in the divisor to make it a whole number; move the same number of places in the dividend. → 4.5 ÷ 0.5 = 45 ÷ 5 = 9

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Section 3.5 — Percentages

"Percent" means per hundred. Any percent can be written as a fraction over 100 or as a decimal.

Example: \(45\% = \dfrac{45}{100} = 0.45\)

Common Conversion Table:

Finding a Percent of a Number:

Example: What is 30% of 80?

• Convert 30% to decimal: 0.30 → 0.30 × 80 = 24

Finding What Percent One Number is of Another:

Example: 15 is what percent of 60?

• Divide: 15 ÷ 60 = 0.25 → Convert to percent: 25%

GED® Language Tip: In word problems: "of" means multiply (×), "is" means equals (=), "what" means the unknown variable (x).

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Section 3.6 — Order of Operations (PEMDAS)

When an expression has multiple operations, follow this order:

• P — Parentheses: solve expressions inside ( ) first
• E — Exponents: compute powers and square roots
• MD — Multiplication & Division: work left to right
• AS — Addition & Subtraction: work left to right

Worked Example 1:

Evaluate: \(3 + 6 \times (5 + 4) \div 3 - 7\)

• Step 1 (Parentheses): \(3 + 6 \times 9 \div 3 - 7\)
• Step 2 (Multiply): \(3 + 54 \div 3 - 7\)
• Step 3 (Divide): \(3 + 18 - 7\)
• Step 4 (Add/Subtract): \(21 - 7 = \mathbf{14}\)

Worked Example 2:

Evaluate: \(2^3 + (12 - 4) \div 4 \times 2\)

• Step 1 (Parentheses): \(2^3 + 8 \div 4 \times 2\)
• Step 2 (Exponents): \(8 + 8 \div 4 \times 2\)
• Step 3 (Divide first): \(8 + 2 \times 2\)
• Step 4 (Multiply): \(8 + 4\)
• Step 5 (Add): \(\mathbf{12}\)

Common Mistake: Many students do Multiplication before Division automatically — this is WRONG. Multiplication and Division have equal priority. Always work from left to right when both appear in the same expression.

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Section 4: Guided Practice

Work through these problems carefully. Full solutions are shown.

Problem 1: Simplify \(|-12| + |5| - |-3|\)

• Step 1: Resolve absolute values → \(|-12| = 12,\; |5| = 5,\; |-3| = 3\)
• Step 2: Calculate → \(12 + 5 - 3 = \mathbf{14}\)

Problem 2: Add \(\dfrac{3}{8} + \dfrac{5}{12}\)

• Step 1: LCD of 8 and 12 = 24
• Step 2: Convert → \(\dfrac{3}{8} = \dfrac{9}{24}\) and \(\dfrac{5}{12} = \dfrac{10}{24}\)
• Step 3: Add → \(\dfrac{9}{24} + \dfrac{10}{24} = \mathbf{\dfrac{19}{24}}\)

Problem 3: Solve \(2\frac{1}{2} \times 1\frac{1}{3}\)

• Step 1: Convert → \(2\frac{1}{2} = \dfrac{5}{2}\) and \(1\frac{1}{3} = \dfrac{4}{3}\)
• Step 2: Multiply → \(\dfrac{5}{2} \times \dfrac{4}{3} = \dfrac{20}{6}\)
• Step 3: Simplify → \(\dfrac{20}{6} = \dfrac{10}{3} = \mathbf{3\frac{1}{3}}\)

Problem 4: Convert 0.625 to a fraction in simplest form

• Step 1: Write as fraction → \(\dfrac{625}{1000}\)
• Step 2: GCF of 625 and 1000 = 125
• Step 3: Simplify → \(\dfrac{625 \div 125}{1000 \div 125} = \mathbf{\dfrac{5}{8}}\)

Problem 5: What is 35% of 140?

• Step 1: Convert percent to decimal → 35% = 0.35
• Step 2: Multiply → \(0.35 \times 140 = \mathbf{49}\)

Problem 6: Evaluate \(4 + 3^2 \times (8 - 5) \div 9\)

• Step 1: Parentheses → \((8 - 5) = 3\)
• Step 2: Exponents → \(3^2 = 9\)
• Step 3: Expression so far: \(4 + 9 \times 3 \div 9\)
• Step 4: Multiply/Divide left to right → \(9 \times 3 = 27;\; 27 \div 9 = 3\)
• Step 5: Add → \(4 + 3 = \mathbf{7}\)

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Section 7: Homework / Self-Study

Complete the following before Session 2 to reinforce what you learned today:

• Khan Academy: Complete practice sets on "Fractions" and "Order of Operations"
• YouTube: Search "GED® Math – Number Sense" and watch at least one full tutorial video
• GED®.com: Take the free Number Sense mini-quiz to benchmark your understanding
• Review any wrong answers from the Activity Sheet — re-work each problem from scratch before looking at the solution

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End of Session 1 · Number Sense & Basic Operations