Integers, PEMDAS & Divisibility - Slides
This chapter covers fundamental numerical concepts essential for success in Philippine college entrance exams and civil service examinations. You'll master operations with integers, the order of operations (PEMDAS), and divisibility rules - skills that form the foundation of numerical ability testing.
Slides
Welcome to Integers, PEMDAS & Divisibility
Notes
Introduction slide establishing the importance and scope of the chapter
Examples
- UPCAT mathematical reasoning
- Civil Service Exam numerical ability
- Daily calculations and problem-solving
This chapter builds your numerical fluency through three interconnected topics that appear frequently in standardized tests across the Philippines.
Main points
- Essential numerical skills for UPCAT, CSE, and other Philippine entrance exams
- Three core topics: Integers, PEMDAS, and Divisibility
- Foundation for advanced mathematical problem-solving
- Practical applications in daily life and academic settings
Diagram
Overview of the three main topics and their subtopics in numerical ability
Understanding Integers
Notes
Foundation slide introducing the concept and properties of integers
Examples
- -5, -1, 0, 7, 15 are all integers
- 1/2, 3.14, √2 are NOT integers
Integers form the foundation of number systems. They extend counting numbers to include negative values and zero, creating a complete system for mathematical operations.
Main points
- Integers include positive numbers, negative numbers, and zero
- Set notation: Z = {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Zero is neither positive nor negative
- Integers are whole numbers without fractions or decimals
Diagram
Visual representation of the integer number line showing negative, zero, and positive integers
Types of Integers
Notes
Detailed classification of integers to establish clear understanding of different types
Examples
- Even: -8, 0, 12, 50
- Odd: -7, -1, 9, 23
- Zero is even because 0 ÷ 2 = 0
Integers can be classified into different types based on their properties. Understanding these classifications helps in applying appropriate mathematical rules and operations.
Main points
- Positive Integers: 1, 2, 3, 4, 5, ... (natural numbers)
- Negative Integers: -1, -2, -3, -4, -5, ...
- Zero: Neither positive nor negative, but still an integer
- Even Integers: ..., -4, -2, 0, 2, 4, ... (divisible by 2)
- Odd Integers: ..., -3, -1, 1, 3, 5, ... (not divisible by 2)
Diagram
Classification of integers showing different types and their characteristics
Addition of Integers - Like Signs
Notes
First rule for integer addition - like signs
Examples
- (+5) + (+3) = +8
- (-7) + (-4) = -11
- 15 + 25 = 40
- (-12) + (-8) = -20
When adding integers with the same sign, add their absolute values and keep the common sign. This rule makes addition straightforward for like signs.
Main points
- Positive + Positive = Positive (add absolute values)
- Negative + Negative = Negative (add absolute values, keep negative sign)
- Sum has the same sign as both addends
- Absolute value: distance from zero, always positive
Diagram
Process for adding integers with the same sign
Addition of Integers - Unlike Signs
Notes
Second rule for integer addition - unlike signs
Examples
- (-15) + 20 = +5
- 15 + (-20) = -5
- (-7) + 7 = 0
- 12 + (-8) = 4
When adding integers with different signs, subtract the smaller absolute value from the larger absolute value. The result takes the sign of the integer with the larger absolute value.
Main points
- Positive + Negative = subtract absolute values
- Result takes the sign of the number with larger absolute value
- If absolute values are equal, result is zero
- Think of it as 'moving' on a number line
Diagram
Decision process for adding integers with different signs
Subtraction of Integers
Notes
Systematic approach to integer subtraction using addition conversion
Examples
- 8 - 5 = 8 + (-5) = 3
- (-6) - 8 = (-6) + (-8) = -14
- (-12) - (-5) = (-12) + 5 = -7
- 18 - (-4) = 18 + 4 = 22
Integer subtraction is simplified by converting it to addition. Change the subtraction sign to addition and change the sign of the second number, then apply addition rules.
Main points
- Key rule: Change subtraction to addition of the opposite
- a - b = a + (-b)
- Two steps: 1) Change minus to plus, 2) Change sign of second number
- Then follow addition rules
Diagram
Step-by-step process for converting subtraction to addition
Multiplication and Division of Integers
Notes
Universal sign rules for integer multiplication and division
Examples
- 6 × 4 = 24
- (-5) × (-3) = 15
- 8 × (-2) = -16
- (-12) ÷ 3 = -4
- 20 ÷ (-5) = -4
Multiplication and division of integers follow simple sign rules. When signs are the same, the result is positive. When signs are different, the result is negative.
Main points
- Same signs: Result is POSITIVE
- Different signs: Result is NEGATIVE
- Positive × Positive = Positive
- Negative × Negative = Positive
- Positive × Negative = Negative
- Division follows the same sign rules
Diagram
Sign rules for multiplication and division of integers
Introduction to PEMDAS
Notes
Introduction to the PEMDAS rule and its importance
Examples
- Without PEMDAS: 2 + 3 × 4 could be 20 or 14
- With PEMDAS: 2 + 3 × 4 = 2 + 12 = 14
PEMDAS ensures everyone gets the same answer when solving mathematical expressions. It provides a standard order for performing operations, preventing confusion and errors in calculations.
Main points
- PEMDAS: Order of Operations rule
- P - Parentheses (grouping symbols first)
- E - Exponents (powers and roots)
- MD - Multiplication & Division (left to right)
- AS - Addition & Subtraction (left to right)
- Also known as BODMAS or GEMDAS in some regions
Diagram
The sequential order of operations in PEMDAS
PEMDAS Step-by-Step Example
Notes
Detailed example showing step-by-step PEMDAS application
Examples
- Final calculation: 2 - (-8) + 4 ÷ 2 × 6 - 3 = 2 + 8 + 2 × 6 - 3 = 2 + 8 + 12 - 3 = 19
Complex expressions require systematic application of PEMDAS. Work from innermost grouping symbols outward, then follow the order strictly.
Main points
- Example: 2 - [4 - 3 × (-3 + 5)²] + 4 ÷ 2 × 6 - 3
- Step 1: Parentheses first (-3 + 5) = 2
- Step 2: Exponent (2)² = 4
- Step 3: Multiplication inside brackets 3 × 4 = 12
- Step 4: Subtraction inside brackets 4 - 12 = -8
- Step 5: Continue with remaining operations left to right
Diagram
Sequential steps in solving a complex PEMDAS expression
Common PEMDAS Mistakes
Notes
Prevention-focused slide highlighting common student errors
Examples
- Wrong: 6 ÷ 2 × 3 = 6 ÷ 6 = 1
- Right: 6 ÷ 2 × 3 = 3 × 3 = 9
- Work left to right for same priority operations
Understanding common errors helps avoid them. Many students struggle with the left-to-right rule for operations of equal priority and proper handling of negative signs.
Main points
- Mistake 1: Not working inside grouping symbols first
- Mistake 2: Doing addition before multiplication
- Mistake 3: Not following left-to-right for same-level operations
- Mistake 4: Forgetting to change signs properly with parentheses
- Remember: Multiplication/Division have equal priority (left-to-right)
Diagram
Common mistakes in PEMDAS and how to avoid them
Divisibility Rules Overview
Notes
Introduction to divisibility rules and their practical importance
Examples
- Is 234 divisible by 6? Check if it's divisible by both 2 and 3
- Is 1,728 divisible by 8? Check the last three digits: 728
Divisibility rules are shortcuts that let you quickly determine if a number is divisible by another without performing actual division. These are crucial for efficiency in standardized tests.
Main points
- Divisibility rules help determine if one number divides another evenly
- No need for long division - use patterns and shortcuts
- Essential for simplifying fractions and factoring
- Common rules: 2, 3, 4, 5, 6, 8, 9, 10, 11, 12
- Save time in exams and mental math
Diagram
Overview of divisibility rules, methods, and benefits
Divisibility Rules: 2, 5, and 10
Notes
Basic divisibility rules focusing on the last digit
Examples
- 124 is divisible by 2 (ends in 4)
- 175 is divisible by 5 (ends in 5)
- 230 is divisible by 2, 5, and 10 (ends in 0)
- 127 is not divisible by 2 (ends in 7)
The easiest divisibility rules focus on the last digit. These patterns emerge from the base-10 number system and are foundational for more complex rules.
Main points
- Rule for 2: Last digit is even (0, 2, 4, 6, 8)
- Rule for 5: Last digit is 0 or 5
- Rule for 10: Last digit is 0
- These rules only require checking the ones digit
- Most basic and frequently used divisibility rules
Diagram
Decision tree for testing divisibility by 2, 5, and 10
Divisibility Rules: 3, 6, and 9
Notes
Digit sum method for divisibility testing
Examples
- 423: 4+2+3=9, divisible by 9 (and 3)
- 312: even and 3+1+2=6 (divisible by 3), so divisible by 6
- 4,578: 4+5+7+8=24, 2+4=6, divisible by 3
These rules require adding all digits in the number. The digital root (repeated digit sum) reveals divisibility patterns. Rule for 6 combines two simpler rules.
Main points
- Rule for 3: Sum of digits is divisible by 3
- Rule for 9: Sum of digits is divisible by 9
- Rule for 6: Must be divisible by both 2 AND 3
- If divisible by 9, automatically divisible by 3
- Keep adding digit sums until you get a single digit
Diagram
Process for testing divisibility by 3, 6, and 9 using digit sums
Divisibility Rules: 4, 8, and 11
Notes
Advanced divisibility rules requiring specific digit pattern checks
Examples
- 536: last two digits 36, 36÷4=9, so 536 is divisible by 4
- 4,928: last three digits 928, 928÷8=116, so divisible by 8
- 4,510: (4+1)-(5+0)=0, divisible by 11
These rules require examining specific digit positions. Rule 11 uses alternating addition/subtraction pattern. These are more complex but very useful for factorization.
Main points
- Rule for 4: Last two digits form a number divisible by 4
- Rule for 8: Last three digits form a number divisible by 8
- Rule for 11: Alternating sum of digits is divisible by 11
- For 11: Add odd position digits, add even position digits, find difference
- Numbers ending in 00, 000 are automatically divisible by 4, 8 respectively
Diagram
Testing process for divisibility by 4, 8, and 11
Factors and Multiples
Notes
Foundation concepts of factors and multiples with clear examples
Examples
- Factors of 12: 1, 2, 3, 4, 6, 12
- Multiples of 4: 4, 8, 12, 16, 20, 24, ...
- 12 is both a factor of 24 and a multiple of 4
Factors and multiples are fundamental concepts in number theory. Understanding their relationship helps with fraction simplification, finding common denominators, and solving word problems.
Main points
- Factors: Numbers that divide evenly into another number
- Multiples: Results of multiplying a number by integers
- Every number is both a factor and multiple of itself
- 1 is a factor of every number
- Prime numbers have only two factors: 1 and themselves
Diagram
Relationship between factors and multiples using 12 as an example
Finding GCF and LCM
Notes
Systematic approach to finding GCF and LCM using multiple methods
Examples
- GCF of 18 and 24: factors of 18 (1,2,3,6,9,18) and 24 (1,2,3,4,6,8,12,24) → GCF = 6
- LCM of 6 and 8: multiples of 6 (6,12,18,24...) and 8 (8,16,24...) → LCM = 24
GCF and LCM are essential for working with fractions and solving real-world problems involving cycles and patterns. Multiple methods exist, each with advantages depending on the numbers involved.
Main points
- GCF (Greatest Common Factor): Largest number that divides all given numbers
- LCM (Least Common Multiple): Smallest number that is a multiple of all given numbers
- Methods: Listing, Prime Factorization, Continuous Division
- GCF is always ≤ smallest number in the set
- LCM is always ≥ largest number in the set
Diagram
Methods for finding GCF and LCM of two numbers
Prime and Composite Numbers
Notes
Clear distinction between prime and composite numbers with examples
Examples
- Prime: 7 (factors: 1, 7)
- Composite: 12 (factors: 1, 2, 3, 4, 6, 12)
- Special: 1 (only one factor: itself)
Prime and composite classification is fundamental in number theory. Primes are building blocks for all other numbers through multiplication, making them crucial for factorization and divisibility.
Main points
- Prime numbers: Have exactly two factors (1 and itself)
- Composite numbers: Have more than two factors
- 1 is neither prime nor composite (special case)
- 2 is the only even prime number
- First few primes: 2, 3, 5, 7, 11, 13, 17, 19, 23...
Diagram
Classification of numbers as prime, composite, or neither
Practical Applications
Notes
Connecting theoretical concepts to practical applications and exam contexts
Examples
- CSE question: 'Find next in sequence: 24, 36, 48, ?'
- Real-life: 'If buses arrive every 12 minutes and trains every 15 minutes, when do they coincide?'
- Budget planning: Finding common factors for expense categories
These concepts appear throughout Philippine standardized testing and have practical applications in daily life. Mastery enables quick problem-solving and builds confidence in mathematical reasoning.
Main points
- UPCAT and entrance exams: Number patterns, sequences, word problems
- Civil Service Exams: Numerical reasoning, logical sequences
- Real-world: Scheduling, budgeting, time management
- Advanced math: Algebra foundations, equation solving
- Mental math: Quick calculations without calculator
Diagram
Type
none
Problem-Solving Strategies
Notes
Strategic approach to problem-solving with emphasis on systematic thinking
Examples
- Word problem: 'A class of 24 students is divided into equal groups. What are the possible group sizes?' → Find factors of 24
- Sequence: '7, 14, 21, ?' → Recognize multiples of 7
Effective problem-solving combines understanding concepts with strategic thinking. Time management is crucial in standardized tests, so developing efficient methods is essential for success.
Main points
- Read problems carefully - identify what's being asked
- Look for key words: sum, difference, product, quotient
- Break complex problems into smaller steps
- Use PEMDAS consistently in multi-step calculations
- Check answers using divisibility rules when possible
- Practice mental math for speed in exams
Diagram
Step-by-step approach to solving numerical problems systematically
Chapter Summary
Notes
Comprehensive summary emphasizing the practical value and interconnected nature of the concepts
Examples
- Master these for success in UPCAT, CSE, NMAT, and other Philippine exams
- Use in daily situations: shopping calculations, time management, budgeting
This chapter provided essential numerical skills that form the foundation for advanced mathematics and standardized test success. Regular practice with these concepts will build both speed and accuracy.
Main points
- Integers: Positive, negative, and zero with specific operation rules
- PEMDAS: Essential order of operations for consistent results
- Divisibility Rules: Quick tests without long division
- Factors and Multiples: Foundation for fraction work and problem-solving
- Applications: Critical for Philippine entrance exams and daily math
Diagram
Summary of key chapter concepts and their interconnections
In summary
Mastery of integers, PEMDAS, and divisibility rules provides the foundation for success in Philippine standardized exams and advanced mathematics. These concepts appear frequently in UPCAT, Civil Service, and other entrance examinations. Regular practice with these fundamentals will build the numerical fluency essential for academic and professional success. Remember to apply these skills systematically, check your work using divisibility rules when possible, and maintain accuracy while building speed through consistent practice.