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1. Algebra and Expressions
1.1.1 Combining Like Terms
1.1.2 Using the Distributive Property
1.1.3 Simplifying Expressions with Brackets
1.1.4 Rearranging Terms to Standard Form
1.1.5 Factoring Common Factors from Terms
1.2.1 Expanding Single Bracket Expressions
1.2.2 Factoring Common Terms from Expressions
1.2.3 Understanding the Reverse Process of Expansion
1.2.4 Creating Equivalent Expressions through Expansion
1.2.5 Solving Problems Involving Bracket Expansion
1.3.1 Creating Algebraic Expressions from Word Problems
1.3.2 Modeling Relationships Using Expressions
1.3.3 Solving Real-Life Problems with Algebra
1.3.4 Writing Multi-Step Expressions from Situations
1.3.5 Checking Solutions for Word-Based Expressions
1.4.1 Substituting Values into Simple Expressions
1.4.2 Substitution with Negative and Decimal Values
1.4.3 Using Substitution in Real-World Formulae
1.4.4 Evaluating Algebraic Formulas with Given Values
1.4.5 Solving Problems Using Substitution Techniques
1.5.1 Understanding the Use of Letters in Math
1.5.2 Defining Constants and Coefficients
1.5.3 Recognizing Like and Unlike Terms
1.5.4 Using Variables in Real-Life Situations
1.5.5 Building Expressions from Verbal Descriptions
1.6.1 Translating Words into Algebraic Expressions
1.6.2 Interpreting Algebraic Expressions in Context
1.6.3 Creating Expressions from Situational Problems
1.6.4 Forming Expressions with Multiple Terms
1.6.5 Constructing Expressions with Brackets
2. Geometry – Properties of Shape
2.1.1 Classifying Triangles by Sides and Angles
2.1.2 Properties of Isosceles, Equilateral, and Right Triangles
2.1.3 Understanding Interior Angles of Polygons
2.1.4 Properties of Squares, Rectangles, Parallelograms, and Trapeziums
2.1.5 Sum of Interior and Exterior Angles in Polygons
2.2.1 Identifying Line Symmetry in Shapes
2.2.2 Understanding Rotational Symmetry
2.2.3 Creating Symmetrical Designs
2.2.4 Tessellating Shapes and Patterns
2.2.5 Exploring Regular and Irregular Tessellations
2.3.1 Understanding Congruent Shapes
2.3.2 Testing for Congruence with Rigid Transformations
2.3.3 Exploring Similar Shapes and Proportionality
2.3.4 Using Scale Factors to Determine Similarity
2.3.5 Applications of Congruence and Similarity in Real Life
2.4.1 Naming Parts of a Circle: Radius, Diameter, Chord, Arc
2.4.2 Understanding Central Angle and Circumference
2.4.3 Drawing Circles with Compass and Ruler
2.4.4 Exploring Tangents and Secants (Introductory)
2.4.5 Symmetry and Angle Properties in Circles
2.5.1 Using Geometric Reasoning to Find Missing Angles
2.5.2 Angle Sum in a Triangle and Quadrilateral
2.5.3 Vertically Opposite and Corresponding Angles
2.5.4 Alternate Interior and Exterior Angles
2.5.5 Solving Problems with Justification and Proof
2.6.1 Identifying Types of Lines and Angles
2.6.2 Complementary, Supplementary, and Vertical Angles
2.6.3 Angles on a Straight Line and Around a Point
2.6.4 Angle Relationships in Parallel Lines and Transversals
2.6.5 Constructing and Measuring Angles with Protractor
3. Ratio, Proportion & Percentages
3.1.1 Understanding Direct Proportion Relationships
3.1.2 Solving Problems Involving Direct Proportion
3.1.3 Understanding Inverse Proportion Relationships
3.1.4 Solving Problems Involving Inverse Proportion
3.1.5 Real-Life Applications of Proportionality
3.2.1 Understanding Percentages as Fractions and Decimals
3.2.2 Finding a Percentage of a Quantity
3.2.3 Solving Percentage Increase and Decrease Problems
3.2.4 Understanding Compound Percentage Growth
3.2.5 Applications of Compound Interest and Depreciation
3.3.1 Calculating Percentage Change
3.3.2 Identifying Percentage Markups and Discounts
3.3.3 Comparing Percentage Gains and Losses
3.3.4 Solving Real-Life Percent Change Scenarios
3.3.5 Using Percentage Multipliers
3.4.1 Understanding the Concept of Reverse Percentages
3.4.2 Finding Original Amounts Before Change
3.4.3 Solving Markup and Discount Reversal Problems
3.4.4 Real-World Applications of Reverse Percentage
3.4.5 Using Equations to Solve Reverse Percentages
3.5.1 Calculating Profit and Loss with Percentages
3.5.2 Using Discounts in Shopping Scenarios
3.5.3 Understanding and Applying Sales Tax
3.5.4 Solving Net Price and Final Cost Problems
3.5.5 Applying Multiple Percent Changes Together
3.6.1 Defining Ratios and Their Notation
3.6.2 Simplifying Ratios to Lowest Terms
3.6.3 Finding Missing Values in Equivalent Ratios
3.6.4 Ratio Word Problems in Real-Life Contexts
3.6.5 Ratios Involving More Than Two Quantities
3.7.1 Sharing Amounts in a Given Ratio
3.7.2 Adjusting Quantities Using Ratios
3.7.3 Comparing Ratios in Real-World Scenarios
3.7.4 Applying Ratios to Scale Models and Maps
3.7.5 Ratio-Based Recipes and Mixtures
4. Patterns, Sequences & Algebraic Thinking
5. Statistics – Averages and Analysis
5.1.1 Grouping Data into Class Intervals
5.1.2 Estimating Mean from Grouped Data
5.1.3 Finding the Modal Class
5.1.4 Calculating Midpoints for Intervals
5.1.5 Using Frequency Tables with Class Intervals
5.2.1 Using the Mean to Find a Missing Value
5.2.2 Solving Problems with Given Total and Frequency
5.2.3 Word Problems Involving Unknowns in Data Sets
5.2.4 Creating Balanced Data Sets
5.2.5 Validating Solutions with Back-Substitution
5.3.1 Understanding the Structure of a Box Plot
5.3.2 Finding the Five-Number Summary
5.3.3 Calculating and Interpreting the Interquartile Range
5.3.4 Drawing Box Plots from Data Sets
5.3.5 Comparing Data Sets Using Box Plots
5.4.1 Recognizing Misleading Graphs
5.4.2 Common Errors in Average Interpretation
5.4.3 Manipulating Data for Persuasion
5.4.4 Critically Evaluating Data in Media
5.4.5 Ethical Use of Statistics
5.5.1 Finding the Mean of a Data Set
5.5.2 Identifying the Median Value
5.5.3 Determining the Mode in a Data Set
5.5.4 Calculating the Range and Interpreting It
5.5.5 Handling Data Sets with Repeated or Missing Values
5.6.1 Comparing Mean, Median, and Mode in Context
5.6.2 Identifying Outliers and Their Effects
5.6.3 Choosing the Most Appropriate Average
5.6.4 Using Averages to Summarize Data
5.6.5 Analyzing Real-Life Data with Averages
6. Number Concepts & Systems
6.1.1 Natural and Whole Numbers
6.1.2 Integers and Their Uses
6.1.3 Even, Odd, and Prime Numbers
6.1.4 Rational vs Irrational Numbers
6.1.5 Real Number System Overview
6.2.1 Defining Rational Numbers
6.2.2 Defining Irrational Numbers
6.2.3 Rational Numbers as Terminating and Repeating Decimals
6.2.4 Irrational Numbers in Square Roots and Pi
6.2.5 Visualizing Rational vs Irrational on a Number Line
6.3.1 Rounding Whole Numbers
6.3.2 Rounding Decimals to Given Places
6.3.3 Estimating Sums and Differences
6.3.4 Estimating Products and Quotients
6.3.5 Significant Figures and Scientific Estimation
6.4.1 Place Value in Large Numbers
6.4.2 Decimal Place Value and Precision
6.4.3 Reading and Writing Numbers in Words
6.4.4 Expanded and Standard Forms
6.4.5 Comparing and Ordering Decimals
6.5.1 Number Line Representation
6.5.2 Ordering Positive and Negative Numbers
6.5.3 Inequalities and Symbols
6.5.4 Rounding Numbers to Nearest 10, 100, 1000
6.5.5 Rounding Decimals to Significant Figures
6.6.1 Prime Numbers up to 100
6.6.2 Composite Numbers and Their Properties
6.6.3 Square Numbers and Their Patterns
6.6.4 Perfect Squares and Their Roots
6.6.5 Applications of Square Numbers in Geometry
6.8.1 Representing Integers on Number Line
6.8.2 Comparing and Ordering Integers
6.8.3 Absolute Value of an Integer
6.8.4 Using Integers in Real-Life Contexts
6.8.5 Opposites and Zero on the Number Line
7. Geometry – Measurement & Calculation
7.1.1 Understanding Surface Area Conceptually
7.1.2 Surface Area of a Cube
7.1.3 Surface Area of a Rectangular Prism
7.1.4 Surface Area of Triangular Prisms
7.1.5 Solving Surface Area Problems in Context
7.2.1 Volume of a Cube
7.2.2 Volume of a Rectangular Prism
7.2.3 Volume of a Triangular Prism
7.2.4 Introduction to Volume of Cylinders
7.2.5 Solving Volume Problems with Units
7.3.1 Breaking Down Compound Shapes into Simple Parts
7.3.2 Area and Perimeter of Compound Shapes
7.3.3 Drawing and Interpreting Nets of 3D Shapes
7.3.4 Surface Area Using Nets
7.3.5 Constructing 3D Shapes from Nets
7.4.1 Understanding Scale in Maps and Blueprints
7.4.2 Creating and Interpreting Scale Diagrams
7.4.3 Reading Bearings and Compass Directions
7.4.4 Measuring Bearings with a Protractor
7.4.5 Solving Problems with Bearings and Distances
7.5.1 Converting Between Metric Length Units
7.5.2 Converting Area Units (cm², m², km²)
7.5.3 Converting Volume Units (cm³, m³, L)
7.5.4 Using Unit Conversion in Word Problems
7.5.5 Common Mistakes in Unit Conversion
7.6.1 Calculating Perimeter of Squares and Rectangles
7.6.2 Finding Perimeter of Irregular Shapes
7.6.3 Area of Rectangles, Triangles, and Parallelograms
7.6.4 Area of Trapeziums and Compound Shapes
7.6.5 Solving Word Problems Involving Perimeter and Area
8. Equations, Inequalities & Formulae
8.1.1 Understanding Inequality Symbols
8.1.2 Solving One-Step Inequalities
8.1.3 Solving Two-Step Inequalities
8.1.4 Graphing Inequalities on a Number Line
8.1.5 Interpreting Inequalities in Real-Life Scenarios
8.2.1 Translating Word Problems into Equations
8.2.2 Solving Problems Involving Age and Time
8.2.3 Using Equations in Geometry Problems
8.2.4 Creating and Solving Multi-Step Word Problems
8.2.5 Checking Solutions in the Context of the Problem
8.3.1 Substituting Values into Given Formulae
8.3.2 Rearranging Formulae to Solve for a Variable
8.3.3 Using Standard Formulae (Area, Volume, Speed)
8.3.4 Solving Problems Involving Real-Life Formulae
8.3.5 Creating Formulae from Contextual Situations
8.4.1 Using the Formula for Area of Shapes
8.4.2 Applying the Speed = Distance ÷ Time Formula
8.4.3 Using Volume and Surface Area Formulas
8.4.4 Solving Finance-Related Formula Problems
8.4.5 Interpreting and Rewriting Real-Life Formulas
8.5.1 Understanding Equality and Balance
8.5.2 Solving One-Step Addition and Subtraction Equations
8.5.3 Solving One-Step Multiplication and Division Equations
8.5.4 Solving Two-Step Linear Equations
8.5.5 Solving Equations with Decimals and Fractions
8.6.1 Expanding Brackets Before Solving
8.6.2 Solving Equations with Variables on Both Sides
8.6.3 Solving Equations Involving Fractions
8.6.4 Clearing Fractions in Equations
8.6.5 Solving Equations with Distribution and Simplification
9. Probability and Outcomes
9.1.1 Defining Theoretical Probability
9.1.2 Conducting Simple Probability Experiments
9.1.3 Comparing Theoretical and Experimental Results
9.1.4 Understanding Variability in Results
9.1.5 Improving Accuracy with Larger Samples
9.2.1 Probability of a Single Event Occurring
9.2.2 Using Frequency to Estimate Probability
9.2.3 Introduction to Independent Events
9.2.4 Calculating Combined Probabilities (Basic)
9.2.5 Solving Real-Life Problems with Combined Events
9.3.1 Probability of Rolling a Die
9.3.2 Probability of Flipping a Coin
9.3.3 Using Spinners to Model Probability
9.3.4 Using Multiple Dice or Coins
9.3.5 Creating Fair and Biased Models
9.4.1 Understanding Tree Diagrams for Two Events
9.4.2 Calculating Basic Probabilities Using Tree Diagrams
9.4.3 Interpreting Simple Venn Diagrams
9.4.4 Using Venn Diagrams to Solve Set Problems
9.4.5 Exploring Overlapping and Disjoint Events
9.5.1 Understanding Probability as a Measure
9.5.2 Probability Scale from 0 to 1
9.5.3 Language of Probability (Certain, Likely, etc.)
9.5.4 Describing Events as Likely or Unlikely
9.5.5 Identifying Impossible and Certain Events
9.6.1 Using Lists and Tables to Find Outcomes
9.6.2 Drawing Sample Space Diagrams
9.6.3 Counting Outcomes for Combined Events
9.6.4 Systematic Listing Methods
9.6.5 Using Tree Diagrams to List Outcomes (Introductory)
10. Geometry – Coordinates and Transformations
11. Data Handling and Representation
11.1.1 Creating Frequency Tables from Raw Data
11.1.2 Using Tally Charts for Counting
11.1.3 Reading and Interpreting Frequency Data
11.1.4 Grouping Data into Class Intervals
11.1.5 Using Frequency Data for Further Calculations
11.2.1 Drawing and Interpreting Bar Charts
11.2.2 Plotting and Analyzing Line Graphs
11.2.3 Constructing Pie Charts with a Protractor
11.2.4 Converting Frequency into Angles for Pie Charts
11.2.5 Choosing the Right Graph for the Data
11.3.1 Understanding Stem-and-Leaf Format
11.3.2 Creating and Interpreting Stem-and-Leaf Diagrams
11.3.3 Identifying Median and Mode from Stem-and-Leaf
11.3.4 Plotting Dot Plots from Small Data Sets
11.3.5 Comparing Data Sets Using Dot Plots
11.4.1 Understanding Strengths of Different Graph Types
11.4.2 Matching Data Types to Appropriate Charts
11.4.3 Avoiding Misleading Graphs
11.4.4 Using Visual Aids for Data Presentation
11.4.5 Justifying Graph Choices in Reports
11.5.1 Understanding Discrete vs Continuous Data
11.5.2 Identifying Data Types in Real-Life Examples
11.5.3 Choosing Appropriate Data Collection Methods
11.5.4 Organizing Data by Type
11.5.5 Comparing Discrete and Continuous Graphs
11.6.1 Creating Effective Survey Questions
11.6.2 Avoiding Bias in Data Collection
11.6.3 Choosing Sample Size and Target Group
11.6.4 Conducting Surveys and Recording Responses
11.6.5 Representing Survey Results Visually
12. Mathematical Modelling and Real-World Applications
12.1.1 Reading Graphs Representing Real-Life Situations
12.1.2 Describing Trends and Patterns in Graphs
12.1.3 Identifying Key Points: Maxima, Minima, and Intercepts
12.1.4 Relating Graph Shape to Real-World Meaning
12.1.5 Comparing Different Graphs Representing the Same Data
12.2.1 Finding the Best Value Among Options
12.2.2 Using Unit Cost to Compare Products
12.2.3 Solving Discount and Offer Problems
12.2.4 Minimizing Waste in Packaging or Cutting
12.2.5 Using Math to Support Financial Decisions
12.3.1 Formulating a Mathematical Investigation Question
12.3.2 Collecting and Organizing Relevant Data
12.3.3 Choosing the Right Representation for the Model
12.3.4 Making Assumptions and Estimations in Modelling
12.3.5 Interpreting Results and Drawing Conclusions
12.4.1 Understanding Algorithms and Flowcharts
12.4.2 How GPS and Navigation Use Math
12.4.3 Exploring Patterns in Digital Media
12.4.4 Applications of Math in Mobile Apps and Games
12.4.5 Ethics and Impact of Data Usage in Society
12.5.1 Creating a Simple Budget Plan
12.5.2 Understanding Income and Expenses
12.5.3 Currency Conversion and Exchange Rates
12.5.4 Solving Word Problems on Spending and Saving
12.5.5 Planning for Short-Term and Long-Term Expenses
12.6.1 Understanding the Relationship Between Speed, Distance, and Time
12.6.2 Using the Formula: Speed = Distance ÷ Time
12.6.3 Solving Problems with Unit Conversions
12.6.4 Using Graphs to Represent Motion
12.6.5 Solving Multi-Step Journey Problems
13. Number Operations and Applications
13.1.1 Understanding the Order of Operations
13.1.2 Applying Rules to Whole Numbers
13.1.3 Applying Rules with Fractions and Decimals
13.1.4 Expressions with Brackets and Exponents
13.1.5 Common Mistakes in BODMAS Problems
13.2.1 Divisibility Rules for Numbers 2–10
13.2.2 Finding Remainders in Division
13.2.3 Prime Divisors and Factorization
13.2.4 Applications of Remainders in Context
13.2.5 Testing for Divisibility in Large Numbers
13.3.1 Addition and Subtraction of Whole Numbers
13.3.2 Multiplication and Division of Whole Numbers
13.3.3 Properties of Operations
13.3.4 Using Brackets in Calculations
13.3.5 Estimating and Checking Calculations
13.4.1 Single-Step Word Problems
13.4.2 Multi-Step Word Problems
13.4.3 Choosing the Correct Operation
13.4.4 Solving with Units and Conversions
13.4.5 Modeling Problems with Diagrams
13.5.1 Place Value in Decimals
13.5.2 Adding and Subtracting Decimals
13.5.3 Multiplying and Dividing Decimals
13.5.4 Converting Between Fractions and Decimals
13.5.5 Real-Life Applications of Decimals
13.6.1 Math in Shopping and Discounts
13.6.2 Banking, Savings, and Transactions
13.6.3 Math in Cooking and Recipes
13.6.4 Using Math in Travel and Time Zones
13.6.5 Estimating in Daily Life Situations
13.7.1 Equivalent Fractions
13.7.2 Simplifying Fractions
13.7.3 Adding and Subtracting Fractions
13.7.4 Multiplying and Dividing Fractions
13.7.5 Fractions in Measurement and Context
13.8.1 Converting Between Mixed and Improper Forms
13.8.2 Adding and Subtracting Mixed Numbers
13.8.3 Multiplying and Dividing Mixed Numbers
13.8.4 Word Problems Involving Mixed Numbers
13.8.5 Applications in Geometry and Measurement
Finding Factors of a Number

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Finding Factors of a Number

Introduction

Understanding how to find factors of a number is a fundamental concept in mathematics, particularly within the curriculum of the IB Middle Years Programme (MYP) for students in grades 1-3. This topic not only lays the groundwork for more advanced mathematical concepts such as multiples, least common multiples (LCM), and highest common factors (HCF), but it also enhances problem-solving and analytical skills critical for academic success in mathematics.

Key Concepts

Definition of Factors

A factor of a number is an integer that can be multiplied by another integer to produce the original number. In other words, if a and b are integers such that a × b = c, then both a and b are factors of c. Understanding factors is essential for simplifying fractions, finding common denominators, and solving equations.

Prime and Composite Numbers

Numbers are classified based on the number of factors they possess:

  • Prime Numbers: These are numbers greater than 1 that have exactly two distinct factors: 1 and the number itself. Examples include 2, 3, 5, 7, 11, and 13.
  • Composite Numbers: These are numbers that have more than two factors. For instance, 4 has factors 1, 2, and 4.

Understanding the difference between prime and composite numbers is crucial for finding factors efficiently.

Methods to Find Factors

There are several systematic methods to determine the factors of a number:

  1. Prime Factorization: Breaking down a number into its prime factors using a factor tree or division method.
  2. Division Method: Dividing the number by integers starting from 1 up to the number itself to identify factors.
  3. Listing Method: Listing all possible pairs of numbers that multiply to give the original number.

Prime Factorization

Prime factorization involves expressing a number as a product of prime numbers. This method is efficient for finding all factors of a number. The process begins by dividing the number by the smallest prime number possible and continues until all factors are prime.

For example, to find the prime factors of 60:

  1. 60 ÷ 2 = 30
  2. 30 ÷ 2 = 15
  3. 15 ÷ 3 = 5
  4. 5 ÷ 5 = 1

Thus, the prime factors of 60 are 2, 2, 3, and 5.

Finding All Factors Using Prime Factors

Once the prime factors are identified, all factors of the number can be determined by finding all possible combinations of these prime factors.

Using the example of 60 with prime factors 2, 2, 3, and 5:

  • 1 (no prime factors)
  • 2 (single 2)
  • 3 (single 3)
  • 4 (2 × 2)
  • 5 (single 5)
  • 6 (2 × 3)
  • 10 (2 × 5)
  • 12 (2 × 2 × 3)
  • 15 (3 × 5)
  • 20 (2 × 2 × 5)
  • 30 (2 × 3 × 5)
  • 60 (2 × 2 × 3 × 5)

Therefore, the factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60.

Common Factors

Common factors are factors that two or more numbers share. For example, the common factors of 12 and 18 are 1, 2, 3, and 6. Identifying common factors is particularly useful when determining the greatest common factor (GCF) or highest common factor (HCF) of numbers.

Applications of Factors

Understanding factors has practical applications in various areas of mathematics and real-life scenarios:

  • Simplifying Fractions: By dividing the numerator and denominator by their GCF, fractions can be simplified to their lowest terms.
  • Solving Equations: Factors are used in solving quadratic and higher-degree equations via factoring methods.
  • Real-world Problems: Factors help in scenarios like distributing items into equal groups, organizing objects, and optimizing resources.

Least Common Multiple (LCM) and Highest Common Factor (HCF)

While not the primary focus, understanding factors is a precursor to finding the LCM and HCF of numbers:

  • LCM: The smallest number that is a multiple of two or more numbers.
  • HCF: The largest number that is a factor of two or more numbers.

These concepts are integral in solving problems involving multiple numbers, such as scheduling events or optimizing space.

Examples of Finding Factors

Let's consider a few examples to illustrate how to find factors:

Example 1: Find all factors of 28.

  1. Prime Factorization of 28:
  • 28 ÷ 2 = 14
  • 14 ÷ 2 = 7
  • 7 ÷ 7 = 1
  • Prime factors: 2, 2, 7
  • All factors: 1, 2, 4, 7, 14, 28

    • Example 2: Find all factors of 45.

    1. Prime Factorization of 45:
    • 45 ÷ 3 = 15
    • 15 ÷ 3 = 5
    • 5 ÷ 5 = 1
  • Prime factors: 3, 3, 5
  • All factors: 1, 3, 5, 9, 15, 45

    • Properties of Factors

    Several properties define the behavior of factors:

    • The number 1 is a universal factor for all integers.
    • A prime number has exactly two factors: 1 and itself.
    • The product of two factors is the original number.
    • Factors come in pairs.

    Factor Pairs

    Factor pairs are two numbers that, when multiplied together, yield the original number. Identifying factor pairs can simplify the process of finding all factors.

    For example, the factor pairs of 12 are:

    • 1 × 12
    • 2 × 6
    • 3 × 4

    Using Factor Trees

    A factor tree is a graphical representation used to break down a number into its prime factors. Each branch of the tree splits a composite number into two factors until only prime numbers remain.

    Example: Factor Tree for 36

    1. 36 can be divided by 2: 36 ÷ 2 = 18
    2. 18 can be divided by 2: 18 ÷ 2 = 9
    3. 9 can be divided by 3: 9 ÷ 3 = 3
    4. 3 is a prime number.

    Thus, the prime factors of 36 are 2, 2, 3, and 3.

    Prime Factorization Using Exponents

    Prime factors can be expressed using exponents to denote repeated multiplication. This notation simplifies the representation of prime factors.

    For example, the prime factorization of 60 is $2^2 \times 3 \times 5$.

    Divisibility Rules

    Divisibility rules are shortcuts that help determine if a number is divisible by another without performing the actual division. These rules are useful in quickly identifying factors.

    Some common divisibility rules include:

    • Divisible by 2: If the number is even.
    • Divisible by 3: If the sum of its digits is divisible by 3.
    • Divisible by 5: If the number ends in 0 or 5.

    Finding Factors Using Divisibility Rules

    Applying divisibility rules can expedite the process of finding factors. For example, to find factors of 30:

    • 30 is even, so it's divisible by 2.
    • The sum of digits (3 + 0 = 3) is divisible by 3.
    • 30 ends with 0, so it's divisible by 5.

    These rules help identify potential factors without exhaustive testing.

    Greatest Common Factor (GCF)

    The Greatest Common Factor (GCF), also known as the Highest Common Factor (HCF), of two or more numbers is the largest factor that all the numbers share. Finding the GCF is essential in simplifying fractions and solving problems involving ratios.

    Example: Find the GCF of 12 and 18.

    1. Factors of 12: 1, 2, 3, 4, 6, 12
    2. Factors of 18: 1, 2, 3, 6, 9, 18
    3. Common factors: 1, 2, 3, 6
    4. GCF: 6

    Least Common Multiple (LCM)

    The Least Common Multiple (LCM) of two or more numbers is the smallest number that is a multiple of each of the numbers. Identifying the LCM is useful in adding and subtracting fractions with different denominators.

    Example: Find the LCM of 4 and 5.

    1. Multiples of 4: 4, 8, 12, 16, 20, 24, ...
    2. Multiples of 5: 5, 10, 15, 20, 25, 30, ...
    3. Common multiples: 20, 40, 60, ...
    4. LCM: 20

    Comparison Table

    Aspect Factors Multiples
    Definition Numbers that divide the original number without a remainder. Numbers obtained by multiplying the original number by integers.
    Examples Factors of 6: 1, 2, 3, 6 Multiples of 6: 6, 12, 18, 24, ...
    Usage Used to simplify fractions, find GCF/HCF. Used to find common denominators, LCM.
    Relation to Prime Numbers Prime numbers have exactly two factors: 1 and themselves. Not directly related; multiples can be prime or composite.
    Calculation Method Identify all possible divisors of the number. Multiply the number by successive integers.

    Summary and Key Takeaways

    • Factors are integers that divide a number without leaving a remainder.
    • Prime factorization simplifies finding all factors of a number.
    • Common factors are essential for determining the GCF.
    • Understanding factors aids in solving real-life mathematical problems.
    • Comparison between factors and multiples clarifies their distinct roles in mathematics.

    Coming Soon!

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    Examiner Tip
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    Tips

    Mastering factors can be easier with these tips:

    • Use Factor Trees: Visualizing the factorization process helps in breaking down numbers efficiently.
    • Memorize Divisibility Rules: Quickly identify potential factors without lengthy calculations.
    • Practice Regularly: Frequent exercises solidify understanding and improve speed in finding factors.
    • Mnemonic for Prime Factors: "Please Find All Prime Factors" can remind you to break down completely.

    Did You Know
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    Did You Know

    Did you know that the concept of factors dates back to ancient civilizations? The Babylonians used factorization for architectural designs and astronomy calculations. Additionally, factorization plays a crucial role in modern cryptography, where large prime factors ensure the security of digital communications. Understanding factors not only helps in mathematics but also contributes to advancements in technology and science.

    Common Mistakes
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    Common Mistakes

    Students often make mistakes when identifying factors, such as:

    • Incorrect: Listing 1 and the number itself as the only factors, ignoring other possible divisors.
      Correct: Systematically checking all integers up to the number to identify all factors.
    • Incorrect: Confusing multiples with factors.
      Correct: Remembering that factors divide the number without a remainder, while multiples are results of multiplying the number by integers.
    • Incorrect: Misapplying prime factorization by not fully breaking down composite numbers.
      Correct: Ensuring all factors are prime by repeatedly dividing until only prime numbers remain.

    FAQ

    What is a factor of a number?
    A factor of a number is an integer that can be multiplied by another integer to exactly produce the original number without any remainder.
    How do you find the prime factors of a number?
    To find the prime factors, repeatedly divide the number by the smallest possible prime number until all factors are prime.
    What is the difference between factors and multiples?
    Factors are numbers that divide another number without a remainder, while multiples are the products obtained by multiplying the number by integers.
    Why is finding the GCF important?
    Finding the Greatest Common Factor (GCF) is important for simplifying fractions, solving ratios, and optimizing problems that involve dividing things into equal parts.
    Can a number have a negative factor?
    Yes, every positive factor has a corresponding negative factor. For example, the factors of 6 include both 2 and -2.
    How are factors used in real-life applications?
    Factors are used in real-life applications such as organizing events, packaging, dividing resources evenly, and in various engineering and architectural designs.
    1. Algebra and Expressions
    1.1.1 Combining Like Terms
    1.1.2 Using the Distributive Property
    1.1.3 Simplifying Expressions with Brackets
    1.1.4 Rearranging Terms to Standard Form
    1.1.5 Factoring Common Factors from Terms
    1.2.1 Expanding Single Bracket Expressions
    1.2.2 Factoring Common Terms from Expressions
    1.2.3 Understanding the Reverse Process of Expansion
    1.2.4 Creating Equivalent Expressions through Expansion
    1.2.5 Solving Problems Involving Bracket Expansion
    1.3.1 Creating Algebraic Expressions from Word Problems
    1.3.2 Modeling Relationships Using Expressions
    1.3.3 Solving Real-Life Problems with Algebra
    1.3.4 Writing Multi-Step Expressions from Situations
    1.3.5 Checking Solutions for Word-Based Expressions
    1.4.1 Substituting Values into Simple Expressions
    1.4.2 Substitution with Negative and Decimal Values
    1.4.3 Using Substitution in Real-World Formulae
    1.4.4 Evaluating Algebraic Formulas with Given Values
    1.4.5 Solving Problems Using Substitution Techniques
    1.5.1 Understanding the Use of Letters in Math
    1.5.2 Defining Constants and Coefficients
    1.5.3 Recognizing Like and Unlike Terms
    1.5.4 Using Variables in Real-Life Situations
    1.5.5 Building Expressions from Verbal Descriptions
    1.6.1 Translating Words into Algebraic Expressions
    1.6.2 Interpreting Algebraic Expressions in Context
    1.6.3 Creating Expressions from Situational Problems
    1.6.4 Forming Expressions with Multiple Terms
    1.6.5 Constructing Expressions with Brackets
    2. Geometry – Properties of Shape
    2.1.1 Classifying Triangles by Sides and Angles
    2.1.2 Properties of Isosceles, Equilateral, and Right Triangles
    2.1.3 Understanding Interior Angles of Polygons
    2.1.4 Properties of Squares, Rectangles, Parallelograms, and Trapeziums
    2.1.5 Sum of Interior and Exterior Angles in Polygons
    2.2.1 Identifying Line Symmetry in Shapes
    2.2.2 Understanding Rotational Symmetry
    2.2.3 Creating Symmetrical Designs
    2.2.4 Tessellating Shapes and Patterns
    2.2.5 Exploring Regular and Irregular Tessellations
    2.3.1 Understanding Congruent Shapes
    2.3.2 Testing for Congruence with Rigid Transformations
    2.3.3 Exploring Similar Shapes and Proportionality
    2.3.4 Using Scale Factors to Determine Similarity
    2.3.5 Applications of Congruence and Similarity in Real Life
    2.4.1 Naming Parts of a Circle: Radius, Diameter, Chord, Arc
    2.4.2 Understanding Central Angle and Circumference
    2.4.3 Drawing Circles with Compass and Ruler
    2.4.4 Exploring Tangents and Secants (Introductory)
    2.4.5 Symmetry and Angle Properties in Circles
    2.5.1 Using Geometric Reasoning to Find Missing Angles
    2.5.2 Angle Sum in a Triangle and Quadrilateral
    2.5.3 Vertically Opposite and Corresponding Angles
    2.5.4 Alternate Interior and Exterior Angles
    2.5.5 Solving Problems with Justification and Proof
    2.6.1 Identifying Types of Lines and Angles
    2.6.2 Complementary, Supplementary, and Vertical Angles
    2.6.3 Angles on a Straight Line and Around a Point
    2.6.4 Angle Relationships in Parallel Lines and Transversals
    2.6.5 Constructing and Measuring Angles with Protractor
    3. Ratio, Proportion & Percentages
    3.1.1 Understanding Direct Proportion Relationships
    3.1.2 Solving Problems Involving Direct Proportion
    3.1.3 Understanding Inverse Proportion Relationships
    3.1.4 Solving Problems Involving Inverse Proportion
    3.1.5 Real-Life Applications of Proportionality
    3.2.1 Understanding Percentages as Fractions and Decimals
    3.2.2 Finding a Percentage of a Quantity
    3.2.3 Solving Percentage Increase and Decrease Problems
    3.2.4 Understanding Compound Percentage Growth
    3.2.5 Applications of Compound Interest and Depreciation
    3.3.1 Calculating Percentage Change
    3.3.2 Identifying Percentage Markups and Discounts
    3.3.3 Comparing Percentage Gains and Losses
    3.3.4 Solving Real-Life Percent Change Scenarios
    3.3.5 Using Percentage Multipliers
    3.4.1 Understanding the Concept of Reverse Percentages
    3.4.2 Finding Original Amounts Before Change
    3.4.3 Solving Markup and Discount Reversal Problems
    3.4.4 Real-World Applications of Reverse Percentage
    3.4.5 Using Equations to Solve Reverse Percentages
    3.5.1 Calculating Profit and Loss with Percentages
    3.5.2 Using Discounts in Shopping Scenarios
    3.5.3 Understanding and Applying Sales Tax
    3.5.4 Solving Net Price and Final Cost Problems
    3.5.5 Applying Multiple Percent Changes Together
    3.6.1 Defining Ratios and Their Notation
    3.6.2 Simplifying Ratios to Lowest Terms
    3.6.3 Finding Missing Values in Equivalent Ratios
    3.6.4 Ratio Word Problems in Real-Life Contexts
    3.6.5 Ratios Involving More Than Two Quantities
    3.7.1 Sharing Amounts in a Given Ratio
    3.7.2 Adjusting Quantities Using Ratios
    3.7.3 Comparing Ratios in Real-World Scenarios
    3.7.4 Applying Ratios to Scale Models and Maps
    3.7.5 Ratio-Based Recipes and Mixtures
    4. Patterns, Sequences & Algebraic Thinking
    5. Statistics – Averages and Analysis
    5.1.1 Grouping Data into Class Intervals
    5.1.2 Estimating Mean from Grouped Data
    5.1.3 Finding the Modal Class
    5.1.4 Calculating Midpoints for Intervals
    5.1.5 Using Frequency Tables with Class Intervals
    5.2.1 Using the Mean to Find a Missing Value
    5.2.2 Solving Problems with Given Total and Frequency
    5.2.3 Word Problems Involving Unknowns in Data Sets
    5.2.4 Creating Balanced Data Sets
    5.2.5 Validating Solutions with Back-Substitution
    5.3.1 Understanding the Structure of a Box Plot
    5.3.2 Finding the Five-Number Summary
    5.3.3 Calculating and Interpreting the Interquartile Range
    5.3.4 Drawing Box Plots from Data Sets
    5.3.5 Comparing Data Sets Using Box Plots
    5.4.1 Recognizing Misleading Graphs
    5.4.2 Common Errors in Average Interpretation
    5.4.3 Manipulating Data for Persuasion
    5.4.4 Critically Evaluating Data in Media
    5.4.5 Ethical Use of Statistics
    5.5.1 Finding the Mean of a Data Set
    5.5.2 Identifying the Median Value
    5.5.3 Determining the Mode in a Data Set
    5.5.4 Calculating the Range and Interpreting It
    5.5.5 Handling Data Sets with Repeated or Missing Values
    5.6.1 Comparing Mean, Median, and Mode in Context
    5.6.2 Identifying Outliers and Their Effects
    5.6.3 Choosing the Most Appropriate Average
    5.6.4 Using Averages to Summarize Data
    5.6.5 Analyzing Real-Life Data with Averages
    6. Number Concepts & Systems
    6.1.1 Natural and Whole Numbers
    6.1.2 Integers and Their Uses
    6.1.3 Even, Odd, and Prime Numbers
    6.1.4 Rational vs Irrational Numbers
    6.1.5 Real Number System Overview
    6.2.1 Defining Rational Numbers
    6.2.2 Defining Irrational Numbers
    6.2.3 Rational Numbers as Terminating and Repeating Decimals
    6.2.4 Irrational Numbers in Square Roots and Pi
    6.2.5 Visualizing Rational vs Irrational on a Number Line
    6.3.1 Rounding Whole Numbers
    6.3.2 Rounding Decimals to Given Places
    6.3.3 Estimating Sums and Differences
    6.3.4 Estimating Products and Quotients
    6.3.5 Significant Figures and Scientific Estimation
    6.4.1 Place Value in Large Numbers
    6.4.2 Decimal Place Value and Precision
    6.4.3 Reading and Writing Numbers in Words
    6.4.4 Expanded and Standard Forms
    6.4.5 Comparing and Ordering Decimals
    6.5.1 Number Line Representation
    6.5.2 Ordering Positive and Negative Numbers
    6.5.3 Inequalities and Symbols
    6.5.4 Rounding Numbers to Nearest 10, 100, 1000
    6.5.5 Rounding Decimals to Significant Figures
    6.6.1 Prime Numbers up to 100
    6.6.2 Composite Numbers and Their Properties
    6.6.3 Square Numbers and Their Patterns
    6.6.4 Perfect Squares and Their Roots
    6.6.5 Applications of Square Numbers in Geometry
    6.8.1 Representing Integers on Number Line
    6.8.2 Comparing and Ordering Integers
    6.8.3 Absolute Value of an Integer
    6.8.4 Using Integers in Real-Life Contexts
    6.8.5 Opposites and Zero on the Number Line
    7. Geometry – Measurement & Calculation
    7.1.1 Understanding Surface Area Conceptually
    7.1.2 Surface Area of a Cube
    7.1.3 Surface Area of a Rectangular Prism
    7.1.4 Surface Area of Triangular Prisms
    7.1.5 Solving Surface Area Problems in Context
    7.2.1 Volume of a Cube
    7.2.2 Volume of a Rectangular Prism
    7.2.3 Volume of a Triangular Prism
    7.2.4 Introduction to Volume of Cylinders
    7.2.5 Solving Volume Problems with Units
    7.3.1 Breaking Down Compound Shapes into Simple Parts
    7.3.2 Area and Perimeter of Compound Shapes
    7.3.3 Drawing and Interpreting Nets of 3D Shapes
    7.3.4 Surface Area Using Nets
    7.3.5 Constructing 3D Shapes from Nets
    7.4.1 Understanding Scale in Maps and Blueprints
    7.4.2 Creating and Interpreting Scale Diagrams
    7.4.3 Reading Bearings and Compass Directions
    7.4.4 Measuring Bearings with a Protractor
    7.4.5 Solving Problems with Bearings and Distances
    7.5.1 Converting Between Metric Length Units
    7.5.2 Converting Area Units (cm², m², km²)
    7.5.3 Converting Volume Units (cm³, m³, L)
    7.5.4 Using Unit Conversion in Word Problems
    7.5.5 Common Mistakes in Unit Conversion
    7.6.1 Calculating Perimeter of Squares and Rectangles
    7.6.2 Finding Perimeter of Irregular Shapes
    7.6.3 Area of Rectangles, Triangles, and Parallelograms
    7.6.4 Area of Trapeziums and Compound Shapes
    7.6.5 Solving Word Problems Involving Perimeter and Area
    8. Equations, Inequalities & Formulae
    8.1.1 Understanding Inequality Symbols
    8.1.2 Solving One-Step Inequalities
    8.1.3 Solving Two-Step Inequalities
    8.1.4 Graphing Inequalities on a Number Line
    8.1.5 Interpreting Inequalities in Real-Life Scenarios
    8.2.1 Translating Word Problems into Equations
    8.2.2 Solving Problems Involving Age and Time
    8.2.3 Using Equations in Geometry Problems
    8.2.4 Creating and Solving Multi-Step Word Problems
    8.2.5 Checking Solutions in the Context of the Problem
    8.3.1 Substituting Values into Given Formulae
    8.3.2 Rearranging Formulae to Solve for a Variable
    8.3.3 Using Standard Formulae (Area, Volume, Speed)
    8.3.4 Solving Problems Involving Real-Life Formulae
    8.3.5 Creating Formulae from Contextual Situations
    8.4.1 Using the Formula for Area of Shapes
    8.4.2 Applying the Speed = Distance ÷ Time Formula
    8.4.3 Using Volume and Surface Area Formulas
    8.4.4 Solving Finance-Related Formula Problems
    8.4.5 Interpreting and Rewriting Real-Life Formulas
    8.5.1 Understanding Equality and Balance
    8.5.2 Solving One-Step Addition and Subtraction Equations
    8.5.3 Solving One-Step Multiplication and Division Equations
    8.5.4 Solving Two-Step Linear Equations
    8.5.5 Solving Equations with Decimals and Fractions
    8.6.1 Expanding Brackets Before Solving
    8.6.2 Solving Equations with Variables on Both Sides
    8.6.3 Solving Equations Involving Fractions
    8.6.4 Clearing Fractions in Equations
    8.6.5 Solving Equations with Distribution and Simplification
    9. Probability and Outcomes
    9.1.1 Defining Theoretical Probability
    9.1.2 Conducting Simple Probability Experiments
    9.1.3 Comparing Theoretical and Experimental Results
    9.1.4 Understanding Variability in Results
    9.1.5 Improving Accuracy with Larger Samples
    9.2.1 Probability of a Single Event Occurring
    9.2.2 Using Frequency to Estimate Probability
    9.2.3 Introduction to Independent Events
    9.2.4 Calculating Combined Probabilities (Basic)
    9.2.5 Solving Real-Life Problems with Combined Events
    9.3.1 Probability of Rolling a Die
    9.3.2 Probability of Flipping a Coin
    9.3.3 Using Spinners to Model Probability
    9.3.4 Using Multiple Dice or Coins
    9.3.5 Creating Fair and Biased Models
    9.4.1 Understanding Tree Diagrams for Two Events
    9.4.2 Calculating Basic Probabilities Using Tree Diagrams
    9.4.3 Interpreting Simple Venn Diagrams
    9.4.4 Using Venn Diagrams to Solve Set Problems
    9.4.5 Exploring Overlapping and Disjoint Events
    9.5.1 Understanding Probability as a Measure
    9.5.2 Probability Scale from 0 to 1
    9.5.3 Language of Probability (Certain, Likely, etc.)
    9.5.4 Describing Events as Likely or Unlikely
    9.5.5 Identifying Impossible and Certain Events
    9.6.1 Using Lists and Tables to Find Outcomes
    9.6.2 Drawing Sample Space Diagrams
    9.6.3 Counting Outcomes for Combined Events
    9.6.4 Systematic Listing Methods
    9.6.5 Using Tree Diagrams to List Outcomes (Introductory)
    10. Geometry – Coordinates and Transformations
    11. Data Handling and Representation
    11.1.1 Creating Frequency Tables from Raw Data
    11.1.2 Using Tally Charts for Counting
    11.1.3 Reading and Interpreting Frequency Data
    11.1.4 Grouping Data into Class Intervals
    11.1.5 Using Frequency Data for Further Calculations
    11.2.1 Drawing and Interpreting Bar Charts
    11.2.2 Plotting and Analyzing Line Graphs
    11.2.3 Constructing Pie Charts with a Protractor
    11.2.4 Converting Frequency into Angles for Pie Charts
    11.2.5 Choosing the Right Graph for the Data
    11.3.1 Understanding Stem-and-Leaf Format
    11.3.2 Creating and Interpreting Stem-and-Leaf Diagrams
    11.3.3 Identifying Median and Mode from Stem-and-Leaf
    11.3.4 Plotting Dot Plots from Small Data Sets
    11.3.5 Comparing Data Sets Using Dot Plots
    11.4.1 Understanding Strengths of Different Graph Types
    11.4.2 Matching Data Types to Appropriate Charts
    11.4.3 Avoiding Misleading Graphs
    11.4.4 Using Visual Aids for Data Presentation
    11.4.5 Justifying Graph Choices in Reports
    11.5.1 Understanding Discrete vs Continuous Data
    11.5.2 Identifying Data Types in Real-Life Examples
    11.5.3 Choosing Appropriate Data Collection Methods
    11.5.4 Organizing Data by Type
    11.5.5 Comparing Discrete and Continuous Graphs
    11.6.1 Creating Effective Survey Questions
    11.6.2 Avoiding Bias in Data Collection
    11.6.3 Choosing Sample Size and Target Group
    11.6.4 Conducting Surveys and Recording Responses
    11.6.5 Representing Survey Results Visually
    12. Mathematical Modelling and Real-World Applications
    12.1.1 Reading Graphs Representing Real-Life Situations
    12.1.2 Describing Trends and Patterns in Graphs
    12.1.3 Identifying Key Points: Maxima, Minima, and Intercepts
    12.1.4 Relating Graph Shape to Real-World Meaning
    12.1.5 Comparing Different Graphs Representing the Same Data
    12.2.1 Finding the Best Value Among Options
    12.2.2 Using Unit Cost to Compare Products
    12.2.3 Solving Discount and Offer Problems
    12.2.4 Minimizing Waste in Packaging or Cutting
    12.2.5 Using Math to Support Financial Decisions
    12.3.1 Formulating a Mathematical Investigation Question
    12.3.2 Collecting and Organizing Relevant Data
    12.3.3 Choosing the Right Representation for the Model
    12.3.4 Making Assumptions and Estimations in Modelling
    12.3.5 Interpreting Results and Drawing Conclusions
    12.4.1 Understanding Algorithms and Flowcharts
    12.4.2 How GPS and Navigation Use Math
    12.4.3 Exploring Patterns in Digital Media
    12.4.4 Applications of Math in Mobile Apps and Games
    12.4.5 Ethics and Impact of Data Usage in Society
    12.5.1 Creating a Simple Budget Plan
    12.5.2 Understanding Income and Expenses
    12.5.3 Currency Conversion and Exchange Rates
    12.5.4 Solving Word Problems on Spending and Saving
    12.5.5 Planning for Short-Term and Long-Term Expenses
    12.6.1 Understanding the Relationship Between Speed, Distance, and Time
    12.6.2 Using the Formula: Speed = Distance ÷ Time
    12.6.3 Solving Problems with Unit Conversions
    12.6.4 Using Graphs to Represent Motion
    12.6.5 Solving Multi-Step Journey Problems
    13. Number Operations and Applications
    13.1.1 Understanding the Order of Operations
    13.1.2 Applying Rules to Whole Numbers
    13.1.3 Applying Rules with Fractions and Decimals
    13.1.4 Expressions with Brackets and Exponents
    13.1.5 Common Mistakes in BODMAS Problems
    13.2.1 Divisibility Rules for Numbers 2–10
    13.2.2 Finding Remainders in Division
    13.2.3 Prime Divisors and Factorization
    13.2.4 Applications of Remainders in Context
    13.2.5 Testing for Divisibility in Large Numbers
    13.3.1 Addition and Subtraction of Whole Numbers
    13.3.2 Multiplication and Division of Whole Numbers
    13.3.3 Properties of Operations
    13.3.4 Using Brackets in Calculations
    13.3.5 Estimating and Checking Calculations
    13.4.1 Single-Step Word Problems
    13.4.2 Multi-Step Word Problems
    13.4.3 Choosing the Correct Operation
    13.4.4 Solving with Units and Conversions
    13.4.5 Modeling Problems with Diagrams
    13.5.1 Place Value in Decimals
    13.5.2 Adding and Subtracting Decimals
    13.5.3 Multiplying and Dividing Decimals
    13.5.4 Converting Between Fractions and Decimals
    13.5.5 Real-Life Applications of Decimals
    13.6.1 Math in Shopping and Discounts
    13.6.2 Banking, Savings, and Transactions
    13.6.3 Math in Cooking and Recipes
    13.6.4 Using Math in Travel and Time Zones
    13.6.5 Estimating in Daily Life Situations
    13.7.1 Equivalent Fractions
    13.7.2 Simplifying Fractions
    13.7.3 Adding and Subtracting Fractions
    13.7.4 Multiplying and Dividing Fractions
    13.7.5 Fractions in Measurement and Context
    13.8.1 Converting Between Mixed and Improper Forms
    13.8.2 Adding and Subtracting Mixed Numbers
    13.8.3 Multiplying and Dividing Mixed Numbers
    13.8.4 Word Problems Involving Mixed Numbers
    13.8.5 Applications in Geometry and Measurement
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