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math | ib-myp-4-5
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1. Graphs and Relations
1.1.1 Translating Graphs Vertically and Horizontally
1.1.2 Stretching and Compressing Graphs
1.1.3 Reflections Across Axes
1.1.4 Combining Multiple Transformations
1.1.5 Describing Graph Changes Algebraically
1.2.1 Reading Distance-Time Graphs
1.2.2 Interpreting Speed-Time (Velocity-Time) Graphs
1.2.3 Identifying Sections of Constant and Changing Rates
1.2.4 Connecting Graphs to Real Situations
1.2.5 Estimating Values and Trends from Graphs
1.3.1 Plotting Points in Four Quadrants
1.3.2 Identifying Coordinates from a Graph
1.3.3 Midpoint and Distance Between Two Points
1.3.4 Real-Life Uses of Cartesian Coordinates
1.3.5 Interpreting Grids and Map Coordinates
1.4.1 Drawing Graphs from y = mx + c
1.4.2 Identifying Slope and Y-Intercept
1.4.3 Table of Values and Line Plotting
1.4.4 Interpreting and Writing Linear Equations from Graphs
1.4.5 Parallel and Perpendicular Lines (Introductory)
1.5.1 Sketching y = ax² + bx + c
1.5.2 Finding Vertex and Axis of Symmetry
1.5.3 Understanding Graph Shape and Turning Points
1.5.4 Graphing Simple Cubic Functions
1.5.5 Graphing and Identifying Exponential Growth and Decay
1.6.1 Gradient of Parallel Lines
1.6.2 Gradient of Perpendicular Lines
1.6.3 Using Slope Relationships in Graphs
1.6.4 Finding Equations of Lines Given Conditions
1.6.5 Verifying Line Relationships Algebraically
2. Statistics and Probability
2.1.1 Conducting Experiments and Simulations
2.1.2 Relative Frequency and Long-Term Trends
2.1.3 Comparing Experimental and Expected Results
2.1.4 Fairness, Bias, and Randomness
2.1.5 Applications in Real-World Risk Assessment
2.2.1 Types of Data: Discrete vs Continuous
2.2.2 Methods of Data Collection and Sampling
2.2.3 Frequency Tables and Two-Way Tables
2.2.4 Designing Surveys and Questionnaires
2.2.5 Organizing Data for Effective Analysis
2.3.1 Defining Independence in Probability
2.3.2 Recognizing and Modeling Dependent Events
2.3.3 Conditional Probability (Introductory)
2.3.4 Using Probability Trees for Multi-Stage Events
2.3.5 Real-Life Applications: Games, Genetics, and Decision-Making
2.4.1 Calculating Mean, Median, and Mode
2.4.2 Choosing the Appropriate Measure
2.4.3 Mean from a Frequency Table
2.4.4 Comparing Data Sets Using Averages
2.4.5 Real-Life Interpretation of Averages
2.5.1 Range and Interquartile Range
2.5.2 Standard Deviation (Introductory Concepts)
2.5.3 Box and Whisker Plots for Spread of Data
2.5.4 Comparing Variability Between Data Sets
2.5.5 Impact of Outliers on Dispersion Measures
2.6.1 Constructing and Interpreting Box Plots
2.6.2 Drawing and Reading Histograms
2.6.3 Using Class Intervals in Histograms
2.6.4 Cumulative Frequency Graphs and Curves
2.6.5 Finding the Median and Quartiles from Cumulative Graphs
2.7.1 Theoretical Probability and Sample Space Diagrams
2.7.2 Complementary Events and Basic Rules
2.7.3 Tree Diagrams for Independent Events
2.7.4 Tree Diagrams for Dependent Events
2.7.5 Calculating Combined Probabilities
3. Trigonometry
3.1.1 Working with Two Sides and Included Angle (SAS)
3.1.2 Solving ASA and AAS Triangle Situations
3.1.3 Using Given Side and Opposite Angle (SSA)
3.1.4 Multiple Solutions in Ambiguous SSA Cases
3.1.5 Real-Life Applications in Engineering and Design
3.2.1 Understanding Bearings as Directional Angles
3.2.2 Solving Bearings Questions Using Trigonometry
3.2.3 Drawing Diagrams from Bearings Descriptions
3.2.4 Multi-Step Navigation Problems
3.2.5 Using Scale and Bearings Together
3.3.1 Sketching Graphs of y = sin x, cos x, tan x
3.3.2 Amplitude, Period, and Phase Shift
3.3.3 Transforming Trig Graphs Algebraically
3.3.4 Graphing Real-Life Periodic Phenomena
3.3.5 Solving Equations Graphically
3.4.1 Defining Sine, Cosine, and Tangent
3.4.2 Using SOH-CAH-TOA to Find Missing Sides
3.4.3 Using Trig Ratios to Find Missing Angles
3.4.4 Solving Word Problems with Right-Angled Triangles
3.4.5 Using Trigonometry with Bearings and Heights
3.5.1 Identifying Right Triangles in 3D Solids
3.5.2 Calculating Diagonals and Angles in Cuboids
3.5.3 Combining Pythagoras and Trig in 3D Geometry
3.5.4 Applications in Architecture and Physics
3.5.5 Word Problems Involving 3D Trigonometric Reasoning
3.6.1 Conditions for Using the Sine Rule
3.6.2 Applying the Cosine Rule to Find Sides or Angles
3.6.3 Choosing Between Sine Rule and Cosine Rule
3.6.4 Finding the Area of Any Triangle Using Trigonometry
3.6.5 Problem Solving with Non-Right Triangles
4. Algebraic Expressions and Identities
4.1.1 Simplifying Algebraic Fractions
4.1.2 Multiplying and Dividing Algebraic Fractions
4.1.3 Adding and Subtracting with Common Denominators
4.1.4 Algebraic Fraction Word Problems
4.1.5 Restrictions and Undefined Values in Expressions
4.2.1 Product, Quotient, and Power Laws
4.2.2 Negative and Zero Exponents
4.2.3 Exponents in Algebraic Equations
4.2.4 Simplifying Complex Exponential Expressions
4.2.5 Word Problems Involving Exponents
4.3.1 Perfect Square Identities
4.3.2 Difference of Squares Identity
4.3.3 Sum and Difference of Cubes (Introductory)
4.3.4 Recognizing and Using Identities in Problems
4.3.5 Deriving Identities from Expansion and Factorization
4.4.1 Simplifying Like Terms
4.4.2 Expanding Single Brackets
4.4.3 Expanding Double Brackets (Binomials)
4.4.4 Simplifying Expressions with Multiple Terms
4.4.5 Combining Like Terms After Expansion
4.5.1 Substituting Numbers into Algebraic Expressions
4.5.2 Evaluating with Negative Numbers and Fractions
4.5.3 Solving Word Problems with Substitution
4.5.4 Using Substitution in Formulas
4.5.5 Multiple Substitutions in Complex Expressions
4.6.1 Factoring Out the Greatest Common Factor
4.6.2 Factoring Trinomials (Quadratic Form)
4.6.3 Factoring Difference of Squares
4.6.4 Grouping and Advanced Patterns
4.6.5 Choosing Appropriate Factorization Methods
5. Geometry and Measurement
5.1.1 Constructing Perpendicular and Angle Bisectors
5.1.2 Constructing Triangles with Given Conditions
5.1.3 Using Compass and Ruler for Geometric Constructions
5.1.4 Loci from Fixed Distances or Rules
5.1.5 Shading Regions Using Multiple Loci
5.2.1 Understanding Bearings in Navigation
5.2.2 Measuring and Drawing Bearings
5.2.3 Using Bearings with Trigonometry
5.2.4 Creating and Interpreting Scale Diagrams
5.2.5 Word Problems Involving Bearings and Maps
5.3.1 Types of Angles and Angle Relationships
5.3.2 Angles on a Line and Around a Point
5.3.3 Parallel Lines and Transversals
5.3.4 Interior and Exterior Angles of Polygons
5.3.5 Properties of Regular and Irregular Polygons
5.4.1 Converting Between Metric Units
5.4.2 Time, Speed, and Distance Problems
5.4.3 Density, Mass, and Volume
5.4.4 Pressure = Force ÷ Area Applications
5.4.5 Multi-Step Problems with Compound Measures
5.5.1 Criteria for Triangle Congruence (SSS, SAS, ASA, RHS)
5.5.2 Using Congruent Triangles in Proofs
5.5.3 Criteria for Similar Triangles
5.5.4 Using Scale Factor in Similar Figures
5.5.5 Real-Life Applications of Similarity and Congruence
5.6.1 Naming Parts of a Circle
5.6.2 Angles at the Centre and in the Circle
5.6.3 Angles in the Same Segment and Alternate Segment Theorem
5.6.4 Cyclic Quadrilaterals and Chords
5.6.5 Tangent Properties and Circle Geometry Proofs
6. Equations, Inequalities, and Formulae
6.1.1 Solving Linear Inequalities in One Variable
6.1.2 Representing Solutions on a Number Line
6.1.3 Using Inequality Symbols Correctly
6.1.4 Graphical Representation of Inequalities
6.1.5 Inequality Word Problems
6.2.1 Solving by Substitution Method
6.2.2 Solving by Elimination Method
6.2.3 Solving Graphically
6.2.4 Applications in Real-Life Problems
6.2.5 Checking Solutions for Both Equations
6.3.1 Solving by Factorization
6.3.2 Solving Using the Quadratic Formula
6.3.3 Solving by Completing the Square (Introductory)
6.3.4 Nature of Roots Using Discriminant
6.3.5 Word Problems Involving Quadratic Equations
6.4.1 Understanding Literal Equations
6.4.2 Rearranging to Make a Variable the Subject
6.4.3 Rearranging with Fractions and Brackets
6.4.4 Applications in Geometry and Physics Formulas
6.4.5 Common Mistakes in Rearranging
6.5.1 Translating Word Problems into Equations
6.5.2 Multi-Step Problems with Variables
6.5.3 Age, Distance, and Geometry-Based Problems
6.5.4 Choosing Appropriate Methods
6.5.5 Interpreting and Validating Solutions
6.6.1 Solving One-Step and Two-Step Equations
6.6.2 Equations with Variables on Both Sides
6.6.3 Equations Involving Brackets and Fractions
6.6.4 Word Problems Involving Linear Equations
6.6.5 Checking Solutions and Substitution
7. Number and Operations
7.1.1 Rounding to Decimal Places and Significant Figures
7.1.2 Upper and Lower Bounds
7.1.3 Calculations Using Rounded Values
7.1.4 Understanding Absolute and Percentage Error
7.1.5 Estimation in Problem Solving
7.2.1 Classifying Rational and Irrational Numbers
7.2.2 Understanding Integers, Fractions, and Decimals
7.2.3 Comparing and Ordering Real Numbers
7.2.4 Set Notation and Venn Diagrams for Number Sets
7.2.5 Number Line Representation of Real Numbers
7.3.1 Simple and Compound Interest
7.3.2 Profit, Loss, and Discount Calculations
7.3.3 Currency Conversion and Exchange Rates
7.3.4 Taxation, Budgeting, and Loan Payments
7.3.5 Solving Real-World Finance Problems
7.5.1 Estimation Techniques in Mental Math
7.5.2 Rounding Large and Small Numbers
7.5.3 Significant Figures in Measurement
7.5.4 Use of Estimation in Checking Results
7.5.5 Approximations in Scientific and Everyday Contexts
7.6.1 Laws of Exponents for Integer Powers
7.6.2 Negative and Zero Exponents
7.6.3 Square Roots and Cube Roots
7.6.4 Simplifying Expressions with Exponents
7.6.5 Rational Exponents and Surds (Introductory)
7.7.1 Writing Large and Small Numbers in Scientific Notation
7.7.2 Converting Between Forms
7.7.3 Multiplying and Dividing Numbers in Scientific Notation
7.7.4 Adding and Subtracting with Powers of 10
7.7.5 Applications in Science and Engineering Contexts
7.8.1 Simplifying Ratios and Comparing Quantities
7.8.2 Solving Direct and Inverse Proportion Problems
7.8.3 Percentage Increase and Decrease
7.8.4 Compound Percentages and Successive Changes
7.8.5 Solving Word Problems Involving Percentages
8. Sequences, Patterns, and Functions
8.1.1 Recognizing and Continuing Arithmetic Sequences
8.1.2 Finding the Common Difference
8.1.3 Deriving the nth Term of an Arithmetic Sequence
8.1.4 Solving Problems Using Arithmetic Sequences
8.1.5 Real-World Applications of Arithmetic Patterns
8.2.1 Understanding Composite Functions Notation
8.2.2 Evaluating Composite Functions
8.2.3 Concept of Inverse Functions
8.2.4 Finding Inverses Algebraically (Intro)
8.2.5 Checking if Two Functions Are Inverses
8.3.1 Recognizing Geometric Sequences
8.3.2 Finding the Common Ratio
8.3.3 General Term of a Geometric Sequence
8.3.4 Solving Word Problems with Geometric Sequences
8.3.5 Sum of a Finite Geometric Series (Introductory)
8.4.1 Distinguishing Between Linear and Non-Linear Sequences
8.4.2 Creating a Rule for the nth Term
8.4.3 Evaluating Terms Using the nth Term Formula
8.4.4 Finding Term Position Given Value
8.4.5 Applications in Tables and Graphs
8.5.1 Understanding Function Notation f(x)
8.5.2 Evaluating Functions for Given Inputs
8.5.3 Interpreting Function Values in Context
8.5.4 Defining Piecewise and Step Functions (Introductory)
8.5.5 Using Function Machines and Diagrams
8.6.1 Identifying Domain and Range from Tables
8.6.2 Determining Domain and Range from Graphs
8.6.3 Writing Inequality Notation for Domain and Range
8.6.4 Identifying Discrete and Continuous Functions
8.6.5 Applications in Real-World Graphs
9. Mensuration
10. Vectors and Transformations
10.1.1 Adding and Subtracting Vectors Graphically
10.1.2 Adding and Subtracting Column Vectors
10.1.3 Scalar Multiplication of Vectors
10.1.4 Vector Arithmetic Word Problems
10.1.5 Simplifying and Combining Vectors
10.2.1 Definition and Use of Position Vectors
10.2.2 Finding Displacement Between Two Points
10.2.3 Using Coordinates to Calculate Displacement
10.2.4 Position Vectors in Geometric Diagrams
10.2.5 Describing Movement in Terms of Vectors
10.3.1 Describing and Performing Translations
10.3.2 Reflections in Lines: x-axis, y-axis, y = x, y = -x
10.3.3 Rotations About a Point and Angle
10.3.4 Enlargement with Positive and Negative Scale Factors
10.3.5 Combining Multiple Transformations
10.4.1 Using Matrices for Simple Transformations (Introductory)
10.4.2 Applying Successive Transformations
10.4.3 Identifying Single Equivalent Transformation
10.4.4 Matrix Representation of Translations
10.4.5 Transforming Coordinates with Matrix Multiplication
10.5.1 Understanding Vectors as Arrows with Direction and Magnitude
10.5.2 Column Vector Notation
10.5.3 Representing Vectors in Diagrams and Coordinates
10.5.4 Notation and Labeling for Displacement Vectors
10.5.5 Unit Vectors and Position Vectors (Introductory)
10.6.1 Proving Collinearity Using Vectors
10.6.2 Midpoints and Parallel Vectors
10.6.3 Geometric Proofs with Vectors
10.6.4 Solving Complex Vector Path Problems
10.6.5 Applications in Navigation and Design
Converting Between Fractions, Decimals, and Percentages

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Converting Between Fractions, Decimals, and Percentages

Introduction

Understanding how to convert between fractions, decimals, and percentages is fundamental in mathematics, especially within the IB MYP 4-5 curriculum. These conversions are essential for solving real-world problems, analyzing data, and performing various mathematical operations. Mastery of these concepts enhances students' numerical fluency and prepares them for more advanced mathematical studies.

Key Concepts

Understanding Fractions, Decimals, and Percentages

Fractions, decimals, and percentages are three different ways to represent parts of a whole. Each form has its unique applications and advantages depending on the context in which it's used.

Fractions

A fraction consists of a numerator and a denominator, separated by a slash. The numerator represents how many parts we have, while the denominator indicates the total number of equal parts the whole is divided into. For example, in the fraction $\frac{3}{4}$, 3 is the numerator, and 4 is the denominator.

Decimals

Decimals are another way to represent fractions, particularly those with denominators that are powers of 10. Decimals use a decimal point to separate the whole number from the fractional part. For instance, $\frac{3}{4}$ can be expressed as $0.75$ in decimal form.

Percentages

A percentage represents a part per hundred. It is denoted by the percent sign (%). Using the previous example, $\frac{3}{4}$ is equivalent to $75\%$.

Converting Fractions to Decimals

To convert a fraction to a decimal, divide the numerator by the denominator. This process involves long division. For example, to convert $\frac{5}{8}$ to a decimal:

$$0.625 = \frac{5}{8}$$

Here, $5 ÷ 8 = 0.625$.

Converting Decimals to Fractions

Converting decimals to fractions involves identifying the place value of the last digit and using it as the denominator. For example, to convert $0.75$ to a fraction:

$$\frac{75}{100} = \frac{3}{4}$$

Simplifying $\frac{75}{100}$ by dividing both numerator and denominator by 25 yields $\frac{3}{4}$.

Converting Fractions to Percentages

To convert a fraction to a percentage, first convert the fraction to a decimal and then multiply by 100. Using $\frac{3}{4}$ as an example:

$$\frac{3}{4} = 0.75 \times 100 = 75\%$$

Converting Percentages to Fractions

To convert a percentage to a fraction, place the percentage value over 100 and simplify. For example, $80\%$ as a fraction is:

$$\frac{80}{100} = \frac{4}{5}$$

Converting Decimals to Percentages

To convert a decimal to a percentage, multiply the decimal by 100 and add the percent sign. For instance, $0.45$ becomes:

$$0.45 \times 100 = 45\%$$

Converting Percentages to Decimals

To convert a percentage to a decimal, divide the percentage value by 100 and remove the percent sign. For example, $60\%$ as a decimal is:

$$60\% = 0.60$$

Real-World Applications

These conversions are not just theoretical; they have practical applications in various fields such as finance, science, and everyday problem-solving. For example, calculating discounts during shopping requires converting percentages to decimals, while interpreting statistical data often involves understanding fractions and percentages.

Common Challenges and Solutions

Students often face challenges in performing these conversions, especially when dealing with repeating decimals or simplifying complex fractions. Practice is key to overcoming these hurdles. Utilizing visual aids like number lines and practicing with real-life examples can enhance comprehension and execution of these conversions.

Strategies for Mastery

To master these conversions, students should:

  • Practice converting between all three forms regularly.
  • Understand the underlying principles of each representation.
  • Use visual tools such as pie charts and number lines to conceptualize fractions, decimals, and percentages.
  • Simplify fractions to their lowest terms to ease conversion to decimals and percentages.

Advanced Concepts

Beyond basic conversions, understanding how these representations interact in mathematical operations like addition, subtraction, multiplication, and division is crucial. For instance, adding percentages involves converting them to the same base, and multiplying decimals requires attention to place values to ensure accuracy.

Importance in Mathematical Literacy

Proficiency in these conversions is a cornerstone of mathematical literacy. It enables students to interpret data accurately, make informed decisions based on numerical information, and apply mathematical concepts to real-world scenarios effectively.

Comparison Table

Aspect Fractions Decimals Percentages
Definition Represents a part of a whole using a numerator and denominator. Expresses numbers using a decimal point to separate whole numbers from fractional parts. Indicates a ratio per hundred.
Applications Used in precise mathematical computations and ratios. Common in financial calculations and measurements. Widely used in statistics, discounts, and probability.
Advantages Provides exact values and is fundamental in fraction operations. Facilitates easier arithmetic operations and is standardized in various measurements. Easy to interpret and compare relative sizes.
Limitations Can be cumbersome for complex calculations without conversion. May lead to rounding errors in certain applications. Less precise without converting back to fractions or decimals.

Summary and Key Takeaways

  • Fractions, decimals, and percentages are interchangeable representations of parts of a whole.
  • Conversion between these forms involves fundamental mathematical operations.
  • Mastery of these conversions is essential for mathematical literacy and real-world problem-solving.
  • Understanding the advantages and limitations of each form enhances their effective application.
  • Regular practice and application of these concepts solidify proficiency and confidence.

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Examiner Tip
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Tips

To effortlessly convert between fractions, decimals, and percentages, remember the acronym F-D-P: Fraction to Decimal to Percentage. For instance, start with a fraction, convert it to a decimal by dividing, then multiply by 100 to get a percentage. Use the "percent means per hundred" trick to simplify understanding percentages. Additionally, practicing with real-life examples, like calculating discounts or interest rates, can reinforce these conversions and aid in retaining the concepts for exam success.

Did You Know
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Did You Know

Did you know that the concept of percentages dates back to ancient Babylonian mathematics, where they used a base of 60? Additionally, understanding these conversions is crucial in fields like medicine, where dosages often require precise calculations between different numerical forms. This fundamental skill also plays a vital role in data analysis, allowing professionals to interpret and present information effectively.

Common Mistakes
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Common Mistakes

One common mistake is confusing the numerator and denominator when converting fractions to decimals, leading to incorrect results. For example, mistakenly dividing the denominator by the numerator in $\frac{2}{5}$ instead of $2 ÷ 5 = 0.4$. Another error is forgetting to simplify fractions before converting, such as leaving $\frac{50}{100}$ instead of simplifying it to $\frac{1}{2}$. Additionally, students often neglect to move the decimal correctly when converting decimals to percentages, resulting in inaccurate percentage values.

FAQ

How do you convert a fraction to a decimal?
To convert a fraction to a decimal, divide the numerator by the denominator. For example, $\frac{3}{4} = 3 ÷ 4 = 0.75$.
What is the easiest way to change a decimal to a percentage?
Multiply the decimal by 100 and add the percent sign. For instance, $0.85 \times 100 = 85\%$.
Can every decimal be expressed as a fraction?
Yes, every decimal can be expressed as a fraction by placing the decimal number over its place value and simplifying. For example, $0.5 = \frac{5}{10} = \frac{1}{2}$.
Why is simplifying fractions important?
Simplifying fractions makes it easier to perform conversions and calculations, reducing the complexity of the numbers involved. It also helps in comparing fractions by having them in their lowest terms.
How do you convert a percentage to a fraction?
Place the percentage number over 100 and simplify the fraction. For example, $50\% = \frac{50}{100} = \frac{1}{2}$.
What common errors should be avoided when converting between these forms?
Avoid confusing the numerator and denominator in fractions, forgetting to simplify fractions, and incorrectly placing the decimal when converting between decimals and percentages. Always double-check your calculations to ensure accuracy.
1. Graphs and Relations
1.1.1 Translating Graphs Vertically and Horizontally
1.1.2 Stretching and Compressing Graphs
1.1.3 Reflections Across Axes
1.1.4 Combining Multiple Transformations
1.1.5 Describing Graph Changes Algebraically
1.2.1 Reading Distance-Time Graphs
1.2.2 Interpreting Speed-Time (Velocity-Time) Graphs
1.2.3 Identifying Sections of Constant and Changing Rates
1.2.4 Connecting Graphs to Real Situations
1.2.5 Estimating Values and Trends from Graphs
1.3.1 Plotting Points in Four Quadrants
1.3.2 Identifying Coordinates from a Graph
1.3.3 Midpoint and Distance Between Two Points
1.3.4 Real-Life Uses of Cartesian Coordinates
1.3.5 Interpreting Grids and Map Coordinates
1.4.1 Drawing Graphs from y = mx + c
1.4.2 Identifying Slope and Y-Intercept
1.4.3 Table of Values and Line Plotting
1.4.4 Interpreting and Writing Linear Equations from Graphs
1.4.5 Parallel and Perpendicular Lines (Introductory)
1.5.1 Sketching y = ax² + bx + c
1.5.2 Finding Vertex and Axis of Symmetry
1.5.3 Understanding Graph Shape and Turning Points
1.5.4 Graphing Simple Cubic Functions
1.5.5 Graphing and Identifying Exponential Growth and Decay
1.6.1 Gradient of Parallel Lines
1.6.2 Gradient of Perpendicular Lines
1.6.3 Using Slope Relationships in Graphs
1.6.4 Finding Equations of Lines Given Conditions
1.6.5 Verifying Line Relationships Algebraically
2. Statistics and Probability
2.1.1 Conducting Experiments and Simulations
2.1.2 Relative Frequency and Long-Term Trends
2.1.3 Comparing Experimental and Expected Results
2.1.4 Fairness, Bias, and Randomness
2.1.5 Applications in Real-World Risk Assessment
2.2.1 Types of Data: Discrete vs Continuous
2.2.2 Methods of Data Collection and Sampling
2.2.3 Frequency Tables and Two-Way Tables
2.2.4 Designing Surveys and Questionnaires
2.2.5 Organizing Data for Effective Analysis
2.3.1 Defining Independence in Probability
2.3.2 Recognizing and Modeling Dependent Events
2.3.3 Conditional Probability (Introductory)
2.3.4 Using Probability Trees for Multi-Stage Events
2.3.5 Real-Life Applications: Games, Genetics, and Decision-Making
2.4.1 Calculating Mean, Median, and Mode
2.4.2 Choosing the Appropriate Measure
2.4.3 Mean from a Frequency Table
2.4.4 Comparing Data Sets Using Averages
2.4.5 Real-Life Interpretation of Averages
2.5.1 Range and Interquartile Range
2.5.2 Standard Deviation (Introductory Concepts)
2.5.3 Box and Whisker Plots for Spread of Data
2.5.4 Comparing Variability Between Data Sets
2.5.5 Impact of Outliers on Dispersion Measures
2.6.1 Constructing and Interpreting Box Plots
2.6.2 Drawing and Reading Histograms
2.6.3 Using Class Intervals in Histograms
2.6.4 Cumulative Frequency Graphs and Curves
2.6.5 Finding the Median and Quartiles from Cumulative Graphs
2.7.1 Theoretical Probability and Sample Space Diagrams
2.7.2 Complementary Events and Basic Rules
2.7.3 Tree Diagrams for Independent Events
2.7.4 Tree Diagrams for Dependent Events
2.7.5 Calculating Combined Probabilities
3. Trigonometry
3.1.1 Working with Two Sides and Included Angle (SAS)
3.1.2 Solving ASA and AAS Triangle Situations
3.1.3 Using Given Side and Opposite Angle (SSA)
3.1.4 Multiple Solutions in Ambiguous SSA Cases
3.1.5 Real-Life Applications in Engineering and Design
3.2.1 Understanding Bearings as Directional Angles
3.2.2 Solving Bearings Questions Using Trigonometry
3.2.3 Drawing Diagrams from Bearings Descriptions
3.2.4 Multi-Step Navigation Problems
3.2.5 Using Scale and Bearings Together
3.3.1 Sketching Graphs of y = sin x, cos x, tan x
3.3.2 Amplitude, Period, and Phase Shift
3.3.3 Transforming Trig Graphs Algebraically
3.3.4 Graphing Real-Life Periodic Phenomena
3.3.5 Solving Equations Graphically
3.4.1 Defining Sine, Cosine, and Tangent
3.4.2 Using SOH-CAH-TOA to Find Missing Sides
3.4.3 Using Trig Ratios to Find Missing Angles
3.4.4 Solving Word Problems with Right-Angled Triangles
3.4.5 Using Trigonometry with Bearings and Heights
3.5.1 Identifying Right Triangles in 3D Solids
3.5.2 Calculating Diagonals and Angles in Cuboids
3.5.3 Combining Pythagoras and Trig in 3D Geometry
3.5.4 Applications in Architecture and Physics
3.5.5 Word Problems Involving 3D Trigonometric Reasoning
3.6.1 Conditions for Using the Sine Rule
3.6.2 Applying the Cosine Rule to Find Sides or Angles
3.6.3 Choosing Between Sine Rule and Cosine Rule
3.6.4 Finding the Area of Any Triangle Using Trigonometry
3.6.5 Problem Solving with Non-Right Triangles
4. Algebraic Expressions and Identities
4.1.1 Simplifying Algebraic Fractions
4.1.2 Multiplying and Dividing Algebraic Fractions
4.1.3 Adding and Subtracting with Common Denominators
4.1.4 Algebraic Fraction Word Problems
4.1.5 Restrictions and Undefined Values in Expressions
4.2.1 Product, Quotient, and Power Laws
4.2.2 Negative and Zero Exponents
4.2.3 Exponents in Algebraic Equations
4.2.4 Simplifying Complex Exponential Expressions
4.2.5 Word Problems Involving Exponents
4.3.1 Perfect Square Identities
4.3.2 Difference of Squares Identity
4.3.3 Sum and Difference of Cubes (Introductory)
4.3.4 Recognizing and Using Identities in Problems
4.3.5 Deriving Identities from Expansion and Factorization
4.4.1 Simplifying Like Terms
4.4.2 Expanding Single Brackets
4.4.3 Expanding Double Brackets (Binomials)
4.4.4 Simplifying Expressions with Multiple Terms
4.4.5 Combining Like Terms After Expansion
4.5.1 Substituting Numbers into Algebraic Expressions
4.5.2 Evaluating with Negative Numbers and Fractions
4.5.3 Solving Word Problems with Substitution
4.5.4 Using Substitution in Formulas
4.5.5 Multiple Substitutions in Complex Expressions
4.6.1 Factoring Out the Greatest Common Factor
4.6.2 Factoring Trinomials (Quadratic Form)
4.6.3 Factoring Difference of Squares
4.6.4 Grouping and Advanced Patterns
4.6.5 Choosing Appropriate Factorization Methods
5. Geometry and Measurement
5.1.1 Constructing Perpendicular and Angle Bisectors
5.1.2 Constructing Triangles with Given Conditions
5.1.3 Using Compass and Ruler for Geometric Constructions
5.1.4 Loci from Fixed Distances or Rules
5.1.5 Shading Regions Using Multiple Loci
5.2.1 Understanding Bearings in Navigation
5.2.2 Measuring and Drawing Bearings
5.2.3 Using Bearings with Trigonometry
5.2.4 Creating and Interpreting Scale Diagrams
5.2.5 Word Problems Involving Bearings and Maps
5.3.1 Types of Angles and Angle Relationships
5.3.2 Angles on a Line and Around a Point
5.3.3 Parallel Lines and Transversals
5.3.4 Interior and Exterior Angles of Polygons
5.3.5 Properties of Regular and Irregular Polygons
5.4.1 Converting Between Metric Units
5.4.2 Time, Speed, and Distance Problems
5.4.3 Density, Mass, and Volume
5.4.4 Pressure = Force ÷ Area Applications
5.4.5 Multi-Step Problems with Compound Measures
5.5.1 Criteria for Triangle Congruence (SSS, SAS, ASA, RHS)
5.5.2 Using Congruent Triangles in Proofs
5.5.3 Criteria for Similar Triangles
5.5.4 Using Scale Factor in Similar Figures
5.5.5 Real-Life Applications of Similarity and Congruence
5.6.1 Naming Parts of a Circle
5.6.2 Angles at the Centre and in the Circle
5.6.3 Angles in the Same Segment and Alternate Segment Theorem
5.6.4 Cyclic Quadrilaterals and Chords
5.6.5 Tangent Properties and Circle Geometry Proofs
6. Equations, Inequalities, and Formulae
6.1.1 Solving Linear Inequalities in One Variable
6.1.2 Representing Solutions on a Number Line
6.1.3 Using Inequality Symbols Correctly
6.1.4 Graphical Representation of Inequalities
6.1.5 Inequality Word Problems
6.2.1 Solving by Substitution Method
6.2.2 Solving by Elimination Method
6.2.3 Solving Graphically
6.2.4 Applications in Real-Life Problems
6.2.5 Checking Solutions for Both Equations
6.3.1 Solving by Factorization
6.3.2 Solving Using the Quadratic Formula
6.3.3 Solving by Completing the Square (Introductory)
6.3.4 Nature of Roots Using Discriminant
6.3.5 Word Problems Involving Quadratic Equations
6.4.1 Understanding Literal Equations
6.4.2 Rearranging to Make a Variable the Subject
6.4.3 Rearranging with Fractions and Brackets
6.4.4 Applications in Geometry and Physics Formulas
6.4.5 Common Mistakes in Rearranging
6.5.1 Translating Word Problems into Equations
6.5.2 Multi-Step Problems with Variables
6.5.3 Age, Distance, and Geometry-Based Problems
6.5.4 Choosing Appropriate Methods
6.5.5 Interpreting and Validating Solutions
6.6.1 Solving One-Step and Two-Step Equations
6.6.2 Equations with Variables on Both Sides
6.6.3 Equations Involving Brackets and Fractions
6.6.4 Word Problems Involving Linear Equations
6.6.5 Checking Solutions and Substitution
7. Number and Operations
7.1.1 Rounding to Decimal Places and Significant Figures
7.1.2 Upper and Lower Bounds
7.1.3 Calculations Using Rounded Values
7.1.4 Understanding Absolute and Percentage Error
7.1.5 Estimation in Problem Solving
7.2.1 Classifying Rational and Irrational Numbers
7.2.2 Understanding Integers, Fractions, and Decimals
7.2.3 Comparing and Ordering Real Numbers
7.2.4 Set Notation and Venn Diagrams for Number Sets
7.2.5 Number Line Representation of Real Numbers
7.3.1 Simple and Compound Interest
7.3.2 Profit, Loss, and Discount Calculations
7.3.3 Currency Conversion and Exchange Rates
7.3.4 Taxation, Budgeting, and Loan Payments
7.3.5 Solving Real-World Finance Problems
7.5.1 Estimation Techniques in Mental Math
7.5.2 Rounding Large and Small Numbers
7.5.3 Significant Figures in Measurement
7.5.4 Use of Estimation in Checking Results
7.5.5 Approximations in Scientific and Everyday Contexts
7.6.1 Laws of Exponents for Integer Powers
7.6.2 Negative and Zero Exponents
7.6.3 Square Roots and Cube Roots
7.6.4 Simplifying Expressions with Exponents
7.6.5 Rational Exponents and Surds (Introductory)
7.7.1 Writing Large and Small Numbers in Scientific Notation
7.7.2 Converting Between Forms
7.7.3 Multiplying and Dividing Numbers in Scientific Notation
7.7.4 Adding and Subtracting with Powers of 10
7.7.5 Applications in Science and Engineering Contexts
7.8.1 Simplifying Ratios and Comparing Quantities
7.8.2 Solving Direct and Inverse Proportion Problems
7.8.3 Percentage Increase and Decrease
7.8.4 Compound Percentages and Successive Changes
7.8.5 Solving Word Problems Involving Percentages
8. Sequences, Patterns, and Functions
8.1.1 Recognizing and Continuing Arithmetic Sequences
8.1.2 Finding the Common Difference
8.1.3 Deriving the nth Term of an Arithmetic Sequence
8.1.4 Solving Problems Using Arithmetic Sequences
8.1.5 Real-World Applications of Arithmetic Patterns
8.2.1 Understanding Composite Functions Notation
8.2.2 Evaluating Composite Functions
8.2.3 Concept of Inverse Functions
8.2.4 Finding Inverses Algebraically (Intro)
8.2.5 Checking if Two Functions Are Inverses
8.3.1 Recognizing Geometric Sequences
8.3.2 Finding the Common Ratio
8.3.3 General Term of a Geometric Sequence
8.3.4 Solving Word Problems with Geometric Sequences
8.3.5 Sum of a Finite Geometric Series (Introductory)
8.4.1 Distinguishing Between Linear and Non-Linear Sequences
8.4.2 Creating a Rule for the nth Term
8.4.3 Evaluating Terms Using the nth Term Formula
8.4.4 Finding Term Position Given Value
8.4.5 Applications in Tables and Graphs
8.5.1 Understanding Function Notation f(x)
8.5.2 Evaluating Functions for Given Inputs
8.5.3 Interpreting Function Values in Context
8.5.4 Defining Piecewise and Step Functions (Introductory)
8.5.5 Using Function Machines and Diagrams
8.6.1 Identifying Domain and Range from Tables
8.6.2 Determining Domain and Range from Graphs
8.6.3 Writing Inequality Notation for Domain and Range
8.6.4 Identifying Discrete and Continuous Functions
8.6.5 Applications in Real-World Graphs
9. Mensuration
10. Vectors and Transformations
10.1.1 Adding and Subtracting Vectors Graphically
10.1.2 Adding and Subtracting Column Vectors
10.1.3 Scalar Multiplication of Vectors
10.1.4 Vector Arithmetic Word Problems
10.1.5 Simplifying and Combining Vectors
10.2.1 Definition and Use of Position Vectors
10.2.2 Finding Displacement Between Two Points
10.2.3 Using Coordinates to Calculate Displacement
10.2.4 Position Vectors in Geometric Diagrams
10.2.5 Describing Movement in Terms of Vectors
10.3.1 Describing and Performing Translations
10.3.2 Reflections in Lines: x-axis, y-axis, y = x, y = -x
10.3.3 Rotations About a Point and Angle
10.3.4 Enlargement with Positive and Negative Scale Factors
10.3.5 Combining Multiple Transformations
10.4.1 Using Matrices for Simple Transformations (Introductory)
10.4.2 Applying Successive Transformations
10.4.3 Identifying Single Equivalent Transformation
10.4.4 Matrix Representation of Translations
10.4.5 Transforming Coordinates with Matrix Multiplication
10.5.1 Understanding Vectors as Arrows with Direction and Magnitude
10.5.2 Column Vector Notation
10.5.3 Representing Vectors in Diagrams and Coordinates
10.5.4 Notation and Labeling for Displacement Vectors
10.5.5 Unit Vectors and Position Vectors (Introductory)
10.6.1 Proving Collinearity Using Vectors
10.6.2 Midpoints and Parallel Vectors
10.6.3 Geometric Proofs with Vectors
10.6.4 Solving Complex Vector Path Problems
10.6.5 Applications in Navigation and Design
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