Math standard E-assessment notes

1. Numerical and Abstract Reasoning

• Number:
    - Absolute Values:
      - Definition: The absolute value of a number is its distance from zero on the number line, disregarding whether it is positive or negative. For example, $|5| = 5$ and $|-5| = 5$.
    - Representing and Solving Inequalities:
      - This involves understanding how to express and solve inequalities, including:
        - Compound Inequalities: Such as a < x < b.
        - Double Inequalities: It is a pair of inequalities that can occur simultaneously.
    - Irrational Numbers:
      - Numbers that cannot be expressed as a simple fraction; their decimal expansion is neither terminating nor repeating. Examples include $ext{√2}$ , $ext{π}$.
    - Surds, Roots and Radicals:
      - A surd is an expression containing a square root, cube root, etc., that cannot be simplified to remove the root. For example, $ext{√3}$ is a surd.
      - Simplifying: Reducing a radical expression to its simplest form, e.g. $ext{√18} = 3 ext{√2}$.
    - Standard Form (Scientific Notation):
      - A way of expressing numbers as a product of a number between 1 and 10, and a power of ten. Example: $6.02 imes 10^{23}$.
    - Laws of Exponents:
      - Rules that describe how to handle mathematical operations involving powers:
        - Product of Powers: $a^m imes a^n = a^{m+n}$.
        - Quotient of Powers: $rac{a^m}{a^n} = a^{m-n}$.
        - Power of a Power: $(a^m)^n = a^{m imes n}$.
        - Negative Exponent: $a^{-n} = rac{1}{a^n}$.
    - Number Systems Notation:
      - Various systems of representing numbers such as decimal, binary, hexadecimal.
    - Direct Proportion:
      - Two variables are directly proportional if one is a constant multiple of the other, expressed as $y = kx$.
    - Inverse Proportion:
      - Two variables are inversely proportional if their product is a constant, expressed as $y = rac{k}{x}$.
    - Number Sequences:
      - A sequence is an ordered list of numbers. Can be used for prediction and description of patterns throughout.

• Algebra:
    - Factorizing Quadratic Expressions:
      - Decomposing a quadratic expression into the product of its linear factors. For example, $x^2 - 5x + 6 = (x-2)(x-3)$.
    - Solving Quadratic Equations:
      - Finding the values of $x$ that satisfy the equation $ax^2 + bx + c = 0$. Methods include factoring, completing the square, and using the quadratic formula: $x = rac{-b ext{±} ext{√}(b^2 - 4ac)}{2a}$.
    - Changing the Subject of an Equation:
      - Rearranging an equation to make a particular variable the subject. For example, from $3x + 4 = 10$ to $x = rac{10 - 4}{3} = 2$.

2. Thinking with Models

• Mappings:
    - Representations that show the relationship between sets of elements, often used in functions.

• Function Notation:
    - Method of expressing functions in a specific way, generally as $f(x)$, where $f$ denotes the function and $x$ is the input.

• Linear Functions:
    - Functions of the form $y = mx + c$, where $m$ is the slope and $c$ is the y-intercept.

• Equation of a Line:
    - The line can be expressed as $y = mx + c$, where a change in $x$ results in a linear change in $y$.

• Parallel and Perpendicular Lines:
    - Parallel Lines: Lines that never intersect and have the same slope (m).
    - Perpendicular Lines: Lines that intersect at right angles, with slopes that multiply to -1, i.e., if $m_1 imes m_2 = -1$.

• Systems of Equations / Simultaneous Equations:
    - Set of equations with multiple variables that are solved together to find common solutions across all equations.

• Quadratic Functions:
    - Functions that can be expressed in the standard form $y = ax^2 + bx + c$.

• Algorithms:
    - Step-by-step procedures for calculations or problem-solving, essential in mathematics and computer science.

3. Spatial Reasoning

• Geometry:
    - Metric Conversions: Changing between measurements in different units, e.g., from meters to centimeters.
    - Volume of Regular Polyhedra: Volume calculations for shapes such as cubes, spheres, cylinders, etc. Example for cube: $V = s^3$ where $s$ is the length of a side.
    - Similarity and Congruence:
      - Similarity: Two shapes are similar if they have the same shape but not necessarily the same size.
      - Congruence: Two shapes are congruent if they have the same shape and size.
    - Coordinate Geometry:
      - A branch of mathematics that connects algebra and geometry using coordinates.
        - Distance Formula: $d = ext{√}((x_2 - x_1)^2 + (y_2 - y_1)^2)$.
        - Midpoint Formula: $M = \bigg( rac{x_1 + x_2}{2}, rac{y_1 + y_2}{2}\bigg)$.
        - Gradient Formula: $m = rac{y_2 - y_1}{x_2 - x_1}$.
    - Movement on a Plane:
      - Coverage of transformations such as isometric transformations (translations, reflections, and rotations), enlargements, and tessellations.
    - Circle Geometry:
      - Studies properties, angles, and relationships within circles including chord lengths, arc lengths, and inscribed angles.
    - Rotation Around a Given Point:
      - The action of turning a shape around a fixed point, which affects its position but not its size.

• Trigonometry:
    - Triangle Properties: Key concepts that include side lengths and angle measures governing triangle characteristics.
    - Bearings: Method of expressing direction often used in navigation, defined as a clockwise angle from the North (e.g., a bearing of $045^ ext{o}$ is 45 degrees clockwise from North).
    - Pythagoras’ Theorem:
      - A fundamental relation in Euclidean geometry regarding right-angled triangles: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.
    - Trigonometric Ratios in Right-Angled Triangles: Ratios that define relationships between the angles and lengths of triangles, including:
      - Sine: $ext{sin}( heta) = rac{ ext{opposite}}{ ext{hypotenuse}}$.
      - Cosine: $ext{cos}( heta) = rac{ ext{adjacent}}{ ext{hypotenuse}}$.
      - Tangent: $ext{tan}( heta) = rac{ ext{opposite}}{ ext{adjacent}}$.

4. Reasoning with Data

• Sampling Techniques:
    - Different methods used to select individuals or observations in a statistical study to represent the larger population.

• Data Manipulation and Misinterpretation:
    - Refers to the distortion or alteration of data presentation affecting analysis and decision-making.

• Graphical Representations:
    - Visual representations of data, including:
      - Bivariate Graphs: Graphs showing the relationship between two variables.
      - Scatter Graphs: Displays values for two variables for a set of data.
      - Box Plots: Displays the distribution of data based on a five-number summary.
      - Cumulative Frequency Graphs: Useful for determining number of observations below a particular value.

• Lines of Best Fit:
    - A line that best represents the data on a scatter plot, indicating the trend of the data.

• Data Processing: Quartiles and Percentiles:
    - Quartiles: Divide a ranked data set into four equal parts.
      - First Quartile (Q1): 25% of data falls below this value.
      - Second Quartile (Q2): Median of the data set.
      - Third Quartile (Q3): 75% of data falls below this value.
    - Percentiles: 100 equal parts of a data set, indicating the relative standing of a value within a dataset.

• Measures of Dispersion:
    - Understanding the spread of data, including:
      - Interquartile Range (IQR): Difference between the third quartile (Q3) and the first quartile (Q1). $IQR = Q3 - Q1$.

• Correlation, Qualitative Handling:
    - Measure of the relationship between two variables, can be positive, negative or zero correlation.

• Relative Frequency:
    - The ratio of the number of times an event occurs to the total number of outcomes in a sample space.

• Response Rates:
    - Measure of how many people responded to a survey (or other sampling methods) out of those who were contacted.

• Sets:
    - Collections of distinct objects or elements, denoted using curly braces, e.g., {1, 2, 3}.
    - Operations include union, intersection, and difference, covering up to three sets.

• Probability with Venn Diagrams, Tree Diagrams and Sample Spaces:
    - Techniques for visualizing and calculating probabilities of events:
      - Venn Diagrams: Used to show relationships between different sets.
      - Tree Diagrams: Illustrate outcomes of events in probability.
      - Sample Spaces: The set of all possible outcomes.

• Mutually Exclusive Events:
    - Events that cannot occur at the same time.

• Combined Events:
    - Refers to events or probabilities calculated together, often using the addition or multiplication rules of probability.

• Factorize:
    - Breaking down an expression into simpler multiplicative components.

• Quadratic Expressions and Equations:
    - As discussed in the earlier section regarding factorizing and solving them.

• Simultaneous Equations:
    - Equations where two or more unknowns are solved together.

• Sequences and Patterns:
    - Investigating relationships and rules among numbers and their sequences.

Functions and Modelling

• Midpoint and Distance: Calculation techniques in geometry and coordinate systems relevant in functions and applications.

• Linear Functions: As described earlier, foundational elements defining straight lines and their relationships.

• Quadratic Function: Explores the parabolic nature of quadratic equations in modelling scenarios relating to projectile motion and other applied sciences.

Geometry and Trigonometry

• Right Angle Trigonometry: Focused on relationships in right triangles, including angles and side relationships.

• Similar Shapes: Investigating properties of shapes that maintain proportional dimensions.

• One Variable: Concepts focusing primarily on single variable equations and relations.

• Bivariate: Involves two variables and can be represented in plots, analysis of relationships, and correlations between these variables.