Year 9 Maths Foundation Tier Edexcel IGCSE Comprehensive Study Guide

Section 1: Number & Operations

• 1.1 Fractions

  • Adding and Subtracting Fractions

  • These operations require a common denominator (the same bottom number) before they can be calculated.

  • For the calculation $\frac{1}{3} + \frac{1}{4}$, you must find the Lowest Common Multiple (LCM) of 3 and 4, which is 12.

  • Convert each fraction to have this denominator: $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

  • Multiplying Fractions

  • Unlike addition, you do not need the same denominator. You multiply the tops (numerators) together and the bottoms (denominators) together.

  • Example: $\frac{2}{3} \times \frac{3}{5} = \frac{6}{15}$. This can be simplified to $\frac{2}{5}$.

  • Dividing Fractions

  • To divide fractions, follow the "Keep, Flip, Multiply" rule.

  • Keep the first fraction, flip the second fraction (this is called the reciprocal), and then multiply them.

  • Example: $\frac{3}{4} \div \frac{2}{5}$ becomes $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$.

  • Mixed Numbers

  • A mixed number is a whole number combined with a fraction.

  • Example: $2 \frac{3}{4} = \frac{11}{4}$. (Calculated by: $(2 \times 4) + 3 = 11$ over the original denominator).

  • You must always convert mixed numbers into improper fractions before performing multiplication or division.

• 1.2 BIDMAS & Order of Operations

  • Calculations must be performed in a specific sequence to achieve the correct result:

  • B: Brackets

  • I: Indices (powers and roots)

  • D/M: Division and Multiplication (calculated from left to right)

  • A/S: Addition and Subtraction (calculated from left to right)

  • Application Examples:

  • For $3 + 4 \times 2$, multiply first ($4 \times 2 = 8$), then add ($3 + 8 = 11$). Calculating $(3+4) = 7 \times 2 = 14$ is incorrect.

  • For $(3 + 4) \times 2$, brackets are solved first: $7 \times 2 = 14$.

  • For $2^3 + 1$, indices are solved first ($8 + 1 = 9$).

• 1.3 Rounding

  • Decimal Places (dp): To round to a specific decimal place, look at the digit to the right of your target position.

  • Example: $3.472$ to 2dp is $3.47$. Since the third decimal is 2, the second decimal stays the same (rounds down).

  • Significant Figures (sf): Count from the first non-zero digit.

  • Leading Zeros: These do not count as significant (e.g., $0.00456$ to 2sf is $0.0046$).

  • Trailing Zeros: These do count if they are after a decimal point (e.g., $0.456$ to 2sf is $0.46$).

  • Rounding Rule: Significant digits 5 and above round the previous digit up; 4 and below round it down.

• 1.4 Algebra - Expanding & Factorising

  • Expanding (One Parenthesis): Multiply every term inside the bracket by the term outside.

  • Example: $3(x + 5) = 3x + 15$.

  • Expanding Double Brackets (FOIL):

  • Use FOIL: First, Outer, Inner, Last.

  • Example: $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.

  • Factorising: The opposite of expanding; finding common factors and putting them into brackets.

  • Single Brackets: $6x + 9 = 3(2x + 3)$.

  • Quadratic Factorisation: For $x^2 + 5x + 6$, find two numbers that multiply to the constant (6) and add to the middle coefficient (5). These numbers are 2 and 3, so the result is $(x + 2)(x + 3)$.

• 1.5 Equations & Inequalities

  • Linear Equations: Use inverse operations on both sides to isolate the variable.

  • For $3x + 7 = 22$, subtract 7 to get $3x = 15$, then divide by 3 to get $x = 5$.

  • Inequalities: Represented by symbols: less than ( < ), greater than ( > ), less than or equal to ($\le$), and greater than or equal to ($\ge$).

  • Solve inequalities exactly like equations with one exception: flip the inequality sign when you multiply or divide by a negative number.

  • Example 1: 2x > 10 \rightarrow x > 5.

  • Example 2: -2x > 10 \rightarrow x < -5 (sign flips from > to < because of the negative division).

• 1.6 Percentages & Standard Form

  • Percentage Change: Calculated using the formula $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$.

  • For a change from 10 to 12: $\frac{12 - 10}{10} \times 100 = 20\%$.

  • Compound Percentage: Repeated multiplication by the same factor.

  • Example: Investing $\pounds 100$ at $5\%$ for 2 years is calculated as $100 \times 1.05^2$.

  • Standard Form: Represented as $a \times 10^n$, where 1 \le a < 10.

  • $45000 = 4.5 \times 10^4$.

  • $0.00056 = 5.6 \times 10^{-4}$.

• 1.7 Ratio & Proportion

  • Ratio: A comparison between quantities. Ratios can be simplified by dividing both sides by a common factor (e.g., $6:8 = 3:4$).

  • Proportion: The relationship remains constant. If a ratio is $3:4$, then sets like $6:8$ and $9:12$ are proportional.

  • Example: If 3 apples cost $\pounds 1$, 9 apples will cost $\pounds 3$ because both parts are multiplied by the same factor (3).

Section 2: Geometry

• 2.1 Angles & Trigonometry

  • Parallel Line Angles:

  • F-pattern (Corresponding angles): These angles are equal.

  • Z-pattern (Alternate angles): These angles are equal.

  • C-pattern (Co-interior angles): These angles add up to $180^{\circ}$.

  • Polygon Angles: The sum of interior angles is found using the formula $(n - 2) \times 180^{\circ}$, where $n$ is the number of sides.

  • Triangle ($n=3$): $180^{\circ}$.

  • Quadrilateral ($n=4$): $360^{\circ}$.

  • Pentagon ($n=5$): $540^{\circ}$.

  • Pythagoras Theorem: Used for right-angled triangles to find a missing side: $a^2 + b^2 = c^2$, where $c$ is the hypotenuse.

  • Example: $3^2 + 4^2 = 9 + 16 = 25$. Therefore, $c = \sqrt{25} = 5$.

  • Trigonometry (SOH CAH TOA):

  • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

  • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

  • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

  • Use these ratios to find an unknown side or angle by involving one known side and the unknown side.

  • Exact Values (Must Memorise):

  • $\sin(30^{\circ}) = \frac{1}{2}$, $\cos(30^{\circ}) = \frac{\sqrt{3}}{2}$, $\tan(30^{\circ}) = \frac{1}{\sqrt{3}}$

  • $\sin(45^{\circ}) = \frac{1}{\sqrt{2}}$, $\cos(45^{\circ}) = \frac{1}{\sqrt{2}}$, $\tan(45^{\circ}) = 1$

  • $\sin(60^{\circ}) = \frac{\sqrt{3}}{2}$, $\cos(60^{\circ}) = \frac{1}{2}$, $\tan(60^{\circ}) = \sqrt{3}$

• 2.2 Coordinates & Graphs

  • Gradient: Refers to the slope of a line, calculated as $\frac{\text{Rise}}{\text{Run}}$. Steiper lines have higher gradients.

  • Formula: $\frac{y_2 - y_1}{x_2 - x_1}$.

  • Equation of a Line: Defined as $y = mx + c$, where $m$ is the gradient and $c$ is the y-intercept.

  • Example: In $y = 2x + 3$, the gradient is 2, and the line crosses the y-axis at 3.

  • Finding Equation from Two Points:

  • 1. Calculate the gradient ($m$).

  • 2. Substitute one point into $y = mx + c$ to solve for $c$.

  • Example: Using points $(1, 5)$ and $(3, 9)$. Gradient = $\frac{9 - 5}{3 - 1} = \frac{4}{2} = 2$. Subst. $(1, 5)$: $5 = 2(1) + c \rightarrow c = 3$. Equation: $y = 2x + 3$.

  • Distance-Time Graphs:

  • Horizontal line: Stationary (not moving).

  • Diagonal line: Moving constant speed.

  • Curved line: Acceleration or deceleration.

  • The gradient of the graph represents speed.

• 2.3 Sequences

  • Nth Term of a Linear Sequence:

  • 1. Find the difference between consecutive terms.

  • 2. For sequence 5, 8, 11: The difference is 3.

  • 3. The formula pattern is $(\text{difference})n + (\text{first term} - \text{difference})$.

  • Calculation: $3n + (5 - 3) = 3n + 2$.

  • Validation: For $n=1$, $3(1)+2=5$. For $n=2$, $3(2)+2=8$.

Section 3: Statistics

• 3.1 Averages & Spread

  • Mean: The sum of all values divided by the number of values.

  • Example for 3, 5, 7: $\text{Sum} = 15$, $\text{Mean} = \frac{15}{3} = 5$.

  • Median: The middle value in an ordered set.

  • Example for 3, 5, 7: The median is 5. If there is an even number of values, average the two middle numbers.

  • Mode: The value that occurs most frequently.

  • Example for 1, 2, 2, 3: The mode is 2.

  • Range: The difference between the highest and lowest values.

  • Example for 1, 5, 8: $\text{Range} = 8 - 1 = 7$.

  • Cumulative Frequency: A running total of the frequencies.

  • If frequencies for values 1, 2, 3, 4 are 2, 3, 4, 1, the cumulative frequencies are 2, 5, 9, 10.

  • This is used to estimate the median ($Q_2$) and quartiles ($Q_1$ and $Q_3$).

  • Interquartile Range (IQR): The spread of the middle $50\%$ of data, calculated as $Q_3 - Q_1$.

• 3.2 Probability

  • Definition: $P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$.

  • Coin Flip: $P(\text{heads}) = \frac{1}{2}$.

  • Dice Roll (1-6): $P(\text{even numbers}) = \frac{3}{6} = \frac{1}{2}$.

  • Tree Diagrams: Visual representations showing all possible outcomes and their associated probabilities.

Section 4: Applied Maths

• 4.1 Compound Measures

  • Speed: $v = \frac{d}{t}$. Example: $120\,m$ in $10\,s = 12\,m/s$.

  • Density: $\rho = \frac{m}{V}$. Example: A $100\,g$ object with volume $50\,cm^3 = 2\,g/cm^3$.

  • Pressure: $P = \frac{F}{A}$. Example: $100\,N$ force over $20\,m^2 = 5\,N/m^2$.

• 4.2 Indices & Surds

  • Indices (Power Laws):

  • Multiplication: $a^m \times a^n = a^{m+n}$. Example: $2^3 \times 2^2 = 2^5 = 32$.

  • Division: $a^m \div a^n = a^{m-n}$. Example: $2^5 \div 2^2 = 2^3 = 8$.

  • Power of a Power: $(a^m)^n = a^{m \times n}$. Example: $(2^2)^3 = 2^6 = 64$.

  • Surds (Irrational Numbers):

  • Numbers like $\sqrt{2}$, $\sqrt{3}$, and $\sqrt{5}$ cannot be simplified to rational numbers.

  • Simplifying: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$.

  • Rationalising: To remove a surd from a denominator: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.

• 4.3 Bounds & Approximation

  • Bounds: Define the range of accuracy for rounded numbers.

  • Example: A number rounded to the nearest 10 is 50.

  • Lower Bound: 45 (the smallest value that rounds up to 50).

  • Upper Bound: 55 (the value below which everything rounds down to 50).

  • Inequality Notation: 45 \le x < 55.

  • Application: Used in calculations to find the maximum possible range of an answer involving rounded data.

Section 1: Number & Operations

• 1.1 Fractions

  • Adding and Subtracting Fractions:

  • Finding a common denominator is crucial for these operations. For example, to calculate $\frac{1}{3} + \frac{1}{4}$, find the Lowest Common Multiple (LCM) of 3 and 4, which is 12. Therefore, convert each fraction:
    $\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{4} = \frac{3}{12}$, giving you $\frac{4}{12} + \frac{3}{12} = \frac{7}{12}$.

  • Multiplying Fractions:

  • No need for a common denominator! Just multiply the tops (numerators) and bottoms (denominators). E.g.,
    $\frac{2}{3} \times \frac{3}{5} = \frac{6}{15}$, which simplifies to $\frac{2}{5}$.

  • Dividing Fractions:

  • Follow the “Keep, Flip, Multiply” rule. For example, $\frac{3}{4} \div \frac{2}{5}$ becomes $\frac{3}{4} \times \frac{5}{2} = \frac{15}{8}$.

  • Mixed Numbers:

  • A mixed number combines a whole number and a fraction. Convert it to an improper fraction before multiplication or division. E.g., $2 \frac{3}{4} = \frac{11}{4}$.

• 1.2 BIDMAS & Order of Operations

  • Remember the order: Brackets, Indices (powers), Division/Multiplication, Addition/Subtraction.

  • For instance, in $3 + 4 \times 2$, perform the multiplication first: $4 \times 2 = 8$, then add: $3 + 8 = 11$. If you group as $(3 + 4) \times 2$, then you do the brackets first, giving you 14.

• 1.3 Rounding

  • Decimal Places: When rounding to specific decimal places, check the next digit. E.g., $3.472$ rounded to 2 decimal places is $3.47$ because the third decimal (2) tells us to round down.

  • Significant Figures: Count from the first non-zero digit. Leading zeros don’t count (e.g., $0.00456$ to 2 significant figures is $0.0046$), but trailing zeros after a decimal do (e.g., $0.456$ to 2 significant figures becomes $0.46$).

• 1.4 Algebra - Expanding & Factorising

  • Expanding: This means multiplying each term inside the bracket by the term outside. For example, $3(x + 5) = 3x + 15$.

  • Expanding Double Brackets: Use FOIL (First, Outer, Inner, Last). For example,
    $(x + 2)(x + 3) = x^2 + 3x + 2x + 6 = x^2 + 5x + 6$.

  • Factorising: This is the reverse of expanding; you find common factors. For instance, $6x + 9$ becomes $3(2x + 3)$.

• 1.5 Equations & Inequalities

  • Linear Equations: Isolate the variable by performing inverse operations. E.g., for $3x + 7 = 22$, subtract 7 to get $3x = 15$, then divide by 3, yielding $x = 5$.

  • Inequalities: Use the same method as equations, but flip the inequality sign if you multiply or divide by a negative number. For example, -2x > 10 becomes x < -5.

• 1.6 Percentages & Standard Form

  • Percentage Change: Use the formula $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$. If something changes from 10 to 12, you calculate $\frac{12 - 10}{10} \times 100 = 20\%$.

  • Compound Percentage: This involves multiplying by the same factor repeatedly. For example, investing $\pounds 100$ at $5\%$ for 2 years is calculated as $100 \times 1.05^2$.

  • Standard Form: Write numbers as $a \times 10^n$ (where 1 \leq a < 10). For example, $45000 = 4.5 \times 10^4$.

• 1.7 Ratio & Proportion

  • Ratio: A way to compare quantities. For instance, $6:8$ can be simplified to $3:4$.

  • Proportion: If a ratio is 3:4, any multiples of that ratio (like $6:8$ or $9:12$) are proportional. E.g., if 3 apples cost $\pounds 1$, 9 apples will cost $\pounds 3$.

Section 2: Geometry

• 2.1 Angles & Trigonometry
  

  • Parallel Line Angles:

  • Corresponding angles are equal (F-pattern).

  • Alternate angles are equal (Z-pattern).

  • Co-interior angles sum up to $180^{\circ}$ (C-pattern).

  • Pythagoras Theorem: For right triangles, use $a^2 + b^2 = c^2$ (where $c$ is the hypotenuse). For example, for a triangle with sides 3 and 4, $3^2 + 4^2 = 25$; therefore, $c = \sqrt{25} = 5$.

  • Trigonometry:

  • Use SOH CAH TOA:
    

    • $\sin(\theta) = \frac{\text{Opposite}}{\text{Hypotenuse}}$

    • $\cos(\theta) = \frac{\text{Adjacent}}{\text{Hypotenuse}}$

    • $\tan(\theta) = \frac{\text{Opposite}}{\text{Adjacent}}$

Section 3: Statistics

• 3.1 Averages & Spread

  • Mean: Average of the values. E.g., for numbers 3, 5, 7, the mean is $\frac{15}{3} = 5$.

  • Median: Middle value in an ordered sequence. For 1, 2, 3, 4, 5, it's 3. For 1, 2, 3, 4, the median is $\frac{2 + 3}{2} = 2.5$.

  • Mode: Most frequently occurring value, e.g., in 1, 2, 2, 3, the mode is 2.

  • Range: The difference between the highest and lowest values. For the set 1, 5, 8, the range is $8 - 1 = 7$.

  • Cumulative Frequency: A running total of frequencies, used to estimate medians and quartiles.

• 3.2 Probability

  • Definition: Probability is defined as $P(\text{event}) = \frac{\text{favorable outcomes}}{\text{total outcomes}}$. E.g., for a coin flip, $P(\text{heads}) = \frac{1}{2}$.

  • Tree Diagrams: Useful tools to visualize all possible outcomes and their corresponding probabilities.

Section 4: Applied Maths

• 4.1 Compound Measures

  • Speed: Calculated using $v = \frac{d}{t}$. If you cover 120 m in 10 seconds, your speed is 12 m/s.

  • Density: $\rho = \frac{m}{V}$. For a 100 g object with a volume of 50 cm³, the density is $2 \, g/cm^3$.

  • Pressure: Calculated as $P = \frac{F}{A}$. A force of 100 N over an area of 20 m² results in a pressure of $5 \, N/m²$.

• 4.2 Indices & Surds

  • Indices (Power Laws):

  • For multiplication: $a^m \times a^n = a^{m+n}$. E.g., $2^3 \times 2^2 = 2^5 = 32$.

  • For division: $a^m \div a^n = a^{m-n}$. E.g., $2^5 \div 2^2 = 2^3 = 8$.

  • Power of a power: $(a^m)^n = a^{m \times n}$. E.g., $(2^2)^3 = 2^6 = 64$.

  • Surds: These are numbers like $\sqrt{2}$ that cannot be simplified to whole numbers.

  • Simplifying a surd: $\sqrt{12} = \sqrt{4 \times 3} = 2\sqrt{3}$.

  • Rationalising the denominator: $\frac{1}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{\sqrt{2}}{2}$.   

• 4.3 Bounds & Approximation

  • Bounds help define ranges for rounded numbers. If a number is rounded to the nearest 10, say 50, the

  • Lower Bound: 45 (smallest value that rounds up to 50).

  • Upper Bound: 55 (largest value that rounds down).

  • This can be expressed as: 45 \leq x < 55.

  • Bounds are practical for estimating maximum possible ranges of results involving rounded data.