Maths Methods - Unit 1 Notes

Maths Methods

• Ms Daniel
• Focus: Surds and Quadratic Functions

Growth Mindset in Maths

• Anyone can learn mathematics.
• Struggle leads to the best learning.
• Challenge, confusion, and frustration are part of the learning process.
• Successful learning = achievement and pride.

Topic 1: Surds

• Definition: Irrational number with a radical (root) symbol.

Learning Intentions:

• Understand surds as irrational numbers represented using a square root or radical sign.
• Simplify square roots of natural numbers containing perfect square factors.
• Rationalise the denominator of fractional expressions involving square roots.

Key Concepts:

• Real Numbers:
  • Rational numbers.
  • Irrational numbers (surds).

Simplifying Surds

• Simplify square roots of natural numbers.
• Rationalise the denominator of fractional surds.

Operations with Surds:

• Adding and subtracting surds: Simplify expressions containing surds.
• Squaring surds: Simplify squared surd expressions.
• Dividing surds: Express answers in the simplest form.
• Rationalising the denominator: Simplify expressions with a rational denominator.

Topic 2: Quadratic Functions

Learning Intentions:

• Solve quadratic equations algebraically using:
  • Factorisation.
  • Quadratic formula (exact and approximate solutions).
  • Completing the square.
  • Technology.
• Use the discriminant to determine the number of solutions.
• Recognise and determine features of parabolas: turning points, axes of symmetry, intercepts.
• Sketch graphs of quadratic functions with or without technology.
• Determine turning points and zeros with and without technology.
• State domain and range.
• Model and solve problems involving quadratic functions.

Key Concepts:

• Null Factor Law: If $ab = 0$, then $a = 0$ or $b = 0$.
  • One side of the equation must be zero, and the other side must be in factorised form.
• Solutions/roots/zeros of the quadratic.

Solving Quadratic Equations

1. Factorisation:

• Rearrange the equation so that all terms are on one side.

2. Completing the Square:

• Transform $x^2 + bx + c$ to $(x ± \frac{b}{2})^2 - (\frac{b}{2})^2 + c$
• Solve by rearranging to isolate x, remembering the ± when taking the square root.

3. Quadratic Formula:

• For $ax^2 + bx + c = 0$, the solutions are given by: $x = \frac{-b ± \sqrt{b^2 - 4ac}}{2a}$.

4. Using the Discriminant:

• $\Delta = b^2 - 4ac$
  • If \Delta < 0, no real solutions.
  • If \Delta > 0, two distinct real solutions.
  • If $\Delta = 0$, one real solution (or two equal solutions).

Graphing Quadratic Functions

• Turning Point Form: $y = a(x - h)^2 + k$ where $(h, k)$ is the turning point.
  • $a$: dilation (if a < 0, reflection in x-axis).
  • $h$: horizontal shift.
  • $k$: vertical shift.
• Standard Form and Factorised Form
• Finding Key Features:
  • Shape (parabola opens upwards or downwards).
  • Turning point.
  • Intercepts (let $x = 0$ for y-intercept, let $y = 0$ for x-intercepts).

Determining the Equation from a Graph

• Identify the equation type.
• Substitute key features (e.g., turning point) into the equation.
• Substitute any other point to solve for unknown coefficients.

Modelling and Problem Solving

• Use quadratic functions to model and solve word problems.
• Maximising or minimising often involves finding the vertex.
• Formulate a mathematical statement from the problem, then solve.