Year 11 Objectives | Mathematics Department

11.1 Sets and Venn Diagrams

Term 1

Sets and Venn Diagrams

1. Identify number sets, use interval notation, and understand subsets and complements.
2. Use set notation for union ($\cup$), intersection ($\cap$), complement ($A'$) and the universal set.
3. Use 2-set Venn diagrams to represent relationships between sets and solve count problems.
4. Use 3-set Venn diagrams and the inclusion–exclusion principle.
5. Determine and describe sample spaces using lists, tables, 2D grids and Venn diagrams.
6. Translate between worded descriptions of events and set notation.
7. Shade regions of a Venn diagram for given set expressions, including De Morgan's laws.

11.2 Probability

Term 1

Probability Basics

1. Understand experimental and theoretical probability and express results as fractions, decimals or percentages.
2. Use compound events with tree diagrams (with and without replacement) and two-way tables.
3. Calculate the probability of unions using $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.

Conditional Probability and Independence

4. Understand and calculate conditional probability using $P(B \mid A) = \dfrac{P(A \cap B)}{P(A)}$.
5. Identify and test for mutually exclusive events ($P(A \cap B) = 0$).
6. Identify and test for independent events ($P(A \cap B) = P(A) \cdot P(B)$).
7. Solve problems combining Venn diagrams, tree diagrams and conditional probability.
8. Understand factorial notation, permutations and combinations.

11.3 Functions

Term 1–2

Functions

1. Understand the definition of a function and a relation; use domain and range.
2. Apply the vertical line test to determine whether a relation is a function.
3. Use function notation: evaluate $f(a)$, solve $f(x) = k$, and interpret in context.
4. Find the largest natural domain of a real-valued function involving fractions and square roots.
5. Determine inverse functions both algebraically and graphically.
6. Use composite functions with $f \circ g$ notation; find the domain of a composition.
7. Recognise and sketch absolute-value and step functions.
8. Explore Euler's number $e$ and the function $e^x$.

11.4 Quadratic Functions

Term 1–2

Solving Quadratics

1. Factorise quadratic expressions (monic and non-monic) and solve by the null factor law.
2. Solve quadratic equations using the quadratic formula.
3. Complete the square and convert between expanded, factorised and vertex forms.

Quadratic Graphs and Modelling

4. Identify key features of a quadratic graph: vertex, axis of symmetry, $x$- and $y$-intercepts.
5. Use the GDC to explore quadratic features, including optimisation.
6. Solve real-world quadratic modelling problems (projectile, area, profit).
7. Solve quadratic inequalities by sketch or sign-table.
8. Use the discriminant $b^2 - 4ac$ to determine the nature of roots.
9. Apply the discriminant to tangency / intersection problems with a line.

11.5 Transformations of Functions

Term 2

Transformations of Functions

1. Describe and perform vertical translations $y = f(x) + c$ and horizontal translations $y = f(x - h)$.
2. Describe and perform vertical dilations $y = a \cdot f(x)$ and horizontal dilations $y = f(ax)$.
3. Describe and perform reflections in the $x$-axis ($y = -f(x)$) and in the $y$-axis ($y = f(-x)$).
4. Apply a sequence of transformations to a graph and state the new key features.
5. Match transformed graphs to their algebraic form, and vice versa.
6. Use vertex form $y = a(x - h)^2 + k$ to read off translations and dilations of a quadratic.
7. Apply transformations to known parent functions (linear, quadratic, exponential).

11.6 Exponential and Logarithmic Functions

Term 2

Exponential Functions

1. Understand the shape of an exponential function $y = a \cdot b^x$ and identify key intercepts and the horizontal asymptote.
2. Explore exponential graph transformations (vertical shift, reflection, dilation).
3. Solve real-life problems involving exponential growth and decay.
4. Solve depreciation and compound-interest problems.
5. Interpret the contextual meaning of the parameters $a$ and $b$ in $y = a \cdot b^x$.
6. Use technology to solve basic exponential equations.

Indices and Logarithms (Extended)

7. Apply index laws for rational indices, including $a^{m/n} = \sqrt[n]{a^m}$.
8. Convert between exponential and logarithmic form: $a^x = y \Leftrightarrow \log_a y = x$.
9. Apply the logarithm laws: product, quotient and power.
10. Solve logarithmic and exponential equations algebraically.

11.7 Non-right-angle Trigonometry

Term 2–3

Non-right-angle Trigonometry

1. Apply right-angled trigonometry and Pythagoras' theorem to multi-step problems, including 3D.
2. Apply the sine rule to find missing sides and angles.
3. Apply the cosine rule to find missing sides and angles.
4. Apply the area-of-a-triangle formula $\tfrac{1}{2}ab\sin C$.
5. Solve problems involving bearings.
6. Combine right-angle and non-right-angle trigonometry in surveying / navigation contexts.
7. Solve problems involving the ambiguous case of the sine rule.
8. Prove the sine rule and the cosine rule.

11.8 Unit Circle

Term 3

Radian Measure and the Unit Circle (Extended)

1. Convert between degrees and radians; understand radian measure as arc length on the unit circle.
2. Use the full unit circle to define $\sin\theta$, $\cos\theta$ and $\tan\theta$ for any angle.
3. Use special triangles to find exact trigonometric values at multiples of $\frac{\pi}{6}$, $\frac{\pi}{4}$, $\frac{\pi}{3}$.
4. Apply negative-angle and complementary-angle identities.
5. Apply the Pythagorean identity $\sin^2\theta + \cos^2\theta = 1$ and $\tan\theta = \dfrac{\sin\theta}{\cos\theta}$.
6. Solve trigonometric equations of the form $\sin x = k$, $\cos x = k$, $\tan x = k$ in radians.
7. Calculate arc length, sector area and segment area in radians.
8. Calculate arc length, sector area and segment area in degrees.

11.9 Trigonometric Modelling

Term 3

Trigonometric Modelling

1. Identify periodic behaviour in real-world contexts (tides, daylight, sound, rotating machinery).
2. Model periodic phenomena using $y = A \sin(B(x - C)) + D$ and $y = A \cos(B(x - C)) + D$.
3. Identify and interpret amplitude, period, mean (vertical shift) and phase shift.
4. Fit a sinusoidal model to data using technology.
5. Use a fitted model to predict and interpret values in context.
6. Investigate transformations of more general set functions and piecewise periodic functions.
7. Explore the tangent function.

11.10 Systems of Equations

Term 3

Systems of Equations

1. Solve simultaneous linear equations by substitution.
2. Solve simultaneous linear equations by elimination.
3. Solve simultaneous linear equations graphically.
4. Solve simultaneous equations using technology (GDC).
5. Recognise systems with no solution or infinitely many solutions.
6. Model word problems with a pair of simultaneous equations and solve them.
7. Apply systems of equations to fit model coefficients using technology.
8. Solve systems with three variables.

11.11 Algebraic Fractions

Term 3

Algebraic Fractions

1. Simplify algebraic fractions where numerator and denominator share a common factor.
2. Simplify algebraic fractions requiring factorisation of quadratics.
3. Review solving linear equations including those with fractional coefficients.
4. Add and subtract algebraic fractions with linear denominators.
5. Multiply and divide algebraic fractions, cancelling where possible.
6. Simplify complex (compound) fractions.
7. Solve rational equations that reduce to linear or quadratic equations, identifying restrictions on the variable.

11.12 Mini IA — DP exploration practice

Term 3

Mini IA Project

1. Choose a personal context that lends itself to mathematical modelling.
2. Formulate a focused research question with a clear mathematical aim.
3. Collect or generate appropriate primary or secondary data.
4. Apply suitable Year 11 mathematics (functions, statistics, trigonometry, modelling) to the question.
5. Use technology (GDC, spreadsheet) to support computation, regression or graphing.
6. Interpret results in the original real-world context, including limitations and validity.
7. Structure a short write-up using the DP IA criteria (presentation, reflection, mathematics, use of technology).
8. Reflect on the choice of model and possible extensions, bridging to the Year 12 Exploration.