Year 10 Objectives | Mathematics Department

Sequences

Term 1

Representing Sequences

1. Represent a sequence using pictures, tables of values, graphs and algebra.
2. Describe a sequence using a term-to-term rule (in words) and a position-to-term rule (using $n$).
3. Identify whether a sequence is arithmetic, geometric, quadratic or neither.
4. Use the general rule of a sequence to predict any $n$th term.
5. Test whether a given number is a term of a sequence by solving an equation.

Arithmetic and Geometric Sequences

6. Determine the general rule $u_n = u_1 + (n-1)d$ for an arithmetic sequence.
7. Determine the general rule $u_n = u_1 \cdot r^{n-1}$ for a geometric sequence.
8. Work backwards from given terms to find $u_1$, $d$, $r$ or $n$ (e.g. given $u_5 = 2$ and $u_8 = 8$, find $r$).
9. Use properties of sequences to solve for unknowns (e.g. consecutive arithmetic terms $k$, $2k+2$, $4k+4$; consecutive geometric terms $2$, $k$, $10$).
10. Write a recursive rule for an arithmetic sequence using $u_{n+1} = u_n + d$.
11. Write a recursive rule for a geometric sequence using $u_{n+1} = u_n \cdot r$.

Shifted Patterns and Series (Extended)

12. Determine the general rule from a shifted square or cube sequence (e.g. one less than the square numbers gives $u_n = n^2 - 1$).
13. Calculate the sum of the first $n$ terms of an arithmetic series.
14. Calculate the sum of the first $n$ terms of a geometric series.
15. Discuss the behaviour of an infinite geometric series and identify when it converges.

Exponents and Indices

Term 1

Index Laws

1. Apply the index laws for multiplication, division and power of a power with positive integer indices.
2. Evaluate expressions involving zero and negative integer indices (e.g. $a^0 = 1$, $a^{-n} = 1/a^n$).
3. Evaluate expressions involving fractional indices (e.g. $a^{1/2}$, $a^{2/3}$).
4. Simplify algebraic expressions that combine the index laws, including negative and fractional indices.

Solving Index Equations

5. Solve equations by writing both sides with the same base and comparing exponents.
6. Solve equations of the form $a^{x} = k$ where the answer is a fraction or negative.
7. Solve equations of the form $a^{-2/3} = 1/25$ using fractional indices.
8. Convert between ordinary numbers and scientific (standard) form, including with negative powers of $10$.
9. Multiply and divide numbers in scientific form, expressing the answer in standard form.

Surds (Extended)

10. Simplify surds by extracting the largest square factor (e.g. $\sqrt{50} = 5\sqrt{2}$).
11. Add, subtract, multiply and divide surds, including expanding brackets containing surds.
12. Rationalise a denominator of the form $1/\sqrt{a}$.
13. Rationalise a denominator of the form $1/(a + \sqrt{b})$ using the conjugate.
14. Apply surds and index laws inside sequence problems and other algebraic contexts.

Algebra

Term 1–2

Expanding and Factorising

1. Expand single and double brackets, including $(a+b)(c+d)$ and special products $(a+b)^2$ and $(a-b)(a+b)$.
2. Factorise expressions by taking out the highest common factor.
3. Factorise quadratic expressions of the form $x^2 + bx + c$ (monic).
4. Factorise quadratic expressions of the form $ax^2 + bx + c$ with $a > 1$ (non-monic).
5. Factorise the difference of two squares, $a^2 - b^2$.

Solving Equations

6. Solve multi-step linear equations, including those with rational coefficients.
7. Solve linear equations with variables on both sides and with brackets.
8. Solve equations involving square and cube roots, recognising when both signs apply.
9. Solve linear equations with irrational coefficients (e.g. coefficients involving $\sqrt{2}$).

Algebraic Fractions and Systems

10. Simplify algebraic fractions where the numerator and denominator share a common factor.
11. Simplify algebraic fractions that require factorising a quadratic first.
12. Simplify algebraic fractions whose numerator or denominator contains surds.
13. Solve a system of two linear equations using substitution and elimination.
14. Model a real-world problem with a system of equations and interpret the solution in context.
15. Solve a system of three linear equations in three unknowns.

Coordinate Geometry

Term 2

Points and Line Segments

1. Plot points on the Cartesian plane and identify the quadrant they lie in.
2. Calculate the distance between two points using $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$.
3. Calculate the midpoint of a line segment using $M = \left(\dfrac{x_1+x_2}{2}, \dfrac{y_1+y_2}{2}\right)$.
4. Find the point that divides a line segment in a given ratio.

Gradient and Equation of a Line

5. Understand gradient as a ratio (rise over run) and as $\tan\theta$ where $\theta$ is the angle with the $x$-axis.
6. Calculate the gradient between two points using $m = \dfrac{y_2 - y_1}{x_2 - x_1}$.
7. Write the equation of a line in the form $y = mx + c$ and identify the gradient and $y$-intercept.
8. Find the equation of a line given a point and the gradient, or given two points.
9. Sketch lines from their equation, including reading off $x$- and $y$-intercepts.

Parallel, Perpendicular and Applications

10. Identify parallel lines from equal gradients.
11. Identify perpendicular lines using $m_1 \cdot m_2 = -1$.
12. Find the equation of a line parallel or perpendicular to a given line through a given point.
13. Solve geometric problems on the Cartesian plane (e.g. show points form a right-angled triangle, parallelogram or rhombus).
14. Interpret gradient and intercept as rates of change in real-world linear contexts.

Functions

Term 2

Function Notation and Representation

1. Understand the definition of a function as a rule that assigns each input exactly one output.
2. Evaluate a function for a given input (e.g. find $f(3)$ given $f(x) = 2x - 5$).
3. Solve equations of the form $f(x) = k$ to find inputs that give a chosen output.
4. Substitute algebraic expressions into a function (e.g. find $f(a+1)$ or $f(2x)$).
5. Move between numerical, algebraic, table and graphical representations of a function.

Domain, Range and Linear Functions

6. Identify the domain and range of a function from its graph.
7. Find a sensible (natural) domain when a function is defined by a formula, including avoiding division by zero or square roots of negatives.
8. Sketch and interpret linear functions, including those written as $f(x) = mx + c$.
9. Sketch and interpret piecewise linear functions defined on different intervals.
10. Use the vertical line test to decide whether a graph represents a function.

Composition and Inverses (Extended)

11. Form a composite function $f \circ g$ and evaluate it for given inputs.
12. Recognise that composition is not commutative ($f \circ g \neq g \circ f$ in general).
13. Find the inverse $f^{-1}(x)$ of a linear function algebraically.
14. Recognise that the graph of $y = f^{-1}(x)$ is the reflection of $y = f(x)$ in the line $y = x$.
15. Verify an inverse by showing $f(f^{-1}(x)) = x$.

Quadratic Equations

Term 2

Solving Quadratic Equations

1. Solve a quadratic equation by factorising and applying the null factor law.
2. Solve a quadratic equation of the form $x^2 = k$ or $(x - a)^2 = k$.
3. Solve rational equations that reduce to a quadratic after clearing fractions.
4. Complete the square to solve a quadratic equation.
5. Solve a quadratic equation using the quadratic formula $x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
6. Give exact (surd) solutions to quadratic equations where appropriate.

Graphs of Quadratic Functions

7. Identify the $y$-intercept of a quadratic function as $f(0)$.
8. Find the $x$-intercepts (roots) of a quadratic by setting $f(x) = 0$.
9. Determine the axis of symmetry and the vertex (maximum or minimum) of a quadratic.
10. Sketch a quadratic from its key features (intercepts, vertex, direction of opening).
11. Convert between expanded, factorised and vertex forms of a quadratic.

Applications and Discriminant (Extended)

12. Solve optimisation problems modelled by a quadratic (e.g. maximum area, maximum profit, projectile motion).
13. Solve a system consisting of a linear and a quadratic equation algebraically.
14. Use the discriminant $b^2 - 4ac$ to determine the nature of the roots (two distinct, one repeated, none).
15. Solve quadratic inequalities using a sketch or sign-table.

Geometry

Term 2–3

Area, Volume and Surface Area

1. Calculate the area of compound shapes built from rectangles, triangles, parallelograms, trapezia and circles.
2. Calculate the volume of prisms, including cylinders.
3. Calculate the volume of pyramids and cones using $V = \tfrac{1}{3} \times \text{base area} \times h$.
4. Calculate the surface area of prisms and cylinders, including curved surfaces.
5. Rearrange a mensuration formula to make a chosen variable the subject.

Right-angle and Non-right-angle Trigonometry

6. Use SOH-CAH-TOA to find missing sides and angles in right-angled triangles.
7. Apply Pythagoras' theorem and right-angle trigonometry inside 3D solids (space diagonals, slant heights).
8. Apply the sine rule to find missing sides and angles in non-right-angled triangles.
9. Apply the cosine rule to find missing sides and angles in non-right-angled triangles.
10. Apply the area-of-a-triangle formula $\tfrac{1}{2}ab\sin C$.

Bearings and Accuracy

11. Read and draw bearings as three-figure angles measured clockwise from north.
12. Solve navigation and surveying problems using bearings together with right-angle and non-right-angle trigonometry.
13. Round answers to a given number of decimal places or significant figures and decide on an appropriate level of accuracy.
14. Calculate the percentage error between an approximate and an exact value, using the absolute-value convention.
15. Combine $\pm$ measurement errors through sums and products, and discuss the effect on the final answer.

Statistics

Term 3

Descriptive Measures

1. Calculate the mean, median and mode of an ungrouped data set.
2. Estimate the mean of a grouped frequency distribution using midpoints.
3. Identify the modal class and median class of a grouped distribution.
4. Calculate measures of variability: range, interquartile range (IQR) and standard deviation (using technology).
5. Discuss the effect of outliers on the mean, median and standard deviation.

Representations: Histograms and Cumulative Frequency

6. Construct a frequency histogram and a relative-frequency histogram for grouped data.
7. Construct a cumulative frequency table from a grouped frequency distribution.
8. Draw a cumulative frequency curve (ogive) and use it to estimate the median, quartiles and percentiles.
9. Construct a box-and-whisker plot from a five-number summary.
10. Use the $1.5 \times \text{IQR}$ rule to identify potential outliers from a boxplot.

Shape, Skew and Comparison

11. Describe the shape of a distribution as symmetric, positively skewed or negatively skewed.
12. Predict the effect of skew on the relative positions of the mean and median.
13. Compare two distributions using measures of centre, spread and shape.
14. Use boxplots side by side to compare two data sets.
15. Justify which measure of centre and which measure of spread are most appropriate for a given data set.

Bivariate Statistics

Term 3

Scatter Plots and Correlation

1. Identify the independent (explanatory) variable and the dependent (response) variable in a bivariate context.
2. Construct a scatter plot from a bivariate data set, with appropriate scales and labels.
3. Describe the form (linear / non-linear), direction (positive / negative) and strength (weak / moderate / strong) of a relationship.
4. Distinguish between correlation and causation, and identify possible confounding variables.
5. Interpret Pearson's correlation coefficient $r$ as a measure of the strength and direction of a linear relationship.

Line of Best Fit

6. Draw a line of best fit by eye through the mean point of the data.
7. Use technology (GDC or spreadsheet) to find the equation of the least-squares regression line $y = mx + c$.
8. Interpret the gradient and $y$-intercept of the regression line in the context of the data.
9. Use the regression equation to make predictions (interpolation), and discuss the danger of extrapolation.

Residuals and Goodness of Fit (Extended)

10. Calculate the residual for a data point as $\text{observed} - \text{predicted}$.
11. Interpret the coefficient of determination $r^2$ as the proportion of variation explained by the model.
12. Compare $r$ and $r^2$ for different bivariate data sets and discuss which model fits best.
13. Discuss the limitations of a linear model when residuals show a clear pattern.

IDU — Aesthetics

Term 3

Mathematical Aesthetics

1. Identify symmetry (reflective, rotational and translational) in a piece of art, architecture or natural form.
2. Use geometric vocabulary (proportion, ratio, tessellation, fractal) to describe an aesthetic choice.
3. Investigate the golden ratio $\varphi$ and where it appears in art, architecture and biology.
4. Construct a geometric pattern, tessellation or fractal that meets a stated aesthetic constraint.

Interdisciplinary Inquiry (Criterion D)

5. Compare how mathematics and another subject (Language, Visual Arts, Music) each describe a chosen aesthetic concept.
6. Bring evidence from both disciplines together to support a single interdisciplinary claim.
7. Reflect on how a mathematical lens changes the way an aesthetic object is read or made.
8. Discuss age-appropriate Theory of Knowledge questions about beauty, pattern and proof.

Presenting the IDU

9. Plan and produce an IDU artefact (poster, slide deck, model or short report) that meets Criterion D expectations.
10. Use consistent terminology and notation across the mathematical and non-mathematical parts of the IDU.
11. Cite sources for images, ideas and data, including images of artworks and architecture.
12. Evaluate the final IDU against the published rubric and reflect on what worked and what to improve.