Home → Curriculum Hub → Maths Lessons → Schemes of Work → Three-Year GCSE Foundation → Patterns and Sequences
This Foundation GCSE scheme of work unit covers patterns and sequences in Year 10. It starts on the coordinate grid and then builds sequences. Students plot points in four quadrants, find a midpoint and complete a function table, then move from term-to-term rules to the nth term of linear, pictorial, quadratic and geometric sequences, finishing with a recurrence formula.
Students have to see n as a position, not as the next term. That distinction is what later straight-line graphs, iteration and proof questions all use.
A student who has secured this unit:
Students should be secure with the following before beginning this unit.
n is the term’s place in the list. Writing n + 3 for 1, 4, 7, 10 describes the jump, not the position-to-term rule.
A constant first difference means the sequence is linear. A constant second difference means it is quadratic.
A growing diagram still has a position. Counting the new sticks, then the total, leads to the same nth term as a number list.
A geometric sequence multiplies by a constant. Its nth term is ar^(n-1), not ar^n.
| Misconception | Teaching focus |
|---|---|
| Treating n as the next term, so 1, 4, 7, 10 is written as n + 3. | Write position above each term and ask what happens when n = 1, before any formula is offered. |
| Using ar^n for a geometric sequence instead of ar^(n-1). | Check the formula on term 1: if n = 1 gives ar instead of a, the power is wrong. |
| Ignoring the linear part of a quadratic sequence once the n² coefficient is found. | Subtract the quadratic piece from the original list and find the leftover linear sequence. |
| Multiplying two negatives incorrectly when a function table or a coordinate has a minus sign. | Complete one signed product as a class before the table is filled in independently. |
| Continuing a picture sequence by copying the last drawing instead of using the nth term. | Ask for the 20th pattern, which cannot be drawn, so the formula has to be used. |
Every lesson in this unit is already built. A Mr Mathematics membership gives you the presentation, worksheet, question generator and video tutorial for all 11 lessons, alongside the full Key Stage 3, GCSE, IGCSE and A-Level libraries.
A school membership covers every teacher in your department, so the whole scheme of work is resourced from one subscription.
Students learn how to plot and read coordinate pairs in the first quadrant.
As learning progresses students use 2D shapes to plot vertices and mid-points.
Students learn to plot and read coordinate pairs in all four coordinates.
As learning progresses, students use coordinates to describe the properties of 2D shapes.
Students learn to find the midpoint of a line segment using the midpoint formula, starting with straightforward coordinate calculations.
Problems then increase in difficulty, including working backwards to find a missing endpoint and applying the midpoint formula within geometric shapes such as rectangles and triangles.
Related Blogs: Midpoint of a Line Segment
Tutorials: Part 1 | Part 2 | Part 3
Students learn how to find the outputs of a two-stage function machine.
Later, they work out the function and inputs as learning progresses.
Students learn how to describe and continue number patterns on a term-to-term basis.
As learning progresses they are challenged to consider picture sequences and create their own logical patterns.
Students learn how to work out the nth term of a linear sequence.
As learning progresses they use the nth term to model picture sequences.
Students learn how to find and apply the nth term of a picture sequence.
Later, as learning progresses, they use the nth term to set up and solve equations.
Related Blog: nth term of picture sequences
Students learn to generate sequences using the nth-term formula.
As learning progresses the sequences involve brackets and powers.
Students learn how to find the nth term of a quadratic sequence.
Later, as learning progresses, they find the nth term from the coordinates of a quadratic graph.
Students learn how to work out the nth term of a geometric progression in the form arn-1.
As learning progresses, they consider progressions with fraction and negative common ratios.
Students learn how to generate a recurrence sequence.
Later, as learning progresses, they work out the formula from the sequence.
Additional practice working out the nth term of arithmetic sequences.
The problem-solving activity challenges students to model a picture sequence using the nth term.
Related Blog:
Students gain additional practice working with geometric progressions.
The problem-solving activity challenges students to use a geometric progression to model a real-life problem..
nth term of a geometric sequence
Students practise generating terms of linear and quadratic sequences, before extending their understanding to more complex non-linear sequences such as geometric, reciprocal and exponential sequences.
Students revise how to calculate missing terms and the nth term of quadratic and geometric sequences.
Later, as learning progresses, they link sequences to algebraic proof and setting up equations.
nth term of a geometric sequence
Students revise how to derive the nth term of an arithmetic sequence.
As learning progresses, they use the nth term to find any sequence term when given its position.
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