Arithmetic Sequences Exploration

Overview

In this lesson, students explore arithmetic sequences by finding the common difference and using it to derive the nth term. They will connect patterns in tables and number lines to an equation form for general term calculations.

Learning intentions

• Students will be able to identify whether a pattern is an arithmetic sequence.
• Students will be able to find the common difference from a sequence or table.
• Students will be able to write the nth term using the form (a_n = a_1 + (n-1)d).
• Students will be able to use the nth term to calculate specific terms and solve simple problems.

Success criteria

• I can explain how to get the common difference (d) from consecutive terms.
• I can write an nth term rule for an arithmetic sequence using (a_1) and (d).
• I can use the rule to find (a_n) for a given (n) and check my answer with the pattern.
• I can distinguish arithmetic sequences from non-arithmetic patterns.

Curriculum links

• Students use number patterns to form and justify generalisations.
• Students represent linear relationships using equations and use them to solve problems.
• Students apply algebraic reasoning to general term formulas for sequences.
• Students communicate mathematical thinking using correct notation.

Lesson structure (60 minutes)

1. 0–8 min: Warm-up pattern check

• Display three short sequences (two arithmetic, one not) and ask the student to compute differences between consecutive terms.
• Students justify whether the differences stay constant, using words like “constant” or “changes.”

2. 8–18 min: Guided discovery with a table

• Provide a table with terms starting at (a_1) and continuing for several (n) values.
• Ask the student to record (d) by subtracting consecutive terms and to state what stays the same.

3. 18–28 min: From differences to an equation

• Work together to connect the table to the rule (a_n = a_1 + (n-1)d).
• Have the student test the formula by substituting the given (n) values and confirming it matches the table.

4. 28–40 min: Independent practice (rule writing)

• Give 3 sequences described by first term and/or a few early terms.
• Student finds (d), identifies (a_1), writes the nth term rule, and calculates (a_5) or (a_8) for each.

5. 40–50 min: Error-checking and reasoning

• Provide one “almost right” rule (e.g., using (a_1 + nd) instead of (a_1 + (n-1)d), or a wrong (d)).
• Student compares outputs for a chosen (n) and explains where the rule fails and how to fix it.

6. 50–57 min: Exit reflection

• Student answers: “How do you know a sequence is arithmetic?” and “Why do we use ((n-1)) rather than (n)?”
• Collect responses to gauge conceptual understanding.

7. 57–60 min: Quick teacher check

• Teacher reviews the student’s rules for consistency of notation and accuracy.
• Confirm next lesson focus by previewing that sequences link to graphing linear patterns.

Resources

• Prepared sequence cards or slides (arithmetic and non-arithmetic examples)
• A table template for (n), (a_n), and consecutive differences
• Whiteboard or digital document for rule building
• Worked example sheet for reference (blank steps for student completion)
• Calculator (optional) for checking larger term values
• Pencil/paper for justification sentences
• “Almost right” rule problem for error analysis
• Exit ticket strip with two reflection prompts

Assessment

• Observation during warm-up: accuracy of computed differences and reasoning about constancy.
• Marking of practice items: correct identification of (d), correct nth term rule, correct term calculation.
• Exit reflection: clear explanation of arithmetic sequences and the role of ((n-1)).

Differentiation

• Support: Provide a partially completed table (students fill in missing differences) and a sentence frame for justification (“It’s arithmetic because…”).
• Support: Offer a checklist for writing the nth term: find (d), confirm constant difference, identify (a_1), then substitute into the formula.
• Extension: Ask for reverse reasoning—given an nth term or two terms, determine (a_1) and (d), or find the value of (n) for a target term.
• EAL/SEN: Use consistent language for “consecutive,” “difference,” and “constant,” and allow responses to include both words and equations; keep tasks in small steps.