The shape of the lesson
By the end of the lesson, learners can
- Spotting the rule: In the lab, set a first term and a common difference and watch the terms build up as bars.
- The nth term formula: In a linear sequence the common difference \(d\) is the number in front of \(n\).
- Using the nth term: Once you have the formula, set it equal to a number (or put it in an inequality) and solve for \(n\).
- Sequences from patterns: Count the objects in the first few patterns, find the nth term, then use it.
- Differences: subtract each term from the next. Constant means linear.
- Start the formula: the common difference \(d\) goes in front of \(n\).
- Adjust: add or subtract so that \(n = 1\) gives the first term.
- Check: test \(n = 2\) as well.
- Use it: set the formula equal to a number; a whole-number \(n\) means it is a term.
One picture every idea returns to
If the terms go up by \(d\) each time, the nth term is \(dn + c\). Find \(d\) from the common difference, then \(c\) from the first term: \(c = \text{first term} - d\).
Forty-five minutes, phase by phase
| Time | Phase | What happens in the room | Grouping |
|---|---|---|---|
| 0 to 5 min | Hook: predict | Open with the question in the title, Same step, every time.. When a sequence goes up or down by the same amount each time, one short formula gives any term you like: the 10th, the 100th, or the first one past 1000. Put the figure on the board: it is the one picture every idea in the lesson returns to. | Whole class, then pairs |
| 5 to 11 min | Teach and check: Spotting the rule | In the lab, set a first term and a common difference and watch the terms build up as bars. Project tab 2 of the lab (Sequences and proportion lab, linear sequences) and let a student drive it while the class predicts. Model the worked example Continue a sequence on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: The nth term formula | In a linear sequence the common difference \(d\) is the number in front of \(n\). Model the worked example Increasing sequence on the board, one line at a time. Your turn: Find the nth term of \(9, 16, 23, 30, \ldots\) and use it to find the 50th term. | Pairs |
| 17 to 23 min | Teach and check: Using the nth term | Once you have the formula, set it equal to a number (or put it in an inequality) and solve for \(n\). Model the worked example Is it in the sequence? on the board, one line at a time. | Pairs |
| 23 to 30 min | Teach and check: Sequences from patterns | Count the objects in the first few patterns, find the nth term, then use it. Model the worked example Matchstick squares on the board, one line at a time. | Anticipation / Reaction Guide |
| 30 to 36 min | Check | The four Stage 1 questions from the student lesson, answered on mini whiteboards or by show of fingers. Re-teach the one most get wrong. | Individual, then whole class |
| 36 to 43 min | Plenary | The exam-style questions from Stage 2, one at a time: thirty seconds alone, then a random call. The feedback lines in the key give the examiner's wording. | Individual, random call |
| 43 to 45 min | Exit and homework | Set the worksheet below (6 questions, 17 marks). Learners self-assess against the objectives before they leave. | Individual |
Protect the parts that carry the learning
The sequence carries about fifty minutes of material, so the lab is driven by one student while the class predicts rather than explored freely. Times are a guide.
Protected: the check at 30 minutes. It is the only individual read on whether the key idea landed, and it is not cut.
Cut first if the build overruns: the last check in Sequences from patterns, never the check or the plenary.
In a 60 minute block: run worksheet questions 1 to 3 in class as a Rally Coach set and model both the hardest worked example and its check live.
Anticipation / Reaction Guide
Before the lesson, students mark five statements true or false. After the teaching they mark them again and justify any change of mind in one sentence each. The statements come from the examiner traps below.
The full structure, with its variations and the other twelve, is on the cooperative-learning page.
What to head off, and how
| Trap learners fall into | Teaching move that pre-empts it |
|---|---|
| A term-to-term rule (add 4) is not an nth term. | To find the 100th term with it you would need all 99 terms before. The nth term formula jumps straight there. |
| For 5, 8, 11, 14 do not write \(n + 3\) (adding the difference) or \(3n + 5\) (adding the first term). | Always check your formula with \(n = 1\) AND \(n = 2\). |
| When the answer to \(n\) is a decimal, do not round it and claim the number is a term. | A decimal \(n\) means the number is NOT in the sequence. |
| A pattern number must be a whole number. | Rounding 66.33 is not about 'nearest': round in the direction that keeps you within the 200 sticks. |
Sequences and the nth term: worksheet
6 questions, 17 marks. Written in the style of the papers; no past paper question is reproduced.
Write down the next two terms and the term-to-term rule: \(31, 25, 19, 13, \ldots\)
Write down the next two terms: \(2, 3, 5, 8, 12, \ldots\)
(a) Find the nth term of \(9, 16, 23, 30, \ldots\) (b) Find the 50th term.
The nth term of a sequence is \(50 - 3n\). Find the first term that is negative.
A sequence begins \(5, 12, 19, 26, \ldots\) Is 245 a term of this sequence? Show your working.
Rectangular tables are pushed together in a row. One table seats 6 people, two tables seat 10 and three tables seat 14.
(a) Find a formula for the number of seats with \(n\) tables. (b) How many seats are there with 12 tables? (c) 75 people come to a party. What is the smallest number of tables needed? (d) Explain why the number of seats is always even.
Answer key and marking notes
- Rule: subtract 6.
- Next terms: 7, 1.
- Differences: 1, 2, 3, 4, so the next differences are 5 and 6.
- \(12 + 5 = 17\), \(17 + 6 = 23\).
- (a) Difference 7, so \(7n\); \(7(1) = 7\), first term 9, so \(7n + 2\).
- (b) \(7(50) + 2 = 352\).
- \(50 - 3n < 0\) gives \(3n > 50\), so \(n > 16.67\) and \(n = 17\).
- \(T_{17} = 50 - 51 = -1\).
- Difference 7, so the nth term is \(7n - 2\).
- \(7n - 2 = 245\) gives \(7n = 247\), so \(n = 35.29...\)
- \(n\) is not a whole number, so 245 is NOT a term.
- (a) Difference 4, first term 6, so \(4n + 2\).
- (b) \(4(12) + 2 = 50\).
- (c) \(4n + 2 \ge 75\), so \(4n \ge 73\) and \(n \ge 18.25\). Round UP: 19 tables (18 tables seat only 74).
- (d) \(4n + 2 = 2(2n + 1)\), which is 2 times a whole number, so it is always even.
6 multiple-choice questions on this lesson
From the topic question bank. Assign them in the app for a marked score in your gradebook, or run the topic as a class game.
Write down the next two terms of the sequence \(31, 26, 21, 16, \ldots\)
- A\(11, 6\)
- B\(11, 7\)
- C\(12, 7\)
- D\(6, 1\)
Find the nth term of the sequence \(2, 9, 16, 23, \ldots\)
- A\(7n - 5\)
- B\(7n + 2\)
- C\(n + 7\)
- D\(2n + 7\)
The sequence \(50, 44, 38, 32, \ldots\) continues in the same way. Find the 20th term.
- A\(-64\)
- B\(-70\)
- C\(-58\)
- D\(64\)
Which of these numbers is a term of the sequence with nth term \(4n + 7\)?
- A131
- B130
- C133
- D128
A row of \(n\) triangles is made from matchsticks. Pattern 1 uses 3 sticks, pattern 2 uses 5 and pattern 3 uses 7. What is the largest pattern that can be made with 150 sticks?
- APattern 74
- BPattern 75
- CPattern 50
- DPattern 301
Sequence P has nth term \(5n - 2\). Sequence Q is \(9, 24, 39, 54, \ldots\) Find the nth term of Q.
- A\(15n - 6\)
- B\(15n - 2\)
- C\(15n + 9\)
- D\(3n + 6\)
Quiz key
Each term is 5 less than the one before: \(16 - 5 = 11\), then \(11 - 5 = 6\). \(6, 1\) skips a term: those are the 6th and 7th terms, not the 5th and 6th.
The difference is 7, so start with \(7n\): \(7, 14, 21, 28\). Each term is 5 less, so the nth term is \(7n - 5\). Check \(n = 2\): \(14 - 5 = 9\). \(7n + 2\) adds the first term and gives 9 for \(n = 1\); \(n + 7\) adds the difference instead of multiplying by it.
The terms go down by 6, so \(-6n\). When \(n = 1\), \(-6n = -6\) but the term is 50, so add 56: \(T_n = 56 - 6n\). \(T_{20} = 56 - 120 = -64\). \(-70\) comes from \(50 - 6n\) (adding the first term); \(-58\) is the 19th term, taking only 19 steps.
\(4n + 7 = 131\) gives \(4n = 124\), \(n = 31\), a whole number, so 131 is the 31st term. 130 gives \(n = 30.75\), 133 gives \(n = 31.5\) and 128 gives \(n = 30.25\). A decimal \(n\) means the number is NOT in the sequence.
Difference 2, first term 3, so the nth term is \(2n + 1\). \(2n + 1 \le 150\) gives \(n \le 74.5\), so round DOWN: pattern 74 uses 149 sticks. Pattern 75 needs 151 sticks, too many. 50 is \(150 \div 3\), treating every triangle as separate; 301 substitutes 150 for \(n\).
P is \(3, 8, 13, 18, \ldots\) Each term of Q is 3 times the term of P, so Q is \(3(5n - 2) = 15n - 6\). Check: difference 15, and \(15 - 6 = 9\). \(15n - 2\) multiplies only the \(5n\) by 3, giving 13 for the first term.
What to project
The student lesson embeds these tabs. Open them full screen on the projector; every slider has a drag handle on the stage.
Slides for this lesson
A projectable deck for every pack is on the way. Until then, present the student lesson full screen: its sections, figure and lab tabs follow the same order as the plan above.