The shape of the lesson
By the end of the lesson, learners can
- Square and cube numbers: In the lab, choose a quadratic sequence and watch the lab stack the first and second differences under the terms.
- Second differences: (Extended) For a general quadratic \(an^2 + bn + c\), the second differences are all \(2a\).
- Multiply each time: (Extended) In the exponential tab of the lab, set a first term and a multiplier and watch the bars shoot up (multiplier above 1) or shrink towards zero (between 0 and 1).
- Mixing sequences: (Extended) Some sequences are a familiar sequence with a twist: one more than a power of 2, or a fraction whose top and bottom are both linear.
- Compare with \(n^2\) (1, 4, 9, 16) and \(n^3\) (1, 8, 27, 64).
- Second differences constant: quadratic, with \(a\) = half the second difference.
- Subtract \(an^2\) and find the nth term of what is left.
- Constant ratio: exponential, \(a \times r^{\,n-1}\).
- Check the formula against at least three terms.
One picture every idea returns to
Constant FIRST differences: linear, \(dn + c\). Constant SECOND differences: quadratic, and the coefficient of \(n^2\) is half the second difference. A constant RATIO between terms: exponential, \(a \times r^{\,n-1}\).
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, When the gaps grow.. Not every sequence goes up in equal steps. 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: Square and cube numbers | In the lab, choose a quadratic sequence and watch the lab stack the first and second differences under the terms. Project tab 3 of the lab (Sequences and proportion lab, quadratic and cubic) and let a student drive it while the class predicts. Model the worked example Close to the squares on the board, one line at a time. | Whole class, then pairs |
| 11 to 17 min | Teach and check: Second differences | (Extended) For a general quadratic \(an^2 + bn + c\), the second differences are all \(2a\). Model the worked example Full quadratic (Extended) on the board, one line at a time. Your turn: (Extended) Find the nth term of \(3, 10, 21, 36, \ldots\) | Pairs |
| 17 to 23 min | Teach and check: Multiply each time | (Extended) In the exponential tab of the lab, set a first term and a multiplier and watch the bars shoot up (multiplier above 1) or shrink towards zero (between 0 and 1). Project tab 4 of the lab (Sequences and proportion lab, exponential sequences) and let a student drive it while the class predicts. Model the worked example Doubling (Extended) on the board, one line at a time. Your turn: (Extended) Find the 8th term of \(2, 6, 18, 54, \ldots\) | Pairs |
| 23 to 30 min | Teach and check: Mixing sequences | (Extended) Some sequences are a familiar sequence with a twist: one more than a power of 2, or a fraction whose top and bottom are both linear. Model the worked example Powers of 2, shifted (Extended) on the board, one line at a time. | Cooperative Concept Mapping |
| 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, 19 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 Mixing sequences, 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.
Cooperative Concept Mapping
Teams receive the key terms on cards and arrange them on a large sheet, drawing and labelling the links. Compare maps between teams before showing the model map.
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 |
|---|---|
| \(n^2\) means \(n \times n\), not \(2n\). | For \(n = 5\), \(n^2 = 25\) but \(2n = 10\). Likewise \(3n^2\) means \(3 \times n^2\), not \((3n)^2\). |
| The coefficient of \(n^2\) is HALF the second difference. | Using the second difference itself (\(4n^2\) for Fig. 1) leaves a leftover that is not linear, and the formula fails at \(n = 2\). |
| \(3 \times 2^{\,n-1}\) is NOT \(6^{\,n-1}\). | The power applies only to the 2; work it out first, then multiply by 3. |
| Do not cancel or simplify the fractions in the sequence before finding the pattern: \(\dfrac{2}{4}\) written as \(\dfrac{1}{2}\) hides the pattern in the numerators and denominators. | Return to Mixing sequences: ask a pair to state the correct version and show it on the lab before moving on. |
Quadratic, cubic and exponential sequences: worksheet
6 questions, 19 marks. Written in the style of the papers; no past paper question is reproduced.
(a) Write down the next two terms of \(1, 8, 27, 64, \ldots\) (b) Write down its nth term.
Find the nth term of \(0, 3, 8, 15, 24, \ldots\)
(Extended) Find the nth term of \(6, 15, 28, 45, 66, \ldots\)
(Extended) A sequence begins \(80, 40, 20, 10, \ldots\) (a) Find the nth term. (b) Find the first term that is less than 0.1.
(Extended) Sequence A has nth term \(n^2\) and sequence B has nth term \(2n + 3\). Each term of sequence C is the term of A minus the term of B.
(a) Write down the nth term of C. (b) Which term of C is equal to 60?
Pattern \(n\) is a rectangle of dots with \(n\) rows and \(n + 1\) columns. So pattern 1 has 2 dots, pattern 2 has 6 and pattern 3 has 12.
(a) Write down an expression for the number of dots in pattern \(n\). (b) How many dots are in pattern 10? (c) Which pattern has 600 dots? (d) Show that no pattern has exactly 500 dots.
Answer key and marking notes
- (a) \(5^3 = 125\), \(6^3 = 216\)
- (b) \(n^3\)
- Compare with \(n^2\): \(1, 4, 9, 16, 25\). Each term is 1 less.
- \(n^2 - 1\)
- First differences 9, 13, 17, 21; second differences 4, so \(a = 2\).
- Subtract \(2n^2\) (2, 8, 18, 32, 50): \(4, 7, 10, 13, 16\), which is \(3n + 1\).
- nth term \(2n^2 + 3n + 1\).
- (a) Ratio \(\dfrac{1}{2}\): \(T_n = 80 \times \left(\dfrac{1}{2}\right)^{n-1}\).
- (b) Keep halving: \(T_{10} = 80 \div 2^{9} = 0.156\,25\), \(T_{11} = 80 \div 2^{10} = 0.078\,125\).
- The first term below 0.1 is \(T_{11} = 0.078\,125\).
- (a) \(n^2 - (2n + 3) = n^2 - 2n - 3\).
- (b) \(n^2 - 2n - 3 = 60\), so \(n^2 - 2n - 63 = 0\).
- \((n - 9)(n + 7) = 0\), so \(n = 9\) (\(n\) cannot be negative).
- Check: \(81 - 18 - 3 = 60\). It is the 9th term.
- (a) \(n(n + 1) = n^2 + n\).
- (b) \(10 \times 11 = 110\).
- (c) Try values near \(\sqrt{600} \approx 24.5\): \(24 \times 25 = 600\). Pattern 24.
- (d) \(21 \times 22 = 462\) and \(22 \times 23 = 506\). 500 lies between two consecutive patterns, so no pattern has 500 dots.
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.
Find the nth term of the sequence \(4, 7, 12, 19, 28, \ldots\)
- A\(n^2 + 3\)
- B\(3n + 1\)
- C\(2n^2 + 2\)
- D\(n + 3\)
Find the nth term of the sequence \(2, 16, 54, 128, \ldots\)
- A\(2n^3\)
- B\((2n)^3\)
- C\(n^3 + 1\)
- D\(14n - 12\)
The nth term of a sequence is \(n^3 - n\). Find the 5th term.
- A120
- B10
- C20
- D125
Find the nth term of the sequence \(0, 7, 18, 33, 52, \ldots\)
- A\(2n^2 + n - 3\)
- B\(4n^2 - 4\)
- C\(2n^2 - 2\)
- D\(7n - 7\)
Find the 7th term of the sequence \(5, 10, 20, 40, \ldots\)
- A320
- B640
- C35
- D1 000 000
Find the nth term of the sequence \(1, 5, 17, 53, 161, \ldots\)
- A\(2 \times 3^{\,n-1} - 1\)
- B\(3^{\,n} - 2\)
- C\(4n - 3\)
- D\(2 \times 3^{\,n} - 1\)
Quiz key
Each term is 3 more than the square numbers: \(1 + 3, 4 + 3, 9 + 3, \ldots\) So the nth term is \(n^2 + 3\). Check \(n = 5\): \(25 + 3 = 28\). \(3n + 1\) fits the first two terms only (it gives 10, not 12, for \(n = 3\)).
Dividing by 2 gives \(1, 8, 27, 64\), the cube numbers. So the nth term is \(2 \times n^3 = 2n^3\). \((2n)^3 = 8n^3\) gives 8 for the first term; \(14n - 12\) fits the first two terms but gives 30, not 54, for \(n = 3\).
\(5^3 - 5 = 125 - 5 = 120\). 10 comes from reading \(n^3\) as \(3n\) (\(15 - 5\)); 20 uses \(n^2\) instead of \(n^3\); 125 forgets to subtract \(n\).
First differences 7, 11, 15, 19; second differences 4, so \(a = 2\). Subtract \(2n^2\) (\(2, 8, 18, 32, 50\)): leftover \(-2, -1, 0, 1, 2\), which is \(n - 3\). So \(2n^2 + n - 3\). Check \(n = 3\): \(18 + 3 - 3 = 18\). \(4n^2 - 4\) uses the second difference itself as the coefficient and gives 12, not 7, for \(n = 2\).
Ratio 2, first term 5, so \(T_n = 5 \times 2^{\,n-1}\) and \(T_7 = 5 \times 2^{6} = 5 \times 64 = 320\). 640 uses \(2^{7}\) (power \(n\) instead of \(n - 1\)); 35 treats the sequence as \(5n\); \(10^{6}\) multiplies 5 and 2 before applying the power.
Adding 1 gives \(2, 6, 18, 54, 162\), an exponential sequence with first term 2 and ratio 3: \(2 \times 3^{\,n-1}\). Take the 1 away again: \(2 \times 3^{\,n-1} - 1\). Check \(n = 4\): \(2 \times 27 - 1 = 53\). \(4n - 3\) fits only the first two terms; \(3^{\,n} - 2\) gives 7 for \(n = 2\).
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.