At a glance. One lesson of about 60 minutes. It assumes the first lesson of this unit and the idea of a quadratic from Year 10. Students compute first and second differences, decide whether a sequence is quadratic and find the number in front of n squared. Lesson 3 finds the whole rule.
Calculator: no calculator. The differences are small whole numbers and the lesson is about structure, so no calculator is needed.
What students will learn
Know
- A quadratic sequence has a constant, non-zero second difference.
- For an² + bn + c the second difference is 2a.
- The first differences of a quadratic sequence form an arithmetic sequence.
Understand
- The second difference measures how fast the gaps change, so it measures the curve.
- b and c move the terms but leave the second difference alone.
- Taking differences is a test: a row that becomes constant tells you the type of sequence.
Do
- Find first and second differences.
- Decide whether a sequence is arithmetic, quadratic or neither.
- Find a from the second difference and extend a sequence from its differences.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: first and second differences and a from the second difference (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the second difference for several a, b and c and state the rule linking it to a.
- Criterion C. Communicating: setting out a difference table with a sentence saying what it shows.
- Criterion D. Applying mathematics in real-life contexts: tiling and growing patterns (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then keep b and c at 0 and move a only, asking what happens to the bottom row. Then change b and c and ask what did not change.
Ask: “If you only knew the bottom row, what could you say about a? What could you not say about b or c?” The second difference fixes a but is blind to b and c.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the tile counts; gaps, then gaps of gaps. |
| 12 min | Figure, key idea, warning card. |
| 8 min | Worked examples; pairs finish the faded example. |
| 5 min | Question 1, the check. If many choose C, return to what the second difference compares. |
| 20 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Something to think about, then the exit ticket, question 8. |
The answers
| Warm-up | 7; n² + 2n + 1; 48. |
|---|---|
| Hook | The gaps are 4, 6, 8, 10 and the second difference is 2. The sixth patio needs 31 + 12 = 43 tiles. After the lesson, show the shape: patio n is a rectangle n tiles by n + 1 tiles plus one step tile, so n(n + 1) + 1 = n² + n + 1. |
| Worked examples | quadratic with second difference 4; a = 4; not quadratic. |
| Faded example | Second difference 4, so a = 2. |
| 1. Reading first and second differences | B. A mistakes steadily rising gaps for equal gaps; C takes the span of the first differences; D confuses growing quickly with a ratio. |
| 2. Finding a second difference | 6. 13 − 7 and 19 − 13. |
| 3. From second difference to coefficient | 7. 14 ÷ 2. |
| 4. Second difference from a rule | 10. 2 × 5. |
| 5. Extending a quadratic sequence | 61. 44 + 17. |
| 6. A later first difference | 35. 7 + 7 × 4. |
| 7. Using two terms and a second difference | 42. 3, 10, 23, 42. |
| 8. Which sequence is quadratic | C. A is arithmetic; B is arithmetic after its first gap; D doubles. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Took steadily rising gaps as equal, used the span of the gaps, or called fast growth geometric. | Ask: are these gaps equal? What is the change from one gap to the next? |
| 7 or 3 in question 2 | Gave the first gap, or halved too early. | Make a two-row table and subtract along the bottom row only. |
| 14 or 28 in question 3 | Gave the second difference, or doubled instead of halving. | Write 2a = 14 and ask what to do to both sides. |
| 5 or 6 in question 4 | Gave a, or doubled the coefficient of n. | Cross out the 3n and the −4 and ask what is left. |
| 44 or 17 in question 5 | Stopped at the 5th term, or gave the next gap. | Count terms and gaps: six terms have five gaps; list them. |
| 31 or 39 in question 6 | Added the second difference one time too few or too many. | List the first differences: 7, 11, 15, … and number them. |
| 23 or 36 in question 7 | Stopped at the 3rd term, or repeated a gap. | Write the gaps as a list and extend by 6 each time. |
| A, B or D in question 8 | Chose an arithmetic sequence, a sequence that settles, or a doubling one. | Ask for the second differences row by row and see which becomes constant. |
Questions worth asking
- Why must we compare the second differences, not just look at the first? Listen for: first differences that change are not enough; the change itself must be constant.
- What happens if the second difference is 0? The first differences are constant, so the sequence is arithmetic.
- Why does changing c not change any difference? Every term moves by the same amount, so the gaps are unaffected.
- What would a negative second difference tell you? The gaps shrink: a is negative and the pattern bends downwards.
Differentiation
A way in for anyone stuck
- Provide the difference table with the empty rows printed so students only fill in numbers.
- Use two colours for first and second differences, as in the figure.
- Give the frame: “The gaps are …. The gaps of the gaps are …, so …”
Stretch
- A sequence has nth term n² + n. Show that its second difference is 2 using differences of the algebra rather than numbers.
- Find a quadratic sequence whose second difference is 6 and whose first term is 10 and whose first difference is 5.
Students learning in English as an additional language
“Difference” means the result of subtracting one term from the next. “First difference” is that result for neighbouring terms. “Second difference” is the difference of the differences. “Constant” means the same each time. “Quadratic” here means the rule has an n squared term.
| Word | Plain meaning | Example |
|---|---|---|
| first difference | a term minus the term before it | from 8, 15, 24: 7 and 9 |
| second difference | the difference between neighbouring first differences | from 7 and 9: 2 |
| constant | staying the same every time | the 5 in 5n + 1 |
| quadratic sequence | a sequence whose nth term has an n squared term | n² + 4 |
| coefficient | the number multiplying a letter | the 9 in 9n² |
Sentence starters for explaining aloud or in writing:
- The first differences are … and the second differences are …
- The second difference is constant, so …
- The number in front of n² is … because …
Two things not to say
“The second difference is the number in front of n squared.” Say instead: it is twice that number.
“Keep subtracting until you get a number.” Say instead: stop when a row is constant, and name the type.
Questions to set
Extra practice in three tiers, including one question from earlier work for spacing. Every answer has been checked.
1Fluency
The idea on its own, with friendly numbers. Set these first.
- FindFind the second difference of 1, 3, 7, 13, 21.
The first differences are 2, 4, 6, 8, so the second difference is 2. - ExpressA quadratic sequence has second difference 18. What is a?
9.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextBeans are stacked in a pyramid in a market in Cusco: the layers hold 1, 4, 9, 16 beans. Find the second difference and the number of beans in the 8th layer.
Second difference 2; the layers are squares, so 8 × 8 = 64. - Earlier work (lesson 1)Find the nth term of 11, 15, 19, 23.
4n + 7.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs 4, 6, 10, 18, 34 a quadratic sequence? Explain.
First differences 2, 4, 8, 16; second differences 2, 4, 8. Not constant, so no. - Working backwardsA quadratic sequence has second difference 8, third term 21 and fourth term 37. Find the first term.
Gap 3 to 4 is 16, gap 2 to 3 is 8, gap 1 to 2 is 0; term 2 = 21 − 8 = 13, term 1 = 13.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
The figure works with the keyboard (tab to a slider, then the arrow keys) and by touch. “Show me the steps” gives the same states in words.
No device? A difference table drawn on squared paper does the same job.