At a glance. One lesson of about 60 minutes. It assumes term-to-term rules from Year 7 and gradient from Year 9. Students find the common difference, write dn + c, find terms, decide which term has a value and recover a rule from two terms.
Calculator: GDC optional. Every number is small enough to work by hand. A GDC table is offered as a way to test a rule, not as a way to find one.
What students will learn
Know
- A sequence with a constant difference d is arithmetic.
- The nth term is a + (n − 1)d, which simplifies to dn + (a − d).
- A value is a term only when the equation for n gives a positive whole number.
Understand
- The terms lie on a straight line: d is its gradient and the constant is its value at n = 0.
- n − 1 jumps separate term 1 from term n.
- Finding a term and finding a position are the same equation solved in opposite directions.
Do
- Find d from consecutive or distant terms.
- Write and check the nth term.
- Solve dn + c = value for n and interpret a non-integer or negative result.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: finding terms and positions from a rule (questions 1 to 8).
- Criterion B. Investigating patterns: the figure. Students record the rule for several a and d, describe how the constant relates to a and d, and justify it.
- Criterion C. Communicating: writing the rule with a check at n = 1, and a short justification of why a value is or is not a term.
- Criterion D. Applying mathematics in real-life contexts: seating, saving plans and counting in rows (hook and extra questions).
The one thing to do with the figure
Ask for the prediction first. Then set a = 2 and slide d, asking what stays the same and what changes about the line of dots. Move to a negative d and ask what the line does.
Ask: “Where would the line cross the vertical axis, and what is the rule's constant?” The constant is the value at n = 0, one jump before the first term, which is why it equals a − d.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 5 min | Hook: the seats in row 25; collect strategies, do not confirm yet. |
| 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 A, return to the warning card and test n = 1. |
| 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 | 21; y = 3x + 2; 39. |
|---|---|
| Hook | Row n has 4n + 2 seats, so row 25 has 102. Adding 24 jumps of 4 to 6 gives the same. |
| Worked examples | 3n + 2; the 32nd term; 46 − 6n. |
| Faded example | 5n + 7. |
| 1. Writing the nth term | C. A uses the first term as the constant; B swaps the two numbers; D takes the difference as the constant. |
| 2. Finding a term | 59. 11 + 8 × 6. |
| 3. Which term has a value | 9. 8n = 72, so n = 9. |
| 4. Common difference from two terms | 5. 40 ÷ 8. |
| 5. First term from two terms | 50. d = −5, then 35 + 15. |
| 6. Where two sequences meet | 7. 21 = 3n. |
| 7. Counting terms above zero | 17. Term 17 is 4; term 18 is −2. |
| 8. Reading a rule | B. A reads the constant as the first term; C drops the minus sign; D adds the 4. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, B or D in question 1 | Used the first term as the constant, swapped the two numbers, or used the difference as the constant. | Ask the student to test their rule at n = 1 and compare with the first term. |
| 65 or 54 in question 2 | Added nine differences, or left out the first term. | Draw the terms as a row of boxes and count the arrows between box 1 and box 9. |
| 72 or 7.75 in question 3 | Stopped before dividing by 8, or subtracted 5 instead of adding. | Write each step: 8n − 5 = 67, then + 5, then ÷ 8. |
| 40 or 4 in question 4 | Did not divide the total change by the number of jumps, or divided by the position. | Ask: how many jumps are there between term 2 and term 10? |
| 40 or 20 in question 5 | Went back one jump only, or went the wrong way with a negative difference. | Write d = −5 and ask what undoing a fall of 5 does. |
| 21 or 43 in question 6 | Stopped before dividing by 3, or gave the shared value. | Ask: what is the question asking for, a position or a value? |
| 16 or 18 in question 7 | Counted jumps instead of terms, or included the first negative term. | Write terms 16, 17 and 18 and look at the signs. |
| A, C or D in question 8 | Read the constant as the first term, dropped the minus, or added the 4. | Substitute n = 1, then n = 2, and list the terms. |
Questions worth asking
- Why does the rule use n − 1 jumps and not n? Listen for: term 1 is already there, so term 5 is four jumps on.
- How can you tell that 60 is or is not a term of 6n + 5? Listen for: solve 6n + 5 = 60 and see if n is a positive whole number (n = 55/6, not a whole number, means no).
- What does a negative difference do to the line? It slopes down; the terms fall and may become negative.
- Why is the constant a − d and not a? It is the value one step before the first term, at n = 0.
Differentiation
A way in for anyone stuck
- Give a table with rows n, jumps made and term, filled for the first five terms before asking for the rule.
- Use a number line with arrows marked +4 to show the jumps.
- Provide the frame: “d = …, so the rule starts …n. At n = 1 it gives …, so I add …”
Stretch
- Show that if the nth term of an arithmetic sequence is dn + c, the sum of any two terms equally far either side of a third is twice that third term.
- For which values of n are 5n + 3 and 3n + 21 within 4 of each other? Describe the pattern.
Students learning in English as an additional language
“Sequence” means a list in a set order. “Term” is one number in the list. “Common difference” means the same amount added each step. “nth” is said “en-th” and means any position. Say “four n plus two” aloud for 4n + 2.
| Word | Plain meaning | Example |
|---|---|---|
| sequence | a list of numbers in a set order | 2, 11, 20, 29 |
| term | one number in a sequence | the 3rd term of 8, 15, 22 |
| common difference | the amount added (or taken away) from one term to the next | 3, 14, 25, 36 has difference 11 |
| arithmetic sequence | a sequence with a common difference | 50, 43, 36, 29 |
| nth term | a rule that gives the term in position n | 9n − 4 |
| position | the place of a term in the list, counted from 1 | n = 12 |
Sentence starters for explaining aloud or in writing:
- The common difference is … because …
- At n = 1 my rule gives …, which matches …
- … is not a term because n would be …
Two things not to say
“Just add the difference on.” Say instead: how many jumps are there from the first term?
“The constant is the first term.” Say instead: the constant is the value one jump before the first term.
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.
- ExpressFind the nth term of 4, 9, 14, 19, …
5n − 1. - FindThe nth term is 2n + 13. Write the 10th term.
33.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cyclist in Chiang Mai rides 15 km on day 1 of training and adds 3 km each day. How far does she ride on day 12?
15 + 11 × 3 = 48 km. - Earlier work (Year 9)A line has gradient −2 and passes through (0, 7). Write its equation.
y = 7 − 2x.
3Challenge
Reasoning, generalizing and working backwards.
- ReasoningIs 150 a term of 6n + 5? Is 155 a term?
150: 6n = 145, n is not a whole number, so no. 155: 6n = 150, n = 25, so yes. - Working backwardsAn arithmetic sequence has 6th term 31 and 15th term 76. Find its nth term.
d = 45 ÷ 9 = 5; first term 31 − 25 = 6; rule 5n + 1.
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? Draw the first six points on squared paper and join them.