At a glance. One lesson of about 60 minutes. Students write a term-to-term rule and a position rule for a linear sequence, use the position rule to find any term, and undo it to test whether a number is a term.
Calculator: no calculator. Nothing here needs one. The multiplications are small, and the structure of the rule is the point.
What students will learn
Know
- The term-to-term rule says how to get the next term from the one before.
- The position rule gives a term from its position n.
- In a linear sequence the number in front of n is the step.
Understand
- A position rule jumps to any term; a term-to-term rule needs the term before.
- The constant in the position rule corrects the first term.
- Undoing the rule tells you whether a number is in the sequence, and where.
Do
- Find a position rule for a linear sequence and test it on the first terms.
- Use a position rule to find a far term.
- Decide whether a given number is a term and give its position.
MYP assessment criteria
Wording paraphrased. What this lesson gives you evidence for, and what to look for:
- Criterion A. Knowing and understanding: using position and term-to-term rules and testing membership (questions 2 to 8).
- Criterion B. Investigating patterns: the first figure. Ask students to say what happens to the position rule when only the first term changes, and when only the step changes, and to predict before they move.
- Criterion C. Communicating: explaining why 3n is not the rule for 7, 10, 13 (question 9) and writing a rule in words and symbols.
- Criterion D. Applying mathematics in real-life contexts: seats in the rows of a stadium stand, where the 100th row is found without listing 99 (the hook and question 4), and distances and counts in the extra questions.
The one thing to do with the figure
Change the step first, then the first term. Ask what happens to the position rule each time, before reading it off. The step goes in front of n; the first term moves the number at the end.
Use the second figure for one question only: “is it a whole number of steps?”. Have students predict yes or no for N before sliding, and say why.
A possible order
| 5 min | Warm-up from the student page, in silence, then answers. |
|---|---|
| 10 min | Hook and the first figure, with predictions first. |
| 10 min | Key idea; the second figure on testing membership. |
| 10 min | Worked example on the 40th term; students finish the faded one in pairs. |
| 5 min | Question 1, the check. If many choose A or C, go back to the first figure and test n = 1. |
| 15 min | Your turn, Levels 1 to 3, at each student's own pace. |
| 5 min | Exit ticket, question 9, on paper or on screen. |
The answers
| Warm-up | 7; 17; 63. |
|---|---|
| Worked example | 5n + 4, so the 40th term is 204. |
| Faded example | 6n − 4, so the 25th term is 146. |
| 1. Position rule from a sequence | B, 4n + 2. A uses the first term as the constant; C swaps the step and the constant; D leaves out the step. |
| 2. Using a position rule | 23. 5 × 4 + 3. |
| 3. Using a term-to-term rule | 37. 2 + 5 × 7 = 37. |
| 4. A far term from the position rule | 199. 4 × 50 − 1. |
| 5. Which term | 23. (117 − 2) ÷ 5 = 23. |
| 6. Is it a term | B, 83. Add 1 and divide by 6: only 84 ÷ 6 is a whole number. 80, 88 and 91 give 13.5, 14.8 and 15.3. |
| 7. First term past a target | 202. The 28th term is 195, the 29th is 202. |
| 8. Working backwards along a sequence | 3. The step is 5. 23 − 4 × 5 = 3. |
| 9. The step is not the whole rule | B. A takes the step as the whole rule; C and D put the wrong number in front of n. |
What each wrong answer tells you
| They give | What it usually means | What to do |
|---|---|---|
| A, C or D in question 1 | Used the first term or the term number as part of the rule, or left out the step. | Ask for the step first, then test n = 1 in every option. |
| 20 or 35 in question 2 | Dropped the +3, or added before multiplying. | Write the rule with the steps in order: multiply n by 5, then add 3. |
| 42 or 30 in question 3 | Multiplied the term number by the step, or stopped one term early. | Number the terms in a row: 1st, 2nd, up to the 6th. How many steps are there? |
| 200 or 203 in question 4 | Left out the constant, or counted 50 steps from the first term. | Use the rule itself on n = 1 to see what it must give. From the 1st term to the 50th is 49 steps. |
| 22 or 17 in question 5 | Found the number of steps, not the term number, or divided by the first term. | Undo the rule step by step in a flow of boxes. |
| A, C or D in question 6 | Tested divisibility by the first term or by the step alone. | Say: subtract the constant, then divide by the step. Is it whole? |
| 195 or 209 in question 7 | Gave the term on either side of the one needed. | Write the 28th, 29th and 30th terms and compare each with 200. |
| 18 or 8 in question 8 | Counted the wrong number of steps back. | Draw the five terms and put the step between each pair of neighbours. |
Questions worth asking
- How do you know the number in front of n? It is the step, the number added each time. Listen for “the gap”.
- Why do you add 1 at the end when testing for a term? Because the first term is at position 1, and the number of steps is one less than the position.
- Can a number be a term of two different sequences? Yes. 17 is in 5, 11, 17 and in 2, 5, 8, 11, 14, 17. Ask for another example.
- What if the step is 0? Then every term is the same. The position rule has no n in it at all.
Differentiation
A way in for anyone stuck
- Give a table with columns n, 4n and the term, so students see the 4n column before the correction.
- Make students test the rule on n = 1 and n = 2 every time, and write the result.
- For testing a term, draw the three boxes (subtract, divide, add 1) on paper and fill them in.
Stretch
- Write the position rule for 20, 17, 14, 11, … (23 − 3n). What happens for large n? (The terms eventually become negative, which is Year 8 work.)
- Find a number that is a term of both 4, 7, 10, 13, … and 5, 9, 13, 17, … (13, then every 12 more: 25, 37, …).
Students learning in English as an additional language
The word position is used in a mathematical sense here. Point at the place under each bar and say “position one, position two”. Term, rule and step also need a picture.
| Word | Plain meaning | Example |
|---|---|---|
| term-to-term rule | says how to get from one term to the next | add 4 |
| position rule | gives a term from its position n | 4n + 1 |
| position | the place of a term in the sequence, counting from 1 | 10 is in position 2 of 5, 10, 15 |
| step | the amount added each time | 4 in 5, 9, 13 |
| linear sequence | a sequence with the same step every time | 3, 7, 11, 15 |
| test | check a rule on a case you already know | 4 × 1 + 1 = 5 |
Sentence starters for explaining aloud or in writing:
- The step is …, so the position rule starts with …n.
- For n = 1 the rule gives …, but the first term is …, so I add / subtract …
- … is / is not a term because … ÷ … is / is not a whole number.
Two things not to say
“The rule is the number you add.” The step is only half the rule. Always test n = 1.
“Just keep adding on your calculator.” That is fine for the 8th term and useless for the 200th. The position rule is the point of the lesson.
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.
- Term-to-termWrite the next two terms of 11, 15, 19, 23, …
27, 31. - Position ruleUse the position rule 6n − 2 to find the 5th term.
28. 6 × 5 = 30, then 30 − 2.
2In context
The same idea inside a situation, where the first job is deciding what to calculate.
- ContextA cyclist rides 12 km on the first day and 5 km more each day than the day before. How far does she ride on day 9?
52 km. 12 + 8 × 5. - Earlier work, powersWrite the first four square numbers, and the gaps between them. Is the sequence linear?
1, 4, 9, 16; gaps 3, 5, 7. Not linear.
3Challenge
Reasoning, generalising and working backwards.
- ReasoningThe 3rd term of a linear sequence is 17 and the 8th term is 42. Find its position rule.
The step is 25 ÷ 5 = 5, and 5n + 2 gives 17 and 42. So 5n + 2. - Is it a termIs 300 a term of 4n + 2? Explain.
No. 300 − 2 = 298, and 298 ÷ 4 = 74.5 is not whole.
Practicalities
Works on a phone or a shared screen. Nothing a student types is saved or sent anywhere.
Both figures work with the keyboard (tab to a slider, then the arrow keys) and by touch. Under each one, “Show me the steps” gives the same states in words.
No device? Write the terms on strips of paper and fold them to show the steps.