Mal3 Teacher Notes Generating Linear Sequences

Generating linear sequences

60 minutes

A Year 8 lesson using spreadsheets to reinforce pupils’ understanding of term-to-term and position-to-term descriptions of sequences, extending to rules that can then be expressed algebraically. The main activity extends to allow pupils to investigate quickly the effects of changing part of the rule for a sequence.
Oral and mental starter /

10 minutes

Objectives
•Generate terms of a linear sequence using term-to-term and position-to-term definitions of the sequence / Key vocabulary
Arithmetic sequence, nth term, rule, term-to-term, position-to-term, justify. / Resources
Counting stick

Starter activity

Use a counting stick to revise sequences:
Count forwards and backwards using the following sequences:
First termTerm-to-term rule
5add 11
23subtract 4
Point to one end of the stick and name it ‘term zero’, then point to the other end of the stick and identify it as ‘term ten’ or ‘tenth term’. Count forwards and backwards, using the following sequences. Then point to random divisions and ask the pupils what the value of that term is.
nth term (Position-to-term rule)
8n – 3
7 + 12n
2n + 1/2
Discuss the differences between ‘position-to-term’ and ‘term-to-term’.

Main teaching

40 minutes

Objectives
•Generate terms of a linear sequence using term-to-term and position-to-term definitions of the sequence, on paper and using a spreadsheet.
•Begin to use linear expressions to describe the nth term of an arithmetic sequence, justifying its form by referring to the activity or practical context from which it was generated. / Key vocabulary
Arithmetic sequence, nth term, difference pattern, predict, relationship, rule, term-to-term, position-to-term, justify. / Resources
Computer suite with Interactive whiteboard + projector
Excel spreadsheet loaded on all computers, including whole-class display.
Teaching/learning activity
On the interactive whiteboard or using a projector, display an Excel spreadsheet. Explain to the class that you are going to use a spreadsheet to calculate two of the sequences from the starter, one using the term-to-term rule and one using the position-to-term rule. Ask the class to discuss in pairs what instructions need to go in the cells of the spreadsheet. Ask one pair to share with the class their instructions, using the interactive whiteboard or computer attached to the projector.
During the next part of the lesson, pupils are asked to investigate a familiar sequence, then to predict what will happen with changes to the sequence. They can check their predictions quickly through the use of the spreadsheet. Being able to explain why the changes have such an effect is key to the pupils’ understanding of sequences.
Position / Term
1 / =A2*2
=A2+1 / =A3*2
=A3+1 / =A4*2
=A4+1 / =A5*2
=A5+1 / =A6*2
=A6+1 / =A7*2
=A7+1 / =A8*2
=A8+1 / =A9*2
=A9+1 / =A10*2
=A10+1 / =A11*2
On their own computers ask pupils to input the numbers 1 to 10, using a term-to-term rule in column A. Then enter the multiples of 2 starting with 2, using a position-to-term rule in column B. (See diagram on the left.)
Pupils explore and record the effect of adding or subtracting a constant number to each term of this sequence of multiples; nth term= 2n + b.
Key questions:
•What stays the same?
•What changes?
It may be useful to hold a mini-plenary part-way through the main part of the lesson to draw together key ideas, for example:
•2n – 1 generates the odd numbers, starting at 1, because each is one less than an even number. The difference between consecutive terms stays the same (2), but the starting point is one less.
•2n + 1 generates the odd numbers, starting at 3, because … .
•2n + 10 generates the even numbers, starting at 12, because … .
Other possible examples:
•nth term = 10n + b (If b is between 0 and 9, this generates numbers with units digit of).
•nth term = b – 3n (This generates a descending sequence).
Take feedback from the pupils, discussing their findings and justifications.
Ask ‘How do the values a and b effect an arithmetic sequence with the rule
nth term = an + b?’

Plenary

10 minutes

Plenary activity

Ask pupils to think of a rule that would generate all the positive multiples of three. What about a rule to give multiples of three that are bigger than 10?
•Ask pupils how would they generate:
–odd numbers bigger than 11;
–multiples of 7 bigger than 100;
–negative integers less than -4.
•In an arithmetic sequence the starting number is 3 and the constant to add on is 5. Will 273 be a term in this sequence? How could we check this using a spreadsheet?
•In the sequence 16, 21, 26, 31, … explain why 60 is not a term. Change the first term, so that 60 is a term in the sequence. How many possibilities are there? What rule connects the possible starting numbers?
•Ask pupils to think of another sequence where the term-to-term difference is 5, and the sequence includes the number 74. How many possibilities are there? What pattern/rule connects the possible starting numbers? What would the starting number be if 74 were the 10th term of this sequence? / Resources
Spreadsheet
Whole-class display

Lesson plan 3: generating linear sequencesKey Stage 3 National Strategy