Lesson plan 3: 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-toterm, position-toterm, justify.
Resources
Counting stick
Starter activity
Use a counting stick to revise sequences:
Count forwards and backwards using the following sequences:
First term
Term-to-term rule
5
add 11
23
subtract 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
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.
ICT in mathematics
Lesson plan 3: generating linear sequences
40 minutes
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 wholeclass display.
© Crown copyright 2004
Key Stage 3 National Strategy
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
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.
=A10+1
•
•
=A11*2
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?’
ICT in mathematics
Lesson plan 3: generating linear sequences
© Crown copyright 2004
Key Stage 3 National Strategy
Plenary
10 minutes
Plenary activity
Resources
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?
Spreadsheet
•
•
•
•
Whole-class
display
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-toterm 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?
ICT in mathematics
Lesson plan 3: generating linear sequences
© Crown copyright 2004
Key Stage 3 National Strategy