Download Teacher Planning and Assessment Pack 5

1
R
Unit objectives
EC
T
ED
PR
O
O
F
1.1 Sequences
Getting things in order
R
• Add, subtract, multiply and divide integers
• Use the function keys for sign change, powers and roots
O
• Recognise squares of numbers to at least 12 x 12 and the corresponding
roots
C
• Use squares, positive and negative square roots, cubes and cube roots, and
index notation for small positive integer powers
• Use known facts to derive unknown facts
N
• Strengthen and extend mental methods of calculation, working with squares
and square roots, cubes and cube roots
U
• Use index notation for integer powers and simple instances of the index laws
• Recognise and use multiples, factors (divisors), common factors, highest
common factors, and lowest common multiples and primes
• Find the prime decomposition of a number
• Use the prime factor decomposition of a number
• Generate and describe integer sequences
• Generate terms of a linear sequence using term-to-term rules
• Generate sequences from practical contexts
• Generate terms of a sequence using term-to-term and position-to-term rules,
on paper and using ICT
• Find the next term of quadratic sequences
2
Getting things in order
Website links
• 1.3 Prime number
reserach and resources
• 1.4 Encyclopedia of
integer sequences
LiveText resources
Notes on context
•
Prime factorisation can form the basis of an attack on the widely used RSA
method of encryption. The full method of encryption is not detailed here as it
involves modular arithmetic; however, it is returned to in the unit plenary.
Aside from this, prime numbers can be a source of fascination to eager pupils –
as well as looking at the largest known prime number, pupils might be interested
in other associated questions, such as the distribution of primes.
• Extended task –
Door 2 door
• Use it!
• Games
• Quizzes
• ‘Get your brain in Gear’
O
O
F
Marie-Sophie Germain (1776–1831) was a mainly self-taught French
mathematician. In her early correspondence with other mathematicians,
she adopted the male pseudonym Louis Le Blanc. Her work on number
theory allowed other mathematicians to show that the possible solutions to
Fermat’s last theorem were restricted. In later life, Germain’s interest turned to
mathematical physics, where she contributed to the mathematical theory of
elasticity.
Awards
ceremony
• Audio glossary
• Skills bank
• Extra questions for
each lesson
Discussion points
• Worked solutions for
some questions
You may also wish to discuss the reasons why Marie-Sophie Germain adopted a
male pseudonym.
Level Up Maths Online
Assessment
ED
PR
Though Euclid’s proof (of the infinity of primes) should not be explored, you may
wish to discuss the general statement that there is no largest prime. You could
link this to a discussion about the search for the prime number with more than
10 million digits. (In particular, you could question the reasons for the challenge.)
Activity A
T
The next five Sophie Germain primes are 3, 5, 11, 23 and 29.
Activity B
EC
There are many possible sequences with these terms, for example:
Add 5
1, 4, 9, 16, 25, ...
Square the term number
0, 4, 9, 15, 22, ...
Add one more than you added last time
−2, 4, 9, 13, 16, ...
Add one less than you added last time
The Online Assessment
service helps identify
pupils’ competencies and
weaknesses. It provides
levelled feedback and
teaching plans to match.
• Diagnostic automarked tests are
provided to match this
unit.
R
−1, 4, 9, 14, 19, ...
• Boosters
b) −12° C
O
1 a) −1° C
R
Answers to diagnostic questions
c) 8° C
2 1, 4, 9, 16, 25
C
3 11, 19, 31
N
4 a) 1, 3, 7, 21
b) 1, 2, 3, 4, 6, 8, 12, 24
c) 1, 2, 4, 5, 10, 20, 25, 50, 100
b) 47, 58
c) 1, −2
U
5 a) 24, 29
Opener
3
1.1 Using negative numbers
Objectives
• Add, subtract, multiply and divide integers
O
O
F
• Use the function keys for sign change
Give each pupil a prepared integer card. Ask pupils
to find the pupil who has the card such that the sum
of their integers is 5. When they have been ‘checked
off’, they can sit down – the objective is to avoid
being the last pair standing!
Starter (2) Introducing the lesson topic
−2
0
−4
3
ED
Magic doughnut.
PR
Starter (1) Oral and mental objective
EC
T
Ask pupils to fill in the blanks so that each outer row
and column adds up to 4. The ‘magic number’ and
integers in the corners can then be varied.
Extension: Explore magic doughnuts where the four
numbers given are in the central boxes on the edges
(these do not have unique solutions).
R
Main lesson
R
– Begin by checking that pupils understand how negative numbers fit into the
number system.
O
Where do we use negative numbers in the real world?
What does −1 mean?
N
C
If Starter (2) was not used, this could be used now to practise mental
calculations involving positive and negative integers. Ensure any difficulties
are explored fully and resolved.
–
1 Addition and subtraction (1)
U
Demonstrate how a number line can be used to answer addition and
subtraction questions.
What is −2 + 5? What is −2 − 3?
–
2 Addition and subtraction (2)
Ask pupils to use a calculator to work out 10 + −5 and 10 − −5. (This will
provide a check on pupils’ ability to use the sign change key on their
calculators.)
What happens when you add a negative number?
What happens when you subtract a negative number?
You could use the analogy that subtracting a negative number is like taking
away a debt – you’re actually giving money, or adding it.
Q1, 4–5
4
Getting things in order
Resources
Starter (1): prepared cards
showing an integer value
(each card pair should
sum to 5, e.g. 7 and −2),
one card per pupil
Functional skills
Examine patterns and
relationships Activity A
Framework 2008 ref
1.2 Y8, 1.2 Y9; 2.2 Y8,
2.7 Y8
Related topics
Use of negative numbers
in real life: temperature,
credit and debt, distance
below sea level.
– Progress to multiplying and dividing positive and
negative integers. Begin by considering patterns:
3 × 1 = 3, 3 × 0 = 0, 3 × −1 = −3, 3 × −2 = −6, and
−3 × −1 = 3, −3 × −2 = 6, and so on.
What is 3 × −3? −4 × −4?
What is −6 ÷ 3? −6 ÷ −3?
–
3 Multiplication and division of positive and
negative integers
O
O
F
What happens when you multiply or divide a
positive number by a negative number?
What happens when you multiply or divide a
negative number by a negative number?
Q2–3, 6–8
PR
– Q9 involves evaluating and expanding brackets.
Give pupils a few minutes to try it and then
generate a class discussion. Ask: Which way is
better? Explain that there is not always a single
correct method in mathematics.
Q9
This provides support for pupils who may be
struggling with the idea of adding and subtracting
negative numbers, and can easily be adapted for
multiplication or division if necessary.
T
Activity B
ED
Activity A
EC
This activity extends the idea of alternating signs and introduces pupils to an
oscillating sequence.
Answers: a) −1, b) 1, c) −1, d) 1, e) −1.
−1n is 1 if n is even, and −1 if n is odd.
R
Plenary
O
R
Play reverse bingo. Display a bingo grid (vary the size depending on how much
time you have) with positive and negative numbers on. Pupils achieve a line
by coming up with calculations that have those numbers as answers. Each
calculation must use at least one negative integer.
C
Homework
Homework Book section 1.1.
N
Challenging homework: Devise an oscillating sequence. Draw a graph of the first
eight terms to prove that it is oscillating.
U
Answers
1 a) −7
b) −4
c) −12
d) 2
e) −12
f) −2
2 a) −364
b) 163
c) 65
d) −17
e) −£110
3 a) −10
b) −12
c) −4
d) −4
e) −3
f) −72
g) −8
h) −9
4 a) 16
b) −1
c) −6
d) −8
e) −5
f) −31
g) 0
h) 8
5 a) −6°C
b) Increase of 7°C.
g) 1
h) −6
6 a) 15
b) −32
c) 4
d) −13
e) 77
f) −12
7 a) −4
b) −6
c) −9
d) −3
e) 2
f) −5
c) −30
d) 77
g) 0
Discussion points
Most European
mathematicians resisted
using negative numbers
until the 17th century
because they didn’t have
a physical meaning – do
you agree?
Common difficulties
Pupils may find any of
the exercises difficult,
depending on their
understanding of the
number line. Be prepared
to draw diagrams or
extend patterns.
Be careful if declaring
‘two negatives make a
positive’ as pupils may
use this in the wrong
context (e.g. −2 + −5 = 7).
h) 5
LiveText resources
Explanations
Booster
Extra questions
Worked solutions
8 7, 4 and −7, −4
9 a) −33
b) −14
Using negative numbers
5
1.2 Indices and powers
Objectives
• Recognise squares of numbers to at least
12 × 12 and the corresponding roots
O
O
F
• Use known facts to derive unknown facts
• Strengthen and extend mental methods of
calculation, working with squares and square
roots, cubes and cube roots
• Use squares, positive and negative square
roots, cubes and cube roots, and index
notation for small positive integer powers
PR
• Use index notation for integer powers and
simple instances of the index laws
Starter (1) Oral and mental objective
T
ED
Select two pupils to challenge each other to a ‘square
off’. The first player begins ‘one squared is one’, the
second player responds with ‘two squared is four’.
This continues until a player makes a mistake and so
loses the game.
EC
Starter (2) Introducing the lesson topic
R
Caesar squares is an old method of coding where a
message is written to have a square number of letters and no spaces. It is then
written into the rows of a square, and the columns are read to give the coded
version.
For example: MATHS CODE becomes M A T and hence ‘MHOASDTCE’.
R
HSC
ODE
O
Challenge pupils to decipher: TNYLANOMOLBDDAUEOIATWAUCYHIRTEISLNIS.
(Today in maths you will learn about indices)
C
Main lesson
N
– Begin with square numbers
and square roots. Ensure pupils are happy with
__
the meaning of the √ and 2 notation.
____
What is √121 ? What is the square root of 121?
U
Discuss how all positive
integers have a positive and negative square root.
__
√
By convention, the
notation always means the positive square root.
What is 152?
Remind pupils that they know 32 and 52. How can you use these to find 152?
Show how 152 = 32 × 52.
___
___
___
What is √40 ? Explain that it must be between √36 and √49 . If appropriate,
pupils could use ICT to explore such estimation.
Q1–5
–
1 Writing in index form
What does 24 mean?
Explain that indices are used for repeated multiplication of the same number.
Use a few examples written in full to see if pupils can condense them. Q6
6
Getting things in order
Resources
Main: ICT to explore
estimation of square and
cube roots (optional),
multilink cubes (optional)
Activity A: scrap paper or
card (optional)
Functional skills
Use appropriate
mathematical procedures.
Decide on the methods,
operations and tools,
including ICT to use in a
situation Q5, 9
Framework 2008 ref
1.1 Y8, 1.4 Y9; 2.2 Y8, 2.2
Y9, 2.5 Y8
– Move on to cubes and cube roots. Sketch the first
few sizes of cubes (or show them with multilink
cubes).
___
3
What is 23? What is √27 ? What is 0.13?
Discuss how pupils can use the cubes and cube
roots they know to mentally work out others.
Introduce and use the cube and cube root keys
on calculators. It may be necessary, depending on
available calculators, to introduce the power key.
O
O
F
Q7–9, 11
2 Multiplying numbers in index form
–
What is 25 × 23?
Show how 25 × 23 = (2 ×2 × 2 × 2 × 2) × (2 × 2 × 2) = 28.
PR
Can you see a link between the question and the
answer? Stress that the base number has to be the
same.
3 Dividing numbers in index form
–
Activity A
T
This activity promotes recall of the basic square
numbers. Pupils can enjoy making the cards and
moving them around physically.
ED
Ask pupils to investigate 25 ÷ 23 and write an ‘index
law’ to explain what happens.
Q10, 12
Algebra – use of algebraic
notation with the index
laws. Q12
Answer: 8-1-15-10-6-3-13-12-4-5-11-14-2-7-9-16
EC
Activity B
This activity is an example of how index notation is useful.
R
Answers: a) two – 0 and 1; b) four – 00, 01, 10 and 11; c) eight – 000, 001, 010,
011, 100, 101, 110 and 111. There are 2n strings with n digits.
R
Plenary
A rematch of ‘square off’ (or ‘cube off’) as Starter (1).
O
Explore some of the links
between the families of
numbers, for example
adjacent triangular
numbers always form a
square number.
C
Homework Book section 1.2.
N
Challenging homework: Find the first four positive integers that are square
numbers and cube numbers.
Answers
b) 5
U
8
2 a) 196
Common difficulties
c) 9
b) 256
d) 121
c) 400
e) 10
4 a) 23
b) 82 or 43 c) 103
b) 4.1
d) 104
5 a) 11
b) 9 or −9
c) 2 or −2
d) 7
6 a) 28
b) 34
c) 74 × 83
d) 53 × 2
7 a) 64
b) 8
c) 3
d) 1000
8 a) 216
b) 512
c) −729
9 Answers (to 1 d.p.): a) 2.1
10 a) 35
f) 81
g) 1
c) 5.7
j) 5
d) 8.6
c) 3.7
e) 53
d) t14
24 = 8? Emphasise to
pupils that a power
indicates how many times
that number is multiplied
by itself.
LiveText resources
e) 1
d) 0.001
b) 2.8
b) 77
c) 66
d) 94
_________
___ 3 ______
2
√
√
√
11 (−5) , 18 ÷ 16 , 12167 , 182
c) z7
i) 49
e) 106
d) 4.5
f) 75
g) 61 (or 6)
e) r6
f) u7
3
b) d6
h) 6
d) 441
3 Actual answers (to 1 d.p.): a) 3.3
12 a) c11
Rapid growth of powers
– sketch a simple graph
of 2x for x = 1 to 5. Liken
this growth to changes in
world population over the
last 2000 years.
Discussion points
Homework
1 a)
Related topics
h) 40 (or 1)
Explanations
Booster
Extra questions
Worked solutions
1.2 Indices and powers
7
1.3 Prime factor decomposition
Objectives
O
O
F
• Recognise and use multiples, factors (divisors),
common factors, highest common factors,
lowest common multiples and primes
• Find the prime decomposition of a number
• Use the prime factor decomposition of a number
• Use index notation for integer powers and
simple instances of the index laws
The Goldbach conjecture: ‘Every even integer greater
than 2 can be written as the sum of two primes.’
Starter (2) Introducing the lesson topic
ED
Ask pupils whether they think this statement is true.
Challenge them to prove it for numbers of increasing
difficulty (e.g. 18 = 11 + 7, 38 = 19 + 19, and so on).
PR
Starter (1) Oral and mental objective
EC
T
Give a number and ask pupils to move to one side of
the room if they think it is a prime or the other side if
they think it is not. After each round reveal the answer
– those who were standing on the wrong side of the
room have to sit down. The winner is the last person
standing. (Include prime numbers such as 199, 233 and 269.)
Main lesson
C
O
R
R
– If Starter (2) was not used, use it now to revise prime numbers and the use of
divisibility tests.
What is a factor? What are the factor pairs of 60?
What is a multiple? What is the lowest common multiple of 7 and 9?
What is the highest common factor of 14 and 21? Why?
Discuss pupil answers and methods, and ensure their basic knowledge is
sound. Q1–3
N
– Discuss how primes are the building blocks of all numbers – much like the
elements in chemistry. Explain that any factor which is a prime number is
called a prime factor and that any number can be written as a product of its
prime factors.
1 Finding the prime factor decomposition
U
–
Show how a factor tree can be used to find the prime factors of 90, and how
prime factors can be written in index notation.
What is 2 × 3 × 3 × 5 in index notation?
Show pupils that the HCF of two numbers must be all of the prime numbers
that the two original numbers have in common. You could use a Venn diagram
to show this clearly.
–
2 Finding the highest common factor
Use prime factor decomposition to find the HCF of 42 and 154.
Develop this idea to explain how the LCM must be all of the prime numbers
multiplied together but only counting the overlap once.
8
Getting things in order
Resources
None
Functional skills
Interpret results and
solutions Q8, 9
Examine patterns and
relationships
Framework 2008 ref
1.2 Y8, 1.2 Y9, 1.4 Y8, 1.5
Y8; 2.2 Y8, 2.2 Y9
Website links
For links to lists of
primes, prime curiosities
and up-to-date details on
the largest known prime
number www.heinemann.
co.uk/hotlinks
–
3 Finding the lowest common multiple (LCM)
Use prime factor decomposition to find the LCM of
42 and 154.
Q4–7
– End with practice work on index notation and the
index laws. Q8 and Q9 introduce zero and negative
indices by encouraging pupils to investigate and
spot patterns. You may like to do this together as a
class, or allow pupils to work individually and then
bring the results together to discuss in detail.
O
O
F
Q8–11
Activity A
PR
This activity provides an alternative approach to HCF
by providing a visual image which pupils can explore
for themselves. It is based around the fact that the
number of coordinates with integer points (excluding
(0, 0) on the line connecting (0, 0) to (a, b) is the HCF
of a and b. Without drawing it, can pupils predict how
many integer coordinates (not including (0, 0)) the line
from (0, 0) to (72, 132) passes through? (12)
Activity B
This activity looks at the connection between LCM
and HCF.
Number b
30
42
30
100
180
210
LCM
210
1260
6
300
3000
10
1260
37 800
30
Discussion points
The Goldbach conjecture
is one of the oldest
unproved statements in
mathematics.
HCF
a×b
EC
Number a
T
Answer:
ED
Answers: 4, 4 – same as HCF, 3.
Homework
Is it twice as difficult to
decompose a number
with twice as many
digits? One of the first
times RSA encryption
was used it had a 129digit number to crack – it
took mathematicians 17
years! Today some primes
used are even longer.
Homework Book section 1.3.
Common difficulties
Challenging homework: Explain, in your own words, why x0 always has a value of 1.
6 = 61: when a quantity is
written without an index
it may be overlooked.
Encourage pupils to write
the power ‘1’.
R
Connecting formula: LCM = a × b ÷ HCF.
R
Plenary
N
C
O
Challenge pupils to decompose numbers of increasing size and see how far they
can get as a class. (As the numbers get larger, pupils will have to devise efficient
ways to use their calculators.)
U
Answers
1 a) 1, 56; 2, 28; 4, 14; 7, 8
c) 1, 48; 2, 24; 3, 16; 4, 12; 6, 8
b) 1, 72; 2, 36; 3, 24; 4, 18; 6, 12; 8, 9
d) 1, 120; 2, 60; 3, 40; 4, 30; 5, 24; 6, 20; 8, 15; 10, 12
2 a) 6
b) 9
c) 6
d) 48
3 a) 60
b) 75
c) 210
d) 363
4 a) 2 × 3 × 5
11
5 a) 4
b) 8
6 a) 6
c) 23 × 32
b) 2 × 3 × 7
6
6
6
c) 6
d) 4
b) 10
c) 30
d) 2
7 a) 210
b) 300
c) 1620
d) 30 800
8 a) 8, 1
b) 20 (or 1) c) 1
9 a) 0.1
10 a) 2−1
11 a) 28 × 3
b) 0.001
c) 0.001
b) 0.5
c) True
e) 3
9
22 × 32 = 64? 24? 34?
Demonstrate how these
answers are incorrect.
d) 32 × 11
f) 7
5
g) 5
h) 23
LiveText resources
Explanations
Booster
d) 0.1
b) 22 × 3 × 5 × 7
Extra questions
c) 22 × 5
d) 22 × 33 × 5 × 7
Worked solutions
1.3 Prime factor decomposition
9
1.4 Sequences
Objectives
• Generate and describe integer sequences
O
O
F
• Generate terms of a linear sequence using
term-to-term rules
• Generate sequences from practical contexts
• Find the next term of quadratic sequences
Starter (2) Introducing the lesson topic
ED
Define a sequence with a non-integer common
difference (e.g. start at 3, add 0.4). Let the sequence
move around the room with each pupil giving the next
term. To add a competitive element, you might either
time the class as a whole, getting them to try and
beat their previous score, or divide the class in two,
one half challenging the other.
PR
Starter (1) Oral and mental objective
EC
T
Display the numbers 1 to 10. Ask pupils to place the
numbers into two sequences. (The obvious answer is
the odd and even numbers.) Now challenge them to
divide up the ten numbers into three sequences.
Main lesson
R
A possible answer consists of the three arithmetic
sequences {1, 4, 7, 10, …}, {2, 5, 8, …} and {3, 6, 9}. More advanced classes
might like to consider the answer {1, 4, 9, …}, {2, 3, 5, 8, …} and {6, 7, 10}.
See if they can work out what the next terms would be for each sequence.
R
O
C
What are the next two terms in the sequence 5, 10, 20, 40, …?
What is the rule? Explain that the correct terminology is ‘term-to-term rule’.
N
Q1–3
1 Identifying an arithmetic sequence
U
–
Explain to pupils that sequences are called ‘arithmetic’ if the difference is
constant. Introduce the first term as a and the common difference as d.
Is 12, 20, 28, 36, … an arithmetic sequence? Why?
If d is negative, what is happening to the sequence?
Q4–5 (Q5 uses a
flow chart to produce a sequence – make sure pupils are happy with its use.)
– Tell pupils that sequences often come from describing real-life situations.
Refer to the context photo (forest fire) in the Pupil Book.
–
2 Working with practical sequences
Begin to explore practical sequences and encourage pupils to describe what
is happening, and give suggestions as to why it is happening.
Q6–7
10
Getting things in order
Activity B: isometric
paper (optional)
Functional skills
– Remind pupils what a ‘sequence’ is and what sets it apart from just a list of
numbers. You might like to give some examples and see if pupils think they
are sequences or not. (Examples: the square numbers, lottery numbers, the
digits of pi, and some arithmetic sequences.)
What is the term-to-term rule of the sequence 5.4, 5.7, 6.0, 6.3, …?
Resources
Examine patterns and
relationships
Change values and
assumptions or adjust
relationships to see the
effects on answers in the
model Activity B
Interpret results and
solutions Activity B
Framework 2008 ref
1.2 Y8, 1.2 Y9, 1.4 Y8, 1.4
Y9; 3.2 Y8, 3.2 Y9
Website links
www.heinemann.co.uk/
hotlinks
– Q8 looks at quadratic sequences, although the
term is not used (pupils are simply asked to identify
the next two terms). Depending on the class, you
may want to explain quadratic sequences in more
Q8
detail and look at the difference patterns.
Activity A
This is a brainteaser activity.
O
O
F
Answer: The rule is in the title. Look at the previous
term and read it out loud, so 21 becomes ‘one two
and one one’ or 1211. Therefore, the next two terms
are 312211 and 13112221.
Activity B
This is an extension of Q7.
ED
PR
Answer: 1, 4, 10, 19, 31, …. This is a slower model
of growth than the squares; physically this could be
interpreted as the trees are further apart, or that it is
wet, so the fire is finding it hard to spread. It could
also have something to do with the geographical
layout of the region, although this is unlikely to be as
homogeneous as the model suggests.
Plenary
Homework
Homework Book section 1.4.
EC
T
Challenge pupils to work backwards and come up
with a non-arithmetic sequence of their own. Share pupil answers with the class
and check together that they are non-arithmetic.
R
Challenging homework: Work out the value of the millionth odd number – explain
how you got this answer.
Answers
Would the decreasing
value of a car over time
follow a sequence? If so,
would it be arithmetic?
What is a quadratic
sequence?
Common difficulties
1
2
3
Squares on fire
1
5
13
LiveText resources
R
Term number
Some pupils have
difficulty in identifying
‘rules’. Tell them to
consider whether the
sequence is increasing
(addition or multiplication)
or decreasing (subtraction
or division). Looking at
the rate of increase or
decrease will guide them
to the correct operation
(e.g. fast increase would
indicate multiplication).
1 a) +4; 27, 31
e) ×2; 32, 64
b) −3; 7, 4
f) ×−1 (or ÷−1); 1, −1
c) +1.5; 11, 12.5
g) ÷2; 12.5, 6.25
3 10, 101
d) +0.1; 2.9, 3.0
1
h) ÷3; _19 , __
27
b) 3, 8, 15, 24
d) 35, 48
O
2 a) Rectangle with 24 squares (4 rows of 6 squares).
c) Add 5, then add the next odd number each time.
C
4 The arithmetic sequences are a) (a = 3, d = 2); c) (a = 10, d = −3); and d) (a = 7, d = 10).
5 4, 17, 30, 43, 56. Yes, it is arithmetic.
N
6 a) 4 houses, made of 21 rods.
b) Number of houses
1
U
Number of rods
6
2
3
4
5
6
11
16
21
26
31
c) Start at 6, term-to-term rule: add 5.
7 a)
Discussion points
d)
Yes; a = 6, d = 5
b) 25, 41
d) Difference is 4, 8, 12, …. Pupil’s own explanation.
e) Start at 1, add multiples of 4 starting with four.
f) No
8 a) 13, 18
d) 66, 112
b) 90, 85
e) 9, 4
c) 25, 35
f) 59, 86
Explanations
Booster
Extra questions
Worked solutions
1.4 Sequences
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1.5 Generating sequences using rules
Objectives
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• Generate terms of a sequence using term-toterm and position-to-term rules, on paper and
using ICT
Display a triangle, square and pentagon. Ask pupils
what the next shape in the sequence is. (Hexagon as
the number of sides is increasing by 1.)
Repeat the activity using more complicated shapes
and/or sequences. For example: square/hexagon/
(octagon); triangle/pentagon/octagon/(dodecagon).
Starter (2) Introducing the lesson topic
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Differentiation: Extend the activity to 3-D shapes and
revise the meanings of the -gon and -hedron suffixes.
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Starter (1) Oral and mental objective
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Display three or four arithmetic sequences.
What’s the next term? Is this an arithmetic sequence?
What’s the term-to-term rule?
Choose one sequence and ask pupils to work out the
100th term in that sequence.
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Main lesson
– If Starter (2) was not used, use it now to recap
previous work on arithmetic sequences and term-to-term rules.
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Explore how to work out any term in a sequence, initially by using the termto-term rule. Explain that there is a second way of describing sequences
– using a position-to-term rule. This is used to find the value of a term by
using the term number. (Explain that the term number indicates the position of
that term in the sequence.)
A good mental picture for a position-to-term rule is to imagine driving along
the sequence. Every mile there is a term in the sequence – so as we pass the
first term our odometer clicks and counts ‘1’, as we pass the second term our
odometer counts ‘2’, and so on. A position-to-term rule connects the number
on the odometer to the term being passed outside.
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1 Using worded position-to-term rules
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Use the position-to-term rule ‘multiply the term number by 2’ to find the first,
second, third and tenth terms of the sequence.
Discuss with pupils that although a position-to-term rule and a term-to-term
rule might look very different, they can describe exactly the same sequence.
You may want to explore some examples of these. For example: term-to-term
rule ‘start at 2, add 2’ and the position-to-term rule ‘2n’.
Q1–5, 7
– To extend this work, explain that it is quicker to write a position-to-term rule
using algebra.
Instead of writing ‘take the term number, multiply by 3 and add 2’, n can be
used for the term number, to give 3n + 2 as the rule.
What does 2n mean? What does 10 − n mean?
What about n2?
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Getting things in order
Resources
Starter (1): 3-D shape
models for extension
activity (optional)
Main: ICT to explore termto-term and position-toterm rules (optional)
Functional skills
Examine patterns and
relationships
Framework 2008 ref
1.3 Y9, 1.4 Y9; 3.2 Y8,
3.2 Y8
Website links
www.heinemann.co.uk/
hotlinks
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2 Using algebraic position-to-term rules
What does 5n + 2 mean?
What is the first term of this sequence? What is the
seventh term?
Q6, 8–9 (Q9 requires pupils to
use their reasoning skills.)
Activity A
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This activity is designed to promote enquiry skills, as
well as getting pupils to think about place value.
Answer: (for example) 15, 26, 37, 48.
Activity B
This activity extends the position-to-term rule into
quadratic sequences.
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Answer: First five terms are 1, 3, 6, 10 and 15. This is
the nth term of the triangular numbers.
Plenary
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Return to the sequence with which you started the
lesson. Give pupils the position-to-term rule and ask
them to calculate the 100th term (and other higher
terms).
Homework
Homework Book section 1.5.
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(You could tie this in with the forest fire modelling
question from lesson 1.4. By predicting how far the
fire will spread, the emergency services can prepare for it and get enough firefighting equipment ready.)
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Challenging homework: An arithmetic sequence has first term a and second
term b. Find the third, fourth and fifth terms.
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Answers
1 a) 3, 7, 11, 15 (a = 3, d = 4)
c) 12, 9, 6, 3 (a = 12, d = −3)
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2 a = 7, d = 4
b) 5, 12, 19, 26 (a = 5, d = 7)
3 a = 22, d = −2
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4 a) 5, 10, 15 and 50
c) 3, 10, 17 and 66
b) 5, 8, 11 and 32
d) 5.5, 6, 6.5 and 10
b) −5, −3, −1 and 193
e) 0.25, 0.5, 0.75 and 25
c) −16, −8, 0 and 776
6 a) 1, 5, 9 and 25
d) − 2, 4, 10 and 34
b) 99, 98, 97 and 93
e) 1, −1, −3 and −11
c) 106, 113, 120 and 148
f) 0.5, 1, 1.5 and 3.5
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5 a) − 3, −6, −9 and −300
d) 9, 8, 7 and −90
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7 a) Double the term number and add 1.
b) Multiply the term number by −3 and add 7.
c) Multiply the term number by 4 and subtract 2.
8 a) 5n + 7: 12, 17, 22, 27, 32
4n − 2: 2, 6, 10, 14, 18
6n + 1: 7, 13, 19, 25, 31
−3n + 2: −1, −4, −7, −10, −13
b) The term-to-term rule uses the same number that precedes the ‘n’ in the
position-to-term rule.
9 a) True
b) True
c) True
d) False
Discussion points
Can pupils spot any
relationships between the
numbers in a position-toterm rule and the way a
sequence is increasing or
decreasing?
Common difficulties
Pupils may find the jump
to position-to-term rules a
little disconcerting. Using
the odometer image and
writing the values of n
above the terms can help.
Pupils may be more
wary of nth terms with a
negative coefficient of n
(e.g. 10 − n, 3 − 2n). Use
repeated demonstration
and practice.
LiveText resources
Explanations
e) False
Booster
Extra questions
Worked solutions
1.5 Generating sequences using rules
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WGM Pages to come
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Getting things in order
Sequences
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Break the bank
Notes on plenary activities
These activities form a narrative which links together
the topics of this unit and returns to the theme of data
encryption.
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Only parts 3 and 6 require the previous answers to be
correct, so pupils can skip a task if they are stuck.
Part 3: A calculator will most likely be required as 23
is one of the prime factors.
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Solutions to the activities
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Part 5: Pupils are asked to complete the positionto-term rule of the sequence. If they are unsure of
how to complete the rule, remind them that the
term number, n, must be included. You may wish to
use the nth term vocabulary here. Can the positionto-term rule tell you if a sequence is increasing or
decreasing?
1 A –5 B 16 C –4 D 35
A and C could be Mr Mann’s accounts since they
have negative values.
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2 The key code is 6425.
4 a) 13, 11, 9, 7, 5
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3 52 × 257
b) Start at 13, subtract 2 each time
c) 8th term
6 In the ninth week.
c) 9th term
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b) 101 − 12n
5 a) 53, 41
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Answers to practice SATs-style questions
−1
4
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(2 marks for all, 1 mark for three correct)
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2 14, √25 , 23, 32, 33 (2 marks; 1 mark for four correct)
3 No. The factors of 20 are 1, 2, 4, 5, 10, 20 so the number could be even or
odd (5). (1 mark for suitable explanation)
4 23 (1 mark)
5 3n + 5, 8n – 4, 3n + 2 (1 mark each)
6 a) p−2
b) a3b9
7 x = 3 (1 mark)
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Getting things in order
c) 2d 2e 3 (1 mark each)
Functional skills
The plenary activity practises the following functional
skills defined in the QCA guidelines:
• Use appropriate mathematical procedures
• Find results and solutions
• Interpret results and solutions
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• Draw conclusions in the light of the situation
Break the bank
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