1.1 Sequences Getting things in order - Pearson … · 2 Getting things in order 1.1 Sequences Unit objectives † Add, subtract, multiply and divide integers ... struggling with

2 Getting things in order

1.1 Sequences

Unit objectives

• Add, subtract, multiply and divide integers

• Use the function keys for sign change, powers and roots

• Recognise squares of numbers to at least 12 x 12 and the corresponding roots

• 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

• Strengthen and extend mental methods of calculation, working with squares and square roots, cubes and cube roots

• 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

Website links

• 1.3 Prime number reserach and resources

• 1.4 Encyclopedia of integer sequences


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Opener 3

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.

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.

Discussion points

Though Euclid’s proof (of the infi nity 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.)

You may also wish to discuss the reasons why Marie-Sophie Germain adopted a male pseudonym.

Activity A

The next fi ve Sophie Germain primes are 3, 5, 11, 23 and 29.

Activity B

There are many possible sequences with these terms, for example:

−1, 4, 9, 14, 19, ... 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

Answers to diagnostic questions

1 a) −1° C b) −12° C c) 8° C

2 1, 4, 9, 16, 25

3 11, 19, 31

4 a) 1, 3, 7, 21 b) 1, 2, 3, 4, 6, 8, 12, 24c) 1, 2, 4, 5, 10, 20, 25, 50, 100

5 a) 24, 29 b) 47, 58 c) 1, −2

LiveText resources

• Awards ceremony

• Extended task –Door 2 door

• Use it!

• Games

• Quizzes

• ‘Get your brain in Gear’

• Audio glossary

• Skills bank

• Extra questions for each lesson

• Worked solutions for some questions

• Boosters

Level Up Maths Online Assessment

The Online Assessment service helps identify pupils’ competencies and weaknesses. It provides levelled feedback and teaching plans to match.

• Diagnostic auto-marked tests are provided to match this unit.

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1.1 Using negative numbers

Starter (1) Oral and mental objective

Give each pupil a prepared integer card. Ask pupils to fi nd 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

Magic doughnut.

−2 0

−4 3

Ask pupils to fi ll 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).

Main lesson

– Begin by checking that pupils understand how negative numbers fi t into the number system.

Where do we use negative numbers in the real world?

What does −1 mean?

If Starter (2) was not used, this could be used now to practise mental calculations involving positive and negative integers. Ensure any diffi culties are explored fully and resolved.

– 1 Addition and subtraction (1)

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

Objectives

• Add, subtract, multiply and divide integers

• Use the function keys for sign change

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 skillsExamine patterns and relationships Activity A

Framework 2008 ref1.2 Y8, 1.2 Y9; 2.2 Y8, 2.7 Y8

Related topicsUse of negative numbers in real life: temperature, credit and debt, distance below sea level.

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– 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

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

– 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

Activity A

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.

Activity B

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.

Plenary

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.

Homework

Homework Book section 1.1.

Challenging homework: Devise an oscillating sequence. Draw a graph of the fi rst eight terms to prove that it is oscillating.

Answers

1 a) −7 b) −4 c) −12 d) 2 e) −12 f) −2 g) 0 h) 5

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.

6 a) 15 b) −32 c) 4 d) −13 e) 77 f) −12 g) 1 h) −6

7 a) −4 b) −6 c) −9 d) −3 e) 2 f) −5

8 7, 4 and −7, −4

9 a) −33 b) −14 c) −30 d) 77

Using negative numbers 5

Discussion pointsMost European mathematicians resisted using negative numbers until the 17th century because they didn’t have a physical meaning – do you agree?

Common diffi cultiesPupils may fi nd any of the exercises diffi cult, 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).

LiveText resourcesExplanationsBoosterExtra questionsWorked solutions

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Objectives

• Recognise squares of numbers to at least 12 × 12 and the corresponding roots

• 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


1.2 Indices and powers


Select two pupils to challenge each other to a ‘square off’. The fi rst 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.


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’.

H S C

O D E

Challenge pupils to decipher: TNYLANOMOLBDDAUEOIATWAUCYHIRTEISLNIS.(Today in maths you will learn about indices)

Main lesson– 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?

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 fi nd 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

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

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– Move on to cubes and cube roots. Sketch the fi rst few sizes of cubes (or show them with multilink cubes).

What is 23? What is 3 √ ___

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.

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.

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

Ask pupils to investigate 25 ÷ 23 and write an ‘index law’ to explain what happens. Q10, 12

Activity A

This activity promotes recall of the basic square numbers. Pupils can enjoy making the cards and moving them around physically.

Answer: 8-1-15-10-6-3-13-12-4-5-11-14-2-7-9-16

Activity B

This activity is an example of how index notation is useful.

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.

Plenary

A rematch of ‘square off’ (or ‘cube off’) as Starter (1).

Homework


Challenging homework: Find the fi rst four positive integers that are square numbers and cube numbers.

Answers 1 a) 8 b) 5 c) 9 d) 121 e) 10 f) 81 g) 1 h) 6 i) 49 j) 5

2 a) 196 b) 256 c) 400 d) 441

3 Actual answers (to 1 d.p.): a) 3.3 b) 4.1 c) 5.7 d) 8.6

4 a) 23 b) 82 or 43 c) 103 d) 104 e) 106

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 e) 1

8 a) 216 b) 512 c) −729 d) 0.001

9 Answers (to 1 d.p.): a) 2.1 b) 2.8 c) 3.7 d) 4.5

10 a) 35 b) 77 c) 66 d) 94 e) 53 f) 75 g) 61 (or 6) h) 40 (or 1)

11 (−5)3, √ _________

182 ÷ √ ___

16 , 3 √ ______

12167 , 182

12 a) c11 b) d6 c) z7 d) t14 e) r6 f) u7

1.2 Indices and powers 7

Related topics

Algebra – use of algebraic notation with the index laws. Q12

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

Explore some of the links between the families of numbers, for example adjacent triangular numbers always form a square number.

Common diffi culties

24 = 8? Emphasise to pupils that a power indicates how many times that number is multiplied by itself.

LiveText resources

ExplanationsBoosterExtra questionsWorked solutions

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1.3 Prime factor decomposition

Objectives

• 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



The Goldbach conjecture: ‘Every even integer greater than 2 can be written as the sum of two primes.’

Ask pupils whether they think this statement is true. Challenge them to prove it for numbers of increasing diffi culty (e.g. 18 = 11 + 7, 38 = 19 + 19, and so on).


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– 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

– 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

Show how a factor tree can be used to fi nd 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 fi nd 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.

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

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– 3 Finding the lowest common multiple (LCM)

Use prime factor decomposition to fi nd 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.

Q8–11

Activity A

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)

Answers: 4, 4 – same as HCF, 3.

Activity B

This activity looks at the connection between LCM and HCF.

Answer:

Number a Number b LCM a × b HCF

30 42 210 1260 6

30 100 300 3000 10

180 210 1260 37 800 30

Connecting formula: LCM = a × b ÷ HCF.

Plenary

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 effi cient ways to use their calculators.)

Homework


Challenging homework: Explain, in your own words, why x0 always has a value of 1.

Answers 1 a) 1, 56; 2, 28; 4, 14; 7, 8 b) 1, 72; 2, 36; 3, 24; 4, 18; 6, 12; 8, 9 c) 1, 48; 2, 24; 3, 16; 4, 12; 6, 8 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 b) 2 × 3 × 7 c) 23 × 32 d) 32 × 11

5 a) 411 b) 86 c) 66 d) 46 e) 39 f) 75 g) 5 h) 23

6 a) 6 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 b) 0.001 c) 0.001 d) 0.1

10 a) 2−1 b) 0.5 c) True

11 a) 28 × 3 b) 22 × 3 × 5 × 7 c) 22 × 5 d) 22 × 33 × 5 × 7

1.3 Prime factor decomposition 9

Discussion points

The Goldbach conjecture is one of the oldest unproved statements in mathematics.

Is it twice as diffi cult to decompose a number with twice as many digits? One of the fi rst times RSA encryption was used it had a 129-digit number to crack – it took mathematicians 17 years! Today some primes used are even longer.


6 = 61: when a quantity is written without an index it may be overlooked. Encourage pupils to write the power ‘1’.

22 × 32 = 64? 24? 34? Demonstrate how these answers are incorrect.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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1.4 Sequences

Objectives

• Generate and describe integer sequences

• Generate terms of a linear sequence usingterm-to-term rules

• Generate sequences from practical contexts

• Find the next term of quadratic sequences


Defi ne 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.


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.

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.

Main lesson

– 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 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’.

What is the term-to-term rule of the sequence 5.4, 5.7, 6.0, 6.3, …? Q1–3

– 1 Identifying an arithmetic sequence

Explain to pupils that sequences are called ‘arithmetic’ if the difference is constant. Introduce the fi rst 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 fl ow 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 fi re) 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

Resources

Activity B: isometric paper (optional)

Functional skills


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

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1.4 Sequences 11

– 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 detail and look at the difference patterns. Q8

Activity A

This is a brainteaser activity.

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.

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 fi re is fi nding 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

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.

Homework


Challenging homework: Work out the value of the millionth odd number – explain how you got this answer.

Answers 1 a) +4; 27, 31 b) −3; 7, 4 c) +1.5; 11, 12.5 d) +0.1; 2.9, 3.0 e) ×2; 32, 64 f) ×−1 (or ÷−1); 1, −1 g) ÷2; 12.5, 6.25 h) ÷3; 1 _ 9 ,

1 __ 27

2 a) Rectangle with 24 squares (4 rows of 6 squares). b) 3, 8, 15, 24 c) Add 5, then add the next odd number each time. d) 35, 48

3 10, 101

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.

6 a) 4 houses, made of 21 rods.b) Number of houses 1 2 3 4 5 6

Number of rods 6 11 16 21 26 31

c) Start at 6, term-to-term rule: add 5. d) Yes; a = 6, d = 5

7 a) Term number 1 2 3

Squares on fi re 1 5 13

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 b) 90, 85 c) 25, 35 d) 66, 112 e) 9, 4 f) 59, 86

Discussion points

Would the decreasing value of a car over time follow a sequence? If so, would it be arithmetic?

What is a quadratic sequence?


Some pupils have diffi culty 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).

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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1.5 Generating sequences using rules

Objectives

• Generate terms of a sequence using term-to-term 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).

Differentiation: Extend the activity to 3-D shapes and revise the meanings of the -gon and -hedron suffi xes.

Starter (2) Introducing the lesson topicDisplay 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.

Main lesson– If Starter (2) was not used, use it now to recap

previous work on arithmetic sequences and term-to-term rules.

Explore how to work out any term in a sequence, initially by using the term-to-term rule. Explain that there is a second way of describing sequences – using a position-to-term rule. This is used to fi nd 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 fi rst 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.

– 1 Using worded position-to-term rules

Use the position-to-term rule ‘multiply the term number by 2’ to fi nd the fi rst, 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?

Resources

Starter (1): 3-D shape models for extension activity (optional)

Main: ICT to explore term-to-term and position-to-term rules (optional)

Functional skills


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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1.5 Generating sequences using rules 13

– 2 Using algebraic position-to-term rules

What does 5n + 2 mean?

What is the fi rst term of this sequence? What is the seventh term? Q6, 8–9 (Q9 requires pupils to use their reasoning skills.)

Activity A

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.

Answer: First fi ve terms are 1, 3, 6, 10 and 15. This is the nth term of the triangular numbers.

Plenary

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).

(You could tie this in with the forest fi re modelling question from lesson 1.4. By predicting how far the fi re will spread, the emergency services can prepare for it and get enough fi re-fi ghting equipment ready.)

Homework


Challenging homework: An arithmetic sequence has fi rst term a and second term b. Find the third, fourth and fi fth terms.

Answers 1 a) 3, 7, 11, 15 (a = 3, d = 4) b) 5, 12, 19, 26 (a = 5, d = 7) c) 12, 9, 6, 3 (a = 12, d = −3)

2 a = 7, d = 4

3 a = 22, d = −2

4 a) 5, 10, 15 and 50 b) 5, 8, 11 and 32 c) 3, 10, 17 and 66 d) 5.5, 6, 6.5 and 10

5 a) − 3, −6, −9 and −300 b) −5, −3, −1 and 193 c) −16, −8, 0 and 776 d) 9, 8, 7 and −90 e) 0.25, 0.5, 0.75 and 25

6 a) 1, 5, 9 and 25 b) 99, 98, 97 and 93 c) 106, 113, 120 and 148 d) − 2, 4, 10 and 34 e) 1, −1, −3 and −11 f) 0.5, 1, 1.5 and 3.5

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 e) False

Discussion points

Can pupils spot any relationships between the numbers in a position-to-term rule and the way a sequence is increasing or decreasing?


Pupils may fi nd 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 coeffi cient of n (e.g. 10 − n, 3 − 2n). Use repeated demonstration and practice.

LiveText resources

Explanations

Booster

Extra questions

Worked solutions

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WGM Pages to come

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Sequences 15

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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.

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.

Part 5: Pupils are asked to complete the position-to-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 position-to-term rule tell you if a sequence is increasing or decreasing?

Solutions to the activities

1 A –5 B 16 C –4 D 35A and C could be Mr Mann’s accounts since they have negative values.

2 The key code is 6425.

3 52 × 257

4 a) 13, 11, 9, 7, 5

b) Start at 13, subtract 2 each time

c) 8th term

5 a) 53, 41 b) 101 − 12n c) 9th term

6 In the ninth week.

Answers to practice SATs-style questions

1−2 3 2

5 1 −3

0 −1 4

(2 marks for all, 1 mark for three correct)

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 c) 2d 2e 3 (1 mark each)

7 x = 3 (1 mark)

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Functional skills

The plenary activity practises the following functional skills defi ned in the QCA guidelines:

• Use appropriate mathematical procedures

• Find results and solutions

• Interpret results and solutions

• Draw conclusions in the light of the situation

Break the bank 17

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Documents

1.1 Sequences Getting things in order - Pearson … · 2 Getting things in order 1.1 Sequences Unit objectives † Add, subtract, multiply and divide integers ... struggling with