Unit 1: Teacher Planning and Assessment Pack - Pearson Schools

2 Pattern perfect

Unit objectives

• Generate and describe simple integer sequences

• Generate terms of a simple sequence, given a rule (e.g. fi nding a term from the previous term, fi nding a term given its position in the sequence)

• Generate terms of a linear sequence using term-to-term defi nitions of the sequence, on paper and using a spreadsheet or graphical calculator

• Generate sequences from practical contexts and describe the general term in simple cases

• Begin to use linear expressions to describe the nth term of an arithmetic sequence

• Express simple functions in words, then using symbols

• Represent simple functions in mappings

• Use letter symbols to represent unknown numbers or variables

• Begin to distinguish the different roles played by letter symbols in equations, formulae and functions

• Know the meanings of the words ‘term’, ‘expression’, ‘equation’, ‘formula’ and ‘function’

• Know that algebraic operations follow the same conventions and order as arithmetic operations

• Use index notation for small positive powers

• Simplify linear algebraic expressions by collecting like terms

Website links

• 1.1 Number patterns



• 1.3 Painting with numbers – patterns in nature

• 1.4 Function machine and formulas

• 1.5 Using inverse operations

• 1.6 Algebra calendars

• To view websites relevant to this unit please visit www.heinemann.co.uk/hotlinks

1 Pattern perfect

Opener 3

Notes on context

Every lighthouse has a distinctive series of signals – different periods of darkness and light produce a unique fl ash pattern for each lighthouse. The individual light sequence of each lighthouse is called its ‘characteristic’.

The fl ash sequences allow ship captains to time intervals between light fl ashes so that they can identify lighthouses, using a publication called the Admiralty List of Lights and Fog Signals (ALL). ALL (produced by the UK Hydrographic Offi ce) is a comprehensive list that includes details of the location and characteristics of all lighthouses.

For details of the location and characteristics of lighthouses in Scotland and the Isle of Man, please visit the relevant unit website at www.heinemann.co.uk/hotlinks.

Discussion points

• Discuss why lighthouses need to have different ‘characteristics’. Why do some lighthouses also use coloured lights?

• Discuss how identifying a lighthouse can help a ship’s captain determine their position at sea in relation to the land.

• Discuss how sequences are used in operating systems of traffi c lights. What problems could be caused if sequences of instructions were incorrectly set?

• Discuss sequences in nature and, in particular, the Fibonacci sequence. Pose the original question: How many pairs of rabbits can be produced in one year if each pair produces a new pair which become productive from the second month?

Activity A

Example answers:

The sequence is a repeated pattern with the light off and then on. The length of time that the light is off is about the same as the length of time that the light is on.

The sequence is a repeated pattern of the light being off for a long period followed by the light on (quick fl ash), then off (same length as the fl ash period), and then on (quick fl ash).

The sequence is a repeated pattern involving coloured lights. A red light is followed by the light being off which is followed by a green light. Each coloured light is on for the same length of time as the light is off.

Activity B

Pupil’s own lighthouse patterns and sequence descriptions.

Answers to diagnostic questions

1 a) 17, 20, 23

b) 0, −2, −4

c) 8.3, 8.6. 8.9

2 5, 10, 20, 40

3 20, 24, 28

LiveText resources

• Mean machine

• Use It!

Games

Audio glossary

Skills bank

• Extra questions − There are extra questions for each lesson on your LiveText CD.

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. Select Year 7. Choose to Assign a Test, then select Medium Term Plans. Select Autumn Term Unit 1 Algebra 1

4 Pattern perfect

1.1 Sequences

Starter (1) Oral and mental objective

Using mini whiteboards, ask pupils to start at a number (e.g. 2) and write the next three numbers when you go up in jumps of 0.3, 0.9, 1 _ 2 ,

1 _ 4 , etc.

Starter (2) Introducing the lesson topic

Set a starting number (e.g. 13) and, using mini whiteboards, ask pupils to write numbers in steps of 6 (e.g. 13, 19, 25, 31). See who can reach the highest number in a set time (e.g. in 20 seconds).

Repeat with steps of, for example, 9, 7, 12.

Differentiation: Go down in jumps of 6, or up in jumps of 16.

Main lesson

– 1 Sequences

– Display the pattern shown on the right. Explain that Shakira has saved £3 pocket money and can earn £2 each time she does the washing up. She arranges her pound coins in a pattern – these are pictures of the different amount she earns.

What are the next two patterns in the sequence? How does the sequence change?

Ask pupils to count the number of coins in each pattern and write this as a sequence (3, 5, 7, 9, 11). Explain that these numbers are called terms. Ask pupils to identify the fi rst term, third term, fourth term, etc.

What is the sixth term? (13) How did you work this out? (add 2.) Explain that this is the term-to-term rule. Q1–2

– Display the following terms: ascending, descending, fi nite, infi nite, term-to-term rule.

What words can you use to describe the sequence? (ascending and fi nite). Explain the terms, if appropriate.

Connor gets £15 pocket money a week, but loses £1.50 each time he oversleeps. Ask pupils to write a number sequence for the different amounts Connor gets when he oversleeps, after receiving his pocket money (15, 13.50, 12, 10.50, 9, 7.50, 6). Which words can be used to describe the sequence? (descending and fi nite). What is the term-to-term rule? (subtract 1.5) Q3

– Display the sequence: 5, 8, 11, 14, 17, ... .

Objectives


• Generate terms of a simple sequence, given a rule (e.g. fi nding a term from the previous term, fi nding a term given its position in the sequence)

• Generate terms of a linear sequence using term-to-term defi nitions of the sequence, on paper and using a spreadsheet or graphical calculator

Resources

Starters: mini whiteboards

Intervention

Level Up Maths 2–3, Lesson 1.1

Functional skills

Find results and solutions Q7, 11

Framework 2008 ref

Process skills in bold type

• 1.2 Y7/8, 1.5 Y7/8, 3.2 Y7/8

PoS 2008 ref


• 1.2b, 2.2d, g, h, 2.4b, 3.1h

Website links

www.heinemann.co.uk/hotlinks

Sequences 5

How would you describe this sequence? (ascending and infi nite) Explain that we use the dots to show the sequence continues.

What is the term-to term rule? (add 3) Why can’t we write an infi nite sequence? (because it would go on for ever).

Give pupils the fi rst term and the term-to-term rule of a sequence (e.g. 5, add 4). Ask them to write the fi rst fi ve terms of the sequence. Repeat for other fi rst terms and term-to-term rules. Q4–11

Activity A

Pupils make up their own sequences to challenge other pupils.

Activity B

Pupils use the same term-to-term rule but different fi rst terms to try to make sequences with given properties.

a) yes – fi rst term any multiple of 3; b) no; c) no; d) yes – fi rst term any multiple of 3 less than 24; e) yes – fi rst term not a whole number.

Plenary

Ask pupils to write a sequence, and then give this sequence to their partner. The partner must describe the sequence and the term-to-term rule.

Homework

Homework Book section 1.1.

Challenging homework: Start with the terms 1, 4. Find four different ways to continue this sequence and describe the rule.

Answers 1 a) 12, 15, 18; multiples of 3, starting at 3

b) 40, 50, 60; multiples of 10, starting at 10c) 28, 35, 42; multiples of 7, starting at 7d) 7, 9, 11; odd numbers, starting at 2

2

3 a) 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0b) 0.2, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2, 2.4, 2.6, 2.8, 3.0c) 3.0, 3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0d) 10.0, 9.6, 9.2, 8.8, 8.4, 8.0, 7.6, 7.2, 6.8, 6.4, 6.0, 5.6, 5.2, 4.8, 4.4, 4.0e) 7.0, 7.6, 8.2, 8.8, 9.4, 10.0, 10.6, 11.2, 11.8, 12.4, 13.0f) 12.0, 11.7, 11.4, 11.1, 10.8, 10.5, 10.2, 9.9, 9.6, 9.3, 9.0

4 a) 26, 31, 36 b) 39, 43, 47 c) 36, 28, 20 d) 3.6, 3.8, 4.0 5 a) 14, 17, 20 b) add 3 c) ascending d) infi nite e) 29 6 a) 2, 7, 12, 17, 22, 27, 32, 37, 42 b) ascending and fi nite 7 a) 15 b) 13, 37 c) 2.5, 3.5 8 a) 5, 11, 23, 47, 95 b) 7, 11, 19, 35, 67 c) 127, 63, 31, 15, 7 9 Pupils’ term-to-term rules and sequences10 a) 1.8, 2.0, 2.2, 2.4, 2.6 b) 1, −2, −5, −8, −11

c) −3, −1, 1, 3, 5 d) 59.2, 59.9, 60.6, 61.3, 62.0e) −10, −45, −80, −115, −150 f) −4, −6, −8, −10, −12

11 Pupil’s own answers

Common diffi culties

When using negative numbers remember to add or subtract correctly.

LiveText resources

Explanations

Extra questions

Worked solutions

a) b) c)

6 Pattern perfect

1.2 Generating sequences

Objectives


• Generate sequences from practical contexts and describe the general term in simple cases

Resources

Starter (2), plenary: mini whiteboards

Activity B: squared paper

Plenary: 0–9 digit cards

Intervention


Functional skills

Examine patterns and relationships Q3, 8

Framework 2008 ref

Process skills in bold type• 1.1 Y7/8, 1.2 Y7/8,

1.3 Y7/8, 1.4 Y7/8, 1.5 Y7/8, 3.2 Y7/8

PoS 2008 ref

Process skills in bold type• 1.1a, c,1.2b, 1.3b, 1.4a,

2.1a–c, 2.2d–f, h, j, k, 2.3a, 2.4a, b, 3.1h

Website links



Display the grid below. Split the class into two teams. Teams take it in turns to choose a square on the grid. The team has to give the complement to 1000 of that number to win that square and then put a O or X in it. The winning team needs to get a row, column or diagonal of three Os or Xs.

93 871 235

560 625 269

304 651 464


Give pupils a mini whiteboard between two. Display the key words: ascending, descending, fi rst term, term-to-term rule.

Ask one pupil to make up a sequence and a second pupil to describe it using the key words. Take some examples to share with the class.

Main lesson

– 1 Generating sequences

Display a growing sequence of a bead necklace, like the one shown here. What are the positions of each of the red beads? Answers can be demonstrated on the board.

Can you describe the sequence? (ascending; fi rst term: 3; term-to-term rule: +4) What position would the tenth red bead be in? (bead 39) How did you work this out? Discuss strategies such as adding on lots of 4, adding on 9 × 4, etc.

Repeat using different bead necklaces. Q1–8

– Ask pupils to design their own necklace (which must have a regular pattern) and explain their strategies for fi nding the position of the tenth chosen colour bead.

Plenary

Pick two digit cards from a 0–9 set. Display one as the position of the fi rst red bead in a necklace. Display the other as the term-to-term rule ‘add �’. Ask pupils to draw the necklace on mini whiteboards. What is the position of the fi fth red bead? The tenth red bead? Repeat for different pairs and colours of beads.

Activity A

Pupils generate different sequences of matchstick patterns from a given fi rst pattern.

Related topics

Square numbers

LiveText resources

Explanations

Extra questions

Worked solutions

Activity B

Pupils investigate sequences in the context of the number of paving slabs needed to surround swimming pools of different sizes. Squared paper maybe useful for the diagrams.

Homework


Challenging homework: This is one way of building up a pattern from one black starting tile.

The sequence that goes with this pattern is 1, 3, 5, ... . The rule is +2.

Make up fi ve different pattern sequences starting with one black tile. Count the tiles in each pattern and write the sequence. Describe each sequence.

Answers1 a) 1, 3, 5, 7, 9, b) odd numbers2 Start at 12. Add 2 each time. Stop after ten terms.

3 a) i) Number of shapes 1 2 3 4

Number of matches 3 6 9 12

ii) Number of shapes 1 2 3 4

Number of matches 4 8 12 16

b) i) Term-to-term rule: +3 ii) Term-to-term rule: +44 a) 1.29 m, 1.37 m, 1.45 m, 1.53 m, 1.61 m, 1.69 m

b) Not a good model as he grows more in some years than others and will stop growing in late teenage years.

5 a) Number of fl owers 1 2 3 4 5

Number of beads 5 9 13 17 21

b) Term-to-term rule: add 46 a) 9, 7, 5, 3, 1, −1,−3 b) Shona owes money.7 a) i)

ii)

b) i) 3, 6, 9, 12, 15ii) 4, 7, 10, 13, 16

c) i) 30 ii) 318 a) £7, £9, £11, £13, £15, £17 b) £25

c) £45 d) After 29 days

Generating sequences 7

8 Pattern perfect

1.3 More sequences

Objectives

• Generate simple sequences from practical contexts

• Find the position-to-term rule

Resources

No special resources required

Intervention


Functional skills

Examine patterns and relationships Q5, 6, 7

Framework 2008 ref


• 1.2 Y7/8, 1.3, 1.4 Y7/8, 1.5 Y7/8, 3.2 Y7/8

PoS 2008 ref


• 1.5a, 2.2a, d, e, j, o, 2.3a, 3.1h

Website links



Practise counting forwards and backwards on a number line with different starting numbers in steps of 0.25, 1 _ 4 , 0.75, 3 _ 4 , 0.4, and so on.


4, 8, ….. How could this sequence continue?

Give pupils a few minutes to discuss this in small groups. Share answers with the class. Elicit that the sequence could continue in different ways. If the sequence continued 4, 8, 12, 16 … and continued to increase by 4 each time, then this is called an arithmetic sequence.

Ask pupils to write a sequence with a term-to-term rule of ‘increases by 5 each time.’

Main lesson

– 1 Square and triangle numbers

Display the sequence 1, 4, 9 from .

Which number is the fi rst term? The third term? The eighth term? Can you explain the term-to-term rule? … the position-to-term rule?

Can you suggest a number larger than 100 that will be in this sequence? Why is it in the sequence?

Display the fi rst three triangle number patterns on . How do the patterns grow? How many rows in the third pattern? How many dots in the bottom row? Q1–3

– 2 Arithmetic sequences

Display the sequence of matchstick patterns on The position-to-term rule .

Pattern 1 Pattern 2 Pattern 3 5 matchsticks 10 matchsticks 15 matchsticks

How many matchsticks will be in shape number 5? 7? What is the term-to-term rule?

Is this an arithmetic sequence? Why? (increases by the same amount each time) Q4

Complete the table to show the number of matchsticks in each shape. Demonstrate how the number of matchsticks can also be found from the position-to-term rule ‘multiply the pattern number by 5’.

Using the position-to-term rule, will there be a shape with 30 matchsticks? 38 matchsticks? How can you test whether your rule works? Q5

1

1

3

Discussion points

Opportunities to discuss sequences that can be found in nature or in practical scenarios.


Ensure pupils understand that 22 = 2 × 2 not 2 + 2, which is a common misconception. Practise fi nding powers of 2 and 3.

LiveText resources

Explanations

Extra questions

Worked solutions

– 4 Using the position-to-term rule to generate a sequence

If the position-to-term rule is 7 × position number, write down the fi rst fi ve terms of the sequence. Repeat for position-to-term rule of +7. Q6–8

Activity A

Pupils investigate digit properties of square numbers. The last digit of a square number will be 0, 1, 4, 5, 6 or 9. 572 will not be square number as its last digit is 2.

Activity B

Pupils investigate which triangle numbers sum to make square ones. When they have found some examples, encourage them to generalise and test their generalisation by drawing. (Consecutive triangle numbers sum to make a square number.)

Plenary

Display the sequence 7, 14, 21, 28 ….

What is the position-to-term rule? If I added 1 to this sequence what would it become? How would the position-to-term rule have changed?

Homework

Homework book section 1.3.

Challenging homework: Ask students to fi nd the relationship between odd numbers and square numbers.

Answers1 a) 3 × 3, 9 b) 4 × 4, 16; 5 × 5, 25

c) Square numbers2 a) 1 + 2 + 3, 6 b) 1 + 2 + 3 + 4, 10; 1 + 2 + 3 + 4 + 5, 15

c) Triangle numbers3 a) 1 × 2 = 2, 2 × 3 = 6, 3 × 4 = 12, 4 × 5 = 20, 5 × 6 = 30

b) length and width increase by 1 each time / multiply the position by the number one more than itself

c) 8 × 9 = 724 a, d, g, h, j5 a)

b) Shape number 1 2 3 4 5 6

Number of squares 2 4 6 8 10 12

c) 12d) Number of squares increases by 2 each time / number of squares is the even numbers.e) Increases by 2f) number of square = shape number × 2g) 10th term has 20 squaresh) correct shape drawn

6 a) 9, 10, 11, 12, 13, 14 b) 2, 4, 6, 8, 10, 12c) 3, 5, 7, 9, 11, 13 d) 1, 4, 7, 10, 13, 16

7 a) a) add 1 b) add 2 c) add 2 d) add 3b) The position is multiplied by the term-to-term rule in the position-to-term rule.

8 The term-to-term rule is add 2; the position-to-term rule is (2 × position number) + 1

More sequences 9

10 squares 12 squares 14 squares

10 Pattern perfect

1.4 Function machines

Objectives

• Express simple functions in words, then using symbols

• Represent simple functions using function machines

Resources

No special resources required

Intervention


Functional skills

Change values and assumptions or adjust relationships to see the effects on answers in the model Q7–8

Framework 2008 ref


• 1.1 Y7, 1.2 Y7, 1.5 Y7, 3.2 Y7/8

PoS 2008 ref


• 2.1b, 2.2f, g, i, 2.4a, b, 3.1h

Website links



Display the following blank multiplication.

�� × � = �Ask pupils to make the largest answer, the smallest answer and the answer nearest to 300 using the digits 4, 6 and 9.

Ask pupils to explain their strategies.

Repeat with different digits such as 2, 0 and 5.


Display a cloud with the numbers 3, 6, 7, 12, 5, 9, 11 in it. Display an operation such as ‘add 29’, −6 or ×7.

Point to numbers in the cloud and ask pupils to calculate the answers using the operation. Repeat with another operation.

If appropriate, repeat using two-step operations such as ×2 + 3, ÷10 + 4.

Main lesson

– 1 Function machines 1

Display a blank function machine such as theone shown. Write an operation (e.g. +5) in themachine. Model putting a number into the machine. Explain that this is an input, and the result after being acted on by the operation is the output. Ask pupils to calculate the output for different inputs. Q1

– The output is 22. What was the input? How did you work it out?

Encourage the use of inverse operations (−5) to fi nd the answer. Repeat using other operations such as −7, ×8, ÷2. Challenge pupils to fi nd the missing input using inverse operations. Q2

– 2 Function machines 2

– Display a blank two-stepfunction machine such as theone shown. Write in an operationsuch as ×2 + 3.

The input is 6. What is the output? (15)

The output is 11. What is the input? (4)

Ask pupils to describe the strategy they used.

Emphasise that because the output is travelling backwards through the function machine, the inverse operations must be done in the order that the output meets them. Hence the inverse of ×2 + 3 is −3 ÷ 2. Q3–5

Related topics

Inverse proportions


Pupils may get the order of calculations wrong when calculating the input of a two-step function machine given the output.

LiveText resources

Explanations

Extra questions

Worked solutions

– Use a one- or two-step function machine to play a ‘What’s my function machine?‘ guessing game. Display a function machine, with the operation covered up. Ask pupils to suggest inputs, and give them the output. Pupils use this information to guess the function.

If you are using a two-step function machine you could ask pupils for strategies for which inputs they chose to work out the function. Q6–8

Activity A

Pupils play a guess the function game in pairs.

Activity B

Pupils try to fi nd as many different two-step function machines as possible for a given input and output.

Plenary

Display a two-step function machine with the operations ×4 − 5. Choose three inputs and ask pupils to fi nd the outputs. Ask pupils to reverse the order of operations. Does the output change?

Repeat with the operations ×4 ÷ 2, and the operations −5 + 3.

(If the operations are × and ÷, or + and − the order of operations does not matter.)

Homework


Challenging homework: I think of a number double it, add 5 then multiply by 3; the answer is 57. Draw a function machine for this puzzle and fi nd the mystery input. Make up three similar problems.

Answers1 a) 3, 1.5, 4 c) 4.5, 1.7, 3.32 a) 5, 4, 35 b) 10, 11, 6 c) 20, 0.6, 0.25 d) 1.5, 2.8, 5.13 a) 8, 17, 23 b) 15, 45, 0 c) 2, 4, −7

d) 12, 11, 8.5 e) 17, 1, 4 f) 10, 12, 604 a)

b)

c)

5 c) The order of the operations does change the output.6 a) +3 b) ×8 c) −7 d) ÷27 a) 3, ×5, +1, 16 b) 7, −4, ×8, 24 c) 15, ÷5, +6, 9 d) 28 −4, ÷68 a) 6 + 2, ×7, 56 b) 6 ×5, −3, 27 c) 6 ÷2, +3, 6 d) 32 − 8 ÷ 4 = 6

Function machines 11

�54 27�7

�211 36�4

�218 2�7

12 Pattern perfect

1.5 Expressions and mappings

Objectives



• Know the meanings of the words ‘term’, ‘expression’, ‘equation’, ‘formula’ and ‘function’

• Know that algebraic operations follow the same conventions and order as arithmetic operations

• Use index notation for small positive powers

• Simplify linear algebraic expressions by collecting like terms

Resources

Starter (2): set of digit cards, mini whiteboards

Main: bag of counters

Intervention


Functional skills

Use appropriate mathematical procedures Q3, 6

Framework 2008 ref


• 1.2 Y7, 1.3, 1.5 Y7, 3.2 Y7/8

PoS 2008 ref


• 2.2h, l–n, 3.1e, f, h


Display a multiplication such as 14 × 36 = 504 and ask pairs of pupils to make up four other multiplications or divisions. Share their answers, explaining strategies.

Emphasise inverse operations, halving and doubling and multiplying either 14 or 36 by powers of ten.


Use a set of digits cards to choose fi ve numbers. Ask pupils to use these numbers and the four operations to make a target number between 50 and 200. Pupils work in pairs. Give them one minute to get as close to the target number as they can. Share solutions.

Main lesson

– 1 Order of operations

Explain the order of operations and practise on calculations involving brackets and indices. Q1

Hold up a bag of counters. Explain that you don’t know how many counter are in the bag – it’s a mystery number’, called a. Take out two counters. How many are there in the bag now? (a − 2). Explain that a − 2 is an expression for the number of counters in the bag.

– What do these expressions mean?

a + a a + 2 a × 2 2 + a a2 2a a × a a + a + 2

What does each expression mean?

Which expressions are equivalent? (a + a, a × 2 and 2a all mean ‘2 bags of counters’; a + 2 and 2 + a, both mean ‘bag + 2 extra’) How can you show that these expressions are equivalent? (By choosing a value for a and substituting it in the expressions. Avoid a = 2 as this could mislead pupils.)

How can each expression be simplifi ed? Take suggestions from pupils and then show how they can be simplifi ed by collecting like terms. Q2

– Ask pupils to simplify the following expressions.

b + b + 3 + b + 5 (3b + 8) 7 × 2b (14b) 3b + a + a − 6 + 8 − b (2a + 2b + 2)

Make sure that pupils know that these expressions cannot be simplifi ed any further. Q3–7

Expressions and mappings 13

Discussion points

In number work, 3 × 5 = 3 lots of 5 = 5 + 5 + 5. So 3 × a = 3 lots of a = a + a + a


Pupils may confuse a + 2 and 2a – use the bag of counters to emphasise the difference.

Pupils may try to simplify a + b to ab.

LiveText resources

Explanations

Extra questions

Worked solutions

– 2 Mapping diagrams

– 3 Mappings 1

– 4 Mappings 2

Demonstrate how the function machine

can be represented as an algebraic mapping x → x + 5. Display an empty mapping diagram and ask pupils to join inputs to complete it. Q8–11

Activity A

Pupils try to make all numbers between 1 and 10 using the digits 1, 2, 3 and 4, the four operations and brackets. Answers:

11: 4 × 2 + 3; 12: 4 × 2 + 3 + 1; 13: 4 × 3 + 1; 14: 4 × 3 + 2; 15: (4 + 1) × 3; 16: 4 × (3 + 1); 17: (4 + 2) × 3 − 1; 18: (4 + 2) × 3; 19: (3 + 2) × 4 − 1; 20: (3 + 2) × 4

Activity B

Pupils create their own multiplication pyramids using letters and numbers, like the pyramids in Q7.

Plenary

Display the following equations.

a2 = 2a a + 4 = 4 − a

5a = a + a + a + a + a a + 3 = a + 2

Ask pupils to decide whether the equations are always true, sometimes true or never true by substituting values into the equations. Pupils can use the numbers 0, 1, 2 and 10 to substitute into the equations. (sometimes, sometimes, always, never)

Homework


Challenging homework: These two expressions are equivalent: n × n n2. Find fi ve more pairs of equivalent expressions.

Answers 1 a) 23 b) 65 c) 12 d) 90 e) 25 f) 8 2 a) 3b b) 3b + 9 c) 2b d) 15b e) 2b + 6 f) 8b+18 3 a) b) c)

4 a) 5a + 2b + 3 b) 4a − 2b c) 8a − 2b − 8 d) 12a + 4b + 9 5 a) 10a b) 32b c) 6y d) 4y

e) 2a f) 5c g) 8b h) z 6 Pupil’s own answer. 7 a) b)

8 a) 1 → 4, 2 → 5, 3 → 6, … b) 1 → 0, 2 → 1, 3 → 2, … c) 1 → 2, 2 → 4, 3 → 6, … d) 1 → 6, 2 → 7, 3 → 8, … e) 1 → 3, 2 → 5, 3 → 7, … f) In each, the mapping lines are all parallel. g) In each, the mapping lines spread out. 9 a) x → x − 3 b) x → 3x c) x → x + 910 a) x → x − 2 b) x → 2x11 a) 15 b) 13 c) 16

200t

5t 40

t 5 8

288y

12 24y

0.5 24 y

3a � 5

2a � 3 a � 2

2a 3 a

8b � 2

4b � 2 4b � 4

b � 2 3b b

4c � 6

2c 2c � 6

c � 4 c � 4 c � 2

�5

14 Pattern perfect

1.6 Constructing expressions

Objectives



• Know the meanings of the words formula and function

• Write algebraic expressions

Resources

Starter (1): mini whiteboards

Intervention


Functional skills

Make an initial model of a situation using suitable forms of representation Q6–7

Framework 2008 ref


• 1.1 Y7, 1.2 Y7/8, 3.1 Y7/8, 3.2 Y7/8

PoS 2008 ref


• 1.4a, 2.1c, 2.2e, 3.1e, f, h

Website links



Display the following blank calculation.

� . � × � = �Give pupils the digits 1, 2 and 3 and ask them to make the largest answer, the smallest answer and the answer nearest 2. Mini whiteboards may be useful.

Repeat with different digits (e.g. 2, 8 and 5).

Ask pupils to explain their strategies. (If the similar starter in 1.4 was used, pupils could suggest links between strategies.)


Ask a pupil to make up a number puzzle (e.g. ? + 3 = 10) and write it on the board. Ask pupils to solve the number puzzle. Repeat with other number puzzles. Ask pupils to explain their strategies.

Main lesson

– Display the function machine shown.Ask pupils to give the outputs for different inputs.

The input is c. What is the output? (c + 6) Explain that they have just used the function machine to construct an expression.

Repeat for different operations in the function machine, for example −4 (c − 4), −78 (c − 78), +0.5 (c + 0.5), ×2 (2c), ÷10 ( c __ 10 ).

– 1 Constructing expressions

Tell pupils that you could also use an expression to represent the number of pupils in the classroom. Discuss with pupils a letter you can use to represent the number (e.g. p).

Two pupils come into the classroom. Can you construct an expression for the number of pupils in the classroom now? (p + 2) Repeat for other numbers of pupils entering or leaving the classroom. Q1–3

– Discuss with pupils expressions for the number of pupils in two identically sized classes. Discuss with the class all the different answers that are equivalent: p + p, p × 2, 2 × p, 2p.

What is the difference between p + 2 and 2p?

Tell pupils that the number of pupils in a lab can be represented by b. What is an expression for the number of pupils in a classroom and in a lab? (p + b)

How many pupils in three classrooms and two labs? (3p + 2b)

�6


Pupils sometimes confuse x + 2 and 2x

LiveText resources

Explanations

Extra questions

Four pupils go into a classroom. What is the expression for the total number of pupils now? (3p + 2b + 4)

Repeat for different examples. Q4–8

Activity A

Pupils describe expressions in words while playing a version of noughts and crosses.

Activity B

Pupils construct expressions from a complex function machine.

Answers: a) 16 b) 40 c) 3b + 14 _______ 2 d) 2d − 14 _______

3

Plenary

Display the following sets of expressions and ask pupils to pick the odd one out and explain why.

a) 5 + x

5x

x + 5 (5x)

b) 6 − b

b − 6

b − 2 − 4

b − 4 − 2 (6 − b)

c) 4y

y + y + y + y

y × y × y × y (y × y × y × y)

Homework


Challenging homework: I think of a number double it, and subtract 3. The answer is 17.

Assume that x is the mystery number and write an expression for this number and write it equal to 17.

Make up three similar problems.

Answers1 a) a + 4 or 4 + a b) b − 3 c) 20 − C

d) D + 7 e) m − 4 f) l + 22 a) w + 3 b) r + 10 c) y − 7 d) w + 8 e) w − 33 a) n + 3 b) n − 21 c) 50 − n d) n + 84 a) 2x b) 5x c) x + y d) xy e) x + y f) 6y5 a) 3g b) 5b c) 7b d) b + g e) y + g + 3

6 a) 3a b) 3a + 6 c) 1 _ 2 b d) b2 − 4 e) b __ 8

7 a) 2b + 9 b) 3g − 6 c) p − b d) b __ 3 e) 6g + 2b + 2

8 a) 5p b) p2 c) 25 ___ p

Constructing expressions 15

16 Pattern perfect

Swimming pool investigation

Notes on plenary activities

Part 4: Show, or ask, pupils how the position-to-term rule given is formed for the swimming pool sequence. Ask them to justify the rule by using the visual pattern sequence.

Part 5: Pupils need to fi nd the pattern (term) number to accommodate the number of available tiles. Discuss with pupils the different methods of approach, e.g. ‘trial and error’ and ‘working backwards’. A link with forming and solving equations could be made at this stage.

Part 6: This question will test pupil understanding of position-to-term rules. Show how the rule given, −2n + 6, can be rewritten as 6 − 2n. Do pupils see that these are equivalent?

Solutions to the activities

1 Design number 1 2 3

Number of surrounding tiles

8 12 16

2 a) Surrounding tiles increase by 4 each time

b) Start with 8, add 4

3 Number of surrounding tiles = 20

Diagram: 16 (4 × 4) blue squares, 20 surrounding tiles

4 a) 44 tiles b) 124 tiles

5 36th pattern

6 Incorrect; the number of surrounding tiles is decreasing

7 a) t → t ÷ 10 or t → t ___ 10

b) 2175 tiles

Answers to practice SATs-style questions

1 a) 10, 14, 18 (1 mark for all three correct)

b) 7, 4, 1 (1 mark for all three correct)

c) 10, 22, 46 (1 mark for all three correct)

2 a) Add 3 (1 mark)

b) Subtract 5 (1 mark)

c) Double or multiply by 2 (1 mark)

d) Divide by 3 (1 mark)

3 a) 13 red triangles (1 mark)

b) Red triangles = 23, blue triangles = 40; 63 triangles in total (2 marks)

4 a) 4b + 3, 4 × b + 3 (1 mark each)

b) There are 4 bags, so 4 × b or 4b; there are 3 sweets left over, so + 3

(1 mark for a suitable explanation)

5 a) Decreasing sequence (1 mark)

The term number is subtracted; as the term number increases the value will decrease

(1 mark for a suitable explanation)

b) 7, −1, −9, −17, −25

6 Input Output

2 1 _ 3

3 1 _ 2

−3 � 1 _ 2

36 6

3 0.5

(1 mark per correct answer)

Functional skills

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

• Examine patterns and relationships

• Change values and assumptions or adjust relationships to see the effects on answers in the model

• Draw conclusions in light of the situation

Plenary 17

Unit 1: Teacher Planning and Assessment Pack - Pearson Schools

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Transcript of Unit 1: Teacher Planning and Assessment Pack - Pearson Schools