Download Report

1
Pattern perfect
Unit objectives
Website links
• Generate and describe simple integer sequences
• 1.1 Number patterns
• Generate terms of a simple sequence, given a rule (e.g. finding a term from
the previous term, finding a term given its position in the sequence)
• 1.2 Number patterns
• Generate terms of a linear sequence using term-to-term definitions of the
sequence, on paper and using a spreadsheet or graphical calculator
• 1.3 Painting with
numbers – patterns in
nature
• 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
• 1.3 Number patterns
• 1.4 Function machine
and formulas
• Express simple functions in words, then using symbols
• 1.5 Using inverse
operations
• Represent simple functions in mappings
• 1.6 Algebra calendars
• Use letter symbols to represent unknown numbers or variables
• To view websites
relevant to this unit
please visit
www.heinemann.co.uk/
hotlinks
• 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
2
Pattern perfect
Notes on context
LiveText resources
Every lighthouse has a distinctive series of signals – different periods of
darkness and light produce a unique flash pattern for each lighthouse. The
individual light sequence of each lighthouse is called its ‘characteristic’.
•
The flash sequences allow ship captains to time intervals between light flashes
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 Office)
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 traffic 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:
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 automarked 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
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 flash), then off (same length as the flash period),
and then on (quick flash).
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
Opener
3
1.1 Sequences
Objectives
• Generate and describe simple integer sequences
• Generate terms of a simple sequence, given a
rule (e.g. finding a term from the previous term,
finding a term given its position in the sequence)
• Generate terms of a linear sequence using termto-term definitions of the sequence, on paper
and using a spreadsheet or graphical calculator
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, _12 , _14 , 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?
Starters: mini whiteboards
Intervention
Level Up Maths 2–3,
Lesson 1.1
Functional skills
Find results and solutions
Q7, 11
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 first term, third term, fourth term, etc.
Framework 2008 ref
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
• 1.2 Y7/8, 1.5 Y7/8,
3.2 Y7/8
– Display the following terms: ascending, descending, finite, infinite, term-toterm rule.
What words can you use to describe the sequence? (ascending and finite).
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 finite). What is the term-to-term rule?
(subtract 1.5) Q3
– Display the sequence: 5, 8, 11, 14, 17, ... .
4
Resources
Pattern perfect
Process skills in bold type
PoS 2008 ref
Process skills in bold type
• 1.2b, 2.2d, g, h, 2.4b,
3.1h
Website links
www.heinemann.co.uk/
hotlinks
How would you describe this sequence?
(ascending and infinite) 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 infinite sequence? (because it would go on
for ever).
Give pupils the first term and the term-to-term rule
of a sequence (e.g. 5, add 4). Ask them to write the
first five terms of the sequence. Repeat for other
first 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
first terms to try to make sequences with given
properties.
a) yes – first term any multiple of 3; b) no; c) no; d) yes
– first term any multiple of 3 less than 24; e) yes – first
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.
Common difficulties
When using negative
numbers remember to
add or subtract correctly.
LiveText resources
Explanations
Extra questions
Answers
1 a)
b)
c)
d)
2 a)
12, 15, 18; multiples of 3, starting at 3
40, 50, 60; multiples of 10, starting at 10
28, 35, 42; multiples of 7, starting at 7
7, 9, 11; odd numbers, starting at 2
b)
Worked solutions
c)
3 a) 0, 0.5, 1.0, 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0
b) 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, 3.0
c) 3.0, 3.3, 3.6, 3.9, 4.2, 4.5, 4.8, 5.1, 5.4, 5.7, 6.0
d) 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, 4.0
e) 7.0, 7.6, 8.2, 8.8, 9.4, 10.0, 10.6, 11.2, 11.8, 12.4, 13.0
f) 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) infinite
e) 29
6 a) 2, 7, 12, 17, 22, 27, 32, 37, 42
b) ascending and finite
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 sequences
10 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, 62.0
e) −10, −45, −80, −115, −150
f) −4, −6, −8, −10, −12
11 Pupil’s own answers
Sequences
5
1.2 Generating sequences
Objectives
• Generate and describe simple integer sequences
• Generate sequences from practical contexts
and describe the general term in simple cases
Starter (1) Oral and mental objective
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
Starter (2) Introducing the lesson topic
Give pupils a mini whiteboard between two. Display
the key words: ascending, descending, first 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; first 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 finding 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 first red
bead in a necklace. Display the other as the term-to-term rule ‘add 䊐’. Ask
pupils to draw the necklace on mini whiteboards. What is the position of the fifth
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 first
pattern.
6
Pattern perfect
Resources
Starter (2), plenary: mini
whiteboards
Activity B: squared paper
Plenary: 0–9 digit cards
Intervention
Level Up Maths 2–3,
Lesson 1.2
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
www.heinemann.co.uk/
hotlinks
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
Homework Book section 1.2.
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 five different pattern sequences starting with
one black tile. Count the tiles in each pattern and
write the sequence. Describe each sequence.
Answers
1
2
a) 1, 3, 5, 7, 9,
b) odd numbers
Start at 12. Add 2 each time. Stop after ten terms.
3
a)
i)
ii)
4
5
6
7
b)
a)
b)
1
2
3
4
Number of matches
3
6
9
12
Number of shapes
1
2
3
4
Number of matches
4
8
12
16
Related topics
i) Term-to-term rule: +3
ii) Term-to-term rule: +4
1.29 m, 1.37 m, 1.45 m, 1.53 m, 1.61 m, 1.69 m
Not a good model as he grows more in some years than others and will stop growing in late
teenage years.
a)
b)
a)
a)
Number of shapes
Number of flowers
1
2
3
4
5
Number of beads
5
9
13
17
21
Square numbers
LiveText resources
Explanations
Extra questions
Worked solutions
Term-to-term rule: add 4
9, 7, 5, 3, 1, −1,−3
i)
b)
Shona owes money.
b)
d)
£25
After 29 days
ii)
b)
8
c)
a)
c)
i) 3, 6, 9, 12, 15
ii) 4, 7, 10, 13, 16
i) 30
ii)
£7, £9, £11, £13, £15, £17
£45
31
Generating sequences
7
1.3 More sequences
Objectives
• Generate simple sequences from practical
contexts
• Find the position-to-term rule
Starter (1) Oral and mental objective
Practise counting forwards and backwards on a
number line with different starting numbers in steps of
0.25, _14 , 0.75, _34, 0.4, and so on.
Starter (2) Introducing the lesson topic
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
1.
No special resources
required
Can you suggest a number larger than 100 that will be in this sequence? Why
is it in the sequence?
Intervention
Display the first three triangle number patterns on
1 . How do the
patterns grow? How many rows in the third pattern? How many dots in the
bottom row? Q1–3
–
Resources
Which number is the first term? The third term? The eighth term? Can you
explain the term-to-term rule? … the position-to-term rule?
2 Arithmetic sequences
Display the sequence of matchstick patterns on
3 The position-to-term rule .
Level Up Maths 2–3,
Lesson 1.3
Functional skills
Examine patterns and
relationships Q5, 6, 7
Framework 2008 ref
Process skills in bold type
• 1.2 Y7/8, 1.3, 1.4 Y7/8,
1.5 Y7/8, 3.2 Y7/8
Pattern 1
5 matchsticks
Pattern 2
10 matchsticks
Pattern 3
15 matchsticks
How many matchsticks will be in shape number 5? 7? What is the term-toterm 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
8
Pattern perfect
PoS 2008 ref
Process skills in bold type
• 1.5a, 2.2a, d, e, j, o,
2.3a, 3.1h
Website links
www.heinemann.co.uk/
hotlinks
4 Using the position-to-term rule to generate a
sequence
–
If the position-to-term rule is 7 × position number,
write down the first five 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.
Discussion points
Challenging homework: Ask students to find the relationship between odd
numbers and square numbers.
Answers
Opportunities to discuss
sequences that can be
found in nature or in
practical scenarios.
1
Common difficulties
2
3
4
5
a)
c)
a)
c)
a)
b)
3 × 3, 9
b) 4 × 4, 16; 5 × 5, 25
Square numbers
1 + 2 + 3, 6
b) 1 + 2 + 3 + 4, 10; 1 + 2 + 3 + 4 + 5, 15
Triangle numbers
1 × 2 = 2, 2 × 3 = 6, 3 × 4 = 12, 4 × 5 = 20, 5 × 6 = 30
length and width increase by 1 each time / multiply the position by the number one more
than itself
c) 8 × 9 = 72
a, d, g, h, j
a)
10 squares
b)
6
7
8
12 squares
14 squares
Shape number
1
2
3
4
5
6
Number of squares
2
4
6
8
10
12
Ensure pupils understand
that 22 = 2 × 2 not 2 + 2,
which is a common
misconception. Practise
finding powers of 2 and 3.
LiveText resources
Explanations
Extra questions
Worked solutions
c) 12
d) Number of squares increases by 2 each time / number of squares is the even numbers.
e) Increases by 2
f) number of square = shape number × 2
g) 10th term has 20 squares
h) correct shape drawn
a) 9, 10, 11, 12, 13, 14
b) 2, 4, 6, 8, 10, 12
c) 3, 5, 7, 9, 11, 13
d) 1, 4, 7, 10, 13, 16
a) a) add 1
b) add 2
c) add 2
d) add 3
b) The position is multiplied by the term-to-term rule in the position-to-term rule.
The term-to-term rule is add 2; the position-to-term rule is (2 × position number) + 1
More sequences
9
1.4 Function machines
Objectives
• Express simple functions in words, then using
symbols
• Represent simple functions using function
machines
Starter (1) Oral and mental objective
Display the following blank multiplication.
䊐䊐 × 䊐 = 䊐
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.
Starter (2) Introducing the lesson topic
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.
Resources
Main lesson
–
1 Function machines 1
Display a blank function machine such as the
one shown. Write an operation (e.g. +5) in the
machine. 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 find the answer. Repeat
using other operations such as −7, ×8, ÷2. Challenge pupils to find the
missing input using inverse operations. Q2
–
2 Function machines 2
– Display a blank two-step
function machine such as the
one shown. Write in an operation
such as ×2 + 3.
Intervention
Level Up Maths 2–3,
Lesson 1.4
Functional skills
Change values and
assumptions or adjust
relationships to see the
effects on answers in the
model Q7–8
Framework 2008 ref
Process skills in bold type
• 1.1 Y7, 1.2 Y7, 1.5 Y7,
3.2 Y7/8
Ask pupils to describe the strategy they used.
PoS 2008 ref
Process skills in bold type
• 2.1b, 2.2f, g, i, 2.4a, b,
3.1h
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
Website links
www.heinemann.co.uk/
hotlinks
The input is 6. What is the output? (15)
The output is 11. What is the input? (4)
10
No special resources
required
Pattern perfect
– 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 find 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 find 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.)
Related topics
Homework
Inverse proportions
Homework Book section 1.4.
Common difficulties
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 find the mystery
input. Make up three similar problems.
Pupils may get the order
of calculations wrong
when calculating the input
of a two-step function
machine given the output.
Answers
1
2
3
4
a)
a)
a)
d)
a)
3, 1.5, 4
5, 4, 35
8, 17, 23
12, 11, 8.5
4
b)
c)
a)
a)
a)
10, 11, 6
b) 15, 45, 0
e) 17, 1, 4
⫹7
c)
c)
4.5, 1.7, 3.3
20, 0.6, 0.25
c)
f)
LiveText resources
d) 1.5, 2.8, 5.1
2, 4, −7
10, 12, 60
⫺2
⫻4
36
18
⫼2
⫺7
2
The order of the operations does change the output.
+3
b) ×8
c) −7
3, ×5, +1, 16 b) 7, −4, ×8, 24
c) 15, ÷5, +6, 9
6 + 2, ×7, 56 b) 6 ×5, −3, 27
c) 6 ÷2, +3, 6
Explanations
Extra questions
Worked solutions
27
11
c)
5
6
7
8
⫻5
b)
d)
d)
d)
÷2
28 −4, ÷6
32 − 8 ÷ 4 = 6
Function machines
11
1.5 Expressions and mappings
Objectives
• 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
Starter (1) Oral and mental objective
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.
Starter (2) Introducing the lesson topic
Use a set of digits cards to choose five 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.
Resources
Main lesson
Main: bag of counters
–
1 Order of operations
Intervention
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
Level Up Maths 2–3,
Lesson 1.5
Functional skills
Use appropriate
mathematical procedures
Q3, 6
Framework 2008 ref
What does each expression mean?
Process skills in bold type
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.)
• 1.2 Y7, 1.3, 1.5 Y7,
3.2 Y7/8
How can each expression be simplified? Take suggestions from pupils and
then show how they can be simplified 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 simplified any
further. Q3–7
12
Starter (2): set of digit
cards, mini whiteboards
Pattern perfect
PoS 2008 ref
Process skills in bold type
• 2.2h, l–n, 3.1e, f, h
–
2 Mapping diagrams
–
3 Mappings 1
–
4 Mappings 2
⫹5
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
Discussion points
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)
In number work, 3 × 5 = 3
lots of 5 = 5 + 5 + 5.
So 3 × a = 3 lots of
a=a+a+a
Homework
Common difficulties
Homework Book section 1.5.
n2.
Challenging homework: These two expressions are equivalent: n × n
Find five more pairs of equivalent expressions.
Answers
1 a)
2 a)
3 a)
23
3b
b)
b)
65
3b + 9
3a ⫹ 5
2a ⫹ 3
2a
a⫹2
t
d)
d)
90
15b
4a − 2b
32b
5c
3b
c)
c)
g)
25
2b + 6
2c
b
d)
d)
h)
12a + 4b + 9
4y
LiveText resources
c⫹2
Extra questions
Worked solutions
z
288y
24y
0.5
8
Pupils may try to simplify
a + b to ab.
Explanations
2c ⫹ 6
c⫹4
12
1 → 4, 2 → 5, 3 → 6, …
1 → 2, 2 → 4, 3 → 6, …
1 → 3, 2 → 5, 3 → 7, …
In each, the mapping lines spread out.
x→x−3
b) x → 3x
x→x−2
15
b) 13
8
8b+18
c⫺4
8a − 2b − 8
6y
8b
b)
f)
f)
4c ⫹ 6
4b ⫺ 4
40
5
e)
e)
c)
8b ⫺ 2
b⫹2
a
5t
12
2b
4b ⫹ 2
3
4 a) 5a + 2b + 3
b)
5 a) 10a
b)
e) 2a
f)
6 Pupil’s own answer.
7 a)
200t
8 a)
c)
e)
g)
9 a)
10 a)
11 a)
c)
c)
b)
Pupils may confuse a + 2
and 2a – use the bag of
counters to emphasise the
difference.
24
y
b)
d)
f)
1 → 0, 2 → 1, 3 → 2, …
1 → 6, 2 → 7, 3 → 8, …
In each, the mapping lines are all parallel.
b)
x → 2x
c)
x→x+9
c)
16
Expressions and mappings
13
1.6 Constructing expressions
Objectives
• 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 formula and
function
• Write algebraic expressions
Starter (1) Oral and mental objective
Display the following blank calculation.
䊐.䊐×䊐=䊐
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.)
Starter (2) Introducing the lesson topic
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.
⫹6
Repeat for different operations in the function machine, for example
c
).
−4 (c − 4), −78 (c − 78), +0.5 (c + 0.5), ×2 (2c), ÷10 (__
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)
14
Pattern perfect
Starter (1): mini
whiteboards
Intervention
The input is c. What is the output? (c + 6) Explain that they have just used the
function machine to construct an expression.
–
Resources
Level Up Maths 2–3,
Lesson 1.6
Functional skills
Make an initial model of
a situation using suitable
forms of representation
Q6–7
Framework 2008 ref
Process skills in bold type
• 1.1 Y7, 1.2 Y7/8, 3.1
Y7/8, 3.2 Y7/8
PoS 2008 ref
Process skills in bold type
• 1.4a, 2.1c, 2.2e,
3.1e, f, h
Website links
www.heinemann.co.uk/
hotlinks
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.
2d − 14
3b + 14 d) _______
Answers: a) 16 b) 40 c) _______
2
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
b) 6 − b
b−6
b−2−4
b−4−2
c) 4y
y+y+y+y
y×y×y×y
(5x)
(6 − b)
Common difficulties
Pupils sometimes confuse
x + 2 and 2x
(y × y × y × y)
LiveText resources
Homework
Explanations
Homework Book section 1.6.
Extra questions
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.
Answers
2
3
4
5
a)
d)
a)
a)
a)
a)
a + 4 or 4 + a
D+7
w+3
b)
n+3
2x
b)
3g
b)
r + 10
c) y − 7
d)
b) n − 21
c) 50 − n
5x
c) x + y
d) xy
5b
c) 7b
d)
6
a)
3a
b)
3a + 6
c)
_1 b
7
a)
2b + 9
b)
3g − 6
c)
p−b
8
a)
5p
1
b)
e)
b)
b−3
m−4
p2
2
c) 20 − C
f) l + 2
w+8
e) w − 3
d) n + 8
e) x + y
f) 6y
b+g
e) y + g + 3
b2 − 4
b
d) __
d)
e)
e)
3
c)
b
__
8
6g + 2b + 2
25
___
p
Constructing expressions
15
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 find 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
t
7 a) t → t ÷ 10 or t → ___
10
b) 2175 tiles
16
Pattern perfect
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
2
Output
3
_1
−3
ⴚ_12
36
6
3
0.5
_1
3
2
(1 mark per correct answer)
Functional skills
The plenary activity practises the following functional skills defined 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