Download Sequences - Pearson Schools and FE Colleges

Sample Pages
CONTENTS
Algebra 1: Sequences, fractions and symbols
Matching chart
1 A1 Sequences, functions and
➔
Units 2, 6 and 10
symbols
(6 hours)
Assumed knowledge
Background
Pupils will have met number sequences in Year 6. This
Unit looks at number sequences in more depth and
explores ways of extending number sequences without
listing all the values. It will be the first time that pupils
use letters to represent unknowns. They will begin to
solve simple linear equations using function machines
and inverse operations.
Before starting this Unit, pupils should:
know that 9 ⫻ 8 ⫽ 8 ⫻ 9
G be able to count on and back in both whole number
multiples and simple decimals
G be able to solve a problem by extracting
information from tables
G
Main teaching objectives
Pupil book sections
1.1 Sequences
1.2
Generating
sequences
1.3
Investigating
sequences
1.4
Function machines
1.5
More function
machines
Using letters to
stand in for unknown
numbers
1.6
Teaching objectives
Generate and describe simple integer sequences
Explore and predict terms in sequences generated by counting in regular steps
Generate terms of a simple sequence, given a rule for finding each term from the
previous term
Generate terms of a simple sequence, given their positions in the sequence
Generate a sequence given a rule for finding each term from its position in the
sequence
Generate sequences from practical contexts
Begin to explore term-to-term and position-to-term relationships
Suggest extensions to problems by asking ‘What if...?’
Use function machines to explore mappings and to calculate input, output and
missing operations
Begin to apply inverse operations where two successive operations are involved
Use letter symbols to represent unknown numbers
Understand algebraic conventions
Use letter symbols to write expressions and construct equations
Oral and mental starters
Starter
27
6
7
16
23
24
Page reference
Multiplication and division facts 1
Numbers in figures and in words
Place value 1
Ordering positive, negative and decimal numbers
Adding and subtracting positive and negative numbers
Adding and subtracting pairs of numbers 1
268
253
254
260
264
265
Common difficulties
G
Pupils may often think that 0.9 ⫹ 0.1 ⫽ 0.10
PB
Pages 2–13
Homeworks 1.1–1.6
Assessment 1B
!
2 Maths Connect 1B
Tournaments (page 286)
Two-step relations (page 289)
Key words
1.1
Sequences
Generate and describe simple integer sequences
Explore and predict terms in sequences generated by counting in regular steps
Links
➔
1.2 Generating sequences
Oral and mental starter
27
Introduction
In this lesson, pupils learn to recognise and
describe sequences. They will practise finding
terms of a sequence following simple rules, and
start to use the language associated with
sequences correctly.
Teaching activity
Outline
The code for a safe is 2, 4, 3, 7, 6. Can I open the safe by pressing 2, 4, 3, 6, 7?
(No)
Explain that a sequence is a set of numbers in a given order, but that
often the order is given by a rule.
Can you give me examples of other number sequences? (Answers could be:
sequences following a simple rule such as square numbers; sequences following
a more complex rule such as time of sunset each day; sequences following an
irregular pattern, such as maximum temperature each day; random sets of
numbers e.g. winning lottery numbers.)
Explain that each number in a sequence is called a term. Explain that
terms next to each other are called consecutive terms and are often
separated by commas.
If the rule for a sequence is ‘Start at 7 and count on in steps of 7’ what would
the sequence look like? (7, 14, 21, 28 ...)
Write the sequence on the board.
What do you notice about the terms in this sequence? (They are multiples of 7.)
Continue for the following three rules below. Write the first five terms
of the sequence on the board each time:
G
Start at 3 and count on in steps of 2. (3, 5, 7, 9, 11) What do you notice
about each term in the sequence? (Each term is an odd number.)
G
Start at 24 and count back in steps of 3. (24, 21, 18, 15, 12) What do you
notice about each term in the sequence? (Each term is a multiple of 3.)
G
Start at 1 and count on in steps of 5. (1, 6, 11, 16, 21) What do you notice
about each term in the sequence? (Each term is four less than a multiple of 5.)
Is the first sequence going up or going down? (Going up)
Explain that this is called an ascending sequence.
Is the second sequence going up or going down? (Going down)
Explain that this is called a descending sequence.
Write on the board:
‘ Odd numbers between 0 and 50
‘ Start at 1 and count on in steps of 2’
G
G
Ask the class if there is any difference between the two sequences
generated following these rules.
4 Maths Connect 1B
sequence
term
consecutive
finite
infinite
Think carefully. Does either sequence contain the number 51? Why doesn’t the
first sequence contain 51? (Because we have indicated where the sequence will
end.)
Explain that a sequence with a stated start and end is called a finite
sequence. A sequence that continues indefinitely is infinite. We can
show this with dots: 1, 3, 5, … 49 is finite. 1, 3, 5, 7, 9 … is infinite.
Variations
Teacher materials:
Resource sheet 1, enlarged to make sets of cards shown below.
For a more visual approach, use cards printed with patterns of dots.
Show the pupils the first four cards:
Tell me how to put the cards in a logical order.
Arrange the cards to illustrate ascending and descending sequences,
meanings of term and consecutive term etc.
Plenary
Questions:
G
G
G
What are examples of sequences you would
find in everyday life? (Telephone numbers,
house numbers, number of pupils absent
each day, hours of daylight each day etc.)
Does a sequence have to go up an equal
amount each time? (No.)
Is this sequence finite or infinite:
6, 4, 2, 0, ⫺2, ⫺4, …?
Key teaching points:
G
G
G
G
G
G
A sequence is a set of numbers in a given
order.
Each number in a sequence is called a term.
Terms next to each other are called
consecutive terms and are often separated by
a comma.
Sequences can be ascending or descending.
Sequences can be finite or infinite.
To find the next few terms of a sequence you
look for a pattern; if there isn’t one you can’t
find the next few terms.
Exercise hints
Questions 1–3
Practice
Question 4–7
Problems
Question 8
Investigation
In Q3d pupils may find it difficult to go
below zero.
In Q3e, a common error is to write: 0.10, 0.11, 0.12.
Remind pupils 11 tenths ⫽ 1.1.
P B
Exercise 1.1, page 2
Homework 1.1, page 73
Answers, page 296
Algebra 1: Sequences 5
Key words
1.2
generate
term
term-to-term rule
Generating sequences
Generate terms of a simple sequence, given a rule for finding each term from
the previous term
Generate terms of a simple sequence, given their positions in the sequence
Links
➔
➔
Introduction
In this lesson, pupils use a starting point and a
rule to generate a sequence. They learn how to
generate the sequence using a calculator.
1.3 Investigating sequences
1.1 Sequences
Oral and mental starter
6
Teaching activity
Teacher materials:
Calculator and projector (or projector and spreadsheet package)
Pupil materials:
Calculators
Outline
Discuss with pupils the meanings of sequence, term, consecutive term and
rule.
To generate a sequence you can use two pieces of information: a
starting point and a rule.
Tell pupils that you are going to generate a sequence given the
following information:
Starting point ⫽ 3
Rule ⫽ ⫹4
Teach pupils how to generate a sequence on a calculator:
Enter 3 into the calculator and press Enter.
Enter ANS ⫹4 and press enter.
Press Enter repeatedly to generate the sequence 3, 7, 11, 15 …
Ask the pupils to work through Q1–8. They may choose a calculator or
non-calculator method for all questions, except Q5. Allow about twenty
minutes, then stop the class. Tell them that you are going to look at
another way of generating terms in the sequence 3, 7, 11, 15 …
What were you adding to get from one term to the next? (4)
Which ‘times table’ are you using in the sequence? (4)
How could you get from the first term to the third term? (Add 8/ add 4 twice)
How could you get from the first term to the fourth term? (Add 12/ add 4 three
times )
How could you get from the second term to the fifth term? (Add 12/ add 4
three times)
Can you spot a pattern that will tell you how to find any term? (It is the four
times table take away 1.)
Repeat this activity with:
Starting Point
7
81
1
6 Maths Connect 1B
Rule
Add 5
Subtract 9
Add 2
Numbers generated
Multiples of 5 add 2
Multiples of 9
Multiples of 2 take away 1
Variations
Ask a volunteer to choose a starting point between 0 and 9 and a termto-term rule (e.g. starting point 8, ‘add 7’). Round the class, each pupil
gives the next term in the sequence.
The first pupil to give a value of 50 or over wins the game and starts it
again.
(For example, if the pupil chose the starting point of 8 and the rule of
‘add 7’, then the class would generate the sequence 8, 15, 22, 29, 36 … .
The pupil who said the number 50 would win.)
Question pupils about individual sequences. For example, for the
sequence above:
How would you work out who will call out the number 78? (Calculate how
many lots of 7 you need to add and count on that many pupils)
Can you work out the third term without working out the second term? (Yes)
How can you do this? (By adding 2 lots of 7 to 8)
Encourage pupils to plan ahead to work out who will win the game.
(For example, with the sequence above, the pupil who starts should
spot that the 7th person to call out a number will win, since starting at 8
and counting up in 7s you need only 6 lots of 7 before you hit 50.)
You can extend this idea so that pupils give more than one rule (e.g.
‘add two, multiply by five’).
Plenary
Questions:
G
G
What information do you need in order to
generate a sequence? (Rule and first term, or
rule and 2nd term etc.)
If you know that the rule for getting from
one term of sequence to the next is ‘add 2’
and your starting point is 5, how would you
work out the tenth term? (3 ⫹ 10 lots of 2).
Note that the first term (5) is represented as
3 ⫹ 1 lot of 2.
Exercise hints
Q1–5
Practice
Q6–9
Problems
Q10
Investigation
Encourage pupils to refer to Example 1 if they
struggle with Q9 and Q10.
Key teaching points:
G
G
You can find a term in a sequence, without
finding all the terms in between, if you
know the relationship between consecutive
terms.
You can generate a sequence given a first
term and a rule.
P B
Exercise 1.2, page 4
Homework 1.2, page 73
Answers, page 296
Algebra 1: Generating sequences 7
Key words
1.3
Investigating sequences
Generate a sequence given a rule for finding each term from its position in the
sequence
Generate sequences from practical contexts
Begin to explore term-to-term and position-to-term relationships
Suggest extensions to problems by asking ‘What if ...?’
Links
➔
➔
sequence
term
term number
rule
Introduction
The activity is based on diagrams and allows
pupils to visualise sequences. The teaching
should focus on generalising (from the patterns
shown) to a rule that will enable any term in the
sequence to be found.
1.4 Function machines
10.2 The general rule
1.1 Sequences 1.2 Generating sequences
Oral and mental starter
7
Teaching activity
Teacher materials:
OHP, OHT of Resource sheet 1, sequences of patterns shown below.
Outline
The sequence 5, 7, 9, 11… is going up in 2s and the first term is 5. The
first term is 3 plus one lot of 2, the second term is 3 plus two lots of 2
and the tenth term is 3 plus ten lots of 2.
Display the pattern on the first OHT.
Draw this table on the board:
Pattern no.
1
2
3
4
No. of
squares
5
9
13
17
5
(1 ⫻ 4) ⫹ 1
(2 ⫻ 4) ⫹ 1
(3 ⫻ 4) ⫹ 1
(4 ⫻ 4) ⫹ 1
6
10
How will this sequence continue? (You add 4 more squares each time.)
How many squares will you need to make the fifth pattern? (17 ⫹ 4 ⫽ 21)
How many squares will you need to make the sixth pattern? (21 ⫹ 4 ⫽ 25)
Explain that you could describe this sequence by saying that the first
pattern has 1 middle square and 4 more squares (1 on each of the 4 ‘arms’).
The second pattern has 1 middle square and then 2 lots of 4 squares.
How many squares do you think you would need to make the tenth pattern?
(10 lots of 4 plus 1 ⫽ 41)
How many squares would there be in the 50th pattern? (50 ⫻ 4 ⫹ 1 ⫽ 201)
How many in the 100th pattern? (100 ⫻ 4 ⫹ 1 ⫽ 401)
Check that the pupils recognise that this method allows us to find any
term in a sequence without calculating every term.
Display the second pattern on the OHT:
Draw this table on the board:
Pattern no.
1
2
3
4
No. of
circles
4
7
10
13
(1 ⫻ 3) ⫹ 1
(2 ⫻ 3) ⫹ 1
(3 ⫻ 3) ⫹ 1
(4 ⫻ 3) ⫹ 1
8 Maths Connect 1B
5
10
15
How will this sequence continue? (You add 3 more circles each time.)
How many circles will you need to make the fifth pattern? (16 ⫽ 5 lots of 3, plus 1)
How many circles will you need to make the sixth pattern? (19 ⫽ 6 lots of 3, plus 1)
Explain that you could describe this sequence by saying that the first
pattern has 1 circle in the middle and 3 more circles (1 on each of the 3
‘arms’). The second pattern has 1 circle in the middle and 2 lots of 3 circles.
The third pattern has a circle in the middle and 3 lots of 3 circles.
How many circles do you think you would need to make the tenth pattern? (10 lots
of 3 plus 1 ⫽ 31)
How many circles would there be in the 15th pattern? (15 lots of 3, plus 1 ⫽ 46)
Variations
You may prefer to omit the diagrams and focus on numerical patterns.
Write the following sequence on the board: 5, 9, 13, 17, 21 …
and the first two rows of Term no. 1
2
3
this table:
Sequence
5
9
13
4
5
17
21
4⫹1
(4 ⫹ 4) ⫹ 1
(4 ⫹ 4 ⫹ 4) ⫹ 1 (4 ⫹ 4 ⫹ 4 ⫹
4) ⫹ 1
(4 ⫹ 4 ⫹ 4 ⫹
4 ⫹ 4) ⫹ 1
(1 ⫻ 4) ⫹ 1
(2 ⫻ 4) ⫹ 1
(3 ⫻ 4) ⫹ 1
(5 ⫻ 4) ⫹ 1
(4 ⫻ 4) ⫹ 1
100
Pupils will find the 6th term, by adding 4 to the 5th term and so on, but
this would be a tedious way of finding the 100th term. Look for a
pattern, and fill in the third and fourth row of the table as a class. Use
coloured pens or chalk to match the term number with the
corresponding number in the rule, if the pupils don’t see the link at first.
The 100th term ⫽ 100 ⫻ 4 ⫹ 1 ⫽ 401.
Repeat this for other sequences such as:
5, 10, 15, 20, 25, …
5, 8, 11, 14, 17, …
Plenary
Questions:
G
G
G
G
G
G
Key teaching points:
Can you think of any sequences which
occur in real life? (E.g. house numbers,
number of pupils absent each day, weight of
an individual each year)
What do you call a sequence which goes
up? (Ascending)
What do you call a sequence which goes
down? (Descending)
How can you describe a sequence? (First
term and a rule)
What do we call two terms that are next to
one another in a sequence? (Consecutive)
Are the following sequences finite or infinite:
Odd numbers bigger than 20? (Infinite)
Even numbers between 0 and 1000? (Finite)
G
You can find any term in a sequence where
consecutive terms are generated by
repeatedly adding the same number. To do
this, you do not need to calculate all the
terms in between if you know the difference
between terms.
P B
Exercise hints
Q1–4
Q5–8
Q9
Practice
Problems
Investigation
Exercise 1.3, page 6
Homework 1.3, page 74
Answers, page 296
Algebra 1: Investigating sequences 9
Key words
1.4
Function machines
Using function machines to explore mappings and to calculate input, output
and missing operations
Links
➔
➔
input
output
function machine
mapping diagram
unknown
Introduction
1.5 More function machines
1.3 Investigating sequences
Oral and mental starter
16
Pupils will already be familiar with the idea of
addition being the inverse of subtraction and
multiplication being the inverse of division. In
the next three lessons pupils will solve simple
equations by finding missing values and
represent algebraic expressions by mappings.
This lesson introduces a range of methods for
solving one-step function machines.
Teaching activity
Pupil materials:
Hand-held whiteboards (optional)
Outline
Write the following on the board:
3 ⫹ 䊐 ⫽ 17
Write the missing number on your whiteboards (14)
We can write this as a function machine:
3 ➝
⫹䊐
➝ 17
You need to identify the missing information in the function machine.
How did you work out what to put in the box? (17 ⫺ 3 ⫽ 14)
We are using an inverse operation.
The inverse of addition is subtraction.
The inverse of multiplication is division.
Explain that a function machine enables us to find outputs for lots of
different inputs. For example, write on the board:
Input ➝ ⫹9 ➝ Output
If the input is 3, what is the output? (12)
If the input is 7, what is the output? (16)
We can also work back through the machine, to find the input, if we
know the output.
If the output is 20, what is the input? (20 ⫺ 9 ⫽ 11)
Repeat with more examples of function machines, using subtraction or
multiplication.
Explain that we can also use a mapping diagram. Start by writing on
the board the function machine:
Input ➝ ⫹4 ➝ Output
When the input is 0, the output is 4.
We write 0 ➝ 4
Also: 1 ➝ 5
2 ➝ 6 etc.
Explain that this is called a mapping. Mappings are usually indicated
by an arrow and can be shown by a mapping diagram.
10 Maths Connect 1B
Adding 4 to the first number gives the second number.
0
0
1
2
3
4
5
6
1
2
3
4
5
6
Now write the following on the whiteboard:
3⫻䊐⫽4⫻3
How did you know that 䊐 stands for 4? (3 ⫻ 4 is the same as 4 ⫻ 3)
Variations
Play a function guessing game. Write some mappings on the board,
explain that each mapping has just one operation, and ask pupils to
write down the hidden rule that connects them.
For example, 3 ➝ 8 (⫹5)
4 ➝ 10 (⫹6) 5 ➝ 12 (⫹7) 10 ➝ 22 (⫹12)
3 ➝ 18 (⫻6 or ⫹15) 12 ➝ 5 (⫺7) 10 ➝ 2 (⫼5 or ⫺8).
Plenary
Questions:
G
G
Key teaching points:
How do we find an unknown number? (By
using inverse operations)
Explain how you would solve the following:
䊐 ⫻ 6 ⫽ 24
(䊐 ⫽ 4)
䉭 ⫹ 5 ⫽ 11
(䉭 ⫽ 6)
䊊⫺7⫽0
(䊊 ⫽ 7)
䊐
ᎏᎏ ⫽ 6
(䊐 ⫽ 30)
5
32 ⫺ 䉭 ⫽ 12
(䉭 ⫽ 20)
Exercise hints
Q1–5
Q6 and 7
Q8
Practice
Problems
Investigation
G
G
G
You can find missing numbers using inverse
operations.
The inverse of multiplication is division, the
inverse of addition is subtraction and vice
versa.
You can represent operations as function
machines and in mapping diagrams.
P B
Exercise 1.4, page 9
Homework 1.4, page 74
Answers, page 297
Algebra 1: Function machines 11
Key words
1.5
operations
inverse
unknown
More function machines
Begin to apply inverse operations where two successive operations are involved
Links
➔
➔
Introduction
1.6 Using letters to stand in for unknown
numbers
1.4 Function machines
This lesson uses the methods that were
introduced last lesson, and applies them to twostep function machines.
Oral and mental starter
23
Teaching activity
Pupil materials:
Hand-held whiteboards
Outline
Remind pupils that if you had this function machine:
and the output was 9, you could work out the input
by working backwards using inverse operations.
9⫺8⫽1
Input
⫹8
⫹8
This can be shown as:
Subtraction is the inverse of addition and vice versa.
Division is the inverse of multiplication and vice versa.
1
9
1
9
inverse
⫺8
Write the following on the board:
Input ➝ ⫻2 ➝ ⫹5 ➝ 11
I thought of a number. I multiplied it by 2 and then added 5, and my answer
was 11. What was the number? Write it on your whiteboards. (3)
How did you work out the answers? (Trial and improvement, or by using
inverse operations and calculating (11 ⫺ 5) ⫼ 2
Demonstrate trial and improvement for a two-step function machine:
Input ➝ ⫻2 ➝ ⫹5 ➝ 11
4 ⫻ 2 ⫹ 5 ⫽ 13
2⫻2⫹5⫽9
3 ⫻ 2 ⫹ 5 ⫽ 11
(too big)
(too small)
So the input number was 3.
12 Maths Connect 1B
Output
Repeat this activity using other operations e.g.
Input ➝ ⫻3 ➝ ⫹4 ➝ 10
(2)
Input ➝ ⫼3 ➝ ⫹5 ➝ 7
(6)
Input ➝ ⫹8 ➝ ⫺12
➝ 6
(10)
Input ➝ ⫻4 ➝ ⫼5 ➝ 20
(25)
You can extend this activity by asking pupils to think of their own
function machines, using more than two operations or using decimals,
fractions and negative numbers.
Variations
Set up a spreadsheet so that column A is the input and column B is the
output. Make sure you hide the formulae.
A
B
1
⫽ A1 ⫹ 5
2
⫽ A2 ⫺ 3
3
⫽ A3 ⫻ 2
4
⫽ A4 ⫼ 2
5
⫽ (A5 ⫻ 2) ⫺ 4
6
⫽ (A6 ⫺ 4) ⫻ 4
7
⫽ (A7 ⫹ 12) ⫼ 2
8
⫽ (A8 ⫻ 2) ⫺ 5
Ask the pupils to work out the rules being used to generate the values in
column B. Encourage them to work systematically, typing in different
whole number values for A1, A2 … etc.
Plenary
Questions:
G
G
I choose a number, add 6 and multiply by 2
and I get 22. What number did I choose? (5)
How did you work out the answer to the
last question? (By trial and improvement, or
by using inverse operations)
Key teaching points:
G
G
G
Exercise hints
Q1–2
Practice
Q3–5
Problems
Q6 & 7
Investigations
Q6 and Q7 are extended questions that involve a
large number of calculations.
If you know the operations that have taken
you from one number to another then by
performing the inverses of these operations
you can get back to your original number.
Multiplication and division, addition and
subtraction are inverses of one another.
You should always check your answers by
substituting the value back in.
P B
Exercise 1.5, page 10
Homework 1.5, page 75
Answers, page 297
Algebra 1: More function machines 13
Key words
1.6
Using letters to stand in for unknown numbers
Use letter symbols to represent unknown numbers
Recognise algebraic conventions
Use letter symbols to write expressions and construct equations
Links
➔
➔
6.1 Using algebra 10.2 The general rule
1.4 Function machines
1.5 More function machines
Oral and mental starter
24
equation
expression
unknown
algebra
Introduction
Pupils will be introduced to the idea of an
algebraic equation in this lesson. They will
practise transforming function machines and
word problems into algebraic equations. The
plenary draws together the learning from this
and the previous two lessons.
Teaching activity
Pupil materials:
Hand-held whiteboards (optional)
Outline
Explain to the class how we can use expressions. For example:
I have some sweets and then I get 4 more.The expression for the number of
sweets I now have is x ⫹ 4.
I think of a number and multiply it by 9, The expression for this is 9x.
I have some apples and give three away. The expression for the number of
apples I have left is n ⫺ 3.
Explain that it does not matter which letter is used for the unknown
number.
Revise the idea of the function machine with pupils.
Input
Draw this on the board:
What is the input? (34)
How did you work this out? (By working backwards)
Explain that you can use a letter (e.g. x) to stand in for the unknown
number in the function machine.
⫹73
107
Input
⫹4
10
Input
⫹8
25
Input
⫺7
22
Input
⫻2
14
x ⫼ 3 ⫽ 6, x ⫽ 6 ⫻ 3 so x ⫽ 18 (This is a good opportunity
Input
x
to introduce the idea of writing x ⫼ 3 as ᎏᎏ)
3
⫹3
6
Write on the board: x ⫹ 73 ⫽ 107
We call this an equation, because it includes an equals sign.
What number you think x represents? (34)
How did you work out the unknown number? (By working backwards using
inverse operations: 107 ⫺ 73)
Draw this function machine on the board:
Write down the input of this function machine. (6)
Write down an equation for the function machine. Use x
for the unknown number. (x ⫹ 4 ⫽ 10)
What is x? (6)
Repeat for the other function machines below:
x ⫹ 8 ⫽ 25, x ⫽ 25 ⫺ 8, x ⫽ 17
x ⫺ 7 ⫽ 22, x ⫽ 22 ⫹ 7, so x ⫽ 29
x ⫻ 2 ⫽ 14, x ⫽ 14 ⫼ 2 so x ⫽ 7 (This is a good
opportunity to introduce the idea of writing x ⫻ 2 as 2x.)
14 Maths Connect 1B
Variations
Use puzzles to help pupils develop equations as a way of finding an
unknown. For example:
1. I think of a number, add 12 and get 42. What number was I thinking
of? (x ⫹ 12 ⫽ 42, x ⫽ 42 ⫺ 12, so x ⫽ 30)
2. I think of a number, subtract 7 and get 12. What number was I
thinking of? (x ⫺ 7 ⫽ 12, x ⫽ 12 ⫹ 7, so x ⫽ 19)
3. I think of a number, multiply it by 5 and get 35. What number was I
thinking of? (x ⫻ 5 ⫽ 35 so 5x ⫽ 35, x ⫽ 35 ⫼ 5 so x ⫽ 7)
(This is a good opportunity to introduce the idea of writing
x ⫹ x ⫹ x ⫹ x ⫹ x ⫽ 5 ⫻ x ⫽ 5x)
4. I think of a number, divide by 10 and get 24. What number was I
x
thinking of? (x ⫼ 10 ⫽ 24 so ᎏᎏ ⫽ 24, x ⫽ 24 ⫻ 10 so x ⫽ 240)
10
(This is a good opportunity to introduce the idea of writing
x
x ⫼ 10 as ᎏᎏ.)
10
Plenary
Questions:
G
Key teaching points:
Write on the board a function machine
involving calculations that pupils will not
easily be able to work out mentally, e.g.
⫻97
Input
G
G
G
⫹343
1701
How would we find the input number? Ask
pupils to suggest methods (trial and
improvement, inverse operations, or
transforming to an equation). Individuals
could demonstrate to the class. Focus on
process rather than final answer, and
discuss the relative merits/disadvantages of
the methods.
G
G
G
We represent an unknown by using a letter.
An expression is a collection of numbers
and letters, such as 2x or 3n ⫹ 2.
An equation is a mathematical expression
containing numbers or letters and an ⫽ sign.
We can transform a function machine into
an equation.
We can write x ⫻ 2 as 2x.
x
We can write x ⫼ 2 as ᎏᎏ.
2
G
w
⫻7
⫻9
126
What is w? (2)
How would the output be changed if the
function machines changed places?
(The output would not be changed.)
G
m
⫹3
⫻8
72
What is m? (6)
Would the output change if the function
machines changed places?
(The output would be 51 not 72.)
Exercise hints
Q1–8
Practice and revision
Q9
Problem
For Q9: tell pupils to start by finding pairs of whole
numbers that sum to 72 and to multiply them
together to see if they get 1100.
P B
Exercise 1.6, page 13
Homework 1.6, page 75
Answers, page 297
Algebra 1: Using letters to stand in for unknown numbers 15
xvi Maths Connect 1B
Understand negative numbers as positions on a number line; order, add and subtract positive and
negative integers in context.
Recognise and use multiples, factors (divisors), common factor, highest common factor and lowest
common multiple in simple cases, and primes (less than 100); use simple tests of divisibility.
Recognise the first few triangular numbers, squares of numbers to at least 12 ⫻ 12 and the
corresponding roots.
Integers, powers and roots
Understand and use decimal notation and place value; multiply and divide integers and decimals
by 10, 100, 1000, and explain the effect.
Compare and order decimals in different contexts; know that when comparing measurements
they must be in the same units.
Round positive whole numbers to the nearest 10, 100 or 1000 and decimals to the nearest whole
number or one decimal place.
Place value, ordering and rounding
Numbers and the number system
Solve word problems and investigate in a range of contexts: number; algebra; shape, space and
measure; handling data; compare and evaluate solutions.
Identify the necessary information to solve a problem; represent problems mathematically making
correct use of symbols, words, diagrams, tables and graphs.
Break a complex calculation into simpler steps, choosing and using appropriate and efficient
operations, methods and resources, including ICT.
Present and interpret solutions in the context of the original problem; explain and justify methods
and conclusions, orally and in writing.
Suggest extensions to problems by asking ‘What if …?’; begin to generalise and to understand the
significance of a counter-example.
Applying mathematics and solving problems
Using and applying mathematics to solve problems
Framework teaching objective
Matching chart 2
84–85
120–121, 200–207
120–121, 206–209
86–89
100–101, 168–175
100–101, 174–177
10.1, 16.1, 16.2,
16.3, 16.4
10.1, 16.4, 16.5
104–107
102–103
22–23, 114–117
18–21, 46–47, 52–53
8–9, 202–203,
230–231
180–197
48–49, 52–53
Teacher book pages
9.2, 9.3
2.3, 9.7, 9.8
9.1
6–7, 170–171,
196–197
1.3, 16.2, 17.7
14–17, 38–39,
44–45
18–19, 96–99
152–163
Unit 15
2.1, 2.2, 4.3, 4.6
40–41, 44–45
Pupil book pages
Units 1, 5, 6, 8, 10,
13, 15 & 17
4.4, 4.6
Throughout
Section no.
25, 27, 28,
33
25, 29, 36
13, 16
18, 19
16, 17, 20
7, 8
Starter ref.
Matching chart 2 xvii
34–39
40–43
44–45, 122–125
126–131
4.1, 4.2, 4.3
4.4, 4.5
4.6, 12.1, 12.2
12.3, 12.4, 12.5
Written methods
Use standard column procedures to add and subtract whole numbers and decimals with up to
two places.
Multiply and divide three-digit by two-digit whole numbers; extend to multiplying and dividing
decimals with one or two places by single-digit whole numbers.
24–25, 178–179
16–17, 90–93,
180–183
2.2, 9.4, 9.5, 16.7
16.8
24–25, 90–93
178–183
2.6, 9.4, 9.5, 16.6,
16.7, 16.8
2.6, 16.6
24, 30, 32,
14–25, 34–45,
122–131
Unit 2, 4, 12
20–21, 108–111,
212–215
28–29, 210–211
28–29, 108–111,
210–215
16–29, 40–53,
144–155
37
14, 24
33, 34, 35
38
26, 33
24–27, 70–71, 76–79
Starters
Mental methods and rapid recall of number facts
Know and use the order of operations, including brackets.
Consolidate the rapid recall of number facts, including positive integer complements to 100 and
multiplication facts to 10 ⫻ 10, and quickly derive associated division facts.
Consolidate and extend mental methods of calculation to include decimals, fractions and
percentages, accompanied where appropriate by suitable jottings; solve simple word problems
mentally.
Make and justify estimates and approximations of calculations.
21, 22, 23,
24
32, 33
40, 41, 43, 44
39
20, 42, 43,
44
Starter ref.
1.5, 2.1, 2.2, 2.4,
See various
2.5, 2.6, 9.3, 9.4,
sections
9.5, 13.2, 13.3,
16.6, 16.7, 16.8
2.4, 2.5, 6.1, 6.4, 6.5
20–21, 24–29,
106–111, 160–163,
210–215
150–155
52–53, 146–149
48–51
42–47
Teacher book pages
Understand addition, subtraction, multiplication and division as they apply to whole numbers and
decimals; know how to use the laws of arithmetic and inverse operations.
Number operations and the relationships between them
Calculations
Fractions, decimals, percentages, ratio and proportion
Use fraction notation to describe parts of shapes and to express a smaller whole number as a
fraction of a larger one; simplify fractions by cancelling all common factors and identify equivalent
fractions; convert terminating decimals to fractions, e.g. 0.23 ⫽ ᎏ120ᎏ30; use a diagram to compare
two or more simple fractions.
Begin to add and subtract simple fractions and those with common denominators; calculate
simple fractions of quantities and measurements (whole number answers); multiply a fraction
by an integer.
Understand percentage as the ‘number of parts per 100’; recognise the equivalence of
percentages, fractions and decimals; calculate simple percentages and use percentages to
compare simple proportions.
Understand the relationship between ratio and proportion; use direct proportion in simple
contexts; use ratio notation, reduce a ratio to its simplest form and divide a quantity in a given
ratio; solve simple problems about ratio and proportion using informal strategies.
Pupil book pages
Section no.
Framework teaching objective
xviii Maths Connect 1B
9.4, 9.5, 12.2, 12.5
16.7, 16.8
Check a result by considering whether it is of the right order of magnitude and by working the
problem backwards.
2–7
4–7, 102–105,
184–187
6–7, 102–105,
184–185
8–13
106–111, 188–189
190–195
1.4, 1.5, 1.6
10.4, 10.5, 10.6,
17.3
17.4, 17.5, 17.6
Express simple functions in words, then using symbols; represent them in mappings.
Generate coordinate pairs that satisfy a simple linear rule; plot graphs of simple linear functions,
where y is given explicitly in terms of x, on paper and using ICT; recognise straight line graphs
parallel to the x-axis or y-axis.
Begin to plot and to interpret the graphs of simple linear functions arising from real-life situations.
224–229
2–9
6–9, 122–125,
218–221
8–9, 122–125,
218–219
10–15
126–131, 222–223
74–75, 156–165,
216–233
62–63, 132–139,
184–199
1.1, 1.2, 1.3
1.2, 1.3, 10.2, 10.3,
17.1, 17.2
1.3, 10.2, 10.3, 17.1
13.2, 13.3, 13.4,
17.7
6.3, Units 13 & 17
6.2, 6.5, 13.1
Starter ref.
2–15, 68–79, 118–131,
156–165, 216–233
2–5, 68–79, 118–131,
156–165, 216–233
72–73, 78–79,
158–159
160–165, 230–231
46, 47
108–111, 148–149,
154–155, 212–215
24–27, 76–79,
208–209
28–29, 114–117
Teacher book pages
2–13, 58–67,
100–111, 184–199
2–13, 58–67,
100–111, 184–199
60–61, 66–67,
132–133
134–139, 196–197
90–93, 124–125,
130–131, 180–183
20–23, 64–67,
176–177
24–25, 96–99
Pupil book pages
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 sequences from practical contexts and describe the general term in simple cases.
Sequences, functions and graphs
Use letter symbols to represent unknown numbers or variables; know the meanings of the words
term, expression and equation.
Understand that algebraic operations follow the same conventions and order as arithmetic
operations.
Simplify linear algebraic expressions by collecting like terms; begin to multiply a single term over
a bracket (integer coefficients).
Construct and solve simple linear equations with integer coefficients (unknown on one side only)
using an appropriate method (e.g. inverse operations).
Use simple formulae from mathematics and other subjects, substitute positive integers in simple
linear expressions and formulae and, in simple cases, derive a formula.
Equations, formulae and identities
Algebra
Checking results
Units 1, 6, 10, 13
& 17
Units 1, 6, 13 & 17
2.4, 2.5, 6.4, 6.5,
16.5
2.6, 9.7, 9.8
Carry out all calculations with more than one step using brackets and the memory; use the square
root and sign change keys.
Enter numbers and interpret the display in different contexts (decimals, percentages, money,
metric measures).
Calculator methods
Section no.
Framework teaching objective
Matching chart 2 xix
7.3, 10.4, 10.5,
10.6, 14.6, 17.3,
17.4, 17.5, 17.7
14.1, 14.2, 14.3,
14.4, 14.5, 14.6,
18.3
14.1, 14.3, 14.5
Measures and mensuration
Use names and abbreviations of units of measurement to measure, estimate, calculate and solve
problems in everyday contexts involving length, area, mass, capacity, time and angle; convert one
metric unit to another (e.g. grams to kilograms); read and interpret scales on a range of measuring
instruments.
Use angle measure; distinguish between and estimate the size of acute, obtuse and reflex angles.
Know and use the formula for the area of a rectangle; calculate the perimeter and area of shapes
made from rectangles.
Calculate the surface area of cubes and cuboids.
168–169, 172–173
176–177
168–173
174–179, 240–241
70–71, 116–117
26–29
32–33, 210–211
3.1, 3.2, 5.1, 5.2,
5.3, 5.4, 7.2, 9.6,
9.7, 9.8, 18.6,
Unit 8
7.2, 11.3
3.1, 3.2
3.4, 18.6
Starter ref.
62–63, 246–247
84–85, 138–139
32–35
30, 31
32–35, 56–63, 84–85, 9, 10, 11, 12,
112–117, 246–247
45
244–245
208–209
26–29, 46–53,
70–71, 94–99,
210–211
138–143, 242–245
116–121, 206–209
106–111, 150–151, , 86–87, 126–131,
188–193, 196–197 178–179, 222–231
140–141, 144–145,
148–149
140–145
146–151, 204–205
36–37, 246–247
30–31, 210–211
3.3, 18.6
82–85
Teacher book pages
112–115, 200–203 134–137, 236–239
68–71
Pupil book pages
11.1, 11.2, 18.1,
18.2
Throughout
7.1, 7.2
Section no.
Use a rule and protractor to: measure and draw lines to the nearest millimetre and angles,
11.3, 11.4, 11.5,
including reflex angles, to the nearest degree; construct a triangle given two sides and the included 18.4, 18.5
angle (SAS), or two angles and the included side (ASA). Explore these constructions using ICT.
Use ruler and protractor to construct simple nets of 3-D shapes, e.g. cuboid, regular tetrahedron
18.5
square-based pyramid, triangular prism.
Construction
Use conventions and notation for 2-D coordinates in all four quadrants; find coordinates of points
determined by geometric information.
Coordinates
Understand and use the language and notation associated with reflections, translations and
rotations.
Recognise and visualise the transformation and symmetry of a 2-D shape; reflection in given
mirror lines, and line symmetry; rotation about a given point, and rotation symmetry;
translation. Explore these transformations and symmetries using ICT.
Transformations
Use correctly the vocabulary, notation and labelling conventions for lines, angles and shapes.
Identify parallel and perpendicular lines; know the sum of angles at a point, on a straight line and
in a triangle and recognise vertically opposite angles.
Begin to identify and use angle, side and symmetry properties of triangles and quadrilaterals,
solve geometrical problems involving these properties, using step-by-step deduction and
explaining reasoning with diagrams and text.
Use 2-D representations to visualise 3-D shapes and deduce some of their properties.
Geometrical reasoning: lines, angles and shapes
Space, shape and measure
Framework teaching objective
xx Maths Connect 1B
Interpreting and discussing results
Use vocabulary and ideas of probability, drawing on experience.
Understand and use the probability scale from 0 to 1; find and justify probabilities based on
equally likely outcomes in simple contexts; identify all the possible mutually exclusive outcomes
of a single event.
Collect data from a simple experiment and record in a frequency table; estimate probabilities
based on this data.
Compare experimental and theoretical probabilities in simple contexts.
Probability
Interpret diagrams and graphs (including pie charts), and draw simple conclusions based on the
shape of graphs and simple statistics for a single distribution.
Compare two simple distributions using the range and one of the mode, median or mean.
Write a short report of a statistical enquiry and illustrate with appropriate diagrams, graphs and
charts, using ICT as appropriate; justify choice of what is presented.
54–57
162–163
50–51, 164–165
166–167
5.3, 15.7
15.8
52–53, 76–77,
156–157
154–155
160–161
5.5, 5.6
15.6
15.2
15.5
5.4, 8.2, 15.3
196–197
60–61, 194–195
64–67
192–193
62–63, 92–93,
186–187
184–185
190–191
92–93, 98–99,
190–191
76–77, 82–83,
160–161
96–97
56–61, 90–91,
182–183
80–81
8.4
94–95
94–95, 188–189
96–97, 188–189
Teacher book pages
46–51, 74–75,
152–153
78–79
78–79, 158–159
80–81, 158–159
Pupil book pages
8.3
8.3, 15.4
8.4, 15.4
Section no.
Calculate statistics for small sets of discrete data: find the mode, median and range, and the modal 5.1, 5.2, 5.3, 8.1,
class for grouped data; calculate the mean, including from a simple frequency table, using a
15.1
calculator for alarger number of items.
Construct, on paper and using ICT, graphs and diagrams to represent data including: bar-line
8.2, 8.5, 15.5
graphs; frequency diagrams for grouped discrete data; use ICT to generate pie charts.
Processing and representing data, using ICT as appropriate
Given a problem that can be addressed by statistical methods, suggest possible answers.
Decide which data would be relevant to an enquiry and possible sources.
Plan how to collect and organise small sets of data; design a data collection sheet or questionnaire
to use in a simple survey; construct frequency tables for discrete data, grouped where appropriate
in equal class intervals.
Collect small sets of data from surveys and experiments, as planned.
Specifying a problem, planning and collecting data
Handling data
Framework teaching objective
Starter ref.
Maths Connect
Teacher Book
Transforming standards at Key Stage 3
Maths Connect Teacher Books will help you deliver interactive whole class teaching
in line with the National Numeracy Strategy’s Framework.
Written and developed by experienced teachers and advisors, Maths Connect
Teacher Books offer you:
●
A practical and realistic route through the Framework and Sample Medium Term
Plans for Mathematics.
●
Practical ideas for whole class teaching based on real Framework practice.
●
Complete lesson plans that include starters, plenaries and teaching ideas.
●
Key words, teaching objectives and common difficulties highlighted for each
lesson.
●
Links showing where you can find relevant pupil resources, homeworks and
assessments.
●
Links between concepts and skills to help you build confidence and
understanding.
Maths Connect - everything you need to deliver effective and
interactive lessons.
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01865 888080
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Maths Connect 1 Blue Teacher Book