1. science
2. mathematics
3. arithmetic

2
G
Teacher Book
Transforming standards at Key Stage 3
Maths Connect Teacher Books will help you deliver interactive
whole class teaching in line with the Framework. Written and
developed by experienced teachers and advisers, 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.
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Maths Connect - everything you need to deliver effective and
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TEACHER BOOK
Sample Pages
Contents of Maths Connect 2G: Teacher Book
Page 2
Algebra 1, Number 1: Integers and sequences
Pages 3-11
2
G
Teacher Book contents
Maths Connect 2G follows the objectives from the teaching programme for
Year 8 as suggested in the support tier of the Sample medium-term plans. It
is written specifically for Year 8 support groups.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
N1/A1
SSM1
HD1
N2
A2
SSM2
A3
N3
SSM3
A4
HD2
N4
A5
N5
SSM4
HD3
Integers and sequences
Angles and shapes
Probability
Fractions, decimals and percentages
Equations and formulae
Measures, area and perimeter
Functions and graphs
Numbers and calculations
Transformations
Solving equations and using formulae
Tables and statistics
Calculations
Sequences and graphs
Ratio and proportion and solving problems
Exploring 2-D and 3-D shapes
Applying skills and analysing data
Features Thinking Maths activities adapted from the King’s College
CAME team – proven to build pupils’ thinking skills and improve
performance across Key Stage 3.
2
Teacher Book 2G
Each unit features an overview page that summarises objectives, outlines assumed
knowledge and common difficulties and links to other components of the course.
1 N1/A1 Integers and
Units 8 and 13
sequences
(6 hours)
Assumed knowledge
Background
Half of this Unit is devoted to Number and half to Algebra.
Lessons 1.1 and 1.2 focus on addition and subtraction of
integers, and thus provide a foundation for collecting like
terms in Algebra. Tests for divisibility are rehearsed again
in Lesson 1.3; they are used frequently throughout the
course, particularly in the study of multiples, factors and
primes in Unit 8.
Lessons 1.4 to 1.6 focus on sequences. Pupils met simple
number sequences in Year 7. Here they explore sequences
generated in practical contexts by making and drawing
sequences of patterns and describing how the pattern
grows, including square and triangular numbers. Pupils
generate sequences given the first term and a term-toterm rule (add, multiply, subtract or divide). In Lesson 1.6
they identify the term-to-term rule of a linear sequence
by looking at the differences between consecutive terms,
and use this pattern to find any term in the sequence.
Before starting this Unit, pupils should:
G be able to recognise positive and negative
numbers
G be able to rapidly recall the halves of numbers up
to 100 and beyond
G be able to follow the order of arithmetic operations
G be familiar with square numbers and the notation
for squaring
G know that 3 3 3 3 4 3
Main teaching objectives
Pupil book sections
1.1 Adding integers
1.2
Subtracting integers
1.3
1.4
Tests for divisibility
Sequences from patterns
1.5
Generating sequences
1.6
Investigating sequences
Teaching objectives
Add positive and negative integers in context
Use the sign change key on a calculator
Subtract positive and negative integers in context
Use the sign change key on a calculator
Use simple tests of divisibility
Generate sequences from practical contexts and describe the general term
in simple cases
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 simple sequence, given a rule for finding a term given
its position in the sequence
Oral and mental starters
Starter 16 20 24 2
3
6
Page 254 256 258 246 246 248
Common difficulties
G
G
When operating on integers, pupils may not appreciate
the difference between the minus sign for subtraction
() and the sign for a negative number ().
Pupils may not realise that a sequence can continue
beyond zero.
PB
Pages 2–13
Homeworks 1.1–1.6
Assessment 1G
!
Thinking Maths
Expressions and equations
2 Maths Connect 2G
Sample page from Maths Connect 2G: Teacher Book
3
Teacher Book 2G
Follows the structure of the Sample medium-term plans, featuring a suggested
starter, main teaching activity and plenary for each lesson.
Key words
1.1
integer
positive integer
negative integer
number line
Adding integers
Add positive and negative integers in context
Use the sign change key on a calculator
Links
Introduction
Pupils were introduced to the addition of positive
and negative integers in Year 7. In this lesson,
they practise addition by visualising it as a move,
or ‘jump’, along a number line. They also practise
using the sign change key on a calculator when
adding a mix of positive and negative integers.
1.2 Subtracting integers
Oral and mental starter
16
Teaching activity
Teacher materials:
OHT of the number line from Resource sheet 1
OHP calculator, or a calculator on a whiteboard
Pupil materials:
Calculators, number lines from Resource sheet 1 (if these are stuck on card, a
rubber band can be wrapped around the number line and used to indicate
positions)
Outline
Display the 20-point number line from Resource sheet 1.
Write these numbers on the board: 4, 3, 5, 0, 6.
What type of numbers are these? (Positive and negative whole numbers, or
integers) If necessary, remind pupils that positive and negative whole
numbers, together with zero, are called integers, and that the raised
signs are used to avoid confusion with the operations of addition and
subtraction.
Choose two numbers on the line to be added, for example 5 and 4.
Write: 5 4 , and invite answers. (1) Remind pupils that they can
use a number line to help them to add integers. The method is as follows:
On the number line, locate the first number in the addition. Start
from that point.
If the second number in the addition is positive, move to the right.
(Adding a positive makes a number higher.)
If the second number in the addition is negative, move to the left.
(Adding a negative makes a number lower.)
G
G
G
Demonstrate by starting at 5, and moving four places to the right to
land on 1:
4
10
5
0
5
10
Does the order in which we add two numbers make any difference to the
answer? (No) Demonstrate that reversing the order in the example
above (4 5) gives the same result.
Demonstrate how to use the sign change key to add these integers
using a calculator, by pressing this sequence:
5
Ⲑⴚ
ⴙ
ⴙ
4
ⴝ
4 Maths Connect 2G
4
Sample page from Maths Connect 2G: Teacher Book
Teacher Book 2G
For each lesson in the Pupil Book, there is a corresponding double page spread in
the Teacher’s book for ease of use.
Discuss how the negative sign is shown on the calculator, and the fact
that there is no equivalent sign to denote a positive number.
Repeat the whole process for different additions, for example 3 6
and 7 5. Pupils can use their number lines to help. Include at least
one ‘missing number’ calculation, for example 1 … 6.
Extend to adding more than two integers, for example 4 2 5.
Pupils can use their number lines to add the first two numbers, and then
add the third number to the result. They use their calculators to check.
Variations
Write each integer from 10 to 10 on separate pieces of card. Give each
of ten pupils one of the integer cards. Select pairs of pupils to hold up
their numbers, and ask the class to decide on the total.
Extend to discussing which pairs will have a total of 1, a total of less
than 6, a total of more than 6, and so on.
Plenary
G
G
G
The result of adding one integer to another
is –3. What could the addition be?
Write these numbers on the board: –12, –3,
–
45. What is the correct name for numbers
like these? (Negative integers) Why do we
use a small, high minus sign in front of the
number? (To avoid confusing the sign for a
negative number with the sign for the
operation of subtraction.)
Can anyone suggest a negative number that
–
is not an integer? ( 12, –0.5, …)
Exercise hints
Q1–4
Q5–6
Q7
Q8
Practice
Problems
Activity
Investigation
Key teaching points:
G
G
G
Positive and negative whole numbers,
together with zero, are called integers.
When we add integers, we can picture
moving to the left or right along a number
line. Start from the position of the first
number and make a move that matches the
second number.
On a calculator, use the sign change key for
additions that involve negative and positive
numbers.
P B
Exercise 1.1, page 2
Homework 1.1
Answers, page 287
Number 1/Algebra 1: Adding integers 5
Sample page from Maths Connect 2G: Teacher Book
5
Teacher Book 2G
Features clear teaching objectives and learning
outcomes to set the scene for each lesson.
Key words
1.2
integer
number line
inverse
Subtracting integers
Subtract positive and negative integers in context
Use the sign change key on a calculator
Links
Introduction
Pupils were introduced to the subtraction of
positive and negative integers in Year 7. Here
they develop and practise the method of
subtraction that is based on converting a
subtraction into an addition of the inverse. They
use calculators to confirm their answers and to
practise the use of the sign change key.
1.1 Adding integers
Oral and mental starter
20
Teaching activity
Teacher materials:
OHT of the number line from Resource sheet 1
OHP calculator, or a calculator on a whiteboard
Pupil materials:
Calculators, number lines from Resource sheet 1 (if these are stuck on card, a
rubber band can be wrapped around the number line and used to indicate positions)
Outline
Display the 20-point number line and write on the board: 7 4 .
Establish that this can be interpreted as ‘Find the difference between 4
and 7’, or ‘What number must be added to 4 to make 7?’ Use the
number line to demonstrate that to get from 4 to 7 requires a move
(or jump) of 3, so 7 4 3.
Write on the board: 4 2 = . Elicit that this can be interpreted as
‘What number must be added to 2 to make 4?’ Choose a pupil to
locate 2 on the number line, and to demonstrate the move required to
reach 4 (moving six places to the left, or adding 6). Complete the
subtraction: 4 2 6.
6
10
5 4
0
2
5
10
Repeat this method for two other subtractions, for example
5 2 3 and 1 5 = 4. Pupils are likely to express surprise that
subtracting a negative from a negative can lead to a positive answer.
Focus their attention on the number line and the move required to find
the difference between the two numbers in the subtraction.
Remind pupils that one method of subtracting integers is to convert the
subtraction into an addition of its inverse.
Write: 4 2 6. Alongside it write: 4 2 = . Explain that 2 is the
inverse of 2. What else can you say about these two calculations? (The
starting number is the same in both cases; one is a subtraction and the other is
an addition; the first calculation involves subtracting 2 and the second
involves adding the inverse of 2, …) If no-one mentions that the answers
are the same, complete the second calculation together and draw
pupils’ attention to that fact.
6 Maths Connect 2G
6
Sample page from Maths Connect 2G: Teacher Book
Teacher Book 2G
Questions and main teaching points provide support for the plenary.
Check that this conversion process also works for the two subtractions
calculated earlier:
5 2 3 is equivalent to 5 2 3
1 5 4 is equivalent to 1 5 4
G
G
Clarify that to subtract an integer, you simply add its inverse.
Demonstrate the first of these calculations using a calculator and the
sign change key. When using a calculator, there is no need to convert the
subtraction into an addition.
4
Ⲑⴚ
ⴙ
ⴚ
2
ⴝ
Ask pupils to confirm the other two subtractions using their
calculators.
Variations
Write a subtraction, e.g. 3 4. Demonstrate that the answer is 7.
Now reverse the calculation, i.e. 4 3. Demonstrate that the answer
to this is 7. One answer is the inverse of the other. Explore to see if this
is true for other calculations.
Plenary
G
G
G
What is the inverse of 2, 4, 0, …?
Explain how to complete this subtraction:
7 1 ...
The result of subtracting one integer from
another is 2. What could the subtraction
be?
Exercise hints
Q1–4
Q5–6
Q7
Practice
Activities
Investigation
Key teaching points:
G
G
We can subtract an integer by adding its
inverse.
On a calculator, use the sign change key for
subtractions that involve negative and
positive numbers.
P B
Exercise 1.2, page 4
Homework 1.2
Answers, page 287
Number 1/Algebra 1: Subtracting integers 7
Sample page from Maths Connect 2G: Teacher Book
7
Teacher Book 2G
Key words
1.3
divisible
divisibility
multiple
Tests for divisibility
Use simple tests of divisibility
Links
Provides
links to other
relevant
sections to help
enrich your
teaching.
8.7 Multiples
8.8 Factors
8.9 Prime numbers
Year 7 work on multiples and factors
Introduction
Pupils studied tests of divisibility in Year 7. This
lesson reviews tests for divisibility by numbers
from 2 to 10, and the relationships between
them. It begins to extend the list of known tests
to include some for numbers greater than 10.
Oral and mental starter
24
Teaching activity
Outline
Start to draw a divisibility test table on the board
or OHP.
Check that pupils can explain the term ‘is divisible
by’. How can we tell whether one number is divisible
by another? (There are different tests to carry out; for
example, if a number is even, it is divisible by 2.)
Write and circle ten numbers on the board or OHP,
e.g. 48, 52, 63, 104, 172, 85, 96, 112, 18, 240.
Which of these numbers are divisible by 2? (All except
63 and 85 are even and so are divisible by 2.) Choose a
pupil to list them in the table.
How can we check whether a number is divisible by 4?
(Halve it and see if the answer is even.) Which of the
numbers are divisible by 4? (48, 52, 96, 104, 112, 172,
240) Choose a pupil to write them in the table.
Establish that an alternative test for divisibility by
4 is to see whether the last two digits of the
number are a multiple of 4. If the last two digits are
divisible by 4, then the whole number is divisible by 4.
(You might like to make the link with testing to see
whether a year will be a leap year or not.)
Divisible
by
Numbers
Test
Numbers
Test
18, 48, 52, 96,
104, 112, 172, 240
Last digit is even
48, 52, 96, 104,
112, 172, 240
Halve it, then
check for
divisibility by 2
2
3
4
5
6
7
8
9
10
Divisible
by
2
3
4
⯗
How can we test for divisibility by 8? (First halve the
number and test the resulting answer for divisibility by
4.) Which of the numbers on the board are divisible by
8? (48, 96, 104, 112, 240)
Next check for divisibility by 3, 6 and 9. How can we tell whether a number is
divisible by 3? (If the sum of its digits is divisible by 3, then the number is
divisible by 3.) What about a divisibility test for 6? (Every even number that is
divisible by 3 will also be divisible by 6.) You may wish to list the first few
multiples of 3 and of 6 to confirm this. Can you tell me the divisibility test for
9? (If the sum of the digits is divisible by 9, then the number is divisible by 9.)
Invite a pupil to list the numbers that are divisible by 5 and by 10 in the
table and ask the class to check they are correct. How do you know?
8 Maths Connect 2G
8
Sample page from Maths Connect 2G: Teacher Book
Teacher Book 2G
Finally, consider divisibility by 7. There is no
simple test to check for this, but one method is to
use near known multiples of 7. For example, 140 is
known as a multiple of 7, and 172 is 32 more; 32 is
not divisible by 7 so 172 is not divisible by 7.
Divisible
by
Numbers
Test
2
18, 48, 52, 96,
104, 112, 172, 240
Last digit is even
3
18, 48, 63, 96,
240
Digit total is a
multiple of 3
4
48, 52, 96, 104,
112, 172, 240
Halve it, then
check for
divisibility by 2
5
85, 240
Last digit is 0 or 5
6
18, 48, 96, 240
Test for divisibility
by both 2 and 3
7
63, 112
Compare with
known multiples of
7
8
48, 96, 104, 112,
240
Halve it, then
check for
divisibility by 4
9
18, 63
Digit total is a
multiple of 9
240
Last digit is 0
Study the completed table.
Discuss the fact that divisibility by 6 or 9 implies
divisibility by 3, and divisibility by 8 implies
divisibility by 2 and by 4.
Discuss tests for divisibility by larger numbers, for
example by 12 (must pass the tests for 3 and for 4),
by 14 (must pass the tests for 2 and for 7) and by
15 (must pass the tests for 3 and for 5).
10
Variations
Ask pupils to find three different numbers, each between 200 and 300,
which pass the test for divisibility by 3. Repeat this process for
divisibility by each of the numbers from 4 to 10.
Variation sections provide
an alternative way of
covering the same teaching
objectives, giving choice in
terms of resources used and
approach.
Plenary
G
G
G
G
If a number is divisible by 15, is it also
divisible by 3? (Yes)
Which numbers are divisible by both 5 and
6? (Numbers ending with 0 that are also
divisible by 3) Give me an example. (30, 60,
90, 120, …)
How could you test for divisibility by 12,
14, 15, …?
Testing 376 for divisibility by 8, for
example, requires good halving skills. We
can split 376 into two parts, 300 and 76.
Half of 300 150 and half of 76 38; these
combine to give 188. What is half of 256?
(128) … 528? (264)
Exercise hints
Q1–5
Q6–8
Practice
Problems
Key teaching points:
G
G
Tests for divisibility are:
a) by 2 – is the last digit even?
b) by 3 – is the digit total a multiple of 3?
c) by 4 – halve it then test for divisibility by
2, or are the last two digits a multiple of 4?
d) by 5 – is the last digit 0 or 5?
e) by 6 – does it pass the tests for 2 and for 3?
f) by 7 – compare it with near known
multiples of 7
g) by 8 – halve it then test for divisibility by 4
h) by 9 – is the digit total a multiple of 9?
i) by 10 – is the last digit 0?
If a number is divisible by 8, for example,
then it must also be divisible by 4 (or any
other factor of 8).
P B
Exercise 1.3, page 7
Homework 1.3
Answers, page 287
Number 1/Algebra 1: Tests for divisibility 9
Sample page from Maths Connect 2G: Teacher Book
9
Teacher Book 2G
Key words
1.4
generate
term
term-to-term rule
Sequences from patterns
Generate sequences from practical contexts and describe the general term in
simple cases
Links
Introduction
In this lesson pupils make and draw sequences
of patterns and describe how the pattern grows,
as preparation for generating sequences in
Lesson 1.5 and following the pattern to find any
term in Lesson 1.6. In the exercise they look at
the special sequences of square and triangular
numbers.
1.5 Generating sequences
1.6 Investigating sequences
13.1 The general term
Oral and mental starter
2
Teaching activity
Teacher materials:
OHP, OHT of Resource sheet 2 (pattern sequences)
Pupil materials:
Counters, squared paper
Outline
Explain to pupils that in this lesson they will be drawing sequences of
patterns.
Show the first pattern from Resource sheet 2 on the OHP.
Ask pupils, working in pairs or small groups, to make the first pattern
with counters. Now make the second pattern. How many counters did you
need to add to make the second pattern? (4) Repeat for the third pattern. Can
you explain how the pattern is growing? (Add 4 counters each time, one to each
‘leg’.) How many counters are there in the third pattern? (13) How many
counters will there be in the fourth pattern? (17) Make the pattern and check.
The numbers of counters in the patterns make a sequence. We can write the
number sequence in a table. Draw the table on the board and ask pupils
for the values to write in. They could make the fifth pattern to check
that it has 21 counters.
Term number
1
2
3
4
5
Number of counters
5
9
13
17
21
Features detailed guidance
for interactive teaching of
concepts and skills – perfect
for the less experienced
teacher.
Show the second pattern from Resource sheet 2 on the OHP.
Repeat the process above, with pupils making the patterns using
counters until they are confident about how the pattern grows. Draw
and complete the table:
Term number
1
2
3
4
5
Number of counters
3
6
9
12
15
How many counters are there in the first pattern? (3) So the first term of the
number sequence is 3. How is the pattern growing? (Add 3 counters each
time, one to each ‘leg’.) So the term-to-term rule for the sequence is ‘add 3’.
10 Maths Connect 2G
10
Sample page from Maths Connect 2G: Teacher Book
Teacher Book 2G
Show the third pattern from Resource sheet 2 on the OHP.
Ask pupils to copy the first three patterns on squared paper, and then
draw the fourth and fifth patterns. Confirm that the number sequence
is 1, 4, 7, 10, 13, … and that the pattern grows by adding 3 counters
each time, one to each ‘leg’. Ask pupils to describe the number
sequence by giving the first term (1) and the term-to-term rule (add 3).
Variations
Teacher materials:
OHT, OHP of Resource sheet 2 (pattern sequences)
Pupil materials:
Squared, plain and isometric paper, felt pens
Instead of using counters, pupils could draw the patterns on squared or
plain paper. The second pattern could be drawn most easily on
isometric paper. The patterns could be displayed in the classroom, and
referred to in Lesson 1.6.
Plenary
G
Each table in a restaurant seats four people:
Key teaching points:
G
G
The number of counters, squares etc. in
each pattern generates a number sequence.
Describing how the pattern grows leads to
the term-to-term rule.
You can join two tables together like this:
G
G
How many people can sit here? (6)
Draw the arrangements for three tables and
for four tables. How many people does each
seat? (8 people, 10 people)
Describe how the sequence grows. (Each
extra table adds one more seat along each
side, i.e. two more seats altogether.)
Exercise hints
Q1–3
Q4–5
Q6
Practice
Problems
Investigation
P B
Includes answers
to all the Pupil
Book exercises,
with a helpful
reference on each
double page.
Exercise 1.4, page 8
Homework 1.4
Answers, page 288
Number 1/Algebra 1: Sequences from patterns 11
Sample page from Maths Connect 2G: Teacher Book
11