Download Maths Connect 2R Teacher Book sample material

2
R
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
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TEACHER BOOK
Sample Pages
Contents of Maths Connect 2R: Teacher Book
Page 2
Algebra 1, Number 1: Integers, powers and roots Pages 3-11
2
R
Teacher Book contents
Maths Connect 2R follows the objectives from the teaching programme for
Year 8 as suggested in the extension tier of the Sample medium-term plans.
It is written specifically for Year 8 extension 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, powers and roots
Angles and shapes
Probability
Fractions, decimals and percentages
Equations and formulae
Measures
Diagrams and graphs
Place value and calculations
Transformations
Solving equations
Analysing statistics
Calculations
Equations and graphs
Ratio and proportion and solving problems
Construction
Collecting, displaying 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 2R
Each unit features
an overview page that summarises objectives, outlines assumed
18/5/04 4:23 pm Page 2
knowledge and common difficulties and links to other components of the course.
A_Teachers.qxd
1 N/A1 Integers, powers and
�
Units 5, 8
and 12
roots
(6 hours)
Background
Assumed knowledge
In Number 1 pupils add, subtract, multiply and divide
positive and negative integers, using the rules of signs
and the sign change key on the calculator. They extend
their knowledge of squares and square roots to include
negative square roots and cubes and cube roots. Index
notation is introduced with powers of 2 and 3 and
beyond, leading to simple index rules for multiplying
and dividing. Pupils apply their calculator skills and
knowledge of square roots to estimate square roots by
the method of trial and improvement. Algebra 1
(Lessons 1.5 and 1.6) deals with sequences, including
finding the general term of linear sequences. Lesson
1.6 considers non-linear sequences, finding the
general term where possible, or using the pattern of
differences to generate the next few terms in the
sequence.
Before starting this Unit, pupils should:
� be able to add and subtract positive and negative
integers using a number line
� understand the concept of squares and positive
square roots
� be able to find the general term of a linear
sequence
Main teaching objectives
Pupil book sections
1.1 Calculations with
integers
1.2
Powers and roots
1.3
Indices
1.4
Square roots
1.5
Sequences
1.6
Special sequences
Teaching objectives
Add and subtract integers
Multiply and divide integers
Use the sign change key to a calculator
Use index notation for small integer powers
Know cubes of 1, 2, 3, 4, 5 and 10 and the corresponding roots
Use a calculator to find squares and cubes
Know that 100 � 102, 1000 � 103, 1 million � 106, 1 billion � 109
Use index notation for integer powers
Use simple instances of the index laws
Use ICT to estimate square roots
Use trial and improvement to find approximate solutions to equations
Use linear expressions to describe the nth term of an arithmetic sequence
justifying its form by referring to the activity or practical context from which it
was generated
Introduce the vocabulary T(n) for the general term
Know that an arithmetic sequence is generated by starting with a number a
and adding a constant number d to the previous term
Continue familiar sequences (square numbers, powers of 10, 2, etc.)
Generate sequences by multiplying or dividing by a constant factor
Oral and mental starters
Starter 1
8
9 12 2 10
Page 245 249 250 252 245 250
Common difficulties
�
Pupils may confuse operation signs of addition and
subtraction with signs indicating that an integer is
positive or negative. For this reason, integers are
recorded like this: �6, �3.
PB
Pages 2–13
Homeworks 1.1–1.6
Assessment 2R
!
Thinking Maths
2 Maths Connect 2R
Sample page from Maths Connect 2R: Teacher Book
3
Teacher Book 2R
Follows the structure
of the Sample medium-term plans, featuring a suggested
18/5/04 4:23 pm Page 4
starter, main teaching activity and plenary for each lesson.
A_Teachers.qxd
Key words
1.1
positive
negative
integer
product
Calculations with integers
Add and subtract integers
Multiply and divide integers
Use the sign change key on a calculator
Links
�
1.3 Indices
Oral and mental starter
1
Introduction
In Year 7, pupils added and subtracted positive and
negative integers using a number line and met the
rules of signs for multiplication and division. In this
lesson they revise these skills. They convert
subtractions into addition of the inverse. For
multiplications and divisions, they calculate the
value for positive integers and find the sign using the
rules. Pupils also use the sign change key on a
calculator.
Teaching activity
Teacher materials:
OHP, an OHP calculator, or a calculator on a whiteboard, OHT of Resource sheet 1
(�10 to �10 number lines), OHT of Resource sheet 2 (multiplication grid)
Pupil materials:
Calculators
Outline
Explain to the pupils that this lesson focuses on calculating with
integers. Remind them that integers are positive and negative whole
numbers, including zero.
Write ‘�4 � �5 �’ on the board.
Emphasise that we use raised signs for positive and negative integers,
to avoid confusion with the operations of addition and subtraction.
Show the �10 to �10 number line from Resource sheet 1. Remind pupils
that we can use a number line to add integers.
The method is (a) start at the first number on the line, then (b) count
right if the second number is �ve, and count left if the second number
is �ve. Demonstrate by starting at �4, and counting five places left to
land on �1.
Next, revise subtraction of integers. Write ‘�4 � �5 �’ on the board and
invite answers. Remind pupils that subtracting an integer is equivalent
to adding its inverse. Choose a pupil to write the converted calculation,
‘�4 � �5 �’. Confirm that the answer is �9.
Next consider multiplication of integers. Show the multiplication table
from Resource sheet 2 and start by doing the multiplications of positive
integers in the bottom right hand corner, writing in the products.
Complete the table together, by following the sequence pattern for
each row/column.
Use the results in the multiplication table to confirm the rules of signs
for multiplication. When multiplying two integers: if they both have
the same sign, the product is positive; if they have different signs, the
product is negative.
Write ‘3 � 2 � 6’ on the board. What division facts can we write from this
multiplication fact? Record the two possibilities: (6 � 3 � 2 and 6 � 2 � 3).
�
�
3
� 3 � 2 �1
�
9
�
6
�
� 2 � 6 � 4 �2
�
1
0
�
3
0
�
2
0
�
0
3 0
0
� 1 �2 � 3
�
3
1 0
�
0
0
0
� 1 � 3 � 2 �1
0
�
6
�
9
� 2 �4 � 6
1
�
2
0
�
3
0
� 1 �2 � 3
�
2
�
6
�
4
�
2 0
�
2
�
4
�
6
�
3
�
9
�
6
�
3 0
�
3
�
6
�
9
4 Maths Connect 2R
4
Sample page from Maths Connect 2R: Teacher Book
Teacher Book 2R
For each lesson
in the Pupil Book, there is a corresponding double page spread in
18/5/04 4:23 pm Page 5
the Teacher’s book for ease of use.
A_Teachers.qxd
Choose other multiplication facts from the multiplication table, and ask
pupils for the related division facts, for example �2 � �3 � �6, leads to
�
6 � �3 � �2, and �6 � �2 � �3
Confirm that the rules of signs for division exactly match those for
multiplication – the answer is positive for like signs, and negative for
unlike signs.
Write calculations on the board, for example, �6 � �2, �7 � �4, �35 �
�
5, �12/�3, �5 � �1, … Choose pupils to explain how they calculated
the answers.
Demonstrate how to use the sign change key on a calculator, and
how to use it to perform these calculations.
6
�
7
�
�
2
�
�
�
�
4
�
�
�
Variations
Teacher materials:
Coin, two sets of 1–9 integer cards from Resource sheet 3
Revise the methods for calculating with integers, as in the Outline. To
generate the integers to use in calculations, pick cards to give integer
values and flip the coin for the sign: heads positive, tails negative.
As an extension to the activity in the Outline, discuss the result of
multiplying together three integers, for example, �5 � �2 � �4.
Plenary
�
�
�
�
The answer to an addition of two integers is
�
5. What could the addition be? (Examples
could be: �10 � �15, �3 � �2, �5 � �10 etc.)
The result of multiplying two integers is
�
10. What could the two integers be?
What is (�4)2, (�2)3? (Answers are 16, �8)
What is �5 � �3 � �2? (Answer is �30)
Key teaching points:
Integers are the positive and negative whole
numbers, together with zero.
•We can add integers by counting on a number
line. Start at the position of the first number,
then count left to add a negative number and
count right to add a positive number.
•To subtract one integer from another, convert
the subtraction into an addition of the inverse,
then use the number line.
•When multiplying two integers, multiply the
whole number parts, and then use the rule of
signs to decide if the result is positive or
negative. The same method applies to
division.
•Rules of signs for multiplication and division:
•If both signs are the same, then the result is
positive.
•If both signs are different, then the result is
negative.
•You can use the sign change key on a
calculator to do calculations with negative
numbers.
�
Exercise hints
Q1–3
Practice
Q4–7
Problems
Q8
Activity
Q9
Investigation
Photocopies of Resource sheet 1, number lines
may be useful for the exercise.
P B
Exercise 1.1, page 3
Homework 1.1
�
Answers, page 276
Number 1, Algebra 1: Calculations with integers 5
Sample page from Maths Connect 2R: Teacher Book
5
Teacher Book 2R
Features clear
teaching objectives and learning
18/5/04 4:23 pm Page 6
outcomes to set the scene for each lesson.
A_Teachers.qxd
Key words
1.2
square
index
power
square root
cube
cube root
Powers and roots
Use index notation for small integer powers
Know cubes of 1, 2, 3, 4, 5 and 10 and the corresponding roots
Use a calculator to find squares and cubes
Know that 100 � 102, 1000 � 103, 1 million � 106, 1 billion � 109
Links
�
1.3 Indices, 1.4 Square roots,
5.1 Index notation
�
Oral and mental starter
8
Introduction
This lesson revises squares and square roots
from Year 7 and extends to cubes and cube
roots, and how to find these using a calculator.
Index notation is extended to powers of 2 and 3,
and beyond for powers of 10, including one
million and one billion.
Teaching activity
Teacher materials:
OHP calculator or a calculator on a whiteboard
Pupil materials:
Calculators
Outline
Revise square numbers and square roots. Invite pupils to state some
square numbers. (1, 4, 9, 16, …)
Write ‘62 � 6 � 6 � 36’. Explain that the raised 2 is called the index or
power of 2. It indicates that we multiply the 6 by itself 2 times. Clarify
different ways of reading this statement: ‘Six squared is thirty-six’ and
‘Six to the power of two is thirty-six’.
� � 6’. Read this statement together: ‘The square root of thirtyWrite ‘�36
six is six’. Use a variety of examples to clarify that finding a square root
is the inverse of finding a square. What is (�6)2? (36) So what is another
�? (�6) Remind pupils that every square root has a positive
solution to �36
and a negative solution, but by convention we take the positive value,
unless told otherwise.
Challenge pupils to calculate the squares of more complex numbers,
2
2
4�
e.g. (0.2)2. You might wish to show that (0.2)2 � �1�0 � �1�0 � �
100 � 0.04.
Demonstrate how to use a calculator to find squares, e.g. 7
and square roots 4 9
.
x2
�
Introduce the concept of cube numbers and cube roots. Write
‘23 � 2 � 2 � 2 � 8’. Explain that the index is 3, and we read it as
‘Two cubed is eight’ and ‘Two to the power of three is eight’.
2
Demonstrate the relationship between two squared and two cubed, by
drawing a 2 � 2 square and a 2 � 2 � 2 cube.
Challenge pupils to state the cubes of 1 (1), 2 (8), 3 (27), 4 (64), 5 (125)
and 10 (1000).
Explain that the inverse of finding a cube is finding a cube root. Clarify
that the cube root of 8 is 2, for example. Challenge pupils to find the
cubes of more complex numbers, for example (�2)3, (0.1)3, …
2
2
2
2
(�2)3 � �8
(0.1)3 � 0.1 � 0.1 � 0.1
� 0.001
6 Maths Connect 2R
6
Sample page from Maths Connect 2R: Teacher Book
Teacher Book 2R
Questions
and main teaching points provide support for the plenary.
4:23 pm Page 7
A_Teachers.qxd
18/5/04
Demonstrate how to use a calculator to find cubes and cube roots,
e.g. 2 7 3
�
Invite pupils to complete this table of base 10.
101 � 10
102 � 10 � 10 � 100
103 � 10 � 10 � 10 �
104 �
105 �
106 �
107 �
108 �
109 �
1010 �
Choose pupils to read each number, confirming that 106 is one million
and 109 is one billion.
Variations
Introduce the lesson by drawing a 2 � 2 square. How do we calculate the
area? (2 � 2) How else can we write this? (22) Explain that we call this
index notation. The index, or power, is the small raised 2. How do we
read 22? (2 squared or 2 to the power 2) Next draw a 2 � 2 � 2 cube. How
do we calculate the volume? (2 � 2 � 2) How do we write this in index
notation? (23) How do we read 23? (2 to the power 3 or two cubed)
Plenary
�
�
�
�
Which number is both a square number and
a cube number? (64)
What is (�3)2, (�5)3? (9, �125)
What is (0.1)2, (0.2)3? (0.01, 0.008)
How many thousands make a million? a
billion? (1000, 1 000 000)
Exercise hints
Q1–4
Q5–7
Q8
Practice
Problems
Investigation
Key teaching points:
72 means 7 � 7, and is read as ‘seven
squared’ and ‘seven to the power of two’.
•The inverse of finding the square of a
number is finding the square root of a
number.
•43 means 4 � 4 � 4, and is read as ‘four
cubed’ and ‘four to the power of three’.
•The inverse of finding the cube of a number
is finding the cube root of a number.
•The first few cube numbers are 1, 8, 27, 64,
125, …
•106 � 1 million, 109 � 1 billion.
•We can use a calculator to find squares,
square roots, cubes and cube roots.
�
P B
Exercise 1.2, page 5
Homework 1.2
�
Answers, page 276
Number 1, Algebra 1: Powers and roots 7
Sample page from Maths Connect 2R: Teacher Book
7
Teacher Book 2R
A_Teachers.qxd
18/5/04
4:23 pm
Page 8
Key words
1.3
index
power
indices
Indices
Use index notation for integer powers
Use simple instances of the index laws
Links
�
�
Provides
links to other
relevant
sections to help
enrich your
teaching.
Introduction
1.4 Square roots, 5.1 Index notation,
12.3 Brackets and powers
1.2 Powers and roots
Oral and mental starter
9
In the previous lesson pupils used index notation
to write squares and cubes and powers of 10. In
this lesson they extend index notation to higher
powers, leading to simple index rules for
multiplying and dividing.
Teaching activity
Outline
Explain that this lesson is about multiplying and dividing quantities
written in index notation.
Write ‘21 � 2’ on the board. Write ‘22 � 4’ underneath.
Choose pupils to extend the list up to the tenth power of 2, by
doubling.
21 � 2
22 � 4
23 � 8
24 � 16
25 � 32
26 � 64
27 � 128
28 � 256
29 � 512
210 � 1012
Explain that we can find many simple patterns when multiplying and
dividing the powers of 2.
Write ‘23 � 24 �’ on the board. Invite pupils to use the table to help
perform the calculation. (8 � 16 � 128) Is this answer also in the table?
(Yes, 27) Repeat for some similar calculations, using a loop, to
demonstrate the patterns.
23 � 24 � 27
22 � 23 � 25
8 � 16 � 128
4 � 8 � 32
Challenge pupils to describe the patterns relating to the powers of 2, for
example ‘when multiplying two powers of 2, the result is also a power
of 2, and the power is the sum of the powers’.
Consolidate this idea by writing out the multiplication in full:
‘23 � 24 � 2 � 2 � 2 � 2 � 2 � 2 � 2 � 27’
Extend to division. Write ‘26 � 24 �’ on the board. Record the answer
using a loop as before. Repeat for ‘27 � 22 �’. Search for the pattern:
‘when dividing two powers of 2, the result is also a power of 2, and the
power is the difference of the powers.
26 � 24 � 22
64 � 16 � 4
8 Maths Connect 2R
8
Sample page from Maths Connect 2R: Teacher Book
Teacher Book 2R
A_Teachers.qxd
18/5/04
4:23 pm
Page 9
Consolidate this by writing out the division in full, and cancelling.
2�2�2
�1 � 2
�1 � 2
�1 � 2
�1
26 � 24 � ��� � 22
�1 � 2
2
�1 � 2
�1 � 2
�1
Will these patterns appear in powers of other numbers? Draw a table of
powers of 3 together, up to 36 � 729.
Use examples to demonstrate that 34 � 32 � 36, 35 � 32 � 33.
Challenge pupils to express the rules more generally, using algebra.
na � nb � na�b
na � nb � na�b
Consolidate the rules by extending to examples using powers of 10.
Finally, demonstrate that for the rules to apply, the numbers must have
the same base. In other words, they must be powers of the same
number. Use some examples to demonstrate this: 23 � 102, 34 � 23.
Variations
After working through the lesson Outline above, extend to developing
the meaning of the power zero, by using the laws to consider
34 � 30 � 34, and 26 � 20 � 26. Consolidate this by writing the
calculations in full, i.e. 3 � 3 � 3 � 3 � ? � 3 � 3 � 3 � 3. Clarify that
any number to the power zero is one. n0 � 1.
Variation sections provide
an alternative way of
covering the same teaching
objectives, giving choice in
terms of resources used and
approach.
Plenary
�
�
�
�
�
What is 102 � 104? (106)
How many thousands make one million?
Demonstrate using powers.
What is (23)2? (26)
What is 24 � 24? (1)
What is 50? 80? n0? (1)
Key teaching points:
�
�
�
�
Exercise hints
Q1–4
Q5–7
Q8
Q9
The plural of ‘index’ is ‘indices’.
When multiplying two powers of a number,
the result is also a power of that number,
given by the sum of the powers.
When dividing two powers of a number,
the result is also a power of that number,
given by the difference of the powers.
The rules can be expressed algebraically:
na � nb � na�b, na � nb � na�b
Practice
Problems
Activity
Investigation
P B
Exercise 1.3, page 7
Homework 1.3
�
Answers, page 276
Number 1, Algebra 1: Indices 9
Sample page from Maths Connect 2R: Teacher Book
9
Teacher Book 2R
A_Teachers.qxd
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Page 10
Key words
1.4
trial and improvement
estimate
square root
Square roots
Use ICT to estimate square roots
Use trial and improvement methods to find approximate solutions to
equations
Links
�
�
8.8 More square roots, 8.9 Cube roots
1.2 Powers and roots
Oral and mental starter
12
Introduction
In Lesson 1.2, pupils used the square root key on
a calculator to find square roots. In this lesson,
pupils are introduced to trial and improvement
methods for finding a square root, using a table
of square roots to help with an initial estimate.
They use calculators to find the squares of
estimated numbers, and compare results with a
targeted square root.
Teaching activity
Teacher materials:
OHP calculator, or a calculator on a whiteboard
Pupil materials:
Calculators
Outline
Explain to the pupils that in this lesson they will be using a method
called trial and improvement to find square roots.
With pupils’ help, construct a table of squares and square
roots on the board.
Number
� �’ on the board. Use the table to confirm that the
Write ‘�21
� � 5’,
root lies between the numbers 4 and 5. Write ‘4 � �21
clarifying the notation and explaining that 4 is the lower
bound and 5 is the upper bound.
Suggest that we make an estimate, for example 4.5.
1
2
3
4
5
6
7
8
9
10
Square
1
4
9
16
25
36
49
64
81
100
Number
10
20
30
40
50
60
70
80
90
100
Square
100
400
900
1600
2500
3600
4900
6400
8100
10 000
Ask pupils to find (4.5)2 using their calculators. (20.25) Too small.
Write ‘4.5 � �21
� � 5’.
Next try (4.6)2. (21.166) Too large.
Write ‘4.5 � �21
� � 4.6’.
Invite suggestions, for example try (4.58)2, giving 20.9764. Too small.
� � 4.6’.
Write ‘4.58 � �21
Invite suggestions, for example try (4.59)2, giving 21.0681. Too large.
Features detailed guidance
for interactive teaching of
concepts and skills – perfect
for the less experienced
teacher.
Write ‘4.58 � �21
� � 4.59’.
Ask pupils to suggest a number between 4.58 and 4.59. (For example
4.582) If necessary, demonstrate this on a 10-point number line, with the
ends labelled 4.58 and 4.59.
Find (4.582)2 using a calculator. (20.994 724) Too small.
Write ‘4.582 � �21
� � 4.59’.
Invite suggestions for the next number to try, for example 4.583.
Try (4.583)2 using their calculators. (21.003 889) Too large.
� � 4.583’.
Write ‘4.582 � �21
10 Maths Connect 2R
10
Sample page from Maths Connect 2R: Teacher Book
Teacher Book 2R
A_Teachers.qxd
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Page 11
Invite suggestions of a number between 4.582 and 4.583. (For example 4.5825) If
necessary, demonstrate this on a 10-point number line, with the ends labelled 4.582
and 4.583.
The pupils try (4.5825)2 using their calculators. (20.999 306)
Confirm that we are very close to finding the square root of 21. Ask pupils to use
� � 4.582 575 6. Discuss the closeness of this to 4.5825.
the square root key to find �21
Challenge pupils to round both solutions to the nearest two decimal places (4.58),
demonstrating that these are equal.
� � 4.583. Ask pupils to
Look back at the results written on the board to 4.582 � �21
round the upper and lower bounds to 2 decimal places. (Both 4.58) Explain that if we
had been asked to find the square root to 2 decimal places we could have stopped
� � 4.583. So, �21
� � 4.58 to 2 decimal places.
the calculation here, as 4.582 � �21
Explain that this method is known as ‘trial and improvement’, because you try a
number, look at the result, and use this to improve the try.
Repeat the whole process to find the square roots of other numbers, for example,
�75
�, �4200
�. Remind pupils that �4200
� can be expressed as �42
�00
� 1� or �42
� � 10.
Variations
Explain the method of trial and improvement, as in the Outline. Write a square root
�. Stress that pupils may not use the square root key
on the board, for example �3100
of their calculators. Pupils work in pairs, and are allowed four iterations of the trial
and improvement method. Stress the importance of making a good first estimate.
Pupils then make an estimate in the form of an 8-digit number, e.g. 55.552 660. Each
pair writes their answer on a whiteboard. Finally show the result obtained, using the
square root key on the board calculator. (55.677 643) Eliminate the results of pupils’
estimates, leaving the closest three as winners. Repeat for a different number.
Plenary
�
�
�
What are upper and lower bounds for �6.7
�,
�350
�, �8600
�,… ?
Try finding �27
� using the square root key
(5.196 152 4), then try squaring the answer
(26.999 999). Discuss these results.
Use the trial and improvement method to
find the cube root of 40.
Key teaching points:
�
�
�
�
Exercise hints
Q1–3
Q4–7
Q8–9
Practice
Problems
Investigations
For square roots of numbers up to 100, it is
possible to state two consecutive whole
numbers as the upper and lower bound of
the root.
For square roots of numbers up to 10 000, it
is possible to state two consecutive
multiples of 10 as the upper and lower
bounds of the root.
By making an estimate of the square root of
a number, and then squaring this estimate,
it is possible, by comparing, to make an
improved estimate.
The method of estimating a solution,
comparing it with the target solution, and
then making an improved estimate is called
the method of ‘trial and improvement’.
P B
Includes answers
to all the Pupil
Book exercises,
with a helpful
reference on each
double page.
Exercise 1.4, page 9
Homework 1.4
�
Answers, page 276
Number 1, Algebra 1: Square roots 11
Sample page from Maths Connect 2R: Teacher Book
11