Transcript Sample_Pages - Chartwell

Teach GCSE Maths
Shape, Space and Measures
The pages that follow are sample slides from the 113
presentations that cover the work for Shape, Space
and Measures.
A Microsoft WORD file, giving more information, is
included in the folder.
The animations pause after each piece of text. To
continue, either click the left mouse button, press
the space bar or press the forward arrow key on the
keyboard.
Animations will not work correctly unless Powerpoint
2002 or later is used.
F4 Exterior Angle of a Triangle
This first sequence of slides comes from a
Foundation presentation. The slides remind
students of a property of triangles that they have
previously met.
These first slides also show how, from time to
time, the presentations ask students to exchange
ideas so that they gain confidence.
We already know that the sum of the angles of
any triangle is 180.
e.g.
57
57 + 75 + 48 = 180
exterior angle
75
48 a
If we extend one side . . .
we form an angle with the side next to it ( the
adjacent side )
a is called an exterior angle of the triangle
We already know that the sum of the angles of
any triangle is 180.
e.g.
57
57 + 75 + 48 = 180
exterior angle
75
a
48 132
Tell your partner what size a is.
Ans: a = 180 – 48
= 132 ( angles on a straight line )
We already know that the sum of the angles of
any triangle is 180.
e.g.
57
57 + 75 + 48 = 180
exterior angle
75
48 132
What is the link between 132 and the other 2
angles of the triangle?
ANS: 132 = 57 + 75, the sum of the other angles.
F12 Quadrilaterals – Interior Angles
The presentations usually end with a basic
exercise which can be used to test the students’
understanding of the topic. Solutions are given to
these exercises.
Formal algebra is not used at this level but angles
are labelled with letters.
Exercise
1. In the following, find the marked angles, giving
your reasons:
(a)
a 60
115
b
37
(b)
40
105
c
30
Exercise
Solutions:
(a) 115
a 60
120
b
37
a = 180 - 60 ( angles on a straight line )
= 120
b = 360 - 120 - 115 - 37
= 88
(angles of quadrilateral )
Exercise
(b)
40
105
c
150
x
30
Using an extra letter:
x = 180 - 30 ( angles on a straight line )
= 150
c = 360 - 105 - 40 - 150
( angles of quadrilateral )
= 65
F14 Parallelograms
By the time they reach this topic, students have
already met the idea of congruence. Here it is
being used to illustrate a property of
parallelograms.
To see that the opposite sides of a parallelogram
are equal, we draw a line from one corner to the
opposite one.
S
R
SQ is a diagonal
Q
P
Triangles SPQ and QRS are congruent.
So,
SP = QR
and
PQ = RS
F19 Rotational Symmetry
Animation is used here to illustrate a new idea.
This “snowflake” has 6 identical branches.
A
When it makes a
complete turn, the shape
fits onto itself 6 times.
The centre of rotation
F
B
E
C
D
The shape has rotational symmetry of order 6.
( We don’t count the 1st position as it’s the same as
the last. )
F21 Reading Scales
An everyday example is used here to test
understanding of reading scales and the opportunity
is taken to point out a common conversion formula.
This is a copy of a car’s speedometer.
Tell your partner what
1 division measures on
It is common to find
each scale.
the “per” written as p
in miles per hour . . .
but as / in kilometres
per
hour.
Ans:
5 mph on the
outer scale and 4
km/h on the inner.
60
40
20
80
100 120
100
80 km/h 140
160
60
40
180
20
200
0
0
mph
120
220
140
Can you see what the conversion factor is between
miles and kilometres?
Ans: e.g. 160 km = 100 miles.
Dividing by 20 gives 8 km = 5 miles
F26 Nets of a Cuboid and Cylinder
Some students find it difficult to visualise the net
of a 3-D object, so animation is used here to help
them.
Suppose we open a cardboard box and flatten it out.
This is a net
Rules for nets:
We must not cut across a face.
We ignore any overlaps.
We finish up with one piece.
O2 Bearings
This is an example from an early Overlap file.
The file treats the topic at C/D level so is useful
for students working at either Foundation or
Higher level.
e.g. The bearing of R from P is 220 and R is due
west of Q. Mark the position of R on the
diagram.
Solution:
Px
x
Q
e.g. The bearing of R from P is 220 and R is due
west of Q. Mark the position of R on the
diagram.
Solution:
Px
.
x
Q
e.g. The bearing of R from P is 220 and R is due
west of Q. Mark the position of R on the
diagram.
Solution:
P x 220
.
x
Q
If you only have a semicircular protractor, you need to
subtract 180 from 220 and measure from south.
e.g. The bearing of R from P is 220 and R is due
west of Q. Mark the position of R on the
diagram.
Solution:
Px
. 40
x
Q
If you only have a semicircular protractor, you need to
subtract 180 from 220 and measure from south.
e.g. The bearing of R from P is 220 and R is due
west of Q. Mark the position of R on the
diagram.
Solution:
P x 220
.
R
x
Q
O21 Pints, Gallons and Litres
The slide contains a worked example. The
calculator clipart is used to encourage students to
do the calculation before being shown the answer.
e.g. The photo shows a milk bottle and
some milk poured into a glass.
There is 200 ml of milk in the glass.
(a) Change 200 ml to litres.
(b) Change your answer to (a) into
pints.
Solution:
(a)
1 millilitre = 1000th of a litre.
200 millilitre = 1  200 = 0·2 litre
1000
(b)
1 litre = 1·75 pints
0·2 litre = 0·2  1·75 pints
= 0·35 pints
O34 Symmetry of Solids
Here is an example of an animated diagram
which illustrates a point in a way that saves
precious class time.
A 2-D shape can have lines of symmetry.
A 3-D object can also be symmetrical but it has
planes of symmetry.
This is a cuboid.
Tell your partner if you can spot some planes of
symmetry.
Each plane of symmetry is
like a mirror. There are 3.
H4 Using Congruence (1)
In this higher level presentation, students use
their knowledge of the conditions for congruence
and are learning to write out a formal proof.
e.g.1 Using the definition of a parallelogram, prove
that the opposite sides are equal.
Proof:
D
A
C
B
We need to
prove that
AB = DC and
AD = BC.
Draw the diagonal DB.
Tell your partner why the triangles are congruent.
e.g.1 Using the definition of a parallelogram, prove
that the opposite sides are equal.
Proof:
D
A
C
x
x
B
We need to
prove that
AB = DC and
AD = BC.
Draw the diagonal DB.
 ABD =  CDB ( alternate angles: AB DC ) (A)
 ADB =  CBD ( alternate angles: AD BC ) (A)
BD is common (S)
ABD are congruent (AAS)
Triangles
CDB
e.g.1 Using the definition of a parallelogram, prove
that the opposite sides are equal.
Proof:
D
A
C
x
x
B
We need to
prove that
AB = DC and
AD = BC.
Draw the diagonal DB.
 ABD =  CDB ( alternate angles: AB DC ) (A)
 ADB =  CBD ( alternate angles: AD BC ) (A)
BD is common (S)
ABD are congruent (AAS)
Triangles
CDB
So, AB = DC
e.g.1 Using the definition of a parallelogram, prove
that the opposite sides are equal.
Proof:
D
A
C
x
x
B
We need to
prove that
AB = DC and
AD = BC.
Draw the diagonal DB.
 ABD =  CDB ( alternate angles: AB DC ) (A)
 ADB =  CBD ( alternate angles: AD BC ) (A)
BD is common (S)
ABD are congruent (AAS)
Triangles
CDB
So, AB = DC and AD = BC.
H16 Right Angled Triangles: Sin x
The following page comes from the first of a set of
presentations on Trigonometry. It shows a typical
summary with an indication that note-taking might
be useful.
SUMMARY
 In a right angled triangle, with an angle x,
sin x = opp
hyp
where,
hyp
opp
x
•
opp. is the side opposite ( or facing ) x
•
hyp. is the hypotenuse ( always the longest
side and facing the right angle )
 The letters “sin” are always followed by an angle.
 The sine of any angle can be found from a
calculator ( check it is set in degrees )
e.g. sin 20 = 0·3420…
The next 4 slides contain a list of the 113 files that
make up Shape, Space and Measures.
The files have been labelled as follows:
F: Basic work for the Foundation level.
O: Topics that are likely to give rise to questions
graded D and C. These topics form the Overlap
between Foundation and Higher and could be
examined at either level.
H: Topics which appear only in the Higher level
content.
Overlap files appear twice in the list so that they can
easily be accessed when working at either Foundation
or Higher level.
Also for ease of access, colours have been used to group
topics. For example, dark blue is used at all 3 levels for
work on length, area and volume.
The 3 underlined titles contain links to the complete files
that are included in this sample.
Teach GCSE Maths – Foundation
F1
F2
O1
Angles
Lines: Parallel and Perpendicular
Parallel Lines and Angles
O2
Bearings
F3
Triangles and their Angles
F4
Exterior Angle of a Triangle
O3
F5
F6
Proofs of Triangle Properties
Perimeters
Area of a Rectangle
F7
F8
F9
F10
Congruent Shapes
Congruent Triangles
Constructing Triangles SSS
Constructing Triangles AAS
F11 Constructing Triangles SAS, RHS
O4
O5
More Constructions: Bisectors
More Constructions: Perpendiculars
F12 Quadrilaterals: Interior angles
F13 Quadrilaterals: Exterior angles
F14 Parallelograms
O6 Angle Proof for Parallelograms
Page 1
F15
O7
F16
O8
F17
F18
Trapezia
Allied Angles
Kites
Identifying Quadrilaterals
Tessellations
Lines of Symmetry
F19
F20
F21
F22
O9
O10
O11
Rotational Symmetry
Coordinates
Reading Scales
Scales and Maps
Mid-Point of AB
Area of a Parallelogram
Area of a Triangle
O12 Area of a Trapezium
O13 Area of a Kite
O14 More Complicated Areas
O15
O16
O17
O18
Angles of Polygons
Regular Polygons
More Tessellations
Finding Angles: Revision
continued
Teach GCSE Maths – Foundation
Page 2
F23 Metric Units
O19 Miles and Kilometres
O33 Plan and Elevation
O34 Symmetry of Solids
O20 Feet and Metres
O35 Nets of Prisms and Pyramids
O36 Volumes of Prisms
O21 Pints, Gallons and Litres
O22
O23
O24
O25
Pounds and Kilograms
Accuracy in Measurements
Speed
Density
O26 Pythagoras’ Theorem
O27 More Perimeters
O28 Length of AB
F24
O29
O30
O31
Circle words
Circumference of a Circle
Area of a Circle
Loci
O32 3-D Coordinates
F25 Volume of a Cuboid and Isometric
Drawing
F26 Nets of a Cuboid and Cylinder
O37 Dimensions
F27 Surface Area of a Cuboid
O38 Surface Area of a Prism and
Cylinder
F28 Reflections
O39 More Reflections
O40 Even More Reflections
F29 Enlargements
O41 More Enlargements
F30 Similar Shapes
O42 Effect of Enlargements
O43 Rotations
O44 Translations
O45 Mixed and Combined Transformations
continued
Teach GCSE Maths – Higher
O1
O2
O3
O4
O5
H1
O6
O7
O8
O9
O10
O11
O12
O13
O14
O15
O16
O17
O18
O19
O20
O21
Parallel Lines and Angles
Bearings
Proof of Triangle Properties
More Constructions: bisectors
More Constructions: perpendiculars
Even More Constructions
Angle Proof for Parallelograms
Allied Angles
Identifying Quadrilaterals
Mid-Point of AB
Area of a Parallelogram
Area of a Triangle
Area of a Trapezium
Area of a Kite
More Complicated Areas
Angles of Polygons
Regular Polygons
More Tessellations
Finding Angles: Revision
Miles and Kilometres
Feet and Metres
Pints, Gallons, Litres
O22
O23
O24
O25
H2
O26
O27
O28
H3
H4
H5
H6
H7
Page 3
Pounds and Kilograms
Accuracy in Measurements
Speed
Density
More Accuracy in Measurements
Pythagoras’ Theorem
More Perimeters
Length of AB
Proving Congruent Triangles
Using Congruence (1)
Using Congruence (2)
Similar Triangles; proof
Similar Triangles; finding sides
O29
O30
H8
H9
H10
Circumference of a Circle
Area of a Circle
Chords and Tangents
Angle in a Segment
Angles in a Semicircle and Cyclic
Quadrilateral
H11 Alternate Segment Theorem
O31 Loci
H12 More Loci
continued
Teach GCSE Maths – Higher
O32
O33
H13
O34
O35
O36
O37
O38
3-D Coordinates
Plan and Elevation
More Plans and Elevations
Symmetry of Solids
Nets of Prisms and Pyramids
Volumes of Prisms
Dimensions
Surface Area of a Prism and Cylinder
O39
O40
O41
O42
More Reflections
Even More Reflections
More Enlargements
Effect of Enlargements
O43 Rotations
O44 Translations
O45 Mixed and Combined
Transformations
H14 More Combined Transformations
H15 Negative Enlargements
H16 Right Angled Triangles: Sin x
H17 Inverse sines
H18 cos x and tan x
H19 Solving problems using Trig (1)
H20
H21
H22
H23
H24
H25
H26
H27
H28
H29
H30
H31
H32
H33
H34
H35
H36
H37
H38
Page 4
Solving problems using Trig (2)
The Graph of Sin x
The Graphs of Cos x and Tan x
Solving Trig Equations
The Sine Rule
The Sine Rule; Ambiguous Case
The Cosine Rule
Trig and Area of a Triangle
Arc Length and Area of Sectors
Harder Volumes
Volumes and Surface Areas of
Pyramids and Cones
Volume and Surface Area
of a Sphere
Areas of Similar Shapes and
Volumes of Similar Solids
Vectors 1
Vectors 2
Vectors 3
Right Angled Triangles in 3D
Sine and Cosine Rules in 3D
Stretching Trig Graphs
Further details of “Teach GCSE Maths” are
available from
Chartwell-Yorke Ltd
114 High Street
Belmont Village
Bolton
Lancashire
BL7 8AL
Tel: 01204811001
Fax: 01204 811008
www.chartwellyorke.co.uk/