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.

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.

A Microsoft WORD file, giving more information, is included in the folder.

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.

57 + 75 + 48 = 180

If we extend one side . . .

a

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.

exterior angle75

57

48

a

We already know that the sum of the angles of any triangle is 180.e.g.

Ans: a 180 – 48= 132 ( angles on a straight

line )

exterior angle

57 + 75 + 48 = 180

75

57

48

Tell your partner what size a is.

132

75

57

48 132

We already know that the sum of the angles of any triangle is 180.e.g.

What is the link between 132 and the other 2 angles of the triangle?

ANS: 132 = 57 + 75, the sum of the other angles.

exterior angle

57 + 75 + 48 = 180

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:

60115

37

a

b

(a)

(b) 105

30

40

c

Exercise

Solutions:

a = 180 60 ( angles on a straight line )

b = 360 120 115 37

= 88(angles of quadrilateral )

= 120

115

37

a

b

(a)

60120

150

Exercise

Using an extra letter:

x = 180 30= 150

(b) 105

30

40

( angles on a straight line )

c = 360 105 40 150

= 65( angles of quadrilateral )

cx

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.

P Q

RS

To see that the opposite sides of a parallelogram are equal, we draw a line from one corner to the opposite one.

SQ is a diagonal

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.

When it makes a complete turn, the shape fits onto itself 6 times.

( We don’t count the 1st position as it’s the same as the last. )

The shape has rotational symmetry of order 6.

A

B

E

D

C

F

The centre of rotation

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.

0

20

40

60 80

100

120

1400

20

60

40

80100 120

140160

180

200

220

mph

km/h

This is a copy of a car’s speedometer.Tell your partner what 1 division measures on each scale.

It is common to find the “per” written as p in miles per hour . . .

Ans: 5 mph on the outer scale and 4 km/h on the inner.

Can you see what the conversion factor is between miles and kilometres? Ans: e.g. 160 km = 100

miles.

but as / in kilometres per hour.

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.

Rules for nets:

We finish up with one piece.

We ignore any overlaps.

We must not cut across a face.

This is a net

O2 Bearing

sThis 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.

P x

Qx

Solution:

P x

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.

.

Qx

Solution:

P x

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.

220.

Solution:

Qx

If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from

south.

P x

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:

Qx

If you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from

south.

40.

P x

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.

220

.

QxR

Solution:

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.

1 millilitre = 1000th of a litre.

1 litre = 1·75 pints

There is 200 ml of milk in the glass.

Solution:

(a) Change 200 ml to litres.(b) Change your answer to (a)

into pints.

200 millilitre = = 0·2 litre

0·2 litre =

0·2 1·75 pints= 0·35

pints

(a)

(b)

20010001

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 3-D object can also be symmetrical but it has planes of symmetry.

This is a cuboid.

Each plane of symmetry is like a mirror. There are 3.

A 2-D shape can have lines of symmetry.

Tell your partner if you can spot some planes of symmetry.

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.

D

B

C

A

e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.

Proof: We need to

prove that AB = DC and AD = BC.

Draw the diagonal DB.

Tell your partner why the triangles are congruent.

D

B

C

A

e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.

Proof:

Draw the diagonal DB.ABD = CDB ( alternate angles: AB DC )

(A)

ADB = CBD ( alternate angles: AD BC ) (A)

BD is common (S)

x

x

Triangles are congruent (AAS)

ABDCDB

We need to prove that AB = DC and AD = BC.

D

B

C

A

e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.

Proof:

Draw the diagonal DB.ABD = CDB ( alternate angles: AB DC )

(A)

ADB = CBD ( alternate angles: AD BC ) (A)

x

BD is common (S)

x

Triangles are congruent (AAS)

ABDCDB

So, AB = DC

We need to prove that AB = DC and AD = BC.

D

B

C

A

e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.

Proof:

Draw the diagonal DB.ABD = CDB ( alternate angles: AB DC )

(A)

ADB = CBD ( alternate angles: AD BC ) (A)

x

BD is common (S)

x

Triangles are congruent (AAS)

ABDCDB

So, AB = DC and AD = BC.

We need to prove that 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,

where,

sin x = opphyp

• opp. is the side opposite ( or facing ) x• hyp. is the hypotenuse ( always the longest side and facing the right angle )

x

opp

hyp

The sine of any angle can be found from a calculator ( check it is set in degrees )e.g. sin 20

= 0·3420…

The letters “sin” are always followed by an angle.

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.

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.

Overlap files appear twice in the list so that they can easily be accessed when working at either Foundation or Higher level.

The 3 underlined titles contain links to the complete files that are included in this sample.

F1Angles

F3 Triangles and their AnglesF4 Exterior Angle of a Triangle

F7 Congruent ShapesF8 Congruent Triangles

F12 Quadrilaterals: Interior anglesF13 Quadrilaterals: Exterior angles

F15Trapezia

F14Parallelograms

F16 Kites

F5Perimeters

F6 Area of a Rectangle

F17Tessellations

F2 Lines: Parallel and PerpendicularO1 Parallel Lines and Angles

O10 Area of a ParallelogramO11 Area of a TriangleO12 Area of a TrapeziumO13 Area of a KiteO14 More Complicated Areas

O2Bearings

O3 Proofs of Triangle Properties

O7 Allied Angles

O8 Identifying Quadrilaterals

O15 Angles of PolygonsO16 Regular Polygons

O6 Angle Proof for Parallelograms

Teach GCSE Maths – Foundation

F18 Lines of SymmetryF19 Rotational Symmetry

continued

F20Coordinates

F21 Reading ScalesF22 Scales and MapsO9 Mid-Point of AB

F9 Constructing Triangles SSS

F11 Constructing Triangles SAS, RHS

F10 Constructing Triangles AAS

O4 More Constructions: BisectorsO5 More Constructions: Perpendiculars

O17 More TessellationsO18 Finding Angles: Revision

Page 1

Teach GCSE Maths – Foundation

F28Reflections

O38 Surface Area of a Prism and Cylinder

O34 Symmetry of Solids

O33 Plan and Elevation

O35 Nets of Prisms and PyramidsO36 Volumes of PrismsO37Dimensions

O32 3-D Coordinates

O39 More Reflections

O44Translations

O41 More Enlargements

O43Rotations

O45 Mixed and Combined Transformations

O42 Effect of Enlargements

O40 Even More Reflections

F27 Surface Area of a CuboidO24 Speed

O25Density

O31Loci

O23 Accuracy in Measurements

F24 Circle wordsO29 Circumference of a

CircleO30 Area of a Circle

F29Enlargements

F30 Similar Shapes

O27 More Perimeters

F25 Volume of a Cuboid and Isometric Drawing

F26 Nets of a Cuboid and Cylinder

O26 Pythagoras’ Theorem

O21 Pints, Gallons and LitresO22 Pounds and Kilograms

O28 Length of AB

F23 Metric UnitsO19 Miles and KilometresO20 Feet and Metres

continued

Page 2

O1 Parallel Lines and Angles

O10 Area of a ParallelogramO11 Area of a TriangleO12 Area of a TrapeziumO13 Area of a Kite

O14 More Complicated Areas

O2Bearings

O3 Proof of Triangle Properties

O7 Allied AnglesO8 Identifying Quadrilaterals

O26 Pythagoras’ TheoremO27 More Perimeters

O15 Angles of PolygonsO16 Regular PolygonsO17 More Tessellations

O6 Angle Proof for Parallelograms

Teach GCSE Maths – Higher

O19 Miles and KilometresO20 Feet and MetresO21 Pints, Gallons, Litres

O22 Pounds and Kilograms

O9 Mid-Point of AB

O28 Length of AB

H2 More Accuracy in Measurements

O24Speed

O23 Accuracy in Measurements

O25DensityH1 Even More

Constructions

O4 More Constructions: bisectorsO5 More Constructions: perpendiculars

H8 Chords and Tangents

H10 Angles in a Semicircle and Cyclic Quadrilateral

H11 Alternate Segment Theorem

H9 Angle in a Segment

O29 Circumference of a CircleO30 Area of a Circle

H3 Proving Congruent TrianglesH4 Using Congruence (1)H5 Using Congruence (2)H6 Similar Triangles; proofH7 Similar Triangles; finding sides

O18 Finding Angles: Revision

continued

Page 3

H12 More LociO31

Loci

Teach GCSE Maths – Higher

H16 Right Angled Triangles: Sin xH17 Inverse sinesH18 cos x and tan

xH19 Solving problems using Trig (1)

H20 Solving problems using Trig (2)H21 The Graph of Sin

xH22 The Graphs of Cos x and Tan

xH24 The Sine

RuleH26 The Cosine

RuleH27 Trig and Area of a Triangle

H25 The Sine Rule; Ambiguous Case

H33 Vectors 1H34 Vectors 2H35 Vectors 3

H36 Right Angled Triangles in 3DH37 Sine and Cosine Rules in 3DH38 Stretching Trig Graphs

H14 More Combined TransformationsH15 Negative Enlargements

O39 More Reflections H29 Harder

VolumesH30 Volumes and Surface Areas of Pyramids and Cones

H31 Volume and Surface Area of a Sphere

H32 Areas of Similar Shapes and Volumes of Similar Solids

O34 Symmetry of Solids

O36 Volumes of PrismsO37Dimensions

O38 Surface Area of a Prism and Cylinder

O44Translations

O41 More Enlargements

O43RotationsO45 Mixed and Combined Transformations

O42 Effect of Enlargements

O40 Even More Reflections

H13 More Plans and Elevations

O32 3-D CoordinatesO33 Plan and Elevation

H23 Solving Trig EquationsO35 Nets of Prisms and

Pyramids

Page 4

H28 Arc Length and Area of Sectors

Further details of “Teach GCSE Maths” are available from

www.chartwellyorke.co.uk/

Chartwell-Yorke Ltd114 High StreetBelmont VillageBoltonLancashireBL7 8AL

Tel: 01204811001 Fax: 01204 811008