# Shape, Space and Measures

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Shape, Space and Measures
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.

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 + 75 + 48 = 180 57 exterior angle a 75 48 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

exterior angle 57 + 75 + 48 = 180 a Ans: a = 180 – 48 = 132 57
We already know that the sum of the angles of any triangle is 180. e.g. 57 + 75 + 48 = 180 57 exterior angle 132 a 75 48 Tell your partner what size a is. Ans: a = 180 – 48 = 132 ( angles on a straight line )

exterior angle 57 + 75 + 48 = 180 57 132 75 48
We already know that the sum of the angles of any triangle is 180. e.g. 57 + 75 + 48 = 180 57 exterior angle 132 75 48 What is the link between 132 and the other 2 angles of the triangle? ANS: 132 = 57 + 75, the sum of the other 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: 115 a (a) 60 b 37 (b) 40 105 c 30

a = 180 - 60 = 120 b = 360 - 120 - 115 - 37 = 88 a 115 120
Exercise Solutions: 115 120 a (a) 60 b 37 a = 180 - 60 ( angles on a straight line ) = 120 b = 360 - 120 - 115 - 37 (angles of quadrilateral ) = 88

x = 180 - 30 = 150 c = 360 - 105 - 40 - 150 = 65 40 105 150
Exercise (b) 40 105 150 x c 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. P Q R S 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.

A B F C E D This “snowflake” has 6 identical branches.
When it makes a complete turn, the shape fits onto itself 6 times. The centre of rotation 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.
20 40 60 80 100 120 140 160 180 200 220 mph km/h Tell your partner what 1 division measures on each scale. It is common to find 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. 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
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 Q x

e. g. The bearing of R from P is 220 and R is due west of 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 . Q x

e. g. The bearing of R from P is 220 and R is due west of 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 220 x . Q x 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
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 40 . Q x 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
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 220 x . R Q x

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.

1 millilitre = 1000th of a litre.
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 1000 1 200 millilitre = = 0·2 litre (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.

Tell your partner if you can spot some planes of symmetry.
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 B C A We need to prove that AB = DC and AD = BC. Draw the diagonal DB. Tell your partner why the triangles are congruent.

ABD = CDB ( alternate angles: AB DC ) (A)
e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D B C A We need to prove that AB = DC and AD = BC. x Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) Triangles are congruent (AAS) ABD CDB

ABD = CDB ( alternate angles: AB DC ) (A)
e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D C We need to prove that AB = DC and AD = BC. x x A B Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) Triangles are congruent (AAS) ABD CDB So, AB = DC

ABD = CDB ( alternate angles: AB DC ) (A)
e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D C We need to prove that AB = DC and AD = BC. x x A B Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) Triangles are congruent (AAS) ABD 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.

0·3420… x SUMMARY In a right angled triangle, with an angle x, opp
hyp sin x = opp hyp where, 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
Page 1 F1 Angles F15 Trapezia F2 Lines: Parallel and Perpendicular O7 Allied Angles O1 Parallel Lines and Angles F16 Kites O2 Bearings O8 Identifying Quadrilaterals F3 Triangles and their Angles F17 Tessellations F4 Exterior Angle of a Triangle F18 Lines of Symmetry O3 Proofs of Triangle Properties F19 Rotational Symmetry F20 Coordinates F5 Perimeters F21 Reading Scales F6 Area of a Rectangle F22 Scales and Maps F7 Congruent Shapes O9 Mid-Point of AB F8 Congruent Triangles O10 Area of a Parallelogram F9 Constructing Triangles SSS O11 Area of a Triangle F10 Constructing Triangles AAS O12 Area of a Trapezium F11 Constructing Triangles SAS, RHS O13 Area of a Kite O4 More Constructions: Bisectors O14 More Complicated Areas O5 More Constructions: Perpendiculars O15 Angles of Polygons F12 Quadrilaterals: Interior angles O16 Regular Polygons F13 Quadrilaterals: Exterior angles O17 More Tessellations F14 Parallelograms O18 Finding Angles: Revision O6 Angle Proof for Parallelograms continued

Teach GCSE Maths – Foundation
Page 2 F23 Metric Units O33 Plan and Elevation O19 Miles and Kilometres O34 Symmetry of Solids O20 Feet and Metres O35 Nets of Prisms and Pyramids O21 Pints, Gallons and Litres O36 Volumes of Prisms O22 Pounds and Kilograms O37 Dimensions O23 Accuracy in Measurements F27 Surface Area of a Cuboid O24 Speed O38 Surface Area of a Prism and Cylinder O25 Density F28 Reflections O26 Pythagoras’ Theorem O39 More Reflections O27 More Perimeters O28 Length of AB O40 Even More Reflections F24 Circle words F29 Enlargements O29 Circumference of a Circle O41 More Enlargements O30 Area of a Circle F30 Similar Shapes O31 Loci O42 Effect of Enlargements O32 3-D Coordinates O43 Rotations F25 Volume of a Cuboid and Isometric Drawing O44 Translations O45 Mixed and Combined Transformations F26 Nets of a Cuboid and Cylinder continued

Teach GCSE Maths – Higher
Page 3 O1 Parallel Lines and Angles O22 Pounds and Kilograms O2 Bearings O23 Accuracy in Measurements O3 Proof of Triangle Properties O24 Speed O4 More Constructions: bisectors O25 Density O5 More Constructions: perpendiculars H2 More Accuracy in Measurements H1 Even More Constructions O26 Pythagoras’ Theorem O6 Angle Proof for Parallelograms O27 More Perimeters O7 Allied Angles O28 Length of AB O8 Identifying Quadrilaterals H3 Proving Congruent Triangles O9 Mid-Point of AB H4 Using Congruence (1) O10 Area of a Parallelogram H5 Using Congruence (2) O11 Area of a Triangle H6 Similar Triangles; proof O12 Area of a Trapezium H7 Similar Triangles; finding sides O13 Area of a Kite O29 Circumference of a Circle O14 More Complicated Areas O30 Area of a Circle O15 Angles of Polygons H8 Chords and Tangents O16 Regular Polygons H9 Angle in a Segment O17 More Tessellations H10 Angles in a Semicircle and Cyclic Quadrilateral O18 Finding Angles: Revision O19 Miles and Kilometres H11 Alternate Segment Theorem O20 Feet and Metres O31 Loci O21 Pints, Gallons, Litres H12 More Loci continued

Teach GCSE Maths – Higher
Page 4 O32 3-D Coordinates H20 Solving problems using Trig (2) O33 Plan and Elevation H21 The Graph of Sin x H13 More Plans and Elevations H22 The Graphs of Cos x and Tan x O34 Symmetry of Solids H23 Solving Trig Equations O35 Nets of Prisms and Pyramids H24 The Sine Rule O36 Volumes of Prisms H25 The Sine Rule; Ambiguous Case O37 Dimensions H26 The Cosine Rule O38 Surface Area of a Prism and Cylinder H27 Trig and Area of a Triangle O39 More Reflections H28 Arc Length and Area of Sectors O40 Even More Reflections H29 Harder Volumes O41 More Enlargements H30 Volumes and Surface Areas of Pyramids and Cones O42 Effect of Enlargements O43 Rotations H31 Volume and Surface Area of a Sphere O44 Translations O45 Mixed and Combined Transformations H32 Areas of Similar Shapes and Volumes of Similar Solids H14 More Combined Transformations H33 Vectors 1 H15 Negative Enlargements H34 Vectors 2 H16 Right Angled Triangles: Sin x H35 Vectors 3 H17 Inverse sines H36 Right Angled Triangles in 3D H18 cos x and tan x H37 Sine and Cosine Rules in 3D H19 Solving problems using Trig (1) H38 Stretching Trig Graphs

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