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Teach GCSE Maths Shape, Space and Measures

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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.

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F 4 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.

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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 angle 75 57 48

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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

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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

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F 12 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.

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Exercise 1.In the following, find the marked angles, giving your reasons: 60 115 37 a b (a) (b) 105 30 40 c

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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) 60 120

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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 ) c x

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F 14 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.

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P Q R S 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

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F 19 Rotational Symmetry Animation is used here to illustrate a new idea.

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This snowflake has 6 identical branches. When it makes a complete turn, the shape fits onto itself 6 times. ( We dont count the 1 st position as its the same as the last. ) The shape has rotational symmetry of order 6. A B E D C F The centre of rotation

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F 21 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.

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This is a copy of a cars 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

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F 26 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.

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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

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O 2 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.

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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 Q x Solution:

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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.. Q x Solution:

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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: Q x If you only have a semicircular protractor, you need to subtract 180 from 220 and measure from south.

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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: Q x If you only have a semicircular protractor, you need to subtract 180 from 220 and measure from south. 40.

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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. Q x R Solution:

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O 21 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.

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e.g.The photo shows a milk bottle and some milk poured into a glass. 1 millilitre = 1000 th 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) 200 1000 1

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O 34 Symmetry of Solids Here is an example of an animated diagram which illustrates a point in a way that saves precious class time.

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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.

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H 4 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.

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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.

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D B C A e.g.1Using 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) ABD CDB We need to prove that AB = DC and AD = BC.

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D B C A e.g.1Using 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) ABD CDB So, AB = DC We need to prove that AB = DC and AD = BC.

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D B C A e.g.1Using 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) ABDABD CDBCDB So, AB = DC and AD = BC. We need to prove that AB = DC and AD = BC.

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H 16 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.

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SUMMARY In a right angled triangle, with an angle x, where, sin x = opp hyp 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.

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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.

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F1Angles F3Triangles and their Angles F4Exterior Angle of a Triangle F7Congruent Shapes F8Congruent Triangles F12Quadrilaterals: Interior angles F13Quadrilaterals: Exterior angles F15Trapezia F14Parallelograms F16Kites F5Perimeters F6Area of a Rectangle F17Tessellations F2Lines: Parallel and Perpendicular O1Parallel Lines and Angles O10Area of a Parallelogram O11Area of a Triangle O12Area of a Trapezium O13Area of a Kite O14More Complicated Areas O2Bearings O3Proofs of Triangle Properties O7Allied Angles O8Identifying Quadrilaterals O15Angles of Polygons O16Regular Polygons O6Angle Proof for Parallelograms Teach GCSE Maths – Foundation F18Lines of Symmetry F19Rotational Symmetry continued F20Coordinates F21Reading Scales F22Scales and Maps O9Mid-Point of AB F9Constructing Triangles SSS F11Constructing Triangles SAS, RHS F10Constructing Triangles AAS O4More Constructions: Bisectors O5More Constructions: Perpendiculars O17More Tessellations O18Finding Angles: Revision Page 1

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Teach GCSE Maths – Foundation F28Reflections O38Surface Area of a Prism and Cylinder O34Symmetry of Solids O33Plan and Elevation O35Nets of Prisms and Pyramids O36Volumes of Prisms O37Dimensions O323-D Coordinates O39More Reflections O44Translations O41More Enlargements O43Rotations O45Mixed and Combined Transformations O42Effect of Enlargements O40Even More Reflections F27Surface Area of a Cuboid O24Speed O25Density O31Loci O23Accuracy in Measurements F24Circle words O29Circumference of a Circle O30Area of a Circle F29Enlargements F30Similar Shapes O27More Perimeters F25Volume of a Cuboid and Isometric Drawing F26Nets of a Cuboid and Cylinder O26Pythagoras Theorem O21Pints, Gallons and Litres O22Pounds and Kilograms O28Length of AB F23Metric Units O19Miles and Kilometres O20Feet and Metres continued Page 2

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O1Parallel Lines and Angles O10Area of a Parallelogram O11Area of a Triangle O12Area of a Trapezium O13Area of a Kite O14More Complicated Areas O2Bearings O3Proof of Triangle Properties O7Allied Angles O8Identifying Quadrilaterals O26Pythagoras Theorem O27More Perimeters O15Angles of Polygons O16Regular Polygons O17More Tessellations O6Angle Proof for Parallelograms Teach GCSE Maths – Higher O19Miles and Kilometres O20Feet and Metres O21Pints, Gallons, Litres O22Pounds and Kilograms O9Mid-Point of AB O28Length of AB H2More Accuracy in Measurements O24Speed O23Accuracy in Measurements O25Density H1Even More Constructions O4More Constructions: bisectors O5More Constructions: perpendiculars H8Chords and Tangents H10Angles in a Semicircle and Cyclic Quadrilateral H11Alternate Segment Theorem H9Angle in a Segment O29Circumference of a Circle O30Area of a Circle H3Proving Congruent Triangles H4Using Congruence (1) H5Using Congruence (2) H6Similar Triangles; proof H7Similar Triangles; finding sides O18Finding Angles: Revision continued Page 3 H12More Loci O31Loci

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Teach GCSE Maths – Higher H16Right Angled Triangles: Sin x H17Inverse sines H18cos x and tan x H19Solving problems using Trig (1) H20Solving problems using Trig (2) H21The Graph of Sin x H22The Graphs of Cos x and Tan x H24The Sine Rule H26The Cosine Rule H27Trig and Area of a Triangle H25The Sine Rule; Ambiguous Case H33Vectors 1 H34Vectors 2 H35Vectors 3 H36Right Angled Triangles in 3D H37Sine and Cosine Rules in 3D H38Stretching Trig Graphs H14More Combined Transformations H15Negative Enlargements O39More Reflections H29Harder Volumes H30Volumes and Surface Areas of Pyramids and Cones H31Volume and Surface Area of a Sphere H32Areas of Similar Shapes and Volumes of Similar Solids O34Symmetry of Solids O36Volumes of Prisms O37Dimensions O38Surface Area of a Prism and Cylinder O44Translations O41More Enlargements O43Rotations O45Mixed and Combined Transformations O42Effect of Enlargements O40Even More Reflections H13More Plans and Elevations O323-D Coordinates O33Plan and Elevation H23Solving Trig Equations O35Nets of Prisms and Pyramids Page 4 H28Arc Length and Area of Sectors

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Further details of Teach GCSE Maths are available from www.chartwellyorke.co.uk/ Chartwell-Yorke Ltd 114 High Street Belmont Village Bolton Lancashire BL7 8AL Tel: 01204811001 Fax: 01204 811008

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