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Teach GCSE Maths

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.

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Teach GCSE Maths

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  1. Teach GCSE Maths Shape, Space and Measures

  2. 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.

  3. 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.

  4. we form an angle with the side next to it ( the adjacent side ) 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 . . . a is called an exterior angle of the triangle

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

  6. 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.

  7. 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.

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

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

  10. 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

  11. 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.

  12. 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

  13. F19 Rotational Symmetry Animation is used here to illustrate a new idea.

  14. A B F C E The centre of rotation D This “snowflake” has 6 identical branches. When it makes a complete turn, the shape fits onto itself 6 times. The shape has rotational symmetry of order 6. ( We don’t count the 1st position as it’s the same as the last. )

  15. 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.

  16. mph 80 60 120 100 40 100 140 80 km/h 160 60 180 40 20 120 200 20 220 0 140 0 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 . . . but as / in kilometres per hour. Ans: 5mph on the outer scale and 4km/h on the inner. Can you see what the conversion factor is between miles and kilometres? Ans: e.g. 160km = 100miles. Dividing by 20 gives 8km = 5miles

  17. 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.

  18. 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.

  19. 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.

  20. e.g. The bearing of R from P is 220 and R is duewest of Q. Mark the position of R on the diagram. Solution: P x Q x

  21. e.g. The bearing of R from P is 220 and R is duewest of Q. Mark the position of R on the diagram. Solution: P x . Q x

  22. 220 e.g. The bearing of R from P is 220 and R is duewest of Q. Mark the position of R on the diagram. Solution: P x . Q x If you only have a semicircular protractor, you need to subtract 180 from 220 and measure from south.

  23. e.g. The bearing of R from P is 220 and R is duewest 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.

  24. 220 e.g. The bearing of R from P is 220 and R is duewest of Q. Mark the position of R on the diagram. Solution: P x . R Q x

  25. 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.

  26. 1millilitre =1000th of a litre. 1  200 1000 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) 200millilitre = = 0·2 litre 1litre = 1·75 pints (b) 0·2 litre = 0·2  1·75 pints = 0·35 pints

  27. O34 Symmetryof Solids Here is an example of an animated diagram which illustrates a point in a way that saves precious class time.

  28. 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.

  29. 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.

  30. D C A B 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.

  31. D C x x A B   ABD=CDB ( alternate angles: AB DC ) (A)   ADB=CBD ( alternate angles: AD BC ) (A) BD is common(S) ABD Triangles are congruent (AAS) CDB 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.

  32.  ABD=CDB ( alternate angles: AB DC ) (A)   ADB=CBD ( alternate angles: AD BC ) (A) ABD Triangles are congruent (AAS) CDB 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. D C x x A B Draw the diagonal DB. BD is common(S) So, AB = DC

  33.  ABD=CDB ( alternate angles: AB DC ) (A)   ADB=CBD ( alternate angles: AD BC ) (A) ABD Triangles are congruent (AAS) CDB 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. D C x x A B Draw the diagonal DB. BD is common(S) So, AB = DC and AD = BC.

  34. H16 Right Angled Triangles: Sinx 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.

  35. opp sinx = hyp hyp opp x SUMMARY • In a right angled triangle, with an angle x, 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. sin20 = 0·3420…

  36. 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.

  37. Page 1 Teach GCSE Maths – Foundation F1 Angles F15 Trapezia F2 Lines: Parallel and Perpendicular O7 Allied Angles O1 Parallel Lines and Angles F16 Kites O8 Identifying Quadrilaterals O2 Bearings F17 Tessellations F3 Triangles and their Angles F18 Lines of Symmetry F4 Exterior Angle of a Triangle F19 Rotational Symmetry O3 Proofs of Triangle Properties 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

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

  39. Page 3 Teach GCSE Maths – Higher 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 H2 More Accuracy in Measurements O5 More Constructions: perpendiculars 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

  40. Page 4 Teach GCSE Maths – Higher O32 3-D Coordinates H20 Solving problems using Trig (2) O33 Plan and Elevation H21 The Graph of Sinx H13 More Plans and Elevations H22 The Graphs of Cosx and Tanx 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 H32 Areas of Similar Shapes and Volumes of Similar Solids O45 Mixed and Combined Transformations H33 Vectors 1 H14 More Combined Transformations H34 Vectors 2 H15 Negative Enlargements H35 Vectors 3 H16 Right Angled Triangles: Sinx H36 Right Angled Triangles in 3D H17 Inverse sines H18 cosx and tanx H37 Sine and Cosine Rules in 3D H38 Stretching Trig Graphs H19 Solving problems using Trig (1)

  41. 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/

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