Presentation on theme: "Shape, Space and Measures"— Presentation transcript:
1Shape, Space and Measures Teach GCSE MathsShape, Space and Measures
2The 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.
3F4 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.
4a 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 = 18057exterior anglea7548If 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
5exterior 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 = 18057exterior angle132a7548Tell your partner what size a is.Ans: a = 180 – 48= 132( angles on a straight line )
6exterior 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 = 18057exterior angle1327548What is the link between 132 and the other 2 angles of the triangle?ANS: 132 = 57 + 75, the sum of the other angles.
7F12 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.
8Exercise1. In the following, find the marked angles, giving your reasons:115a(a)60b37(b)40105c30
9a = 180 - 60 = 120 b = 360 - 120 - 115 - 37 = 88 a 115 120 ExerciseSolutions:115120a(a)60b37a = 180 - 60( angles on a straight line )= 120b = 360 - 120 - 115 - 37(angles of quadrilateral )= 88
10x = 180 - 30 = 150 c = 360 - 105 - 40 - 150 = 65 40 105 150 Exercise(b)40105150xc30Using an extra letter:x = 180 - 30( angles on a straight line )= 150c = 360 - 105 - 40 - 150( angles of quadrilateral )= 65
11F14 ParallelogramsBy 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.
12To see that the opposite sides of a parallelogram are equal, we draw a line from one corner to the opposite one.PQRSSQ is a diagonalTriangles SPQ and QRS are congruent.So, SP = QRand PQ = RS
13F19 Rotational SymmetryAnimation is used here to illustrate a new idea.
14A 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 rotationThe shape has rotational symmetry of order 6.( We don’t count the 1st position as it’s the same as the last. )
15F21 Reading ScalesAn everyday example is used here to test understanding of reading scales and the opportunity is taken to point out a common conversion formula.
16This is a copy of a car’s speedometer. 20406080100120140160180200220mphkm/hTell 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
17F26 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.
18Suppose we open a cardboard box and flatten it out. This is a netRules for nets:We must not cut across a face.We ignore any overlaps.We finish up with one piece.
19O2 BearingsThis 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.
20e. 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:PxQx
21e. 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:Px.Qx
22e. 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:P220x.QxIf you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.
23e. 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:Px40.QxIf you only have a semicircular protractor, you need tosubtract 180 from 220 and measure from south.
24e. 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:P220x.RQx
25O21 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.
261 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.20010001200 millilitre == 0·2 litre(b)1 litre = 1·75 pints0·2 litre =0·2 1·75 pints= 0·35 pints
27O34 Symmetry of SolidsHere is an example of an animated diagram which illustrates a point in a way that saves precious class time.
28Tell 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.
29H4 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.
30e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof:DBCAWe need to prove that AB = DC and AD = BC.Draw the diagonal DB.Tell your partner why the triangles are congruent.
31ABD = CDB ( alternate angles: AB DC ) (A) e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.Proof:DBCAWe need to prove that AB = DC and AD = BC.xDraw 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)ABDCDB
32ABD = CDB ( alternate angles: AB DC ) (A) e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.Proof:DCWe need to prove that AB = DC and AD = BC.xxABDraw 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)ABDCDBSo, AB = DC
33ABD = CDB ( alternate angles: AB DC ) (A) e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal.Proof:DCWe need to prove that AB = DC and AD = BC.xxABDraw 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)ABDCDBSo, AB = DC and AD = BC.
34H16 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.
350·3420… x SUMMARY In a right angled triangle, with an angle x, opp hypsin x =opphypwhere,opp. is the side opposite ( or facing ) xhyp. 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…
36The 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.
37Teach GCSE Maths – Foundation Page 1F1 AnglesF15 TrapeziaF2 Lines: Parallel and PerpendicularO7 Allied AnglesO1 Parallel Lines and AnglesF16 KitesO2 BearingsO8 Identifying QuadrilateralsF3 Triangles and their AnglesF17 TessellationsF4 Exterior Angle of a TriangleF18 Lines of SymmetryO3 Proofs of Triangle PropertiesF19 Rotational SymmetryF20 CoordinatesF5 PerimetersF21 Reading ScalesF6 Area of a RectangleF22 Scales and MapsF7 Congruent ShapesO9 Mid-Point of ABF8 Congruent TrianglesO10 Area of a ParallelogramF9 Constructing Triangles SSSO11 Area of a TriangleF10 Constructing Triangles AASO12 Area of a TrapeziumF11 Constructing Triangles SAS, RHSO13 Area of a KiteO4 More Constructions: BisectorsO14 More Complicated AreasO5 More Constructions: PerpendicularsO15 Angles of PolygonsF12 Quadrilaterals: Interior anglesO16 Regular PolygonsF13 Quadrilaterals: Exterior anglesO17 More TessellationsF14 ParallelogramsO18 Finding Angles: RevisionO6 Angle Proof for Parallelogramscontinued
38Teach GCSE Maths – Foundation Page 2F23 Metric UnitsO33 Plan and ElevationO19 Miles and KilometresO34 Symmetry of SolidsO20 Feet and MetresO35 Nets of Prisms and PyramidsO21 Pints, Gallons and LitresO36 Volumes of PrismsO22 Pounds and KilogramsO37 DimensionsO23 Accuracy in MeasurementsF27 Surface Area of a CuboidO24 SpeedO38 Surface Area of a Prism and CylinderO25 DensityF28 ReflectionsO26 Pythagoras’ TheoremO39 More ReflectionsO27 More PerimetersO28 Length of ABO40 Even More ReflectionsF24 Circle wordsF29 EnlargementsO29 Circumference of a CircleO41 More EnlargementsO30 Area of a CircleF30 Similar ShapesO31 LociO42 Effect of EnlargementsO32 3-D CoordinatesO43 RotationsF25 Volume of a Cuboid and Isometric DrawingO44 TranslationsO45 Mixed and Combined TransformationsF26 Nets of a Cuboid and Cylindercontinued
39Teach GCSE Maths – Higher Page 3O1 Parallel Lines and AnglesO22 Pounds and KilogramsO2 BearingsO23 Accuracy in MeasurementsO3 Proof of Triangle PropertiesO24 SpeedO4 More Constructions: bisectorsO25 DensityO5 More Constructions: perpendicularsH2 More Accuracy in MeasurementsH1 Even More ConstructionsO26 Pythagoras’ TheoremO6 Angle Proof for ParallelogramsO27 More PerimetersO7 Allied AnglesO28 Length of ABO8 Identifying QuadrilateralsH3 Proving Congruent TrianglesO9 Mid-Point of ABH4 Using Congruence (1)O10 Area of a ParallelogramH5 Using Congruence (2)O11 Area of a TriangleH6 Similar Triangles; proofO12 Area of a TrapeziumH7 Similar Triangles; finding sidesO13 Area of a KiteO29 Circumference of a CircleO14 More Complicated AreasO30 Area of a CircleO15 Angles of PolygonsH8 Chords and TangentsO16 Regular PolygonsH9 Angle in a SegmentO17 More TessellationsH10 Angles in a Semicircle and Cyclic QuadrilateralO18 Finding Angles: RevisionO19 Miles and KilometresH11 Alternate Segment TheoremO20 Feet and MetresO31 LociO21 Pints, Gallons, LitresH12 More Locicontinued
40Teach GCSE Maths – Higher Page 4O32 3-D CoordinatesH20 Solving problems using Trig (2)O33 Plan and ElevationH21 The Graph of Sin xH13 More Plans and ElevationsH22 The Graphs of Cos x and Tan xO34 Symmetry of SolidsH23 Solving Trig EquationsO35 Nets of Prisms and PyramidsH24 The Sine RuleO36 Volumes of PrismsH25 The Sine Rule; Ambiguous CaseO37 DimensionsH26 The Cosine RuleO38 Surface Area of a Prism and CylinderH27 Trig and Area of a TriangleO39 More ReflectionsH28 Arc Length and Area of SectorsO40 Even More ReflectionsH29 Harder VolumesO41 More EnlargementsH30 Volumes and Surface Areas of Pyramids and ConesO42 Effect of EnlargementsO43 RotationsH31 Volume and Surface Area of a SphereO44 TranslationsO45 Mixed and Combined TransformationsH32 Areas of Similar Shapes and Volumes of Similar SolidsH14 More Combined TransformationsH33 Vectors 1H15 Negative EnlargementsH34 Vectors 2H16 Right Angled Triangles: Sin xH35 Vectors 3H17 Inverse sinesH36 Right Angled Triangles in 3DH18 cos x and tan xH37 Sine and Cosine Rules in 3DH19 Solving problems using Trig (1)H38 Stretching Trig Graphs
41Further details of “Teach GCSE Maths” are available from Chartwell-Yorke Ltd114 High StreetBelmont VillageBoltonLancashireBL7 8ALTel: Fax: