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Transcript
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. We already know that the sum of the angles of any triangle is 180. e.g. 57 57 + 75 + 48 = 180 exterior angle 75 48 a 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 We already know that the sum of the angles of any triangle is 180. e.g. 57 57 + 75 + 48 = 180 exterior angle 75 a 48 132 Tell your partner what size a is. Ans: a = 180 – 48 = 132 ( angles on a straight line ) We already know that the sum of the angles of any triangle is 180. e.g. 57 57 + 75 + 48 = 180 exterior angle 75 48 132 What is the link between 132 and the other 2 angles of the triangle? ANS: 132 = 57 + 75, the sum of the other angles. 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: (a) a 60 115 b 37 (b) 40 105 c 30 Exercise Solutions: (a) 115 a 60 120 b 37 a = 180 - 60 ( angles on a straight line ) = 120 b = 360 - 120 - 115 - 37 = 88 (angles of quadrilateral ) Exercise (b) 40 105 c 150 x 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. S R SQ is a diagonal Q P 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. A When it makes a complete turn, the shape fits onto itself 6 times. The centre of rotation F B E C D 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. Tell your partner what 1 division measures on It is common to find each scale. 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. 60 40 20 80 100 120 100 80 km/h 140 160 60 40 180 20 200 0 0 mph 120 220 140 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. Mark the position of R on the diagram. Solution: Px x 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 . x 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 220 . x Q 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. Mark the position of R on the diagram. Solution: Px . 40 x Q 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. Mark the position of R on the diagram. Solution: P x 220 . R x Q 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. 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 millilitre = 1 200 = 0·2 litre 1000 (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. 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 A C B We need to prove that AB = DC and AD = BC. Draw the diagonal DB. Tell your partner why the triangles are congruent. e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D A C x x B We need to prove that AB = DC and AD = BC. Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) ABD are congruent (AAS) Triangles CDB e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D A C x x B We need to prove that AB = DC and AD = BC. Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) ABD are congruent (AAS) Triangles CDB So, AB = DC e.g.1 Using the definition of a parallelogram, prove that the opposite sides are equal. Proof: D A C x x B We need to prove that AB = DC and AD = BC. Draw the diagonal DB. ABD = CDB ( alternate angles: AB DC ) (A) ADB = CBD ( alternate angles: AD BC ) (A) BD is common (S) ABD are congruent (AAS) Triangles 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. SUMMARY In a right angled triangle, with an angle x, sin x = opp hyp where, hyp opp x • 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 F1 F2 O1 Angles Lines: Parallel and Perpendicular Parallel Lines and Angles O2 Bearings F3 Triangles and their Angles F4 Exterior Angle of a Triangle O3 F5 F6 Proofs of Triangle Properties Perimeters Area of a Rectangle F7 F8 F9 F10 Congruent Shapes Congruent Triangles Constructing Triangles SSS Constructing Triangles AAS F11 Constructing Triangles SAS, RHS O4 O5 More Constructions: Bisectors More Constructions: Perpendiculars F12 Quadrilaterals: Interior angles F13 Quadrilaterals: Exterior angles F14 Parallelograms O6 Angle Proof for Parallelograms Page 1 F15 O7 F16 O8 F17 F18 Trapezia Allied Angles Kites Identifying Quadrilaterals Tessellations Lines of Symmetry F19 F20 F21 F22 O9 O10 O11 Rotational Symmetry Coordinates Reading Scales Scales and Maps Mid-Point of AB Area of a Parallelogram Area of a Triangle O12 Area of a Trapezium O13 Area of a Kite O14 More Complicated Areas O15 O16 O17 O18 Angles of Polygons Regular Polygons More Tessellations Finding Angles: Revision continued Teach GCSE Maths – Foundation Page 2 F23 Metric Units O19 Miles and Kilometres O33 Plan and Elevation O34 Symmetry of Solids O20 Feet and Metres O35 Nets of Prisms and Pyramids O36 Volumes of Prisms O21 Pints, Gallons and Litres O22 O23 O24 O25 Pounds and Kilograms Accuracy in Measurements Speed Density O26 Pythagoras’ Theorem O27 More Perimeters O28 Length of AB F24 O29 O30 O31 Circle words Circumference of a Circle Area of a Circle Loci O32 3-D Coordinates F25 Volume of a Cuboid and Isometric Drawing F26 Nets of a Cuboid and Cylinder O37 Dimensions F27 Surface Area of a Cuboid O38 Surface Area of a Prism and Cylinder F28 Reflections O39 More Reflections O40 Even More Reflections F29 Enlargements O41 More Enlargements F30 Similar Shapes O42 Effect of Enlargements O43 Rotations O44 Translations O45 Mixed and Combined Transformations continued Teach GCSE Maths – Higher O1 O2 O3 O4 O5 H1 O6 O7 O8 O9 O10 O11 O12 O13 O14 O15 O16 O17 O18 O19 O20 O21 Parallel Lines and Angles Bearings Proof of Triangle Properties More Constructions: bisectors More Constructions: perpendiculars Even More Constructions Angle Proof for Parallelograms Allied Angles Identifying Quadrilaterals Mid-Point of AB Area of a Parallelogram Area of a Triangle Area of a Trapezium Area of a Kite More Complicated Areas Angles of Polygons Regular Polygons More Tessellations Finding Angles: Revision Miles and Kilometres Feet and Metres Pints, Gallons, Litres O22 O23 O24 O25 H2 O26 O27 O28 H3 H4 H5 H6 H7 Page 3 Pounds and Kilograms Accuracy in Measurements Speed Density More Accuracy in Measurements Pythagoras’ Theorem More Perimeters Length of AB Proving Congruent Triangles Using Congruence (1) Using Congruence (2) Similar Triangles; proof Similar Triangles; finding sides O29 O30 H8 H9 H10 Circumference of a Circle Area of a Circle Chords and Tangents Angle in a Segment Angles in a Semicircle and Cyclic Quadrilateral H11 Alternate Segment Theorem O31 Loci H12 More Loci continued Teach GCSE Maths – Higher O32 O33 H13 O34 O35 O36 O37 O38 3-D Coordinates Plan and Elevation More Plans and Elevations Symmetry of Solids Nets of Prisms and Pyramids Volumes of Prisms Dimensions Surface Area of a Prism and Cylinder O39 O40 O41 O42 More Reflections Even More Reflections More Enlargements Effect of Enlargements O43 Rotations O44 Translations O45 Mixed and Combined Transformations H14 More Combined Transformations H15 Negative Enlargements H16 Right Angled Triangles: Sin x H17 Inverse sines H18 cos x and tan x H19 Solving problems using Trig (1) H20 H21 H22 H23 H24 H25 H26 H27 H28 H29 H30 H31 H32 H33 H34 H35 H36 H37 H38 Page 4 Solving problems using Trig (2) The Graph of Sin x The Graphs of Cos x and Tan x Solving Trig Equations The Sine Rule The Sine Rule; Ambiguous Case The Cosine Rule Trig and Area of a Triangle Arc Length and Area of Sectors Harder Volumes Volumes and Surface Areas of Pyramids and Cones Volume and Surface Area of a Sphere Areas of Similar Shapes and Volumes of Similar Solids Vectors 1 Vectors 2 Vectors 3 Right Angled Triangles in 3D Sine and Cosine Rules in 3D Stretching Trig Graphs 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/