Teach GCSE Maths Shape Space and Measures The

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.

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.

We already know that the sum of the angles of any triangle is 180. e. g. 57 75 57 + 75 + 48 = 180 exterior 48 a angle 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 75 57 + 75 + 48 = 180 exterior a angle 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 75 57 + 75 + 48 = 180 exterior 48 132 angle What is the link between 132 and the other 2 angles of the triangle? ANS: 132 = 57 + 75 , the sum of the other angles.

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.

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 = 65 ( angles of quadrilateral )

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.

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

F 19 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 1 st position as it’s the same as the last. )

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.

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 0 mph 80 100 120 80 km/h 140 160 60 100 40 180 20 200 0 220 140 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

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.

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.

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.

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

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.

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 = 1000 th 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

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.

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.

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.

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.

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.

SUMMARY Ø 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 ) Ø The letters “sin” are always followed by an angle. Ø 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 F 1 F 2 O 1 O 2 Angles Lines: Parallel and Perpendicular Parallel Lines and Angles Bearings F 3 F 4 O 3 F 5 F 6 F 7 F 8 F 9 F 10 F 11 Triangles and their Angles Exterior Angle of a Triangle Proofs of Triangle Properties Perimeters Area of a Rectangle Congruent Shapes Congruent Triangles Constructing Triangles SSS Constructing Triangles AAS Constructing Triangles SAS, RHS O 4 O 5 F 12 F 13 F 14 O 6 More Constructions: Bisectors More Constructions: Perpendiculars Quadrilaterals: Interior angles Quadrilaterals: Exterior angles Parallelograms Angle Proof for Parallelograms Page 1 F 15 O 7 F 16 O 8 F 17 F 18 Trapezia Allied Angles Kites Identifying Quadrilaterals Tessellations Lines of Symmetry F 19 F 20 F 21 F 22 O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16 O 17 O 18 Rotational Symmetry Coordinates Reading Scales and Maps 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 continued

Teach GCSE Maths – Foundation F 23 Metric Units O 19 Miles and Kilometres O 20 Feet and Metres O 21 O 22 O 23 O 24 O 25 Pints, Gallons and Litres Pounds and Kilograms Accuracy in Measurements Speed Density O 26 O 27 O 28 F 24 O 29 O 30 O 31 Pythagoras’ Theorem More Perimeters Length of AB Circle words Circumference of a Circle Area of a Circle Loci O 32 3 -D Coordinates F 25 Volume of a Cuboid and Isometric Drawing F 26 Nets of a Cuboid and Cylinder O 33 O 34 O 35 O 36 Page 2 Plan and Elevation Symmetry of Solids Nets of Prisms and Pyramids Volumes of Prisms O 37 Dimensions F 27 Surface Area of a Cuboid O 38 Surface Area of a Prism and Cylinder F 28 Reflections O 39 More Reflections O 40 Even More Reflections F 29 Enlargements O 41 More Enlargements F 30 Similar Shapes O 42 Effect of Enlargements O 43 Rotations O 44 Translations O 45 Mixed and Combined Transformations continued

Teach GCSE Maths – Higher O 1 O 2 O 3 O 4 O 5 H 1 O 6 O 7 O 8 O 9 O 10 O 11 O 12 O 13 O 14 O 15 O 16 O 17 O 18 O 19 O 20 O 21 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 O 22 O 23 O 24 O 25 H 2 O 26 O 27 O 28 H 3 H 4 H 5 H 6 H 7 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 O 29 O 30 H 8 H 9 H 10 Circumference of a Circle Area of a Circle Chords and Tangents Angle in a Segment Angles in a Semicircle and Cyclic Quadrilateral H 11 Alternate Segment Theorem O 31 Loci H 12 More Loci continued

Teach GCSE Maths – Higher O 32 O 33 H 13 O 34 O 35 O 36 O 37 O 38 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 O 39 O 40 O 41 O 42 O 43 O 44 O 45 More Reflections Even More Reflections More Enlargements Effect of Enlargements Rotations Translations Mixed and Combined Transformations More Combined Transformations Negative Enlargements Right Angled Triangles: Sin x Inverse sines cos x and tan x Solving problems using Trig (1) H 14 H 15 H 16 H 17 H 18 H 19 H 20 H 21 H 22 H 23 H 24 H 25 H 26 H 27 H 28 H 29 H 30 H 31 H 32 H 33 H 34 H 35 H 36 H 37 H 38 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; Ambiguous Case The Cosine Rule Trig and Area of a Triangle Arc Length and Area of Sectors Harder 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 3 D Sine and Cosine Rules in 3 D Stretching Trig Graphs

Further details of “Teach GCSE Maths” are available from Chartwell-Yorke Ltd 114 High Street Belmont Village Bolton Lancashire BL 7 8 AL Tel: 01204811001 Fax: 01204 811008 www. chartwellyorke. co. uk/