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Mathematics Intermediate Tier

Mathematics Intermediate Tier. Shape and space. GCSE Revision. Intermediate Tier – Shape and space revision. Contents : Angle calculations Angles and polygons Bearings Units Perimeter Area formulae Area strategy Volume Nets and surface area

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Mathematics Intermediate Tier

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  1. Mathematics Intermediate Tier Shape and space GCSE Revision

  2. Intermediate Tier – Shape and space revision Contents : Angle calculations Angles and polygons Bearings Units Perimeter Area formulae Area strategy Volume Nets and surface area Spotting P, A & V formulae Transformations Constructions Loci Pythagoras Theorem Similarity Trigonometry Circle angle theorems

  3. Use the rules to work out all angles “Z” angles are equal f g Angles in a half turn = 1800 Angles in a full turn = 3600 420 b a 720 210 1350 1620 Opposite angles are equal Angles in a quadrilateral = 3600 1530 c e k d 730 l 980 “F” angles are equal Angles in a triangle = 1800 Angles in an isosceles triangle 570 120 j m 80 350 h i Angle calculations

  4. There are 3 types of angles in regular polygons c e c Angles at = 360 the centre No. of sides c e c c c e Exterior = 360 angles No. of sides To calculate the total interior angles of an irregular polygon divide it up into triangles from 1 corner. Then no. of x 180 e i Angles and polygons Interior = 180 - e angles Calculate the value of c, e and i in regular polygons with 8, 9, 10 and 12 sides Answers: 8 sides = 450, 450, 1350 9 sides = 400, 400, 1400 10 sides = 360, 360, 1440 12 sides = 300, 300, 1500 Total i = 5 x 180 = 9000

  5. N Bristol N Bath 0560 2360 Bearings A bearing should always have 3 figures. A bearing is an angle measured in a clockwise direction from due North What are these bearings ? Here are the steps to get your answer 2360 Notice that there is a 1800 difference between the outward journey and the return journey 560 What is the bearing of Bristol from Bath ? What is the bearing of Bath from Bristol ?

  6. x 1000 x 100 x 10 kl km kg m l m cl cm cg ml mm mg Length ÷ 1000 ÷ 100 ÷ 10 Capacity Weight Units Learn these metric conversions Imperial  Metric 5 miles  8 km 1 yard  0.9 m 12 inches  30 cm 1 inch  2.5 cm Learn these rough imperial to metric conversions

  7. Perimeter = 4 x L of a square Circumference =  x D of a circle Perimeter = 2(L + W) of a rectangle 6.5m 15cm Circumference =  x D of a semi-circle 2 2m 5m 7.2m Perimeter = ? 1m 3m 1m Perimeter Be prepared to leave answers to circle questions in terms of  especially in the non-calculator exam The perimeter of a shape is the distance around its outside measured in cm, m, etc. 26m Perim = D + ( x D)  2 Perim = 15 + ( x 15)  2 Perim = 15 + 7.5 31.4m 7.85m 4.71m 18.4m = 7.85 + 4.71 + 1 + 1 = 14.56m

  8. The area of a 2D shape is the amount of space covered by it measured in cm2, m2 etc. Area of = L x W square Area of = b x h parallelogram 4m 5m Area of = (a + b) x h Trapezium 2 10m 6m 7m Area of = b x h triangle 2 4m 5m 2m Area of = L x W rectangle 9m 8m Area of =  x r2 circle 2m 6m 10cm 9m 7m Area of = b x h rhombus 3m 8m 5m 6m 7m Area formulae Be prepared to leave answers to circle questions in terms of  especially in the non-calculator exam 49m2 40m2 16m2 Area = ( x r x r)  2 Area = ( x 5 x 5)  2 Area = 12.5 18m2 24m2 42m2 50.24m2 7.5m2

  9. 2. 2m 10m 4m 7m 3. 1.5m 1. 3m 5. 4. 9m 1.5m 6m 6m 2m 6m 8m Area strategy What would you do to get the area of each of these shapes? Do them step by step!

  10. Volume = L x L x L of cube Volume of = Area at end x L a prism 3m A = 14m2 4m Volume of = L x W x H cuboid Volume of = ( x r2) x L cylinder 2m 3m 7m 7m 10m Volume The volume of a 3D solid shape is the amount of space inside it measured in cm3, m3 etc. 27m3 56m3 42m3 384.65m3

  11. 6 2. 3. 1. 6 by 6 by 1 5 by 4 by 3 5 by 5 by 5 Nets and surface area 12cm2 12cm2 4cm2 2 2 4cm2 12cm2 Cuboid 2 by 2 by 6 Net of the cuboid 12cm2 Volume = 2 x 2 x 6 = 24cm3 Total surface area = 12 + 12 + 12 + 12 + 4 + 4 = 56cm2 To find the surface area of a cuboid it helps to draw the net Find the volume and surface area of these cuboids: V = 5 x 4 x 3 = 60cm3 V = 6 x 6 x 1 = 60cm3 V = 5 x 5 x 5 = 125cm3 SA = 94cm2 SA = 96cm2 SA = 150cm2

  12. Spotting P, A & V formulae r(+ 3) 4rl P A • Which of the following • expressions could be for: • Perimeter • Area • Volume r(r + l) A 1d2 4 4r2 3 4r3 3 A A r + ½r V 4l2h P 1r2h 3 1rh 3 V r + 4l A V 1r 3 P rl 3lh2 4r2h P V V A

  13. y x Transformations 1. Reflection Reflect the triangle using the line: y = x then the line: y = - x then the line: x = 1

  14. y x Transformations Describe the rotation of A to B and C to D 2. Rotation • When describing a rotation always state these 3 things: • No. of degrees • Direction • Centre of rotation • e.g. a rotation of 900 anti-clockwise using a centre of (0, 1) C B A D

  15. What happens when we translate a shape ? The shape remains the same size and shape and the same way up – it just……. . 3 -4 1. A to B -3 4 -3 -1 6 5 6 0 2. A to D 3. B to C 4. D to C Transformations slides 3. Translation Horizontal translation Use a vector to describe a translation Give the vector for the translation from…….. Vertical translation D C A B

  16. Enlarge this shape by a scale factor of 2 using centre O y x O Transformations 4. Enlargement

  17. Perpendicular bisector of a line 900 Triangle with 3 side lengths 600 Bisector of an angle Constructions Have a look at these constructions and work out what has been done

  18. Draw the locus to show all that grass he can eat 1. A A B 1.5m 1.5m 2. Draw the locus to show all that grass he can eat 1.5m Loci A locus is a drawing of all the points which satisfy a rule or a set of constraints. Loci is just the plural of locus. A goat is tethered to a peg in the ground at point A using a rope 1.5m long A goat is tethered to a rail AB using a rope (with a loop on) 1.5m long

  19. Shapes are congruent if they are exactly the same shape and exactly the same size Triangle C Triangle B Triangle A Similarity Shapes are similar if they are exactly the same shape but different sizes How can I spot similar triangles ? These two triangles are similar because of the parallel lines All of these “internal” triangles are similar to the big triangle because of the parallel lines

  20. Same multiplier x 2.1 x 2.1 Multiplier = 15.12  7.2 = 2.1 Similarity Triangle 2 These two triangles are similar.Calculate length y y = 17.85  2.1 = 8.5m 15.12m 17.85m y 7.2m Triangle 1

  21. Calculating the Hypotenuse D D A ? ? ? 11m 21cm 3cm AC = 11.6m F F E E B C 45cm 6cm 16m DE = 45 DE = 2466 Calculate the size of DE in surd form Calculate the size of DE to 1 d.p. Calculate the size of AC to 1 d.p. Longest side & opposite 135 = AC DE = 35 cm DE = 49.7cm Pythagoras Theorem Hyp2 = a2 + b2 How to spot a Pythagoras question DE2 = 212 + 452 Be prepared to leave your answer in surd form (most likely in the non-calculator exam) DE2 = 441 + 2025 DE2 = 2466 Right angled triangle Hyp2 = a2 + b2 DE = 49.659 DE2 = 32 + 62 No angles involved in question DE2 = 9 + 36 DE2 = 45 Hyp2 = a2 + b2 Calculating a shorter side 162 = AC2 + 112 DE = 9 x 5 256 = AC2 + 121 256 - 121 = AC2 How to spot the Hypotenuse 135 = AC2 11.618 = AC

  22. Finding lengths in isosceles triangles y Find the distance between 2 co-ords x x x Finding lengths inside a circle 1 (angle in a semi -circle = 900) Finding lengths inside a circle 2 (radius x 2 = isosc triangle) O O Pythagoras Questions Look out for the following Pythagoras questions in disguise:

  23. Calculating an angle   = 0 D D ? 11m 26cm H = m F E B C 53cm Calculate the size of  to 1 d.p. Calculate the size of BC to 1 d.p. 730 Trigonometry SOHCAHTOA Tan  = O/A How to spot a Trigonometry question H Tan  = 26/53 Tan  = 0.491 O Right angled triangle A An angle involved in question Calculating a side SOHCAHTOA Sin  = O/H O A Sin 73 = 11/H • Label sides H, O, A • Write SOHCAHTOA • Write out correct rule • Substitute values in • If calculating angle use • 2nd func. key H = 11/Sin 73 H

  24. Circle angle theorems Rule 1 - Any angle in a semi-circle is 900 A F Which angles are equal to 900 ? c B C E D

  25. Circle angle theorems Rule 2 - Angles in the same segment are equal Which angles are equal here? Big fish ?*!

  26. Circle angle theorems c c Rule 3 - The angle at the centre is twice the angle at the circumference c c c An arrowhead A little fish A mini quadrilateral Look out for the angle at the centre being part of a isosceles triangle Three radii

  27. Circle angle theorems Rule 4 - Opposite angles in a cyclic quadrilateral add up to 1800 D A + C = 1800 C A and B B + D = 1800

  28. Circle angle theorems Rule 5 - The angle between the tangent and the radius is 900 c A tangent is a line which rests on the outside of the circle and touches it at one point only

  29. c Circle angle theorems Rule 7 - Tangents from an external point are equal (this usually creates a kite with two 900 angles in….. …… or two isosceles triangles) 900 900

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