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Shape and Space 2 PGCE Seminar Dr David Bolden d.s.bolden@durham.ac.uk 0191 334 8325 1

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Aims To explore perimeter & area of 2-D shapes; To explore conservation of area; To explore volume of 3-D shapes. Some questions well consider: What exactly is pi ( )? When is a circle a triangle? How can we convince ourselves of the formulae we use? 2

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Some Definitions Perimeter is the length around the edge of a polygon or closed curve. Area is the amount of 2-D surface (in square units, e.g. cm 2 ) within a given perimeter. Volume is the amount of 3-D space (in cubic units, e.g. cm 3 ) within a given object. Capacity is the amount of liquid (usually expressed in ml) that a given container can hold. 3

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Perimeter The perimeter of a circle is called the circumference ……………. To calculate the circumference of any circle we need pi ( ) 4

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The area of rectangles & triangles Area of any rectangle = base x height (or length x width ) Therefore, the area of the triangle = ½ base x height height base 5

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Does this works for any triangle? Area of any triangle = ½ base height base perpendicular height YES 6

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The area of parallelograms 7

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We already know how to calculate the area of a rectangle. Well, a parallelogram is simply a sheared rectangle. Area of the rectangle= base × height parallelogram Area of the rectangle base height perpendicular height 8

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Area of any parallelogram = base height Perpendicular height Does this works for any parallelogram? base height YES 9

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The area of trapeziums 10 cm 3 cm 4 cm 6 cm Area of trapezium = area of parallelogram + area of triangle = (base x height) + (½ base x height) = (4 x 3) + (½ 6 x 3) = (12 cm 2 ) + (9 cm 2 ) = 21 cm 2 Or = height x (base1 + base2) 2 10

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Question: What is pi ( )? is a ratio (the circumference of the circle to the diameter of the circle). 11

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12 Pi is the number of times you must travel straight across the circle to go the same distance as all the way round the circle. Once across Twice across So is a bit more than 3. Three times across And a bit further!

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13 How can we be sure that is a bit more than 3? For a regular hexagon, the distance all the way round is exactly 3 times the distance straight across the middle. 12 3

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14 And all the way round the circle is a little bit more than all the way round the hexagon. So all the way round the circle is a little bit more than 3 times straight across the middle. Circumference = × d or 2r (2 r)

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15 Imagine a circle made out of strands of beads. We could open it out. Area of circles

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16 circumference radius (half the diameter) Its a triangle! base = circumference height = radius (half the diameter)

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17 circumference radius (half the diameter) = circumference × radius 2 Area of the triangle circle We already know how to find the area of a triangle. = base × height 2 = 2 r × radius 2 = r 2

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Volume The amount of 3-D space within a given object: 1cm 1 cm 1 cm 1 cm = 1 cm 3 1 cm 18

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The linear ratio of your cubes is 1:2 The cubic ratio of your cubes is 1 3 :2 3 or 1:8 19

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Volume of prisms Prisms are simply 3-D shapes comprising two congruent parallel polygons at each end, joined by straight edges: Volume of a prism = area of cross-section x length 20

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Calculate the volume of these: 8 cm 20 cm 10 cm 4 cm 12 cm 8 cm 12 cm 5 cm Answers: 1.800 cm 3 2.240 cm 3 3.502.7 cm 3 1. 2. 3. 21

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References/Further Reading Dickson, L., Brown, M. & Gibson, O. (1984) Children Learning Mathematics: A Teachers Guide to Recent Research London: Cassel Haylock, D. (2001) Mathematics Explained for Primary Teachers 2 nd Ed. London: Paul Chapman Publishing Suggate, J., Davis, A. & Goulding, M. (1998) Mathematical knowledge for Primary teachers London: David Fulton Publishers The Mathematics Framework site has some interactive teaching programs (Primary Framework ITPs)Primary Framework ITPs Teachers TV has some useful ideas for classroom activities concerning shape. Type a keyword into the search box at Teachers TVTeachers TV Click on this link for several ideas for a shape-themed lesson (Teachers TV Video 37880)Teachers TV Video 37880 BBC Bitesize Maths site has some fun activities (BBC Bitesize Maths Activities)BBC Bitesize Maths Activities Thanks are due to Tandi Clausen-May from NFER for some of the more interactive slides used in this presentation. These can be downloaded free from the ATM website at ATMATM 22

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