Shape and Space 2 PGCE Seminar Dr David Bolden

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

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

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

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

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

The area of parallelograms 7

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

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

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

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

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!

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

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)

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

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

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

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

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

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

Calculate the volume of these: 8 cm 20 cm 10 cm 4 cm 12 cm 8 cm 12 cm 5 cm Answers: cm cm cm

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 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