Shape and Space 2

PGCE SeminarDr David Bolden

d.s.bolden@durham.ac.uk0191 334 8325

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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 we’ll consider:• What exactly is pi ()?• When is a circle a triangle?• How can we convince ourselves of the formulae we use?

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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. cm2) within a given perimeter.

Volume is the amount of 3-D space (in cubic units, e.g. cm3) within a given object.

Capacity is the amount of liquid (usually expressed in ml) that a given container can hold.

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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

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

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Does this works for any triangle?

Area of any triangle = ½ base heightbase

perpendicular height

YES

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The area of parallelograms

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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

parallelogramArea of the rectangle

base

heightperpendicular

height

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Area of any parallelogram = base height

Perpendicular height

Does this works for any parallelogram?

base

height

YES

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The area of trapeziums

10 cm

3 cm

4 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 cm2) + (9 cm2) = 21 cm2

Or = height x (base1 + base2) 2

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

the circle).

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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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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.

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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 (2r)

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Imagine a circle made out of strands of beads.

We could open it out.

Area of circles

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circumference

radius (half the diameter)

It’s a triangle!

base = circumferenceheight = radius (half the diameter)

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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

= 2r × radius 2

= r2

Volume

The amount of 3-D space within a given object:

1cm

1cm

1 cm

1 cm 1 cm 1 cm = 1 cm3

1 cm1 cm

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• The linear ratio of your cubes is 1:2• The cubic ratio of your cubes is 13:23 or 1:8

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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 length20

Calculate the volume of these:

8 cm

8 cm

20 cm

10 cm

4 cm

12 cm

8 cm

12 cm

5 cm

Answers:1.800 cm3

2.240 cm3

3.502.7 cm3

1.2.

3.

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References/Further ReadingDickson, L., Brown, M. & Gibson, O. (1984) Children Learning Mathematics: A Teacher’s

Guide to Recent Research London: Cassel

Haylock, D. (2001) Mathematics Explained for Primary Teachers 2nd 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)

Teachers TV has some useful ideas for classroom activities concerning shape. Type a keyword into the search box at Teachers TV

Click on this link for several ideas for a shape-themed lesson (Teachers TV Video 37880)

BBC Bitesize Maths site has some fun 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 ATM

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