shape and space 2 pgce seminar dr david bolden [email protected] 0191 334 8325 1
TRANSCRIPT
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.
123
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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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