module 2 space, shape and orientation: summary (levels 2
TRANSCRIPT
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 1 APRIL 2020 T SWANEPOEL/ts
MODULE 2
Space, Shape and Orientation:
SUMMARY
(Levels 2, 3 and 4)
VOCABULARY
Perimeter
The distance around a flat (two-dimensional) shape, measured in units
such as mm, cm, m and km
Diagonal
A line segment joining two points that are not adjacent to each other
Area
The measurement of the size of a surface that is covered by the shape
and measured in units such as mm2, cm2, m2 and km2
Angle
The space (usually measured in degrees) between two intersecting lines
Diameter
The line that passes through the centre of a circle from one side to the other
Radius
A line segment from the centre of a circle to the circle boundary.
The radius is always equal to half the diameter.
Circumference
The distance around a circle. The perimeter of a circle is called it’s circumference,
which is measured in units such as km, m, cm and mm.
Volume
The space occupied by a shape. When dealing with liquids, the volume of a
container can also be called the capacity of the container. Capacity is measured in
units such as mm3, cm3, l and kl.
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 2 APRIL 2020 T SWANEPOEL/ts
Point of intersection
Where lines or surfaces meet
Perpendicular
When lines intersect at an angle of 90ᵒ
Parallel
When lines are an even distance from each other at any given position.
Parallel lines never intersect (cross each other or meet).
TWO-DIMENSIONAL SHAPES
Rectangle
A shape with four sides at 90ᵒ angles to each other.
The opposite sides are the same length to form two breadths and
two lengths. The opposites sides are parallel.
Perimeter:
2 x length + 2 x breadth
(2 x l) + (2 x b)
Area:
length x breadth
l x b
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 3 APRIL 2020 T SWANEPOEL/ts
Square
A shape with four equal sides at 90ᵒ angles to each other.
The opposite sides are parallel.
Perimeter:
4 x side length
4 x s
Area:
side length x side length
s x s
Triangle
A basic shape with three corners and three sides.
Perimeter:
Add the three side lengths
Area:
½base x perpendicular height
½b x h
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 4 APRIL 2020 T SWANEPOEL/ts
Circle
A simple shape consisting of joined points that are at an equal distance
from the centre.
Perimeter:
(called circumference)
2 x pi x radius
2 x л x r
Area:
pi x radius x radius
л x r2
Semicircle
Half a circle. The straight line is the diameter of the circle.
Perimeter:
Diameter + pi x radius
d + л x r
Area:
2
radiusradiuspi
2
2r
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 5 APRIL 2020 T SWANEPOEL/ts
THREE-DIMENSIONAL SHAPES
Rectangular prism
An object that has six faces that are all rectangles
Surface area:
Add the areas of the 6 sides
or
2lb + 2lh + 2bh
Volume:
Area of base x height
or
length x breadth x height
Cube
A special rectangular prism with six square faces
Surface area:
6 x area of a side
or
6 x side length x side length
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 6 APRIL 2020 T SWANEPOEL/ts
or
6(side length)2
Volume:
Area of base x height
or
side length x side length x side length
or
(side length)3
Cylinder
Sphere
A basic geometrical shape with a circular base
Surface area:
2 x area of base + height x circumference of base
or
2лr2 + 2лrh
Volume:
Area of base x height
or
лr2h
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 7 APRIL 2020 T SWANEPOEL/ts
Sphere
A three-dimensional object shaped like a ball.
Every point on the surface is the same distance from the centre.
Surface area:
4лr2
Volume:
3
4лr3
Cone
A shape with a circle at the bottom and sides that narrow to a point.
Surface area:
pi x radius x side + pi x radius x radius
or
лrs + лr2
Volume:
3
1 x pi x radius x radius x height
or
3
1лr2h
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 8 APRIL 2020 T SWANEPOEL/ts
Triangular prism
An object with two triangular bases and three rectangular sides
Surface area:
2 x area of base + length x base + 2 x side x length
or
bh + bl + 2sl
Volume:
Area of base x height
or
½bhl
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 9 APRIL 2020 T SWANEPOEL/ts
The Theorem of Pythagoras
For triangles where one of the angles is 90ᵒ, the theorem of Pythagoras can be used
to calculate the length of the hypotenuse.
The theorem of Pythagoras states that the square of the hypotenuse equals the sum
of the squares of the two opposite sides.
Manipulation of Pythagoras leads to the calculation of the lengths of the right-angled
sides.
XZ2 (XZ is the hypotenuse) = XY2 + YZ2
Therefore XZ = )( 22 YZXY
XY2 = XZ2 – YZ2
Therefore XY = )( 22 YZXZ
YZ = )( 22 XYXZ
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 10 APRIL 2020 T SWANEPOEL/ts
SUMMATIVE ASSESSMENT
1. Determine the perimeter of the shape below.
2. Use the diagram below to answer the following questions.
2.1 Determine the area of the shape.
2.2 Calculate the perimeter of the shape.
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3. Calculate the area of the red section of the diagram below, if
the diameter of the circle below is 8,2 cm.
4. John wants to build a fireplace in the shape below.
He wants to cover the whole surface with fireproof tiles.
The dimensions of a tile are 8 cm by 8 cm.
4.1 Calculate exactly how many tiles are needed to cover the surface.
Add an additional 5% tiles for possible breakage
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 12 APRIL 2020 T SWANEPOEL/ts
5.
.
The diagram shows a concrete tank on Mr Mudau’s farm.
The height of the tank is 85 cm and the width of the wall of the tank is 20 cm.
5.1 Calculate the outside diameter of the tank.
5.2 Calculate the volume of the inside of the tank.
5.3 Calculate th volume of the wall of the tank.
5.4 Mr Mudau wants to paint the outside of the tank.
Determine the surface area of the outside of the tank.
(formula: outside area = 2лrh)
5.5 One litre of paint is needed to paint an area of 28 000 cm2.
What will it cost to paint the outside area, if 1 litre of paint costs R49,95?
6.
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A brick has a length of 20 cm, a breadth of 8 cm and a height of 8 cm.
There are two circular holes in the brick, each with a diameter of 3 cm.
Calculate the volume of the clay that the brick is made of.
7. Calculate the quantity of sheet metal needed to manufacture 50 cylindrical jugs,
each with a diameter of 20 cm and a height of 30 cm. The jugs are open at the
top without lids. The metal sheets can only be bought in sheets of 1 m2.
8. The area of a semi-circle is 5 024 cm2.
8.1 Calculate the length of the straight lined side.
8.2 What is the circumference of the semi-circle?
Give your answer to the nearest centimetre.
9.
9.1 The area of a netball court is 465,125 m2. If the breadth of the court is 15,25 m,
what is the length?
9.2 A netball court is divided into thirds. What will the area of a third be?
Give your answer to the nearest square metre.
10.
10.1 The volume of a cylindrical jug is 2 000 cm3.
What is the diameter of the jug if the height is 200 mm?
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11. Calculate the height of a rectangular prism with a base area of 120 cm2 and
capacity of 48 l. (1 l = 1 000 cm3)
12 If the volume of a cube is 64 cm3, what is the length of its side?
13. The area of a circle is 706,5 cm2.
13.1 What is the radius of the circle?
13.2 If the area is doubled, what would the radius be?
14.
Given:
Area of a circle = лr2
Volume of a cylinder = area of base x height
1 m3 = 1 kl
If the cement wall of the dam is 35 cm thick, calculate the volume of the water in
the dam if the dam is 80% full. Give the answer in kilolitres.
15.
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The height of the triangle above is 8 cm and the area is 1 800 mm2.
Calculate the base length.
16.
The diameter of a swimming area, including the pool and a paved area of
1,5 m wide around the pool, is 15 m.
Formulae: Area of circle = л x r2
Circumference of circle = 2 x л x r
Volume of a cylinder = л x r2 x h
Use л = 3,14
16.1 Determine the area of the paving.
16.2 Determine the circumference of the pool.
16.3 Determine the volume of the water in the pool if the depth is 1,2 m and the
pool is 85% full.
16.4 Calculate the number of bricks required for the paving, if 52 bricks cover an
area of one square metre.
Make provision for breakages and cutting by adding an extra 10%.
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 16 APRIL 2020 T SWANEPOEL/ts
17
17.1 Calculate the area of the concrete slab below.
17.2 The concrete is mixed by combining 1 unit of cement with 4 units of sand and
8 units of gravel. To produce 0,25 cubic metre of concrete, 2 bags of cement
are needed. How many bags of cement are required to lay this slab if it is 5 cm
thick?
17.3 How many bags of sand are required?
17.4 For every two bags of cement, 37 l of water is used. How much water will be
used to lay the slab?
18 The area af the mirror with a semi-circular shape is 502 400 mm2.
18.1 Calculate the length of the straight lined side.
18.2 What length of frame is needed for the mirror? Give the answer to the nearest
centimetre.
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19 Calculate the length of AC.
20. Calculate the length of BC.
21. The diagram below shows a kite, ABCD. The diagonals cut at right angles and
intersect at O. Calculate the length of the diagonal AC.
(All lengths are given in centimetres.)
MATHEMATICAL LITERACY LEVELS 2,3 AND 4 P a g e | 18 APRIL 2020 T SWANEPOEL/ts
[Some extracts were made from the text book: Mathematical Literacy HANDS-ON
TRAINING by Cecile Bruwer & Salome Voges]