GM1 Consolidation Worksheet

Cambridge Essentials Mathematics Core 8 GM1 Consolidation Worksheet
GM1 Consolidation Worksheet
1
Calculate the size of each angle marked by a letter.
Give reasons for your answers. or example, angles on a straight line add up to 180°.
Work out the angles in alphabetical order.
Remember
a
Angles on a straight line
add up to 180°.
Vertically opposite angles
are equal.
b
Alternate angles are
equal.
c
Corresponding angles are
equal.
d
The angles of a triangle
add up to 180°.
e
Isosceles triangles have
two equal sides and two
equal angles.
The four angles of a
quadrilateral add
up to 360°.
Original material © Cambridge University Press 2009
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Cambridge Essentials Mathematics Core 8 GM1 Consolidation Worksheet
2
A square has four right angles, four equal sides and two pairs of parallel sides.
Name the quadrilateral that has the following properties. You can use a name more than once.
a
Four right angles and two pairs of equal sides
b
Four equal sides, but no right angles
trapezium
c
Opposite sides parallel, but no right angles
rhombus
d
Just one pair of parallel sides
e
Just one line of symmetry
parallelogram
f
Rotational symmetry of order 2
arrowhead
Choose from:
rectangle
3
Draw a line AB, 7.5 cm long. Use a pair of compasses to
construct the perpendicular bisector of AB.
kite
Leave all your
construction lines and
arcs in the diagram.
Draw an angle similar to this.
4
Use a pair of compasses to construct the bisector of your angle.
5
The diagram below shows the right angled triangle ABC.
a
Construct the perpendicular bisector of the side BC.
b
Construct the perpendicular bisector of the side AB.
c
Put your compass point where the two bisectors meet.
Turn the page
round so AB is
horizontal.
Put the pencil point on B.
Draw a circle.
Does it go through A and C?
If it does, well done! Your constructions
must be very accurate.
Original material © Cambridge University Press 2009
2
Cambridge Essentials Mathematics Core 8 GM2 Consolidation Worksheet 1
GM2 Consolidation Worksheet 1
1
Area tells you the amount of space inside a flat shape. It is measured in square units.
cm2 , m2 and mm2
are all examples
of square units.
Area of a rectangle = length × width
Calculate the area of each rectangle.
a
b
5 cm
1m
2 cm
c
30 cm
Remember to check
that the length and
width are in the same
units. If not, convert
one of the lengths.
7 cm
15 mm
2
Area of a triangle = (base × height) ÷ 2
Calculate the area of each triangle.
a
b
11 cm
9 cm
20 mm
12 mm
14 cm
7 mm
3
Volume measures the total space inside a three-dimensional (3-D) shape.
Volume is always measured in cubic units.
A cubic centimetre is a cube that is
1 cm
1 cm long, 1 cm wide and 1 cm high.
This cube has a volume of 1 cm3.
1 cm
1 cm
Volume of a cuboid = length × width × height
Calculate the volume of each cuboid.
a
b
2m
c
10 mm
3 cm
5m
6 cm
2 cm
10 m
4 mm
4 mm
Original material © Cambridge University Press 2009
1
Cambridge Essentials Mathematics Core 8 GM2 Consolidation Worksheet 1
4
8 cm
An 8 cm cube is not the same as a shape that has a volume of 8 cm3.
8 cm
An 8 cm cube is a cube with length, width and height each 8 cm.
Its volume is 8 cm × 8 cm × 8 cm = 512 cm3.
8 cm
Look at this cuboid.
1 cm
Its volume is 4 cm × 2 cm × 1 cm = 8 cm3.
2 cm
4 cm
a
What is the side length of a cube with volume 8 cm3?
b
Draw a different cuboid with volume 8 cm3.
Mark the length of each edge.
5
Surface area is the total area of all the faces of the shape.
A cuboid has six faces. Top and bottom have the same area.
Front and back have the same area. The two sides have the same area.
Remember to
include the faces
you cannot see. It
helps to write the
length of each edge
on the diagram.
a Find the surface area of these cuboids.
B
A
2 cm
9m
4 cm
6 cm
3m
2m
b A net of a 3-D shape is a 2-D shape that can be folded to make the 3-D shape.
The surface area is also the area of the net.
P
This is a net for cuboid A.
ii
What is the length TU?
Top
V W
Side
What is the length QR?
iii Work out the total area of the net.
iv Does your answer match part a?
c i
ii
Back
R
Bottom
Side
i
Q
Front
U
S
T
Draw a net of cuboid B. Label the faces and show the edge lengths.
Find the surface area. Check that it matches your answer to part a.
Original material © Cambridge University Press 2009
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Cambridge Essentials Mathematics Core 8 GM2 Consolidation Worksheet 2
GM2 Consolidation Worksheet 2
1
2-D drawings can be used to represent 3-D shapes.
The 3-D shapes are drawn from different views.
The plan is the view of the 3D shape from above.
An elevation is a view from the side, front or back.
Look at these solid shapes.
a
c
b
For each shape, draw
i
2
the plan
ii
the front elevation
When converting from one unit to another,
kilo- always means × 1000 so 1 km = 1000 m and 1 kg = 1000 g
centi- always means
1
100
so 1 cm =
milli- always means
1
1000
so 1 mm =
1
m or 100 cm = 1 m
100
1
m
1000
or 1000 mm = 1m
Change 6 mm to centimetres.
Millimetres are smaller than centimetres, so divide by 10: 6 ÷ 10 = 0.6 cm.
Change 9 m to centimetres.
Metres are bigger than centimetres, so multiply by 100:
9 × 100 = 900 cm.
When changing from a smaller unit to a bigger unit, divide.
When changing from a bigger unit to a smaller unit, multiply.
Change each quantity to the unit given in brackets.
Show what you did to get the answer.
a
14 km (m)
b
36 mm (cm)
c
425 cm (m)
d
3265 g (kg)
e
1200 mg (g)
f
42 kg (g)
g
8260 ml (litres)
h
466 kg (tonnes)
Original material © Cambridge University Press 2009
1
Cambridge Essentials Mathematics Core 8 GM3 Consolidation Worksheet
GM3 Consolidation Worksheet
1
Congruent shapes are exactly the same shape and size.
These triangles are congruent.
These are not congruent.
Which two of these shapes are congruent?
2
To reflect a shape in a mirror line find the image of
each vertex (corner) of the shape in the mirror line.
The image of each vertex will be the same distance
from the mirror line as the original.
The shape and its image will be congruent.
Sometimes the shape being
reflected is known as the object.
Reflect the shape in the mirror
line x = 2.
(The line x = 2 joins all points
with an x coordinate of 2.)
Check your image using a mirror
placed on the line x = 2.
Original material © Cambridge University Press 2009
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Cambridge Essentials Mathematics Core 8 GM3 Consolidation Worksheet
3
When you rotate a shape it helps to use tracing paper.
Trace the shape and the centre of rotation. Pin the tracing paper to the centre of rotation
using a pencil point. Turn the tracing paper round through the required angle. Mark the
position of the corners of the image on the square grid. Join the corners with a ruler and
pencil to show the new shape.
Use tracing paper to rotate the triangle
in this diagram through 90° clockwise.
Use the point (1, –2), marked, as the
centre of rotation.
4
When you translate a shape find the image of each vertex of the shape.
Use the vector to count along, then up or down to find the image.
The vector is written
along
up/down
.
A minus sign before the top number
means you move left.
A minus sign before the bottom
number means you move down.
Translate the L-shape in this diagram
⎛− 8⎞
using the vector ⎜⎜ ⎟⎟ .
⎝ 6⎠
Along 8 to the
left, then up 6.
Original material © Cambridge University Press 2009
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Cambridge Essentials Mathematics Core 8 GM3 Consolidation Worksheet
5
An enlargement of scale factor 3 gives an image with sides 3 times as long as the original
shape. Each vertex of the image will be 3 times as far from the centre of enlargement as
the original vertex and in the same direction.
You can do this more easily on squared paper by counting squares.
Enlarge this shape by a scale factor of 2.
Use the point P as the centre of enlargement.
Original material © Cambridge University Press 2009
Remember to go twice as far from P to the
image vertex. Count twice the distance
along and twice the distance down.
3
Cambridge Essentials Mathematics Core 8 GM4 Consolidation Worksheet
GM4 Consolidation Worksheet
1
Look at the diagrams.
State which triangles are
a
equilateral
b
right-angled
c
isosceles
S
Explain how you know.
2
A locus is a set of points that obey a set of conditions. It can be a line or an area.
Complete the following statements. The first letter of each missing word has been shown.
a
The locus of all points 4 cm from a point X would be a c____ ___________ with a
radius of 4 cm.
b
The locus of all the points that are equidistant from two lines OA and OB is the
b______________ of the a________________ between the two lines.
c
To find the locus of all the points which are the same distance from two points X and
Y, construct the p________
3
__ b__________ of the line joining X and Y.
Describe the shaded region in each diagram.
Explain the difference between the two diagrams.
a
Original material © Cambridge University Press 2009
b
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Cambridge Essentials Mathematics Core 8 GM4 Consolidation Worksheet
4
An angle measures an amount of turn. Angles can be measured using a protractor.
A bearing is an angle measured from North in a clockwise direction.
It describes the direction of one point relative to another.
Remember
North is always
up the page.
Bearings have 3 digits
so a bearing of 20° is
written as 020°.
The bearing of Q from P is x.
The bearing of P from Q is y.
For each diagram below, write down the three-figure bearing of Q from P.
a
5
b
c
d
In a scale drawing, the real length is either decreased or increased in a certain ratio.
A ratio of 1:200 means 1 cm as scale drawing represents 200 cm (2m) of real length.
The scale factor is 200.
To change from the scaled length to the real length you multiply by the scale factor.
To change from the real length to the scaled length you divide by the scale factor.
a
A plan is drawn to a scale of 1:30. Calculate the actual length in metres.
i Scaled length 10 cm
b
ii Scaled length 3.8 cm
A plan is drawn to a scale of 1:250. The scaled length is 12 mm.
What is the actual length in metres?
c
The scale used on a plan is 1:500. The real length is 100 m.
Check that
your answers
are in the
correct units.
1000 mm = 1 m
What will the scaled length be in centimetres?
Original material © Cambridge University Press 2009
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