Download Grade England 3 TWO-DIMENSIONAL SHAPES （P.80

Grade
3
England
TWO-DIMENSIONAL SHAPES （P.80）
TRIANGLES：2-D shapes with three straight edges are called triangles.
POSITION AND DIRECTIO（P.88）
Find the position of a square on a grid and use four compass directions.
The position of a point on a grid is given by its co-ordinates.
The across co-ordinate always comes first.
ANGLE （P.90）
Recognize whole, half and quarter turns.
11 12
11 12
1
2
10
9
4
7
6
2
8
4
9
3
8
5
N
3
7
6
S
5
N
E W
W
1
10
E
S
4
AREA AND PERIMETER （P.82）
The area of a shape is the amount of surface it covers.
It is measured in squares, usually square meters（ m 2 ）or square
centimeters（ cm 2 ）.
The perimeter of a shape is the distance around its edges.
It is a length and is measured in units of length such as meters or
centimeters.
To understand the difference between area and perimeter think of a field.
The perimeter is the fence.
The area is the field itself.
6cm
4cm
Area = length × breadth ( l × b )
= ( 6 × 4 ) cm 2 = 24cm 2
Perimeter = 2 × ( l + b )
= 2 × (6 + 4) cm = 2 × 10 cm = 20 cm
Measure shape and work out the perimeter
TWO-DIMENSIONAL SHAPES （ P.92 ）
2-D shapes with three straight edges are called triangles.
Equal lines are shown with dashes and equal angles are marked.
A three-sided polygon is a triangle.
right-angled triangle
Isosceles triangle
equilateral triangle
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right-angled triangle
isosceles triangle
equilateral triangle
CO-ORDINATES （ P.102 ）
Learn to use co-ordinate to find the position of a point on a grid.
LINES （P.104）
The horizon is in the far distance where the land meets the sky.
A horizontal line is a flat line drawn in the direction of the horizon.
A vertical line is at right angles to a horizontal line.
A diagonal line goes from one corner to another.
Examples：
Horizontal lines –– top and bottom of the flag.
Vertical lines –– sides of the flag.
Diagonal lines –– cross on the flag.
horizontal
v
e
r
t
i
c
a
l
ANGLES （ P.108 ）
An angle measures the amount something turns or rotates.
Angles are measured in degree （ ° ）.
A whole turn is 360° .
A half turn is 180° .
A right angle is 90° .
The minute hand of a clock turns:
360° in one hour.
180° in 30 minutes.
90° in 15 minutes.
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11 12
1
10
2
8
4
9
3
7
5
6
Turning in a clockwise direction from:
N to S is 180° .
N to E is 90° .
N to NE is 45° .
N
NE
NW
W
E
SE
SW
S
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Clockwise and Anti-clockwise
AREA AND PERIMETER （P.84）
The area of a shape is the amount of surface it covers.
It is measured in squares, usually square meters（ m 2 ）or square
centimeters（ cm 2 ）.
The perimeter of a shape is the distance around its edges.
It is a length and is measured in units of length such as meters or
centimeters.
To understand the difference between area and perimeter think of a field.
The perimeter is the fence.
The area is the field itself.
6cm
4cm
Area = length × breadth ( l × b )
= ( 6 × 4 ) cm 2 = 24cm 2
5
Perimeter = 2 × ( l + b )
= 2 × (6 + 4) cm = 2 × 10 cm = 20 cm
Measure shape and work out the perimeter
CO-ORDINATES （P.92）
Use co-ordinates to find the position of
a point on a grid.
The position of a point on a grid is given
by its x and y co-ordinates.
LINES （P.94）
＊ to recognize parallel and perpendicular
lines.
Parallel lines are lines that are the same
distance apart for all their length.
Railway lines are parallel lines.
Perpendicular lines cross or meet at right
angles.
＊ to recognize diagonal lines.
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Diagonal lines go from one vertex （corner）
of a shape to another.
TWO-DIMENSIONAL SHAPES （P.96）
scalene triangle.
scalene triangle
right-angled triangle
isosceles triangle
equilateral triangle
ANGLES（P.107）
A whole turn 360° .
A right angle 90° .
An acute angle less than 90° .
An obtuse angle greater than 90° and less than 180° .
A whole turn 360
A right angle 90
An acute angle Less than 90
An obtuse angle Greater than 90
and Less than 180
USING A PROTRACTOR （P.107）
Most protractors have two scales, a clockwise outer scale and an
anti-clockwise inner scale. It is important to use the correct scale.
COMMON MISTAKES
1. Using wrong scale.
2. Reading the scale in the wrong direction.
The Mark
（
∧
）,
∧
example AOE
ANGLES IN A STRAIGHT LINE （P.113）
Example:
45
x
x ° + 45° = 180°
x ° = 135°
6
AREA AND PERIMETER （P.84）
To calculate the area of a right-angled triangle.
Example
Think of the triangle as half of a rectangle.
4 cm
5 cm
Think of the triangle as half of a rectangle.
Area of rectangle = (5 × 4) cm 2 = 20 cm 2
Area of the triangle = 20 cm 2 ÷ 2 = 10 cm 2
LINES （P.90）
＊ to recognize parallel and perpendicular
lines.
Parallel lines are lines that are the same
distance apart for all their length.
Railway lines are parallel lines.
Perpendicular lines cross or meet at right
angles.
to recognize intersecting lines and intersections.
Two lines that cross each other are called intersecting lines.
The point at which they cross is an intersection.
Point A is the intersection of the
diagonal of rectangle.
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A
ANGLES （P.105）
A REFLEX ANGLE greater than 180° .
A whole turn 360
An acute angle Less than 90
An obtuse angle Greater than 90
and Less than 180
A REFLEX ANGLE
greater than 180
MISSING ANGLES
Example:
․ANGLES ON A STRAIGHT LINE
The sum of the angles on a straight line is 180°
․ANGLES AT A POINT.
A whole turn is 360°
80
y
y + 80° = 360°
y = 280°
․ANGLES IN A TRIANGLE
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The sum of the angles in a triangle is 180° .
#7 Angle （P.47 7T/7）
understand acute、obtuse、reflex and right angles.
An angle is made when a line turns from one position to another.
full turn
half turn
a quarter turn
A quarter turn is also called a right angle.
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A. Comparing angles
B. Measuring angles
C. Drawing angles
D. Angles round a point
E. Calculating angles
Angles round a point
Angles on a line
Vertically opposite angles
#23 Tringles （P.164 7C/7）
※draw triangles with particular lengths and angles.
※identify special sorts of triangle and use heir properties.
※calculate with angles of a triangle.
A. Drawing a triangle accurately
arc、radius、constructions、vertices
some new words to use
◎Part of a circle is called an arc.
◎The distance between the points of a pair of compasses is called the
radius of the arc or circle you draw.
◎Drawing a shape accurately with a pencil, ruler and compasses is
sometimes called constructing it. So drawings done this way are called
constructions.
◎An other word for the corner of a shape is vertex. The plural of vertex
is vertices
（pronounced verty-seas）.
B. Equilateral triangles
A triangle with three sides the same length is called an equilateral
triangle.
Tetrahedron （4 triangles） 、Hexahedron （6 triangles） 、Icosahedron
（20 triangles）
C. Isosceles triangles
A triangle with two or more sides the same length is called an isosceles
triangle.
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D. Scalene triangles
all sides of different lengths
E. Using angles
Need to know at least one angle of the triangle.
F. Angles of a triangle
◎ 180°
◎A triangle with one of its angles 90° is called a right-angled triangle.
G. Using angles in isosceles triangles
#26 Area and perimeter （P.195 7T/13、7C/17）
What is the area of each triangle?
#35Perpendicular and parallel lines
（P.250 7T/20、7T/22、7C/24、7C/30）
※identify perpendicular and parallel lines.
※draw perpendicular and parallel lines.
#51 Functions and graphs
※draw a graph based on a rule.
※find the equation of a straight-line graph.
C. You can think of the equation y = 2 x + 1 as a rule, or function, linking
x
and y .
For each value of x , you can find the value of y .
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#2 Ratio （P.6）
※use notation for ratio.
※understand equal ratios.
※share a quantity in a given ratio.
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※compare ratios.
#6 Fractions, decimals and percentages （P.37）
※revise fractions, decimals and percentages.
#8 Area （P.114）
These formulas all say the same thing.
Area of triangle =
A=
base × height
2
bh
1
bh ， A =
2
2
Think of them as meaning
‘work out base height and halve it.’
Remember that b can be the length of any side of the triangle, but h must
be measured at right angles to b.
#16 Squares, cubes and roots （P.160）
※work with square and cube number
※find square roots （positive and negative） and cube roots.
#22 Scaling （P.152）
※spot shapes that have been scaled.
※use scale factors.
※use and make scale drawings.
#24 Bearings （P.170）
※measure and record a direction as a three-figure bearing.
※fix the position of a point by using its bearing form two other points.
#29 Constructions （P.207）
※draw triangles accurately.
※use drawing methods involving ruler and compasses only.
#32 Rules and coordinates（P.222）
※graph functions in all four quadrants.
※find the mid-point of
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a line segment, given its end points.
#5 Working with rules （P.37）
※solve simple equations.
※substitute numbers into a formula to give an equation and solve it.
※change the subject of a simple formula.
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#6 Circumference of a circle
※calculate a circle＇s circumference from its diameter or radius.
※calculate its diameter or radius from its circumference.
※solve problems that involve these measurements.
The circumference of a circle is the distance all round it.
Formulas for circumference（P.50）
The exact number to multiply by is called π .
Formulas for circumference
let r be the radius of a circle, d be the diameter, and C the circumference.
Then C= π d.
Also, because d=2r, it follows that
C= π ×2r which C=2 π r
radius r
diameter d
#8 Enlargement （P.55）
※enlarge a shape from a center.
※solve enlargement problems.
C. Scale factors
#10 Straight-line graphs （P.63）
※draw straight-line graphs from algebraic equations.
※work out the gratient and y-intercept of a straight line.
※use the gradient and y-intercept of a straight line to work out its
equation.
#11 Point, lines and arcs （P.73）
※understand the idea of a locus.
（a set of points that follow some rule）
※make accurate drawings and get information from them.
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#13 Ratio and proportion (P.85)
※write a ratio as a single number.
※use ratios to decide if two quantities are in direct proportion.
※solve problems involving direct proportion using algebra.
This rectanglar window is 50 cm high and 20 cm wide.
height
50 cm
width
20 cm
The ratio height：width is 50:20 or 5:2
Another way to write the ratio is as a division：
So, for this window, the ratio
height
width
height 50
=
= 2.5
width 20
#14 Angle of a polygon （P.91）
※relate the angles of a polygon to the number of sides it has.
※work out and use the angles of regular polygons.
If you extend a side of a polygon, this angle is called an exterior angle.
#21 Angles （P.151）
※revise earlier work on angles.
※learn about angles and parallel line.
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#1 Pythagoras’s theorem （P.4）
※find the length of one side of a right-angled triangle if you know
the lengths of the other two sides.
※solve problems involving the lengths of sides right-angled triangles.
A. Areas of tilted squares
B. Squares on right-angled triangles
Pythagoras’s theorem
In a right-angled triangle the side
opposite the right angle is called
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the hypotenuse.
The area of the square on the hypotenuse equals the total of the areas of
the squares on the other two sides.
C. Squares roots
D. Using Pythagoras
#20 Using a calculator
※what brackets mean in a calculation.
※how to round to a number of decimal places or significant figures.
※use a calculator for complex calculations.
※work with squares, square roots and negative number.
#25 Parallel lines and angles （P.222）
｀Angle
DAB＇means the angle A at its vertex and D and B along its
arms.
n
You can also write ∠ DAB or DAB
#35 Gradient（P.297）
※calculate positive and negative gradients
※interpret gradients as rates
Example
Water is added to a container over a period of 8 minutes.
Water flows in at a slow steady rate.
This graph shows the volume of water in the container during these 8
minutes.
90
+ 60 liters
60
30
0
0
2
6 minutes
4 6 8
Time (minutes)
For a horizontal increase of 6 minutes
There is a vertical increase of 60 liters.
The gradient is
So the rate of
60
= 10
6
flow of the water is 10 liters per minute.
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#36 Maps and plans （P.309）
※use simple scales.
※use bearings.
※use four-figure grid references.
※use different scales on maps.
A. Working to plans
Use simple scales.
Scale：1 cm represents 2m
B. Grid references
C. Scales
D. Bearings （P.309）
◎
◎
Bearings tell you which direction to go towards a given point.
Bearings are always measured
clockwise from North.
◎
Vertical lines on maps usually point to North.
◎
Bearings are always given as three figures, for example 065° .
#37. Ratio （P.312）
Revise using ratios given in the form a：b
and a：b：c
#38. Similar shapes
※Deal with decimal scale factors.
※Understand similar shapes and deal with ratios within shapes.
Similar triangles （P.324-P.348）
◎
If tow triangles are exact scaled copies of each other they are
called similar triangles.
◎
If the same scale factor is used to enlarge each corresponding
side of two triangles then the triangles are similar.
◎
Any two triangles with all the same angles must be similar.
#42 The tangent function （P.364）
A. Finding an opposite side
◎
In a right-angled triangle.
When the angle is 35° , use the rule：
10
adjacent side
× 0.7
opposite side
This is another way of stating the rule：
opposite　side
= 0.7
adjacent side
◎Draw a horizontal line 10 cm long with a vertical line at the right hand
end of it.
◎Draw angles of 10° 、 20° 、 30° ,…, 70° .
◎Measure the opposite side for each triangle and record the results in a
tale like the following：
Angle
10°
20°
30°
40°
Adjacent
×
side
10cm
10cm
10cm
10cm
Opposite
Work out the
numbers that
write them in
this table.
◎Use calculators to find tan10° 、 tan 20°
,…, tan 70° , to 3 d.p.
Compare these results with what got in the above table.
B. Finding an adjacent side
C. Finding an angle
example：
Find tan 23° and the angle whose tangent is 0.78 in the tangent
simple table.
D. Mixed questions and problems
Summary：
The tangent of angle can be thought of as a multiplier or as a ratio：
opposite　tan a =
adjacent 10
#45 Triangles and polygons exterior angle （P.391）
◎An exterior angle of a triangle is equal to the sum of the other two
interior angles.
◎The sum of the interior angles of a polygon with n sides is
180(n-2)° .
◎A regular polygon has all its sides equal and all its angles equal.
#49 Sine and cosine （P.431）
A. The tangent function – a reminder
opposite　side
tan a =
adjacent side
B. The sine function
In a right-angled triangle.
When the angle is 40° , use the rule：
hypotenuse
× 0.64
opposite side
This is another way of stating the rule：
side
opposite　= 0.64
htpotenuse
The number you multiply the hypotenuse by to get the opposite side is
called the sine of the angle.
So the sine of 40° is about 0.64
◎Work out
sin 10° 、 20° 、 30° ,…, 60° 、 70°
the following table:
Angle
Hypotenuse
10°
10cm
10cm
10cm
10cm
20°
30°
40°
×
Work out the
numbers that
write them in
this table.
Opposite
10
C. Using sine to find the hypotenuse
D. Finding an angle
E. The cosine function