Download KS3 Shape 3b Vector and Transformational



3D Shapes

Names of 3D solids
Prisms
Volume
Surface Area
Plans, elevations and nets
Isometric Drawing
Circles
Parts of
Circumference
Area
Circle geometry
Equation of a circle
Units of measure
Length, area and volume
Capacity / Mass
Metric and imperial measure
Conversion / conversion graphs
Real-life graphs
Speed, distance time


Right-angled triangles
Finding missing side lengths
Angles and sine, cosine and
tangent ratios
KS3 Shape, Space and
Measure
Constructions
Triangles
Similar & Congruent Shapes
Bisectors
Loci
Shape 3b: Vector and Transformational Geometry
Angles

Angles on a straight line
Angles at a point
Parallel lines and transversals alternating / corresponding /opposite angles
Supplementary angles
Polygons: interior & exterior angles
Extension work: Bearings
Lines & Angles
Line Segments
Vertical / Horizontal
Perpendicular / Parallel
Types of angle
Estimating measuring and drawing
Direction of turn
Compass directions
Describing angles (90o= ¼ of turn)
Pythagoras and
Trigonometry
Perimeter and Area
Dimensions (including volume)
Units of measure (including volume)
Counting squares
Intrinsic and Extrinsic information
Rectangles, triangles and compound shapes
Using Formulae (Rectangle, Triangle,
Trapezium, Parallelogram)



Properties of 2D Shapes
Names of polygons up to 10 sides
Special Quadrilaterals and Triangles
Geometric Properties
Tessellation
Transformations

a) Basic congruent & similar shapes
Coordinate geometry
Reflection (include lines of symmetry)
Rotation (include rotational symmetry)
Translation (including vector notation)
Enlargement
b) Vector analysis
Transforming functions
All topics can be covered by the end
of year 8.

Vector and Transformational Geometry (B to A* GCSE content)
Must
Understand the term, ‘translation’.
Should
Could
Understand the difference between
Understand that the sum of two
scalar quantities and vector quantities vectors AB and BC is equal to
Describe a translation using vector Multiply vectors by scalar quantities vector AC.
notation.
and understand that the resultant is a Describe vectors in component
form ai + bj where i and j are unit
𝑎 parallel vector.
Describe the inverse of a vector ( )
vectors in the horizontal and
𝑏 Consider whether two vectors are
𝑎
−𝑎
vertical directions respectively.
as equivalent to -1 × ( ) or as ( ) parallel by checking whether one
𝑏
−𝑏
vector is a scalar multiple of the other Derive a perpendicular vector and
Use vector notation, eg a to describe verify whether vectors are
perpendicular to each other
⃗⃗⃗⃗⃗ and –a to describe 𝐴𝑂
⃗⃗⃗⃗⃗ and
𝑂𝐴
describe line segments by adding
Use vector notation to define
vectors
parallelism and to define shapes
Use vector notation to describe
by their properties
proportions of a vector
Understand the equivalence between Describe a translation of a line using a Relate an equation to another
describing equations as y = ... and as vector in the form (𝑎)
equation using a vector for
𝑏
functions; f(x) = ... such that a
equations where one is a
function defines a dependency on an
translation of the other; eg y = (x
𝑎
Understand
that
a
translation
of
(
)
independent variable, x.
+ 1)2 + 2 is a translation of y = x2
𝑏
−1
will have a resulting equation such
by vector ( )
2
that f(x) becomes f(x – a) – b
Understand how circles centre (a,b) Use this information to sketch
a known equation,
have equations of the form (x – a)2 + lines based on
2
such
as
y
=
x
(y – b)2 = r2 by reference to a circle of
radius r, centre O.
Apply transformational geometry to
trigonometric graphs hence showing,
for example, that Sin (x) ≡ Cos (x
∓ 90o)
Understand the difference between Sketch graphs of equations which
Consider combinations of
enlargement and stretch (in either have been stretched in either the x or y transformations involving
the x direction or the y direction)
direction, including trig functions (eg translation and stretch and
y = (2x)2 and y = sin 2x or y = 2sinx) consider the order of
transformations when sketching
the graph
Key Words: Transform, translate, reflect, rotate, enlarge, map, mapping, congruent, similar, scale factor,
vector, object, image, centre of rotation, centre of enlargement, ray lines, clockwise, anticlockwise, symmetry,
vertex, vertices
Starters:
Identify congruent shapes
Double-doodling exercise (pen in both hands, each hand draws the mirror image of the other at the same time)
Drawing coordinate grids with accurate labelling
Shape 3b: Vector and Transformational Geometry
Activities:
Battleships game using vector notation
Using vector notation to navigate along a course, through a maze etc
Describing combinations of transformations as a single transformation or expressing a single transformation
as a combination of transformations
Plenaries:
Learning framework questions:
- What does ‘transformation’ mean?
- What clues can help us identify what transformation has been applied?
- Is the mirror image of a shape congruent?
- What is the difference between a vector and a coordinate?
- Where would you need to place the centre of rotation to create an image further away / closer to the
object / in the bottom left quadrant / overlapping the object etc
Resources:
10 ticks worksheets
Possible Homeworks:
Identify similar shapes around the home
Research similar shapes which follow the golden ratio
Teaching Methods/Points:
Transformations: objects and images
Students should understand that transformations are used to change shapes by translating, reflecting, rotating
or enlarging. A shape before a transformation is called the ‘object’, and a shape after a transformation is
called the ‘image’. There are two conventional ways that students should label the object and the image and
it is important that they are able to use both.
Each shape can be labelled with a capital letter as follows;
A
90o rotation
clockwise
Object
Object A is
transformed into
Image A’
A’
Image
or by labelling each vertex as follows;
A
Shape 3b: Vector and Transformational Geometry
A’
C
B
Object
C’
B’
Image
Reflection in
mirror line
Object ABC is
transformed into
Image A’B’C’
Congruent shapes
Shapes that are congruent are exactly the same size and shape. Usually they will appear in different
orientations. The following shapes are all similar;
In the topic of transformations, students must understand that rotations, translations and reflections will all
produce shapes of exactly the same size and therefore the object and the image will always be congruent.
Translations
To translate an object means to move it without rotating or reflecting it.
Every translation has a direction and a distance. They can be written using words or vectors;
So, for example, “move two spaces to the right and 3 spaces down” can be written in vector form for
transformations on coordinate grids as;
 2 
  which means ‘move 2 in the x direction and -3 in the y direction.’
  3
Encourage students to consider translating vertices on the shape (as these have coordinate locations) rather
than trying to move the entire shape which often leads to mistakes when counting squares.
Students should feel comfortable with translating shapes on squared paper (and even on blank paper by
referring to movement as specific measurements) and on coordinate grids, the drawing of which is an
essential prerequisite and this work provides an opportunity to further practise and consolidate the skill.
Consider the following examples of translations;
and
6 y
4
A’
2
A
x
-6
-4
-2
2
Shape 3b: Vector and Transformational-2 Geometry
-4
4
6
Object A has been
translated 3 spaces
left and 1 space up.
As a vector, the
translation is written;
  3
 
 1 
Working on basic vector geometry (regardless of ability level) is another opportunity to develop geometric
reasoning skills. So, introduce students to combinations of translations and ask them to describe the single
transformation that maps the object onto the final image as follows;
 3
A is mapped onto A’ using vector   . A’ is then translated onto A’’ using vector
 4
 3  2  5
   
The single transformation which maps A onto A’’ is; 
 4  1   3
2
  .
 1
Understanding a general conclusion that a single translation can be described as any combination of
translations, leads into more advanced work on vector analysis and reasoning; for example;
A
OA is described as vector a
OB is described as vector b
a
OC is described as vector c
O
b
B
Therefore, ½ BA = ½( BO  OA )
= ½(- b  a )
c
= ½ a -½ b
C
Rotations
The topic of rotation requires students to know the terms; ‘clockwise’ and ‘anticlockwise’ as describing
directions of rotation; ‘quarter turns’, ‘half turns’ and ‘three quarter turns’ and their equivalent ‘degrees of
turn’ to describe the size of a turn; and ‘centre of rotation’ to describe the fixed point about which a rotation
takes place.
Students need to be able to draw the image of an object after rotation (where there is no centre of rotation)
For example;
Object:
90o rotation
clockwise
Image
Progress onto rotations using centres of rotation;
Object
90o rotation
anticlockwise about
the centre of rotation
as shown
Shape 3b: Vector and Transformational Geometry
Centre of rotation
A rotation of a shape can be
drawn with tracing paper,
rotating the sheet about the
centre of rotation
Image
While tracing paper is convenient and relatively simple to use when drawing quarter, half, and three quarter
rotations, using ‘ray lines’ to show how each vertex of the objects maps onto each corresponding vertex of
the image helps deepen the understanding of rotation about a centre, and the effect of moving the centre of
rotation to other positions in relation to the shape.
For example;
C
A’
Object
Image
C’
A
90o rotation clockwise
about the centre of
rotation as shown
B’
B
The ray lines map each vertex
from the object onto each
corresponding vertex on the image
such that the distance between
each vertex and the centre of
enlargement is the same for both
object and image, and the angle
formed is 90o.
This level of understanding empowers students to perform rotations of angles other than 90, 180 or 270
degrees, and performing these more advanced rotations is a means to practise and consolidate skills in
measuring and drawing angles, and using ‘construction’ lines.
It is important that students practise rotating shapes on a coordinate grid, using their skills to plot vertices
given their coordinates, and performing rotations using centres of rotation also expressed as coordinates.
Consider examples a) and b) below;
a) rotate shape A, 90o clockwise about the origin and label the image A’
b) rotate shape A 180o clockwise about the centre of rotation (1,2) and label the image A’’
6 y
4
A
2
A’’
x
-6
-4
-2
2
-2
4
The ray lines for the rotation
of one vertex have been
shown, although the
reflections can equally be
performed using tracing
paper
6
A’
-4
-6
Reflections
Reflections can be drawn using ‘mirror lines’ or lines of symmetry that are vertical, horizontal, or (for the
purposes of KS3 and KS4 work) at 45o to the horizontal. Similarly, mirror lines can be located at a distance
from the object, touching the object at one or more parts, or even intersecting (travelling through) the object.
Here are examples of
reflections that students
should be able to perform
Shape 3b: Vector and Transformational Geometry
The key principle that students should adhere to in order to avoid careless errors is
to reflect each vertex of the object at 90o (perpendicular) to the mirror line and
to ensure the reflected vertex is an equal distant away from the mirror line on both the object and the
image. While ray lines have been shown for a selection of points on the examples above, there is no limit to
the number of ray lines that can be drawn.
Students will also need to reflect shapes (and points) on coordinate grids. Students should be prepared to plot
coordinates, reflect in the y-axis and x-axis, understand the equivalence of y-axis and x= 0, and x axis and y =
0, and reflect in the lines y = x or y = -x. More advanced students may attempt reflections in lines of the form
y = x + c or y = -x + c. These skills are explained fully in the learning plans, Arithmetic 9. Nonetheless, the
process of reflection follows the same guidelines as previously explained.
Enlargement, scale factors and similarity
Similar shapes and the relationship between the ratio of side lengths and the ratio of area and volume is
discussed in the learning plans, ‘Shape 6: Perimeter and Area’. This section describes the process of
‘enlargement’ as a transformation. There is a distinct difference between enlargement and the other three
transformations (translation, reflection and rotation) such that an enlargement produces a ‘similar’ shape
while other transformations produce ‘congruent’ shapes, as previously defined.
Explain to students that shapes that are similar have corresponding angles of the same size and
corresponding sides are in the same ratio. A scale factor describes the value that each side of the original
shape has been multiplied by. Consider the following examples;
eg
A
E
D
B
C
F
Shapes A and B are similar because the corresponding angles are the same size and all of the lengths of
Shape B are exactly twice the size of the lengths of Shape A. The ratio of the lengths is 1 : 2. This means
that Shape B is an enlargement of Shape A by scale factor 2.
Shapes C and D are similar because the corresponding angles are the same size and all of the lengths of
Shape D are three times the size of the lengths of Shape C. The ratio of the lengths is 1 : 3. This means that
Shape D is an enlargement of Shape C by scale factor 3.
Shapes E and F are similar because the corresponding angles are the same size and all of the lengths of
Shape F are 2.5 times the size of the lengths of Shape E. The ratio of the lengths is 1 : 2.5
[this can be
written as 2 : 5 by finding an equivalent ratio]. This means that Shape F is an enlargement of Shape E by
scale factor 2.5.
It is important to clarify that an ‘enlargement’ can lead to a reduction in size as much as it can lead to an
increase in size, and enlargements with a scale factor between 0 and 1 will lead to a smaller image. For
example, a scale factor of ½ creates an image which is half the size of the original object such that the
Shape 3b: Vector and Transformational Geometry
lengths are in the ratio 1 : ½. It is also worth discussing the fact that enlargements by scale factor 1 leads to a
congruent shape (as every corresponding length is in the ratio 1: 1 in a like for like way.)
Similar Shapes and Length, Area and Volume
Similar shapes have corresponding lengths in the same ratio. This means their areas and volumes are also in
corresponding ratios. This is discussed in greater detail in Shape 6: Perimeter and Area, but the general rule
that for an enlargement of scale factor n, the side lengths will correspond in the ratio 1 : n, while the area of
the shapes will correspond in the ratio 1 : n2 and the volume of the shapes will correspond in the ratio 1 : n3.
This can be investigated with the students.
Performing an enlargement (examples)
Corresponding
lengths are in the
ratio 1:2
The location of the
enlargement is
unimportant
Draw an enlargement of the following shape
scale factor 2
Draw an enlargement of the following shape
Using the centre of enlargement as indicated
By scale factor 3
a) Enlarge triangle A by scale factor ½, centre of enlargement: (5,5) and label the enlargement A’
b) Enlarge triangle A by scale factor 2, centre of enlargement: (5,5) and label the enlargement A’’
6 y
4
A’
This is the original
shape: Object A
A
A’’2
x
-6
-4
-2
2
-2
4
6
It is not always
necessary to draw
ray lines through
all of the vertices
-4
The position of the centre of enlargement
and the impact on the location of the image in comparison to the
-6
object following an enlargement can be explored.
Asking students what if questions, such as;
 What if the centre of enlargement is located on one of the sides of the shape?
 What if the centre of enlargement is located inside the shape?
 What if the centre of enlargement is located further away?
 What if I wanted the image to appear in a particular position?
helps to develop an understanding of how enlargements can be used and controlled to transform shapes.
For more able students, considering the effect of a negative scale factor (starting with negative 1, and
Shape 3b: Vector and Transformational Geometry
therefore producing a congruent shape with
in inverted positions) will further embed
and extend their understanding. Negative scale
corresponding images leads to interesting
conversations about the parallels with ‘sight’
received by the eyes and therefore the need
for the brain to re-invert the images it receives.
corresponding vertices
factors and their
and the way light is
Transformations: Translations, Reflections and Rotations Help Sheet
Transformations are used to change shapes by translating, reflecting, rotating or enlarging.
A shape before a transformation is called the ‘object’, and a shape after a transformation is called the ‘image’.
Congruent means exactly the same size and shape.
A vector still follows the rule of
Translations, reflections and rotations produce congruent shapes.
along the corridor first then up the
stairs, as with coordinates
Translations
To translate an object means to move it without rotating or reflecting it.
  3
Every translation has a direction and a distance. They can be written using words or vectors;
 
So, for example, “move three spaces to the left and 1 space up” can be written in vector form as;  1 
When translating shapes,
it is a good idea to
translate each vertex
(corner point) of the
shape to avoid making
mistakes
Object A has been
translated 3 spaces left
and 1 space up to form
Image A’
6 y
4
A’
2
Object A and Image A’
are congruent shapes
A
x
-6
-4
-2
2
4
6
Reflections
-2
A shape is reflected in a ‘mirror line’ so that every vertex (corner point) is an equal distance from the mirror
line and the reflection is at 90o (perpendicular) to the mirror line.
-4
Here are some examples of reflections
-6
Rotations
A rotation about a centre of rotation (COR) can be drawn using tracing paper or by using ‘ray lines’ and
measuring the angle between each vertex (corner point) the COR and the corresponding vertex on the reflection
Object
90o rotation
anticlockwise about
the centre of rotation
as shown
Shape 3b: Vector and Transformational Geometry
A rotation of a shape can be
drawn with tracing paper,
rotating the sheet about the
centre of rotation
Image
Object
Image
A
A’
90o rotation clockwise
about the centre of
rotation as shown
The ray lines map each vertex
from the object onto each
corresponding vertex on the image.
The distance between
each vertex and the centre of
enlargement is the same for both
object and image, and the angle
formed is 90o between them is 90o.
Transformations: Enlargements Help Sheet
An enlargement can make a shape bigger or smaller. All of the lengths are multiplied by the same scale factor.
Similar shapes have corresponding angles of the same size and corresponding sides are in the same ratio.
A scale factor describes the value that each side of the original shape has been multiplied by.
Consider the following examples;
eg
A
C
E
B
D
F
Shapes A and B are similar because the corresponding angles are the same size and all of the lengths of Shape
B are exactly twice the size of the lengths of Shape A. The ratio of the lengths is 1 : 2. This means that Shape B
is an enlargement of Shape A by scale factor 2.
Shapes C and D are similar because the corresponding angles are the same size and all of the lengths of Shape
D are three times the size of the lengths of Shape C. The ratio of the lengths is 1 : 3. This means that Shape D is
an enlargement of Shape C by scale factor 3.
Shapes E and F are similar because the corresponding angles are the same size and all of the lengths of Shape
F are 2.5 times the size of the lengths of Shape E. The ratio of the lengths is 1 : 2.5 [this can be written as 2 : 5
by finding an equivalent ratio]. This means that Shape F is an enlargement of Shape E by scale factor 2.5.
Drawing enlargements
Example 1: Draw an enlargement using the
centre of enlargement by scale factor 3
The dotted lines are
‘ray lines’ which
must be shown as part
of the construction of
the enlargement
Example 2: Enlarge triangle A by scale factor ½, centre of enlargement: (5,5) and label the enlargement A’
Example 3: Enlarge triangle A by scale factor 2, centre of enlargement: (5,5) and label the enlargement A’’
Shape 3b: Vector and Transformational Geometry
A’
6 y
This is the original
shape: Object A
A
A’’
4
2
x
-6
-4
-2
2
4
6
These lines are called, ‘ray
lines’.
It is not always necessary to
draw ray lines through all of the
vertices
-2
The
changes
-4
-6
Shape 3b: Vector and Transformational Geometry
position of the centre of enlargement
the position of the enlarged shape.