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Crown copyright 200800366-2008PDF-EN-01

Pupils should learn to: As outcomes, Year 7 pupils should, or example:

202 The National Strategies | SecondaryMathematics exempliication: Y7

GEOMETRY AND MEASURES Transormations and coordinates

Understand and use the language and

notation associated with refections,

translations and rotations

Recognise and visualise transormations

and symmetries o shapes

Use, read and write, spelling correctly:transormation image, object, congruent

refection, mirror line, line o symmetry, line symmetry,

refection symmetry, symmetricaltranslation rotate, rotation, rotation symmetry,

order o rotation symmetry, centre o rotation

Refection

Understand refection in two dimensions as a transormation o aplane in which points are mapped to images in a mirror line or axiso refection, such that:

the mirror line is the perpendicular bisector o the line joiningpoint A to image A;

the image is the same distance behind the mirror as the originalis in ront o it.

Know that a refection has these properties:

Points on the mirror line do not change their positionater the refection, i.e. they map to themselves.

A refection which maps A to A also maps A to A,

i.e. refection is a sel-inverse transormation.

Relate refection to the operation o olding. For example:

Draw a mirror line on a piece o paper. Mark point P on one sideo the line. Fold the paper along the line and prick through thepaper at point P. Label the new point P. Open out the paper andjoin P to P by a straight line. Check that PP is at right angles tothe mirror line and that P and P are the same distance rom it.Repeat or other points.

Explore refection using dynamic geometry software.

For example:

Construct a triangle and a line to act as a mirror line. Constructthe image o one vertex by drawing a perpendicular to themirror line and nding a point at an equal distance on theopposite side. Repeat or the other vertices and draw the imagetriangle.Observe the eect o dragging vertices o the original triangle.What happens when the triangle crosses the mirror line?

A A

axis of reflection or mirror line

object image

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As outcomes, Year 8 pupils should, for example:

Crown copyright 2008 00366-2008PDF-EN-01

As outcomes, Year 9 pupils should, for example:

203The National Strategies | SecondaryMathematics exempliication: Y8, 9

GEOMETRY AND MEASURES Transformations and coordinates

Use vocabulary rom previous year.

Combinations of two transformations

Transorm 2-D shapes by repeated refections, rotationsor translations. Explore the eect o repeated refectionsin parallel or perpendicular lines. For example:

Refect a shape in one coordinate axis and then theother. For example, refect the shape below rst inthex-axis and then in the y-axis. What happens?What is the equivalent transormation?Now refect it rst in they-axis and then in the

x-axis. What happens? What is the equivalenttransormation?

Investigate refection in two parallel lines.For example, nd and explain the relationshipbetween the lengths AA2 and M1M2.

Investigate how repeated refections can be used togenerate a tessellation o rectangles.

Explore the eect o repeated rotations, such as halturns about dierent points. For example:

Generate a tessellation o scalene triangles (orquadrilaterals) using hal-turn rotations about themidpoints o sides.

Explain how the angle properties o a triangle (orquadrilateral) relate to the angles at any vertex o thetessellation.

Use vocabulary rom previous years and extend to:plane symmetry, plane o symmetry

axis o rotation symmetry

Combinations of transformations

Transorm 2-D shapes by combining translations,rotations and refections, on paper and using ICT.

Know that refections, rotations and translationspreserve length and angle, and map objects on tocongruent images.

Link to congruence (pages 1901).

Use mental imagery to consider a combination otransormations and relate the results to symmetry andother properties o the shapes. For example:

Say what shape the combined object and image(s)orm when:a. a right-angled triangle is refected along its

hypotenuse;b. a square is rotated three times through a quarter

turn about a corner;c. a scalene triangle is rotated through 180 about

the midpoint o one o its sides.

Working practically when appropriate, solve problemssuch as:

Refect this quadrilateralin they-axis.Then refect both shapesin thex-axis.In the resulting pattern,

which lines and which anglesare equal in size?

Flag A is refected in the line y=xto give A. A is thenrotated through 90 centre (0, 3) to give A.

Show that A could also be transormed to A bya combination o a refection and a translation.Describe other ways o transorming A to A.

03

3

2

2

11

1

2

3

4 1 2 3 4

A1 A2M1 M2A

3

03 2

2

11

1

2

3

1 2 3

x

y

A

A'

A''

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Crown copyright 200800366-2008PDF-EN-01

Pupils should learn to: As outcomes, Year 7 pupils should, or example:

204 The National Strategies | SecondaryMathematics exempliication: Y7

GEOMETRY AND MEASURES Transormations and coordinates

Refection (continued)

Refect a shape in a line along one side. For example:

Refect each shape in the dotted line.What is the name o the resulting quadrilateral?Which angles and which sides are equal?Explain why.

Refect a shape in parallel mirrors and describe what is happening.Relate this to rieze patterns created by refection.

Construct the refections o shapes in mirror lines placed atdierent angles relative to the shape. For example:

Which shapes appear not to have changed ater a refection?

What do they have in common?

In this diagram, explain why rectangle R is not the refection orectangle R in line L.

Recognise and visualise transormations

and symmetries o 2-D shapes

(continued)

F N

F

horizontal line

F

A

N

S

F

vertical line

A

N

S

F

F

sloping line

SA

L

R'R

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As outcomes, Year 8 pupils should, for example:

Crown copyright 2008 00366-2008PDF-EN-01

As outcomes, Year 9 pupils should, for example:

205The National Strategies | SecondaryMathematics exempliication: Y8, 9

GEOMETRY AND MEASURES Transformations and coordinates

Combinations of transformations (continued)

Understand and demonstrate some general results

about repeated transormations. For example: Refection in two parallel lines is equivalent to a

translation.

Refection in two perpendicular lines is equivalent toa hal-turn rotation.

Two rotations about the same centre are equivalentto a single rotation.

Two translations are equivalent to a singletranslation.

Explore the eect o combining transormations.

Draw a 1 by 2 right-angled triangle in dierentpositions and orientations on 5 by 5 spotty paper.Choose one o the triangles to be your original.Describe the transormations rom your original tothe other triangles drawn.Can any be done in more than one way?

Triangles A, B, C and Dare drawn on a grid.

a. Find a single transormation that will map: i. A on to C; ii. C on to D.

b. Find a combination o two transormations thatwill map: i. B on to C; ii. C on to D.c. Find other examples of combined

transformations, such as:A to C: with centre (0, 0), rotation o 90, ollowed

by a urther rotation o 90;A to C: refection in the y-axis ollowed by

refection in thex-axis;B to C: rotation o 90 centre (2, 2), ollowed by

translation (0, 4);C to D: refection in they-axis ollowed by

refection in the linex= 4;C to D: rotation o 270 centre (0, 0), ollowed by a

rotation o 90 centre (4, 4).

Use ICT, or plastic or card shapes, to generatetessellations using a combination o refections,rotations and translations o a simple shape.

Combinations of transformations (continued)

ABC is a right-angled triangle.

ABC is refected in the line ABand the image is thenrefected in the line CA extended.State, with reasons, what shape is ormed by thecombined object and images.

Two transormations are dened as ollows: Transormation A is a refection in thex-axis.

Transformation C is a rotation of 90 centre (0, 0).Does the order in which these transormations areapplied to a given shape matter?

Some congruent L-shapes are placed on a grid in thisormation.

Describe transormations rom shape C to each othe other shapes.

Some transormations are dened as ollows:

P is a refection in the x-axis.

Q is a refection in the y-axis.

R is a rotation o 90 centre (0, 0).

S is a rotation o 180 centre (0, 0).

T is a rotation o 270 centre (0, 0).

I is the identity transormation.

Investigate the eect o pairs o transormations and

nd which ones are commutative.

Investigate the eect o a combination o refections

in non-perpendicular intersecting mirror lines, linking

to rotation symmetry, the kaleidoscope eect and the

natural world.

Use dynamic geometry software to explore equivalences

o combinations o transormations, or example:

to demonstrate that only an even number o refections

can be equivalent to a rotation;

to demonstrate that two hal turns about centres C1

and C2 are equivalent to a translation in a direction

parallel to C1C2 and o twice the distance C1C2.

A

y

x

B

C D

A

C

B

x

y

F

A

E

H

G

C

D

B