Mathematics with ICT in Key Stage...

Mathematics with ICT

in Key Stage 3

Geometry lessons

Mathematics with ICT in Key Stage 3 | Geometry lessons 2

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IntroductionThe introduction to the document Integrating ICT into mathematics in Key Stage 3

states:

Computers offer powerful opportunities for pupils to explore mathematical

ideas, to generalise, explain results and analyse situations, and to receive fast

and reliable, and non-judgemental, feedback. Their use needs careful planning

– not just of the organisation of hardware and appropriate software but also of

activities that allow for off-computer mathematical thinking as well as on-

computer exploration.

Mathematics with ICT in Key Stage 3 comprises sets of teaching notes, lesson

plans and resources. Lesson content has been developed through a flexible use of

the Sample medium-term plans for mathematics. Objectives are drawn from both

the Framework for teaching mathematics: Years 7, 8 and 9 and the Framework for

teaching ICT capability: Years 7, 8 and 9.

These geometry lessons use Logo and dynamic geometry software to explore

transformations (Years 7 and 8), and angles in polygons (Year 9).

In planning these lessons:

• the mathematics objectives are drawn from ‘Shape, space and measures’ and

‘Using and applying mathematics to solve problems’ in the mathematics

Framework;

• the ICT objectives are from ‘Models and modelling’ in the ICT Framework.

Using ICT in the mathematics classroom

Integrating ICT into mathematics in Key Stage 3 comments on the use of Logo:

Logo is a powerful computer programming language that is easily accessible to

pupils. Programming in Logo uses simple everyday words. Logo can be ‘taught’

to do new things by defining new words (procedures). The most recognisable

feature of Logo is the microworld of turtle geometry, where pupils can explore

angle, shape and space by programming turtle movement around the screen.

Logo can be used to explore and develop understanding of other areas of

mathematics, including number and algebra; it can also be used to develop and

test mathematical models, and to set up and manage control, monitoring and

modelling activities. Logo can be used in a variety of ways in Key Stage 3:

• exploring and learning about the properties of shapes;

• generating number patterns and sequences;

• developing the concept of a function;

• developing simple programming skills.

The benefits of using dynamic geometry software are also discussed in the

booklet:

Dynamic geometry software allows pupils to explore and learn geometrical

facts through experimentation and observation. Pupils can construct figures on

the screen and then explore them dynamically. When an independent point or

line is dragged with the mouse, all dependent constructions remain intact. They


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can be used to understand what stays the same and what changes under

different conditions. They can motivate pupils to explain and prove. Dynamic

geometry software can be used in a variety of ways in Key Stage 3:

• exploring and learning about the properties of shapes;

• studying geometric relationships and learning geometrical facts;

• transforming shapes;

• working with dynamic images to make and test hypotheses about

properties of shapes;

• making and exploring geometric constructions;

• constructing and exploring loci.

The report ICT and mathematics: a guide to learning and teaching mathematics

11–19 contains further relevant information – see especially page 34 on Logo and

pages 41–43 on the use of dynamic geometry software.

Lesson outlines

Recognising rotation symmetry (7S4)

This lesson is designed to be used as part of the teaching programme for Year 7 in

the summer term. It uses Logo to explore transformations and should form part of a

series of lessons which may or may not use ICT, as appropriate. The objectives

are taken from the unit ‘Shape, space and measures 4’ in the Sample medium-

term plans for mathematics.

Transformations and tessellations (8S3)


the spring term. It uses dynamic geometry software to explore transformations and

tessellations and should form part of a series of lessons which may or may not use

ICT, as appropriate. The objectives are taken from the unit ‘Shape, space and

measures 3’ in the Sample medium-term plans for mathematics.

Generalising about polygons (9S1)


the autumn term. It uses Logo to explore angles in polygons and should form part

of a series of lessons which may, or may not, use ICT, as appropriate. The

objectives are taken from the unit ‘Shape, space and measures 1’ in the Sample

medium-term plans for mathematics.


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Overview of lessons

Recognising rotation symmetry


the summer term. It uses Logo to explore transformations and should form part of a

series of lessons which may or may not use ICT, as appropriate. The objectives

are taken from the unit ‘Shape, space and measures 4’ in the Sample medium-

term plans for mathematics.

Lesson objectives

Mathematics objectives

• Present and interpret solutions in the context of the original problem; explain

and justify methods and conclusions, orally and in writing.

• Recognise and visualise the transformation and symmetry of a 2-D shape:

– rotation about a given point, and rotation symmetry.

ICT objectives

• Use software to investigate and amend a simple model by:

– entering rules or formulae and checking their appropriateness and accurate

working;

– explaining the rules governing a model.

Resources

Main ICT resources

• Computer and data projector for teacher demonstration and pupil presentation

• Logo software

• Computer room or networked laptop computers (minimum of one computer per

two pupils)

Other resources

• Whiteboard or flipchart

• Resources to be displayed or printed

SH Logo helpsheet

7S4.1 Rotation challenge

Prerequisites

Pupils’ prior learning

Pupils should already know:

• the language associated with reflections, translations and rotations;

• how to save files to, and import files from, a designated area on the school

network.

Lesson 7S4


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ICT expertise needed by teacher

To teach these lessons you will need to know how to:

• use simple Logo commands such as forward (FD), right (RT), clear screen

(CS), left (LT) and repeat;

• build and edit simple procedures in Logo;

• save files to, and import files from, the shared area of the school network.

Preparation and planning

Before the lesson, set up and save the procedures FLAG and RTRIANGLE (see

the teaching notes in the lesson plan) on the shared area of the network.

Provide a wall display of the key vocabulary.

Print copies of any of the resources you need.

Key vocabulary

angle of rotation

centre of rotation

diagonal

equal angles and sides

parallel

right angle

rotate

rotation

rotation symmetry

Transformations and tessellations


the spring term. It uses dynamic geometry software to explore transformations and

tessellations and should form part of a series of lessons which may or may not use

ICT, as appropriate. The objectives are taken from the unit ‘Shape, space and

measures 3’ in the Sample medium-term plans for mathematics.

Objectives


• Use logical argument to establish the truth of a statement.

• Transform 2-D shapes by simple combinations of rotations, reflections and

translations, on paper and using ICT; identify all the symmetries of 2-D shapes.

ICT objectives

• Develop ICT-based models and test predictions by changing variables and

rules.

Lesson 8S3


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Resources

Main ICT resources

• Dynamic geometry software such as Cabri GeometryTM or The Geometer’s

Sketchpad®

• Interactive whiteboard or data projector for teacher demonstration and pupil

presentation


two pupils)

Other resources


• Resources to be displayed or printed

8S3.1 Hexagon and equilateral triangle tessellation

8S3.2 Tessellating quadrilaterals

Prerequisites

Pupil prior learning


• how to save files and load files.


To teach these lessons you will need to know how to:

• use basic geometric objects (e.g. points, lines and circles) to construct simple

geometrical figures (such as different types of triangles);

• build simple macros with Cabri or scripts with The Geometer’s Sketchpad (see

teaching notes in the lesson plan);

• save and load files.


Practise building and saving simple Cabri macros or scripts in The Geometer’s

Sketchpad.

Provide a wall display of key vocabulary.

Key vocabulary

reflect

reflection

rotation

tessellate

tessellation

transformation

translate

translation


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Generalising about polygons


the autumn term. It uses Logo to explore angles in polygons and should form part

of a series of lessons which may, or may not, use ICT, as appropriate. The

objectives are taken from the unit ‘Shape, space and measures 1’ in the Sample

medium-term plans for mathematics.

Objectives


• Present a concise, reasoned argument, using symbols, diagrams, graphs

and related explanatory text.

• Explain how to find, calculate and use:

– the interior and exterior angles of regular polygons.

ICT objectives


rules (Year 8).

Resources

Main ICT resources

• Logo software

• Data projection for teacher demonstration and pupil presentation


two pupils)

Other resources


• Resources to be displayed or printed:

SH Logo helpsheet

Prerequisites

Pupil prior learning


• angle and side properties of equilateral, isosceles and right-angled triangles

and special quadrilaterals;

• how to use simple Logo commands such as forward (FD), right (RT), clear

screen (CS), left (LT) and repeat;

• how to save files to, and import files from, the shared area of the school

network.


To teach these lessons you will need to how to:

• use simple Logo commands and build and edit simple procedures in Logo;

Lesson 9S1


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• save files to, and import files from, the shared area of the school network.


Practise creating and saving simple Logo procedures (such as the ones suggested

in the lesson plan).

Provide a wall display of the key vocabulary.

Key vocabulary

equal sides and angles

exterior angle

interior angle

irregular

polygon

regular


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Recognising rotationsymmetry

Objective

• Visualise and describe 2-D shapes.

Using a computer and a data projector, create a procedure in Logo that will

generate a square:

TO SQUARE

REPEAT 4 [FD 100 RT 90]

END

Ask pupils to explain why this produces a square.

Q How many sides and angles are there?

Q What happens if any of the numbers in the procedure are changed?

Q How could the square be made larger or smaller?

Point out the efficiency of procedures and the usefulness of the repeat command.

Now produce the diagram on the left:

REPEAT 8 [SQUARE RT 45]

Ask pupils to describe the resulting shape. Focus on the use of precise geometrical

language, particularly the language of rotation and rotation symmetry.

Extension

Ask pupils to talk in pairs and write down the instructions to create (a) an

equilateral triangle and (b) a regular hexagon. Confirm their instructions on the

computer or ask a pupil to verify by ‘acting turtle’.


• Present and interpret solutions in the context of the original problem; explain

and justify methods and conclusions, orally and in writing.

• Recognise and visualise the transformation and symmetry of a 2-D shape:

– rotation about a given point, and rotation symmetry.

ICT objectives

• Use software to investigate and amend a simple model by:

– entering rules or formulae and checking their appropriateness and accurate

working;

– explaining the rules governing a model.

7S4

Starter

10 minutes

Vocabulary

centre of rotation, equalangles and sides, parallel,rotate, rotation symmetry,

square

Resources

Logo software and dataprojector for whole-classprojection

Main teaching

40 minutes

Vocabulary

angle of rotation, centre ofrotation, diagonal, rightangle, rotate, rotation,

rotation symmetry

Resources

Logo software and dataprojector for whole-class

projection

Helpsheet SH

Resource 7S4.1


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Create for the class a Logo procedure to draw a flag:

TO FLAG

FD 60


BK 60

END

Ask pupils to explain what is happening. Focus their answers by asking some of

the following questions.

Q Where does the drawing start?

Q How long is each line?

Q By how much does the pen turn at each corner?

Q What is the use of the command REPEAT?

Clear the screen using the command CS, and type:

REPEAT 4 [RT 90 FLAG]

Ask pupils to predict the resulting shape. Confirm by using the computer. Ask them

to discuss the following questions in pairs:

Q How many times did the flag shape rotate? What is the order of rotation

symmetry? Where is the centre of rotation?

Point out the connections between their answers and the Logo procedure. Ask:

Q What if RT 90 is changed to RT 60? To RT 45?

Show pupils how to access the two procedures FLAG and RTRIANGLE previously

set up on the shared area on school network (see the teaching notes below).

Remind them of the command CS to clear the screen. Make sure that for each

procedure pupils note where the turtle starts and ends and the direction that it is

pointing.

Give out copies of resource 7S4.1. Ask pupils to work in pairs to complete the two

problems. Where appropriate, provide copies of helpsheet SH. After a suitable

time, bring the class together to share solutions. For example, one possible

solution to the second problem is:

REPEAT 4 [RTRIANGLE FD 60 RT 90]

Demonstrate the use of the simple procedure:

REPEAT 4 [RTRIANGLE FD 60 RT 90]

Discuss how the top shape on the right has been constructed,

making particular reference to the size of angles and sides.

Use the procedure to create the lower shape; ask pupils to

describe and justify its geometric properties. Focus on the use

of accurate geometrical language.

FLAG

RTRIANGLE

Plenary

10 minutes

By the end of the lessonpupils should be able to:

• rotate a shape;

• explain the result of

rotating a shape;

• explain how to embedLogo procedures to createnew procedures.


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Extension

Demonstrate the procedure:

RT 45

REPEAT 4 [RTRIANGLE RT 90]

Ask pupils to present a convincing argument to a partner that the shape is a square

by identifying equal sides and the property that the diagonals bisect each other at

right angles. Could they convince the class?


Before the lesson, set up and save the procedures FLAG and RTRIANGLE on the

shared area of the network.

TO FLAG

FD 60


BK 60

END

TO RTRIANGLE

FD 60 RT 135

FD SQRT 7200 RT 135

FD 60 RT 90

END

Main activity

Working with the whole class and using the whole-class display, take pupils

through the process of recognising and visualising rotation symmetry and rotation

about a given point. Make sure pupils are given the opportunity to share their

solutions to the two challenges. Pupils should be asked to describe the completed

shapes. You should focus on pupil use of correct geometrical language.

Teaching notes


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Transformations andtessellations

Objectives

• Make a general statement about familiar shapes.

• Explore transformations and symmetries using ICT.

The purpose of this part of the lesson is to consider how 2-D tessellations can be

described using the language of geometrical transformations.

Show pupils a tessellation such as the one on the right

(from resource 8S3.1):

Ask them to rehearse, in pairs, a clear description of the tessellation. Tell them that

their description should include the basic shape that is being tessellated and the

transformation.

Ask a pair to share their description and ask other pupils to help refine their use of

correct geometrical language. Ask if any pair have a description using a different

transformation. Aim to get two or even three different descriptions.

[If the basic shape is a regular hexagon, the possible transformations are:

translation; reflection in a side; rotation through 120° about a vertex.]


• Use logical argument to establish the truth of a statement.

• Transform 2-D shapes by simple combinations of rotations, reflections and

translations, on paper and using ICT; identify all the symmetries of 2-D shapes.

ICT objectives


rules.

This activity focuses on simple regular tessellations based on reflections. Tell the

class that they are going to investigate and then justify which regular and irregular

shapes will tessellate.

Working with the whole class, use Cabri or

The Geometer’s Sketchpad (GSP) to model

the construction of a triangle as shown here.

Point out that A and B are the centre points

of the two circles.

Ask pupils to define the properties of the

triangle and use their explanations to justify

why the triangle is equilateral.

G28S3

Starter

10 minutes

Vocabulary

reflect, reflection, tessellate,tessellation, transform,transformation, translate,

translation

Resources

Whiteboard or flipchart

OHP or data projector

Resource 8S3.1

Regular polygon mats/tiles,posters of tiling patterns

Main teaching

40 minutes

Resources

Cabri or The Geometer’s

Sketchpad software

Whole-class projection

Resource 8S3.1


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Hide the circles and then model the creation of a tessellation formed by the

continued reflection of the triangle in one of its sides. This pattern is also shown on

resource 8S3.1.

[You can model the use of the Cabri macro or GSP custom tool facility to define a

construction for an equilateral triangle on a given base (use the two vertices of the

base as the initial objects and the triangle as the final object). Save it as EquiTri,

say. Then model the use of EquiTri and reflection to create the tessellation.]

Ask pupils to work in pairs with a computer to replicate this activity [using a

macro/tool] to construct:

• a square and its associated tessellation;

• a regular hexagon and its associated tessellation.

Bring pupils together and ask them to refer to the tessellations of the regular

triangles, squares and hexagons to explain why these shapes tessellate. Ask them

to consider the interior angles of each tessellating shape.

Pose the question:

Q Will every regular polygon tessellate? Why? Why not?

Using dynamic geometry software, demonstrate that

triangles and quadrilaterals (regular and irregular)

tessellate by rotating the chosen polygon through 180°

about the midpoints of each of the figure’s sides.

If time permits, you may wish pupils to investigate tessellations using other

polygons as starting points. For example, can they explain why a regular pentagon

does not tessellate? There is an opportunity to explore some of the 14 different

types of convex pentagons that tile the plane. More information can be found on:

http://www.mathpuzzle.com/tilepent.html

You may wish to encourage some pupils to find out about the classification system

and try to find a further type of convex pentagon that tiles the plane.

[The regular pentagon does not tessellate. So not every pentagon will tessellate –

but there are some that will, like the one on the left. The question that has puzzled

mathematicians is: ‘How many pentagons will tessellate?’ Prior to 1968, it was

thought that all tessellating pentagons could be classified into five types but in that

year three more types were found. By 1975, no more types had been found but an

article in Scientific American sparked interest in its readers and led to the discovery

of a further type. By 1977, Marjorie Rice, who had no formal training in

mathematics, had found four more types. In 1985, a fourteenth type of tessellating

pentagon was found. The question facing mathematicians now is: ‘Are all the types

of tessellating pentagon now known?’]

Plenary

10 minutes

Resources

Resource 8S3.2


• transform 2-D shapes bysimple combinations ofrotations, reflections andtranslations, on paper andusing ICT.


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Creating macros using dynamic geometry software

Using The Geometer’s Sketchpad, here is the script for an equilateral triangle:

The corresponding macro for Cabri is:

Screen shots from Cabri Géomètre® II are reprinted by courtesy of Texas Instruments and with permission fromCabrilog

Teaching notes


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Generalising aboutpolygons

Objectives

• Make a general statement about familiar shapes.

• Explain how to calculate and use the sums of the interior and exterior angles of

regular polygons.

The focus for this activity is the precise use of geometrical language, particularly

the language of exterior angle and interior angles.

Show the Logo procedure


Using a diagram, equipment or pupil turtle, illustrate the resulting picture, pointing

out the angle of turn. (Is this the exterior or interior angle?)

Using appropriate diagrams, equipment or pupil turtles, discuss the properties of

the following regular polygons: triangle, square, pentagon, hexagon, octagon and

decagon. Tabulate the information as follows.

Polygon No. of sides Angle of turn

triangle

square 4 90

Make the point that in order to draw a closed shape the turtle must make a total

turn of 360°.

Ask pupils, in pairs, to predict the resulting shape from the procedures:



Tell them that they will have the opportunity to test out their predictions in the main

part of the lesson.


• Present a concise, reasoned argument, using symbols, diagrams, graphs

and related explanatory text.

• Explain how to find, calculate and use:

– the interior and exterior angles of regular polygons.

ICT objective


rules (Year 8).

By the start of the main activity, pupils will have understood that an efficient way of

drawing a square in Logo is with the following procedure:

G39S1

Starter

10 minutes

Vocabulary

equal sides and angles,interior and exterior angle,

irregular, polygon, regular

Resources

Whiteboard or flipchart

Main teaching

40 minutes

Resources

Logo software

Data projector

Computers or networkedlaptops sufficient for

pupils to work in pairs

Helpsheet SH


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TO SQUARE


END

Suggest that pupils work in pairs at the computer. Tell them that their task is to

change this procedure to generate other regular polygons. Some pupils may find

that Logo helpsheet SH provides a useful guide.

Encourage pupils to limit their exploration to one variable at a time and to make

use of the properties of shapes discussed in the starter. Tell them to test out their

suggestions.

In a mini-plenary draw out the conflict that may arise between the angle of turn and

the resulting polygon, emphasising the relationship between the number of turns

and the size of angles when drawing regular polygons.

Encourage the more able pupils to write and test procedures to generate any

regular polygon. Emphasise the relationship between the total turn 360°, the angle

of turn of each polygon and the number of sides.

Pupils are asked in this part of the lesson to give a reasoned argument on the

relationship between the angles, both interior and exterior, and the number of sides

of the polygon.

Ask pupils to predict the procedure for a 12-sided polygon.

Q What will be the repeated angle of turn?

Q What about a 20-sided polygon? An n-sided polygon?

Now demonstrate the procedure below, noting that the variable :sides represents

the number of sides of the polygon:

TO POLYGON :sides

REPEAT :sides [FD 100 RT 360/:sides]

END

Test the procedure for some simple regular polygons – triangle, square and

hexagon.

Using all available data ask pupils to deduce the exterior angles and then interior

angles of the following regular polygons:

triangle

square

pentagon

hexagon

octagon

decagon

20-sided

n-sided

Plenary

10 minutes


• explain how to find,calculate and use thesums of the interior andexterior angles ofquadrilaterals, pentagonsand hexagons;

• explain how to find,calculate and use theinterior and exteriorangles of regular

polygons;

• use variables whenwriting simple Logoprocedures.


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Main activity

Working with the whole class and using a whole-class display, take the pupils

through the process of writing a simple procedure to generate any regular polygon

and ask them to describe the angle properties of the completed shapes. You

should focus on pupil use of correct geometrical language.

In creating procedures to draw an equilateral triangle and a regular pentagon,

pupils should produce something like the following:

TO TRIANGLE :length

REPEAT 3 [FD :length RT 120]

END

TO SQUARE :length


END

TO PENTAGON :length


END

TO POLYGON :length :sides

REPEAT :sides [FD :length RT 360/:sides]

END

Examples:

POLYGON 100 3

POLYGON 100 4

POLYGON 100 5

Teaching notes


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Logo

Useful commands

Click in the bottom box to enter commands to tell the turtle what to do.

Press Enter after the commands to get the turtle to move.

FD 100 Go forward 100 steps.

BK 100 Go backwards 100 steps.

RT 90 Turn right through 90 degrees.

LT 50 Turn left through 50 degrees.

REPEAT 4 [FD 100 RT 90] This command repeats the instructions in thesquare brackets the number of times given by thenumber before the bracket.

CS Clear screen.

PU Pen up − takes the turtle off the page so that youcan move it without drawing a line.

PD Pen down − puts the turtle back on the page.

Note. Remember to put a space after the commands FD, BK, RT and LT.

Resource 7A2.1Helpsheet SH

The turtle drawsin this window

Previous commandsare shown here

New commandsare entered here


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Writing procedures

To start writing a procedure (which is a like a small program), type EDIT "

and then the name for your procedure. For example, EDIT "SQUARE

A window will pop up. Type the Logo instructions between the TO and

END commands. For example:

TO SQUARE


END

Click File and then Save and exit.

To run your procedure, type the procedure name. For example, type

SQUARE.

Using variables

To add variables to your procedure, use a letter or word with a colon

before it. For example:

TO SQUARE :side

REPEAT 4 [FD :side RT 90]

END

Everywhere the variable appears in the procedure you must put a colon

before it.

To run the procedure type the procedure name followed by a number for

the variable. For example, type SQUARE 100.


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Logo challenge

Use the procedure FLAG to create this diagram.

Use the procedure RTRIANGLE to create this diagram.

Resource 7S4.1


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Two tessellations Resource 8S3.1


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More tessellations Resource 8S3.2


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References and furtherreading

Framework for teaching mathematics: Years 7, 8 and 9 (DfES 0020/2001)

Framework for teaching ICT capability: Years 7, 8 and 9 (DfES 0321/2002)

ICT and mathematics: a guide to learning and teaching mathematics 11–19

(produced for the Teacher Training Agency by The Mathematical Association,

July 2002)

Integrating ICT into mathematics in Key Stage 3 (DfES 0332/2003)

Sample medium-term plans for mathematics (DfES 0504/2001)

Acknowledgements

Logo screen shots were produced using MSW Logo (www.softronix.com).

Screen shots from The Geometer’s Sketchpad® are reprinted by permission from

Key Curriculum Press (www.keypress.com/sketchpad).

Screen shots from Cabri Géomètre® II are reprinted by courtesy of Texas

Instruments and with permission from Cabrilog, the French publisher of Cabri®

geometry software family (www.cabri.com).

Disclaimer

The Department for Education and Skills wishes to make clear that the Department

and its agents accept no responsibility for the actual content of any materials

suggested as information sources in this document, whether these are in the form

of printed publications or on a website.

In these materials, icons, logos, software products and websites are used for

contextual and practical reasons. Their use should not be interpreted as an

endorsement of particular companies or their products.

The websites referred to in these materials existed at the time of going to print.

Teachers should check all website references carefully to see if they have changed

and substitute other references where appropriate.


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The content of this publication may be reproduced free of charge by schools and

local education authorities provided that the material is acknowledged as Crown

copyright, the publication title is specified, it is reproduced accurately and not used

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The permission to reproduce Crown copyright protected material does not

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