;

National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

This page, and all sections of the National Curriculum Resource Tool, will be removed in summer 2024. Find out more

KS3 - Geometry and Measures

New Curriculum

Pupils should be taught to:

  • derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)
  • calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes
  • draw and measure line segments and angles in geometric figures, including interpreting scale drawings
  • derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle); recognise and use the perpendicular distance from a point to a line as the shortest distance to the line
  • describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric
  • use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles
  • derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
  • identify properties of, and describe the results of, translations, rotations and reflections applied to given figures
  • identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grids
  • apply the properties of angles at a point, angles at a point on a straight line, vertically opposite angles
  • understand and use the relationship between parallel lines and alternate and corresponding angles
  • derive and use the sum of angles in a triangle and use it to deduce the angle sum in any polygon, and to derive properties of regular polygons
  • apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides, including Pythagoras’ Theorem, and use known results to obtain simple proofs
  • use Pythagoras’ Theorem and trigonometric ratios in similar triangles to solve problems involving right-angled triangles
  • use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D
  • interpret mathematical relationships both algebraically and geometrically

Non-Statutory Guidance

Links and Resources

Making connections to other topics within Key Stage 3

Number

Pupils will need to be conversant with a whole range of number topics to support their learning and development within this area. In particular they will need to be secure in operations with integers, fractions, and decimals.

Ratio and proportion

Pupils should be encouraged to see the connections between transformations, enlargements and similar figures in both 2D and 3D and calculations applying the concepts of ratio and proportion.

Algebra

Pupils should be encouraged to see the links between aspects of substitution of numerical values into formulae used within geometry.

Making connections to this topic in Year 6

In year 6 measurement pupils should be taught to:

  • solve problems involving the calculation and conversion of units of measure, using decimal notation up to three decimal places where appropriate
  • use, read and convert between standard units, converting measurements of length, mass, volume and time from a smaller unit of measure to a larger unit, and vice versa, using decimal notation up to three decimal places
  • convert between miles and kilometres
  • recognise that shapes with the same areas can have different perimeters and vice versa
  • recognise when it is possible to use formulae for area and volume of shapes
  • calculate the area of parallelograms and triangles
  • calculate, estimate and compare volume of cubes and cuboids using standard units, including cubic centimetres (cm3) and cubic metres (m3), and extending to other units [for example, mm3 and km3].

Notes and guidance on measurement (non-statutory) suggest

  • Pupils connect conversion (for example, from kilometres to miles) to a graphical representation as preparation for understanding linear/proportional graphs.
  • They know approximate conversions and are able to tell if an answer is sensible.
  • Using the number line, pupils use, add and subtract positive and negative integers for measures such as temperature.
  • They relate the area of rectangles to parallelograms and triangles, for example, by dissection, and calculate their areas, understanding and using the formulae (in words or symbols) to do this.
  • Pupils could be introduced to compound units for speed, such as miles per hour, and apply their knowledge in science or other subjects as appropriate.

In year 6 Geometry - properties of shapes pupils should be taught to:

  • draw 2-D shapes using given dimensions and angles
  • recognise, describe and build simple 3-D shapes, including making nets
  • compare and classify geometric shapes based on their properties and sizes and find unknown angles in any triangles, quadrilaterals, and regular polygons
  • illustrate and name parts of circles, including radius, diameter and circumference and know that the diameter is twice the radius
  • recognise angles where they meet at a point, are on a straight line, or are vertically opposite, and find missing angles

Notes and guidance on Geometry - properties of shapes (non-statutory) suggest

  • Pupils draw shapes and nets accurately, using measuring tools and conventional markings and labels for lines and angles
  • Pupils describe the properties of shapes and explain how unknown angles and lengths can be derived from known measurements
  • These relationships might be expressed algebraically for example, d = 2 × r; a = 180 – (b + c)

In year 6 Geometry – position and direction pupils should be taught to:

  • describe positions on the full coordinate grid (all four quadrants)
  • draw and translate simple shapes on the coordinate plane, and reflect them in the axes

Notes and guidance on Geometry – position and direction (non-statutory) suggest:

  • Pupils draw and label a pair of axes in all four quadrants with equal scaling. This extends their knowledge of one quadrant to all four quadrants, including the use of negative numbers.
  • Pupils draw and label rectangles (including squares), parallelograms and rhombuses, specified by coordinates in the four quadrants, predicting missing coordinates using the properties of shapes. These might be expressed algebraically for example, translating vertex (a, b) to (a – 2, b + 3); (a, b) and (a + d, b + d) being opposite vertices of a square of side d.

Cross-curricular and real life connections

Finding distances outside the classroom, e.g. the height of a tree or building given the distance away and the angle of elevation.

Calculation of the distance to the horizon on a ship (see the NCETM micro-site ‘Maths at Work’)

The mathematics of the race track

Environmental Geometry

(You must register in the National STEM Centre website to access the “Environmental Geometry” document – free registration)

This guide delves deeper in to how the study of the environment can lead to geometrical work. The range of activities discussed in this guide extend beyond experiences with objects in the classroom which the pupil can touch or handle.

  • Jones, K. (2010) The role of the teacher in teaching proof and proving geometry, BSRLM

Through a consideration of an example of classroom teaching, this paper scrutinises a selection of theoretical frameworks that may afford greater insight into, and greater explanatory power on, the role of the teacher in the teaching of proof and proving in geometry.

  • Ding, L and Jones, K. (2006) Teaching geometry in lower secondary school in Shanghai, China, BSRLM

This paper reports on a study of geometry teaching at the lower secondary school level in Shanghai, China. Through an analysis of data from observing a variety of Year 9 (Grade 8) lessons, and utilising data from the students’ performance in school examinations, the study suggests that teachers in this region of China use classroom strategies that attempt to reinforce visual and deductive approaches in order to develop students’ thinking in the transition to deductive geometry.

  • Forsythe, S. and Jones, K. (2009) Tasks that support the development of geometrical reasoning at KS3, BSRLM

This paper describes an attempt to devise a task which would encourage students to develop a deeper understanding of how shapes can be defined by considering the properties of two internal perpendicular lines.

Cuboid Challenge

In this activity pupils investigate the volume of a box, without a lid, made from a square sheet of paper. It extends into maximum volumes and investigating the relationships between the size of the paper, the size of the corner cut and the maximum volume.

This activity enables pupils to:

  • derive and apply formulae to calculate and solve problems involving: perimeter and area of triangles, parallelograms, trapezia, volume of cuboids (including cubes) and other prisms (including cylinders)

Pick’s Theorem

This activity invites pupils to find a relationship between the perimeter, area, the number of dots on the perimeter and inside shapes drawn on square dotty paper. The shapes are generated by joining points with straight lines.

This activity enables the pupils to:

  • calculate and solve problems involving: perimeters of 2-D shapes (including circles), areas of circles and composite shapes

Triangle Tantaliser

This activity invites pupils to draw all the possible distinct triangles on a 4 × 4 dotty grid. In doing so they use and apply concepts of rotation, reflection, translation and congruence.

This activity enables the pupils to:

  • use the standard conventions for labelling the sides and angles of triangle ABC, and know and use the criteria for congruence of triangles

Property Chart

While doing this activity pupils draw and work collaboratively to identify quadrilaterals with two given properties.

This activity enables the pupils to:

  • derive and illustrate properties of triangles, quadrilaterals, circles, and other plane figures [for example, equal lengths and angles] using appropriate language and technologies
  • describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles, regular polygons, and other polygons that are reflectively and rotationally symmetric

Combining transformations

Pupils investigate the outcome of repeating and combining transformations and inverse transformations, including reflections, rotations and transformations both geometrically and algebraically.

This activity enables the pupils to:

  • identify properties of, and describe the results of, translations, rotations and reflections applied to given figures

Who is the fairest of them all?

This activity requires pupils to draw enlargements with scale factors k and 1/k with different centres of enlargement and to investigate the single transformation that maps from one enlargement to the other.

This activity enables the pupils to:

  • identify and construct congruent triangles, and construct similar shapes by enlargement, with and without coordinate grid

Weekly problem 3 - 2012

This activity requires learners to apply Pythagoras’ Theorem to decide which pieces of hardboard can be passed through a window frame.

This activity enables the pupils to:

  • use Pythagoras’ Theorem … to solve problems involving right-angled triangles

3-D shapes –Short Problems

These short activities support pupils to develop their skills to solve problems in 3-D.

This activity enables the pupils to:

  • use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3-D

Bull’s Eye and Algebra from geometry

These two linked activities enable the pupils to:

  • interpret mathematical relationships both algebraically and geometrically

Example A

Write down the names of three quadrilaterals whose diagonals cross at right angles.

For which of these quadrilaterals do the diagonals also bisect each other?

Example B

Quadrilateral

ABCD is a quadrilateral.

Work out the size of the largest angle in the quadrilateral.

Example C

The size of each exterior angle of a regular polygon is 24°.

Work out the number of sides the polygon has.

Example D

Semi-circle

N.B. The diagram is not drawn accurately

A semicircle has a diameter of 20 cm.

Work out the perimeter of the semicircle.

Take the value of 𝜋 to be 3.14

Example E

The perimeter of this triangle is 22 cm.

All lengths on the diagram are in centimetres.

Triangle

N.B. The diagram is not drawn accurately

  1. Work out the value of t
  2. What is the length of the longest side?

Example F

Cuboid

N.B. The diagram is not drawn accurately

The diagram represents a large tank in the shape of a cuboid.
The tank has a base.
It does not have a top.
The width of the tank is 2.6 metres.
The length of the tank is 3.5 metres.
The height of the tank is 4.2 metres.

The outside of the tank is going to be painted.

1 litre of paint will cover 2.5 m2 of the tank.
The cost of paint is £5.99 per litre.

Calculate the cost of the paint needed to paint the outside of the tank.

Example G

Triangle

N.B. The diagram is not drawn accurately

PQ is a straight line.

  1. Work out the size of the angle marked x°.

    Give a reason for your answer.
  2. Work out the size of the angle marked y°.

    Explain how you worked out your answer.

Example H

Parallelograms

N.B. The diagrams are not drawn accurately

Parallelogram B is an enlargement of parallelogram A.
The scale factor of the enlargement is 3.

  1. Calculate the length of the side marked x.
  2. Work out the length of side y.
  3. Work out the size of angle z.

Example I

Here is a solid cube of side 2 cm.

Cuboid

N.B. The diagram is not drawn accurately

  1. Write down

    the number of faces of the cube,

    the number of vertices of the cube,

    the number of edges of the cube.

  2. Draw an accurate net of this cube.

Example J

The diagrams show the circular top of a cylinder and a piece of plastic in the shape of a right-angled triangle. The two shorter sides of the triangle have lengths 76 cm and 77 cm.

Can the triangular piece of plastic fit on the top of the cylinder without overlapping?

Circle and triangle

N.B. The diagrams are not drawn to scale

Example K

Irregular shaped car park illustration

N.B. The diagrams is not drawn accurately

The diagram shows a car park.

Mrs Roberts is selling the car park. She will accept any offer that is more than £30 per square metre.

  1. Mr Brown offers £239 900 for the car park.

    Will Mrs Roberts accept Mr Brown’s offer for the car park? You must show how you reached your decision.
  2. When the car park is sold the new owner intends to put a fence around the entire perimeter.

    What is the length of fencing required?

Fencing is sold in 5m lengths and costs £38.95 per 5m section. Sections can only be bought in 5 metres (no parts)

What is the total cost of the perimeter fence?

How many peas fill the classroom?

(Part of Teachers TV ‘Great Secondary Lesson Ideas’ series – free registration).

This involves measuring, estimating and calculating. Pupils work out how many chickpeas might fit into their classroom.

Demonstrating Dynamic Geometry

From Teachers Media (free registration)

The teacher demonstrates how dynamic geometry software can transform the teaching of mathematical concepts:

  • a demonstration and proof of the sum of the angles in a triangle
  • how to use the software to explore reflections, rotations and symmetry
  • how to do some classic geometrical constructions
  • a visual illustration of Pythagoras’ theorem

3-D Awareness

(part of Teachers TV ‘Maths for all’ series – free registration)

A boys’ school in south London improves their students’ 3D awareness by visiting local buildings, making paper shapes, using student mentors, and looking at the algebra of origami.