National Curriculum Resource Tool

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National Curriculum Resource Tool

Materials to support teachers and schools in embedding the National Curriculum

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National Curriculum Tool
KS3 - Geometry and Measures

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.