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Surd Delights

Surd Spirals
Federation Square
Golden Rectangles
Surds with Nspire


These activities provide delightful ways to enrich the teaching of surds. Explorations with spiral and other geometric patterns, supported by dynamic geometry, are used to link surds with length through Pythagoras’ theorem, emphasise their relative size and to provide attractive contexts for simple and complex surd calculations. The exact value mode of CAS may support students’ work with surds, allowing them to construct patterns and share a world of beautiful mathematics

There are four activites for exploring surds. The 2005 MAV paper by Kaye Stacey and Beth Price titled Surds, Spirals, Dynamic Geometry and CAS provides a detailed description of each activity, practical teaching ideas and possible benefits for student learning of mathematics.

CAS can be used to scaffold students' early learning of symbolic manipulations of surds. The dynamic geometry is used to present the problem situations and to enable measurements to link surds with their numerical approximations.

Activities 1 and 2 - Surd Spirals

These two introductory activities involve students in finding numerical approximations to common surds and simple manipulations with surds.

There are two files for the activities:

SurdSpiralsWorksheet and SurdSpirals.gsp

Both files can be downloaded in this zip file:

Activity 3 - Federation Square

This activity involves students in more complicated, but still relatively simple, surd calculations in an interesting real context.

Everything needed for this activity is in the pages of the Geometer's Sketchpad file .

Activity 4 - Golden Rectangles

This activity makes use of the Goldern Ratio and requires students to add and subtract surds, and optionally, multiply surds and rationalise surd denominators.

There are three files for the activity:




All three files can be downloaded in this zip file:

Activity 5-Surds with Nspire

In this 100-minute lesson (or 2×50 minutes) for Year 10 students, the spiral generated by adjacent right-angled triangles is used as a stimulus for studying patterns in surd expressions. Students use the Rule of Pythagoras and measurement formulae to find patterns in the exact values for the perimeter and area of successive triangles. Generalisations are used to create a spreadsheet of these exact values, which is used to support by-hand skills of simplifying surd expressions.  This lesson was created and refined through the Lesson Study approach and was trialled at two Victorian schools in 2009.

Surds_Lessonplan (pdf)  Full details of technology requirements, assumed background knowledge and technology skills, key concepts and suggested time allocations for the six main activities are included. There are also suggestions of extension and assessment options related to the concept knowledge and skills developed during the lesson, as well as reference to pre- and post-lesson tests used in the research process.

Surds_Pre- and post-tests (pdf)  Each of these 3-minute (technology-free) tests consists of 10 surd expressions which are to be expressed in either simplified or entire surd form. Teachers may administer these tests before and after the lesson to monitor any improvement in students’ skill level in surds work after teaching the lesson.

Surds__StdntWksht (pdf) Each student uses this multiple-page document to record manual calculations for perimeter and area values. They also use it to formulate generalisations and make observations throughout the lesson.

Surds_Nspire (tns) The teacher and students use this file for performing any calculations, generalising values for the various measurements using the spreadsheet, and checking approximate values using a pre-drawn spiral.

All four files can be downloaded in this zip file: Surds with




For information about this page, contact: Kaye Stacey
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Last modified: Fri 11 December 2020

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