Mathematics Developmental Continuum
Surds: 5.5
Supporting materials
- Related Progression Point
- Developmental Overview of Working Mathematically (PDF - 31Kb)
- Developmental Overview of Structure (PDF - 35Kb)
Indicator of Progress
At this level students can simplify surds such as √12 = 2√3 and they can perform computations such as addition, subtraction and multiplication with simple surds without using technology.
Before this level students may not work with surds as exact arithmetic but convert them immediately to decimal form.
Development of surd ideas gives an overview of surd skills and understandings.
Illustration 1: The shift from approximate thinking to exact thinking
For students who are used to working with scientific or graphics calculators, working with surds requires a shift from approximate to exact thinking. For example, if √3 is entered into a calculator a decimal approximation 1.732… is displayed. Students won’t see the exact number on their screen, because it has an infinite number of decimal places.
The progression for students here is the ability to recognise and calculate with the exact value as a mathematical object, rather than only consider its decimal approximation.
Illustration 2: Working with surds extends other number work
Working with surds provides additional practice with fractions, order of operations and writing equivalent expressions. Difficulties working with whole number and fraction calculations will affect students’ ability to calculate correctly with surds. In particular it is likely that students who cannot correctly work with surds also experience difficulty with fractions.
Working with surds also extends algebra rules e.g. √12 = 2√3 is only meaningful if students know √3 + √3 = 2√3 etc.
Teaching Strategies
It is useful to provide examples that use previous knowledge associated with four operations on fractions, order of operations and calculations with integers. In order to develop facility with surds students need a lot of practice with a range of problems.
Activity 1: Exact thinking is an initial exploration where students develop an understanding that a surd is an exact number.
Activity 2: Relationships between surds is an exploration using a calculator to find numerical relationships between surds, leading to rules for simplifying surds.
Activity 3: Calculating with surds is an activity where students analyse hypothetical students’ work and which involves classroom discussion.
Activity 4: Surd spirals uses a geometric pattern and Pythagoras’ Theorem to generate surds, and find relations between them. Many similar spiral patterns can be created by students to vary the activity.
Activity 1: Exact thinking
Step 1: Have students enter √529 into a calculator, to give 23. Ask students to note that 529 is a perfect square. Discuss that in this case √529 can be written exactly as 23. Now ask students to find another number greater than 200 that is a perfect square.
Step 2: Get students to enter √7 into their calculators and record the decimal output.
2.64575
Now square the decimal approximation for √7 (i.e. 2.64575) and discuss the fact that the square of the decimal approximation does not give the exact value √7. In this case the decimal output provided by the calculator (to a specified number of decimal places) does not provide an exact value. We write √7 if we want to refer to the exact value.
Step 3: Sometimes a calculator will give 7 (apparently exactly) as the square of the decimal approximation. For example, my best calculator gave the thirty digit square root √7 = 2.6457513110 6459059016 157536393 and gave its square as 7 exactly. However, students can easily see that this is not exactly correct by beginning to square the thirty digits using by-hand calculation. The last digit should be zero if the answer is exactly 7, but instead it is 9 (last digit 3 by last digit 3).
Step 4: This is a good time to discuss calculator technique. Technology output is an approximation for the exact value. Most calculators will store more decimal places than are displayed on the screen. If students retype the value displayed on the screen and square it, the result will be further from 7 than if they put the output into the memory and square that. Moreover, the calculator may return 7 as it rounds the answer. Discuss the need to work with the exact value rather than a decimal approximation.
Activity 2: Relationships between surds
Step 1: Have students use their calculators to fill in a table such as that shown.
|
|
Calculator output |
Surd? |
|---|---|---|
|
√1 |
1 |
No |
|
√2 |
|
yes |
|
√3 |
|
yes |
|
etc. |
|
|
Step 2: Discuss which numbers are surds.
Step 3: Students look for, and record, numerical relationships between the square roots (surds and natural numbers). For example, they may observe from the numerical values that
√20 = 2 × √5 and that √6 = √2 × √3
Encourage students to formulate general rules and test them.
Step 4: Discuss students’ rules, and explain why they work by examining factors. For example, the square root of 20 can be rewritten as √4 × 5 = √(22 × 5) = 2√5 . Get students to start thinking about the possibility of writing a surd in simplest form.
Step 5: Students list the general rules that have been tested and explained, and use them to find equivalent surd expressions.
Activity 3: Calculating with surds
In this activity students demonstrate their understanding of calculations with surds through analysis of hypothetical students’ work. Prior to this activity students would have practised surd calculations, so this activity provides informal feedback about the ability to work with surds. This activity complements textbook work and provides examples, other than students’ own work, to discuss possible difficulties with work on surds. It may highlight some difficulties with fractions, understanding of expressions and order of operations.
‘Calculating with surds (PDF - 21Kb)’ – distribute the student resource sheet. Students should indicate whether each response is correct or incorrect, and if neither of the given responses is correct, they should provide the correct answer.
Discuss the sources of common errors such as identifying ‘like terms’ for addition (e.g. item 7), confusing multiples and powers (e.g. item 4), having a limited view of equivalent expressions (e.g. item 6), incorrect cancelling (e.g. item 8).
Students can check the answers numerically (e.g. see which expression evaluates to the correct number), but they should also be able to deal with these surds using the algebra of like and unlike terms and properties of surds.
Activity 4: Surd spirals
In this activity surds give exact lengths of lines in some spirals. Students work exactly with surds and in parallel, they calculate decimal approximations for surds and measure the lengths of the lines. The activity promotes the move to exact arithmetic, whilst emphasising that a surd is above all a number.
Begin by showing the spirals and discussing the geometry of the spiral and in particular to point out the right triangle that enables Pythagoras’ theorem to be used.
Surd Spirals Worksheet (PDF - 35Kb) – distribute the student resource sheet by Beth Price. A CAS calculator can support students’ exact calculations, and a dynamic geometry file is also available here (link to website reference). If students do not have access to CAS calculators, they can do the calculations by-hand, but the activity may take longer.
The dynamic geometry file can be downloaded from RITEMATHS. Download these files prior to the lesson and either store them on your school intranet or provide students with a copy of the files on CD. Students can work from the computer file, use photocopies of the spirals or draw the spirals themselves.
This spiral begins with a right-angled isosceles triangle, with two sides of length 1 unit. A new right-angled triangle is built on the hypotenuse with the shortest side 1 unit. The process continues making a spiral. Students have to find the lengths of all the lines.
This spiral begins with a right-angled isosceles triangle, with two sides of length 1 unit. A new right-angled isosceles triangle is built on the hypotenuse. The process continues making a spiral. Students have to find the lengths of all the lines.
References
The Surd Spirals activity, resource sheet and GSP dynamic geometry files are from the RITEMATHS project website at The University of Melbourne: (http://extranet.edfac.unimelb.edu.au/DSME/RITEMATHS/)
Stacey, K., & Price, E. (2005). Surds, spirals, dynamic geometry and CAS. In J. Mousley, L. Bragg, & C. Campbell (Eds.), Mathematics - Celebrating Achievement . Proceedings of the 41st Annual Conference of the MAV (pp. 298-306). Melbourne: Mathematical Association of Victoria .