Golden Rectangle

Task Description

Students create a number of rectangles (including the golden rectangle) and survey peers on the most aesthetically appealing rectangle.

Length of Task

100 minutes

Materials

·  Golden Rectangle worksheet, internet access, paper, ruler, calculators.

Using the Activity

Introductory

The teacher reveals to the class three rectangles of differing proportions. 1:1, 1:2, 1:1.618. The teacher asks the students to look at each rectangle and note in their workbooks which rectangle they consider is more pleasing to the eye. Note: Typically, most people will select the golden rectangle which is the rectangle that has a ratio of 1:1.618.

Main Activity – Part One: Survey

The students create a series of 3 - 4 rectangles of different ratios including one golden rectangle. The students ask the survey question “Which do you think looks more pleasing to the eye?” to children and teachers throughout the school. The students note the responses and report back to the whole class.

The students complete the attached worksheet and explore the use of the golden rectangle in their environment.

Reflection

In a whole class discussion, the students reflect on the following discussion questions:

What is special about the golden ratio?

Why do you think most people prefer the golden rectangle to other rectangles?

What examples of the golden rectangle or ratio have you seen in our classroom?

Key Mathematical Concepts

·  Recognition of golden ratio representations in the environment. Appreciation for the aesthetic appeal of the golden ratio.

Prerequisite Knowledge

·  Understanding simple ratios.

Links to VELS

Dimension / Standard
Number (Level 4) / Students use decimals, ratios and percentages to find equivalent representations of common fractions (for example, 3/4 = 9/12 = 0.75 = 75% = 3 : 4 = 6 : 8).
Working mathematically (Level 4) / Students recognise and investigate the use of mathematics in real and historical situations.

Assessment

To be working at Level 4, students should be able to:

·  Create a golden rectangle.

·  Recognise representations of the golden rectangle in the environment.

Extension Suggestions

For students who would benefit from additional challenges:

·  Through manipulation of a Java applet, students can resize the sides of a golden rectangle to view the different lengths that create the golden rectangle while keeping the ratio constant. http://www.mathopenref.com/rectanglegolden.html

Teacher Advice and Feedback

The teachers noted that those students who did not have a sound understanding of percentages and simple ratios where struggling to complete the task. The teachers found that a couple of refresher sessions on percentages and ratio assisted in students having greater access to the task.

The teachers also noticed that the tasks related to ratio helped identify students’ previous misconceptions of fractions, decimals and percentages. This allowed the teachers to address these difficulties in the context of real-life ratio tasks.

Potential Student Difficulties

Students may cut a golden rectangle out of mounting board to assist in discovering examples of the golden rectangle existing in their school environment.


Students may profile different rectangles in their classroom and identify what changes would need to be made for it to be considered a golden rectangle. Masking tape might be applied to different rectangles (e.g. a window, cupboard door) to assist in visualising the golden rectangle within these items.

References / Acknowledgements

Source: http://math.rice.edu/~lanius/Geom/golden.html

Thank you to the teachers and students from Lloyd Street PS, for providing valuable feedback on the use of this activity.

Student work samples

Example 1: Working at Level 4

This student has correctly ranked the items in order of the closest to the golden ratio. While it is helpful to do the calculations, it is also possible for students to eye-ball the items for ranking purposes once a student is familiar with the dimensions of the golden rectangle.

This activity is intended for use by teachers for research purposes only, as part of the Task Types and Mathematics Learning (TTML) project at Monash University. No authority is granted for persons to use these activities beyond the scope of this project, without express permission of the TTML Project Leader.