by Katherine LaVange

Discovering the Golden Ratio

Goals and Objectives

NC SCOS Objectives

> Use models and pictures to relate concepts of ratio, proportion, and percent; record results.
> Construct congruent segments, congruent angles, bisectors of line segments and bisectors of angles.
> Calculate areas of triangles, parallelograms and circles.
> Estimate solutions to problems involving geometry and measurement. Determine when estimates are sufficient for the measurement situation.

Teacher-specified Objectives

Students will:

  • Demonstrate knowledge of geometric shapes by drawing them on Geometer's Scetchpad
  • Explain the concept of ratio by participating in a pre-assesment class discussion
  • Set up ratios by comparing parts of geometric figures on Geometer's Scetchpad
  • Discover the Golden Ratio by completing the Golden Ratio Worksheet
  • Appreciate technology by using Geometer's Scetchpad and a calculator to help solve problems
  • Appreciate the significance of the Golden Ratio by making connections to real-life situations



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Materials and Equipment

For this lesson, we will use The Golden Ratio activity sheet and The Golden Ratio Calculator Investigation sheet. Both activities were obtained from


Safety Issues

Because we will be in the computer lab, students need to be aware of the rules and regulations having to do with operating the equipment. Make sure students know how to respect the computers and handle them safely. Some rules for students to follow include: no running in the computer lab, do not move any equipment or the machines, be aware of your surroundings so as not to trip over any cords, etc.

Notes/Reminders to self

This lesson requires students to move from activity to activity and into different classrooms. Make sure students move quickly and quietly through the halls. Also make sure students are aware of the policies when working in the computer lab. Remind the students to be respectful of others and of the equipment as they work. When working on the calculator activity, remind students to be respectful of that equipment as well.

Key Vocabulary/Concepts

  • regular pentagon: a five-sided polygon with interior angles each measuring 108 degrees
  • isosceles triangle: a triangle with two congruent sides
  • ratio: a comparison of two quantities with the same unit of measure
  • proportion: synonym of ratio

Instructional Strategy

Focus and Review

As students walk into the classroom, have the attached PowerPoint displayed for students to see and think about. Only display the first slide, the second slide will be shown during Lesson 3 after the two Golden Ratio activities have been completed.

Statement of Objective

After the class has all arrived and they have looked at and thought about the pictures on the PowerPoint, give a brief statement of objective: "Today we are going to learn about what these four objects, as well as many others in the world, have in common. There is a special number that is derived from math that we will discover through an activity on the computer, and using calculators. Then we will discuss the significance of this number, and why it is important."

Sequence of Instruction

After the brief introduction of the lesson, some review will take place in order for students to activate prior knowledge that is needed for the upcoming activities.  Complete these review and instructional activities by asking the questions that are listed with the activity.

  • Review regular polygons.  What is a regular polygon?  How is it different from an irregular polygon? 
  • Have a student give the definition.
  • Have a student draw examples.
  • Review regular pentagon.  What is a regular pentagon? How is it different from an irregular pentagon?
  • Have a student give the definition.
  • Have a student draw an example.
  • Review isosceles triangle.  What is an isosceles triangle?  How is it different from a regular triangle? 
  • Have a student give a definition.
  • Have a student draw an example.
  • Review ratios.  What is a ratio? 
  • Have a student give a definition.
  • Have a student write an example, given two measurements to compare.

Move the class into the lab, and have the students each sit at their own computer.

  • Brief explanation of Geometer's Scetchpad
  • Show the students the basic tools and functions of the program.
  • Pass out the Golden Ratio worksheet (one per person).
  • Briefly go over the worksheet so the students know what is expected of them.
  • Have the students work on the activity, recording their answers on the worksheet.
  • Walk around the room and monitor work as students complete the activity.

Move back into the classroom for the calculator activity.

  • After students are settled back in the classroom, briefly discuss their findings in he lab. Have students offer results of the activities for the class to discuss all together for a few minutes.
  • Pass out calculators, or have students take out their own.
  • Post the calculator activity instructions on the overhead for everyone to see.
  • Walk through the steps as a class.
  • Talk about the answer that the students should have gotten, and why this answer is important.


Closing Activity

  • Put the PowerPoint presentation (slide 1) back up for students to see.
  • Ask students, "What did we discover today?  Who can tell me what this special number is called?  Why do you think it appears so often?"
  • Have the students generate ideas of how what they learned today about the Golden Ratio is connected to these objects, and how it ties them together.   
  • Have students write down ideas silently first in their journals, and then share their ideas with the class
  • Let the students know that tomorrow we will discover how the Golden Ratio is related to these objects.


Reflection on Lesson


Include here how lesson could be improved or changed and why. What was effective and what was not? What worked and did not work?