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Regular Pentagons, "Star Polygons," and the Golden Ratio
7, 8
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Professional Commentary

This learning unit helps students explore the golden ratio and golden rectangles and to find the golden ratio in regular pentagons and in the Fibonacci sequence. Students use Geometer's Sketchpad® to rotate, translate, and dilate various figures to solve problems. Activity sheets and complete solutions are included. An extension of the lesson using a spreadsheet is also suggested. (author/sw)

Common Core State Standards for Mathematics
Standards for Mathematical Practice
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Grade 7
Ratios and Proportional Relationships
Analyze proportional relationships and use them to solve real-world and mathematical problems.
Recognize and represent proportional relationships between quantities.
  1. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.
  2. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
  3. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.
  4. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.
Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
Draw, construct, and describe geometrical figures and describe the relationships between them.
Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.
Ohio Mathematics Academic Content Standards (2001)
Measurement Standard
Benchmarks (8–10)
Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates.
Geometry and Spatial Sense Standard
Benchmarks (8–10)
Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.
Grade Level Indicators (Grade 10)
Make, test and establish the validity of conjectures about geometric properties and relationships using counterexample, inductive and deductive reasoning, and paragraph or two-column proof, including: a. prove the Pythagorean Theorem; b. prove theorems involving triangle similarity and congruence; c. prove theorems involving properties of lines, angles, triangles and quadrilaterals; and d. test a conjecture using basic constructions made with a compass and straightedge or technology.
Principles and Standards for School Mathematics
Geometry Standard
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
Expectations (9–12)
explore relationships (including congruence and similarity) among classes of two- and three-dimensional geometric objects, make and test conjectures about them, and solve problems involving them;