## Georgia Learning Connections

Creating Fibonacci’s Golden Rectangle

This is an extension of an existing unit on Fibonacci.  Students will have already seen examples of art and architecture that employs the Golden Rectangle.  Students will have the opportunity to make a Golden Triangle and “see” for themselves how the ratio of the sides approaches 0.618 and is formed by:                            width  =      length_____

## length    (width + length)

The Primary Learning Outcome, other than simply creating an example of a Golden Rectangle will be that students can construct a rectangle with Golden Rectangle proportions and understand how ratios are used to determine its proportions.

Assessed QCC:  Strand Problem Solving # 2 Topic  Vocabulary:  Describes orally and in writing, using the appropriate mathematical vocabulary, mathematical concepts and procedures, such as the reasoning involved in solving problems or computing  AND  Strand  Number Sense & Numeration; Fractions & Decimals  # 37 Topic:  Equivalence, Equivalent Representations  Standard:  Expresses equivalent ratios as a proportion

## Standards: Local and/or National:  Gwinnett County AKS # 14 Use a ratio to compare two quantities #15 use proportions to solve problems

Total Duration: This component of the lesson should take 60 minutes

## Materials and Equipment: Photos of examples of Golden Rectangles in art and architecture

Procedure:

This construction activity comes from Geometry Activities for Middle School Students with The Geometer’s Sketchpad by Karen Windham Wyatt, Ann Lawrence and Gina M. Foletta.

Steps:

1.     Construct a square using the GSP features for construction.

2.     Construct a midpoint and connect it to the point at the top of the right adjacent side.

3.     Use that segment as a radius and construct a circle.

4.     Extend the original line segment to a point on the circle.

5.     Construct a perpendicular segment to the side of the square with a point on the circle.

6.     Extend the top side of the original square to the previously constructed segment.

7.     Hide all points and lines leaving only the newly constructed rectangle.

8.     Measure base and height of new rectangle.

9.     Substitute these measurements into the ratio that defines a Golden Rectangle.  (see above formula)

10.   Have students “stretch” the rectangle and note that the ratio always remains the same.

Example:

Assessment:  Students will print their work showing the ratios after measuring the sides.  They will write a paragraph explaining what they did and what they learned.