The Golden Rectangle
Problem: What is the Golden Ratio and why does it work. The actual Golden Ratio can be seen here.
Discussion: The Golden Ratio appears often in nature such as the proportions in our bodies and faces. A rectangle whose sides are in the Golden Ratio is called a Golden Rectangle. If you cut a square off the Golden Rectangle you are left with a smaller rectangle similar to the first which is also golden. If you add a square onto the long side of a Golden Rectangle you will also get a Golden Rectangle. Since Golden Rectangles are all throughout nature, I will try to construct one first to see how close I can get to an actual Golden Rectangle. Then I will construct an actual Golden Rectangle. In the investigation, I will explore why the ratio of the short side to the long side is equal to the ratio of the long side to the sum of the sides.
Step 1: Construct a square ABCD.
Step 2: Extend two sides by constructing rays AD and BC.
Step 3: Construct E on ray BC and a line through E, perpendicular to ray AD.
Step 4: Construct F, the intersection of the line and ray AD.
Step 5: Measure BE, EF, and FD.
I moved point E around until rectangle ABEF looked like the perfect rectangle to me. Next, I calculated BE/EF and EF/FD. The ratios ended up being the same, 1.53. My Golden Ratio came out to be 1.53.
Hiding everything except square ABCD and using the following steps, I constructed an actual Golden Rectangle.
Step 1: Construct G, the midpoint of AD
Step 2: Construct circle GC.
Step 3: Construct H, the intersection of ray AD and the circle.
Step 4: Construct a line perpendicular to ray AD through H.
Step 5: Construct J, the intersection of this line and ray BC.
ABHJ is a Golden Rectangle, as is CJHD. Next, I measured BJ, JH, and HD and calculated HD+JH. How does BJ compare to HD+JH and why?
The Golden Ratio came out to be about 1.62. I was not too far off from my prediction of a perfect rectangle. BJ is equal to HD+JH because BJ=BC+CJ. BC is equal to JH since ABCD is a square and CJ=HD because they are opposite sides of a rectangle.
Complete the ratio: X/BJ=BJ/Y=JH/HD. BJ/JH=JH/HD is about 1.62 which is the Golden Ratio.
Let the short side of a Golden Rectangle have length 1 and the long side length x. Use the quadratic formula to calculate an exact value for the Golden Ratio.
To find the golden ratio using the quadratic formula, let the short side of a golden rectangle be 1 and the long side be x. The proportion (1+x)/x=x/1 must hold. Cross multiplying and gathering terms on one side gives x^2-x-1=0. Completing the square will yield the exact value of the Golden Ratio: (1+sqr(5))/2. The other root, (1-sqr(5))/2 is the reciprocal, as well as the conjugate, of the Golden Rectangle.
A Golden Rectangle can be continually broken down smaller and smaller by taking out squares.
Implications for Instruction:
I think this activity is very interesting because it has to do with nature and real life. Students should be motivated to do the activity and explore the different relationships.
The lesson itself has many algebraic concepts in the questions. A student must use ratios and understand how they work. They also must understand the properties of squares and rectangles in order to prove that BJ=HD+JH. The students must also know how to set up a quadratic equation and solve the equation by using either completing the square or using the quadratic formula.
I think this would be a good lesson to review certain concepts. It does not go into detail enough to introduce any new concepts but it would be good to enforce concepts and review others.