Graphing Bases to Integral Exponents

 

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This activity helped me understand the relationship between exponents and bases better by showing me visually.  I learned that in our equation, the bigger X is, the smaller Y will be as it is turned into a fraction; if X is bigger than 1, Y will be in fraction form.  This is because if the exponent is negative, the equation has to be a fraction in order for the exponent to be positive. On the X axis, the points get closer and closer to 0 as X grows, without ever actually reaching 0.

When we compared our group to group 5’s, we realized that our graphs are mirror images of each other. This is because they had the same equation to represent, without the negative sign in front of the exponent, so when our X was negative, the other negative sign cancelled it out, making a positive. This is the opposite of group 5, as when X was negative, the exponent was negative also (this means that we had small fractions on opposite ends of the graphs on the X-axis.)

The Golden Ratio

The golden ratio is something very commonly used in math, art, architecture, geometry, etc.

It is represented by the Greek number phi (φ or ϕ) which is equal to 1.6180339887.., that goes on infinitely, making it an irrational number.

The golden ratio is when the ratio of 2 quantities is the same as the ratio of the larger quantity to the sum of the 2. (a+b/a = a/b).

The golden ratio is related very closely to the Fibonacci sequence; when 2 consecutive numbers from the sequence are taken, their ratio is very close to that of the golden ratio.

It’s unknown who exactly discovered the golden mean/the number phi, since it’s been said to have existed forever.  Although, ancient Greeks Phidias, Plato, and Euclid have played a great role in the study of the golden ratio.

Math Football Problem

1. First, our group brainstormed before coming up with the idea to make several pentagons which would connect together to make one sphere shape.  We continued making pentagons by connecting the pasta with marshmallows.

2.  We learned that the bigger the pentagon, the harder it is to keep the shape intact so we made them smaller.

3. Because this is a problem of geometry, and it involves problem solving and knowledge of geometry (math) in order to create a ball shape.

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