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TRANSCRIPT
Creating a Square Root Spiral
The Art of Mathematics
High School
Week 3
Overview Students will execute the square root spiral activity from Exploring Geometry with Geometer’s Sketchpad. Essential Question: Does art start with an aesthetic idea or with a mathematical composition? Objectives Students will be able to…
1. Utilize modeling software to generate the square root spiral 2. Identify the Fibonacci numbers within the spiral 3. Use the spiral as a basis for an art piece.
Activities Students will use the square root spiral activity sheet to create the square root spiral. Compare and contrast the square root spiral and the Fibonacci spiral they created by hand in small groups. During the activity the following will be discussed:
• Observe that the hypotenuse of the previous triangle becomes the base of the next triangle in the spiral.
• Recognize the significance of the use of the circle in the construction.
Adaptations Adapt and use The Ordered Distribution of Natural Numbers on the Square Root Spiral to identify and explore the relationship between the Fibonacci numbers and the square root spiral. Students using Geometer’s Sketchpad can create a custom tool which will allow them to create the square root spiral by re-‐creating the triangle and placing it in the proper location. Evaluation
• Have the students save or print the model. • Observe the students work throughout the class session to ensure
that students are not having trouble using the software or understanding the mathematical explorations.
• Take last ten – fifteen minutes to discuss findings and explore the mathematics.
Materials
Students will need access to a computer lab or a class set of computers. Modeling software and applications will be implemented in this unit. § Geometer’s Sketchpad or
GeoGebra § Student worksheet
Standards CCSS.Math.Content.HSG.CO.A.2: Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). CCSS.Math.Content.HSG.CO.A.5: Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. CCSS.Math.Content.HSG.CO.D.12: Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). NYS The Arts Standard 2: Visual Arts: Use the computer and electronic media to express their visual ideas and demonstrate a variety of approaches to artistic creation
Name: __________________________________ Date: ____________________
Creating a Square Root Spiral
1. Make sure the software your using is set to measure in inches with precision set at thousandths. 2. Create an isosceles right triangle a. Create a point A and translate it at 0° by 1 inch to
create 𝐴’
b. Rotate point 𝐴’ 90° around point A.
c. Connect the points with segments to create
∆𝐴𝐴’𝐴’’
3. Calculate the hypotenuse of ∆𝐴𝐴’𝐴’’
The following steps will allow you to create the rest of the spiral, starting with an isosceles triangle with a hypotenuse of 3.
1. Construct a line passing through 𝐴’ and ⊥to 𝐴’𝐴’’
2. Construct a circle with a point at A and center at point 𝐴’
3. Construct a point where the circle and perpendicular line intersect and label it B. Then hide the circle and the line.
4. Construct a triangle using points 𝐵,𝐴’ 𝑎𝑛𝑑 𝐴’’
5. Calculate the hypotenuse of ∆𝐴𝐴’𝐴’’ Predict what the value the next hypotenuse will have.
6. Repeat steps 1-‐3 to continue making the spiral, calculate the hypotenuse of each new triangle.
7. Construct interiors for the triangles whose bases are the square roots of the Fibonacci numbers.