Investigation #4 - Post by April 5th

This investigation will be focusing on measurement. Measurement is a large portion of the K-8 mathematics curriculum and can also be found spread throughout the secondary mathematics curriculum.

Try all of these investigations and choose two for your write up.

1) Polygon measurements

Part 1. Use a dynamic geometry software to show that the sum of the interior angles of any triangle always equal 180 degrees.

Part 2. Using the information from part 1, show that the sum of the interior angles:

• Any quadrilateral is 360

• Any pentagon is 540

• Any hexagon is 720

• Any octagon is 1080

Part 3. Now use the software to show the measurements for the exterior angles of all the polygons stated above. Make at least one conjecture about the exterior angles for an n-sided polygon, regular and irregular. Make sure to support your conjectures with reasoning and proof when needed.

2) Constructable angles and regular polygons

Part 1) An angle is considered constructable if you can make it precisely using a straight edge and compass (like in dynamic geometry software). Here is a list of a few constructable angles. Explain how to construct them using only ruler and compass (GSP or geogebra).

• 90⁰

• 60⁰

• 30⁰

• 45⁰

• 15⁰

• 72⁰

^{Find at least one other constructable angle.}

^{Part 2) There are several constructable regular polygons. Use the angles from part one to help you construct these regular polygons.}

• ^Triangle

• ^Square

• ^Pentagon

• ^Hexagon

• ^Decagon

^{Make sure to explain how you constructed these polygons.}

3) Exact Squares

Part 1. Download the exact squares GSP file from the bottom of the page. Go through the activity (including the extension) and post your findings. Discuss advantages and/or disadvantages you see from using this tool.

Part 2. Examine how exact squares can be used to explain the Pythagorean Theorem. Make sure to discuss how using exact squares with the Pythagorean focuses more on the area of the squares then the side length.

Part 3. Examine how exact squares can be used to help students understand proportional reasoning. For example, if you double the area of a square, the new squares sides are not double the length of the old square.

4) Weights and Values in GSP

Part 1. Download Balance.gsp from the attachments.

• Investigate how the balance works and determine the weights of each of the objects.

• How did you figure out the weights? Explain your thought process. Are there other ways to determine the weights?

• Reflect on how you can use this activity with students. What mathematical topics/skills does an activity like this provide for students?

• How can you change this activity to be more or less challenging for students?

Part 2. Download Circles and Squares.gsp from the attachments.

• Investigate how the activity and determine the values for a circle and a square.

• How did you figure out the values? Explain your though process. Are there other ways to determine the values?

• Reflect on how you can use this activity with students. What mathematical topics/skill does an activity like this provide for students?

• How can you change this activity to be more or less challenging for students?

Part 3. Investigate on your own.

Go onto the Geometer's Sketchpad Activity Webpage and investigate other activities that have to do with measurement. Make a write up similar to part 1 and 2 for one of the activities found on that website.

5) The Spiral of Theodorus

The spiral of Theodorus is a spiral constructed from contiguous right triangles. This is also known as the square root spiral.

Part 1. Use GSP or Geogebra to construct this spiral.

• The spiral above only goes up to the square root of 17. Construct a spiral that goes beyond the square foot of 17.

• Provide one reason you think that this spiral is useful in mathematics.

• Do some research on this spiral and include some history in your write up.

Part 2. It is said that the angle with its vertex at the center of the spiral for each right triangle can be found by using this formula:

Where n is the referring to the triangle number in the construction. For example, n = 1 refers to the first triangle made in the construction (base is 1 and height is 1). The second triangle (n=2) refers to the triangle with base 1 and height square root of 2.

• Use GSP or Geogebra to investigate this claim.

• If it seems to be true, use reasoning or proof to show the equation is correct. If the equation is not true, provide a counter example.

6) The Archimedean Spiral

Watch the following Youtube video on constructing the Archimedean Spiral

Part 1: In GSP or geogebra construct an Archimedian Spiral that goes one revolution of the circle.

Part 2: In GSP or geogebra construct an Archmedian Spiral that goes one and a half revolutions of the circle.

Investigate this claim on either part 1 or part 2.

The radius of the spiral and the rotation angle are linearly proportional, meaning that r, the distance from the origin to any point on the curve, increases at the rate of a*θ. (a is related to the linear proportionality)

7) Baravelle Spiral

What is a Baravelle Spiral and how are they made?

Part 1)

Go to this web page and learn about Baravelle Spirals and work through the activity guide.

Part 2)

On your investigation explain what Baravelle Spirals are and create at least two of them using different regular polygons.

Part 3)

Explain how are Baravelle Spirals connected to fractions and geometric series. Determine the area of each of the spirals that you created in Part 2.

8) Tires

Part 1. On her 18" bicycle, Lauren pedaled as fast as she can and then coasts until the bike stops. After Lauren stops pedaling, the bike goes 500 feet. How many revolutions did the bike wheel do while coasting?

• Use a dynamic geometry software to model this problem and compare the algebraic answer to this problem to the answer that comes from the model.(Constructing a cycloid can be helpful in this problem)

Part 2. Carlton has a 16" bicycle and coasts just like Lauren. His bike goes the same distance (500 ft). Do you think that the bike wheel had to go more, less, or the same amount of revolutions as Lauren's 18" bicycle? Explain your reasoning.

• Use a dynamic geometry software to model this problem and compare the algebraic answer to this problem to the answer that comes from the model.

  • (Constructing a cycloid can be helpful in this problem)

Part 3. Many people like to pimp their rides by getting a cool paint job, TV's with game systems, and larger rims. Do you think it would make a big difference in how much faster a car is going with bigger rims?

If you had a car with 16" rims and upgrade to 18" rims, how much faster would the car go?