Title: Introduction to Spheres



Title: Introduction to Spheres

Name: Elaine Hebert

Date of Lesson: Thursday of Week 2

Description of Class: High School Geometry

Grade Level: High School Geometry

Length of Lesson: 50 minutes

TEKS Addressed:

Geometry:

b.1(C): The student compares and contrasts the structures and implications of Euclidean and non-Euclidean geometry.

c.1: The student uses numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons, ratios in similar figures and solids, and angle relationships in polygons and circles.

d.1: The student analyzes the relationship between three-dimensional objects and related two-dimensional representations and uses these representations to solve problems.

e.1 (D): The student finds surface areas and volumes of prisms, pyramids, spheres, cones and cylinders in problem situations.

e. 2(D): The student analyzes the characteristics of three-dimensional figures and their component parts.

I. Overview:

Students will explore properties of spheres through a hands-on investigation. This will precede the topic of spherical geometry.

II. Learner Objectives:

Students will be able to:

-find the surface area and volume of a sphere when only the circumference is known.

-measure distances on a sphere.

- estimate surface areas of figures on a sphere’s surface.

III. Materials:

Medium sized Styrofoam balls (one per group)

String to measure around the Styrofoam balls (one per group)

Rulers (one per group)

Attached Handout

Paper (one per group)

Engagement:

|Teacher Does |Expected Student Response |Ongoing Evaluation |

|“We are going to get into our groups of 3 |Students get into their assigned groups of | |

|and play a word association game.” |3. | |

|“Today our team building exercise will be a|Students listen. | |

|game where I say a word and your group goes| | |

|around writing a word associated with my | | |

|word.” | | |

|“So take turns in your group. You will have|Students write words in their groups | |

|2 minutes then we will compare our words. |associated with the word ‘sphere’. | |

|The word to associate with is ‘sphere’.” | | |

|“Ok, so let’s go around and each group can |Students read their words to the class. | |

|read their words. Take note of the ones | | |

|that repeat.” | | |

|“Ok, I noticed that a lot of you guys said |Students listen. | |

|words like ‘ball’ and ‘round’ and even some| | |

|circle words like ‘diameter’. This is a | | |

|very good start. Today’s activity deals | | |

|with properties of spheres.” | | |

Exploration:

|Teacher Does |Expected Student Response |Ongoing Evaluation |

|“I’m going to pass out a Styrofoam ball and|Students work on the first problem and | |

|a ruler to each group. Each person will get|discover that finding the diameter and | |

|a worksheet that they must turn in at the |circumference is hard with just the ruler. | |

|end of class. You will work in groups but |They suggest a string. | |

|everyone must turn in the worksheet. For | | |

|right now, I want your group to only be | | |

|working on the first problem dealing with | | |

|surface area and volume. Can you figure | | |

|this out with just the ruler? Is there an | | |

|easier way to do this problem?” | | |

|‘Ok, so we figured out that using a string |Students work in their groups to complete |“So the first problem requires us to find |

|would be easier. I will pass out a string |the worksheet. |the diameter, but we know the circumference|

|to each group and you can now, hopefully, | |by measurement. How are we going to find |

|get the rest of the worksheet done.” I go | |the diameter to find the surface area and |

|around helping groups. | |volume?” |

|About 10 minutes before the end of class- |Students write in their journals until the | |

|“Now I would like you guys to get your log |end of class. | |

|books out and write what you thing was easy| | |

|and challenging about this activity. Also, | | |

|how were you a good group member today? Did| | |

|you learn something new?” | | |

|“Tomorrow we will be exploring properties | | |

|of other 3-D shapes.” I pick up the | | |

|journals and materials. | | |

Explanation:

“Spherical geometry is different from planar geometry. We discovered that measuring distances and surface areas on a sphere is a little more difficult than on a 2-D plane due to the curvature of a sphere. As we all know, our world is a sphere, which is one reason learning about them is important. We do not notice the curvature of our world because it is so big. For our final project we will be working with transformations of spheres to a 2-D plane, like maps of the world, so it is important to remember these properties we have learned today.”

Elaboration:

If students are interested there is more information about spherical geometry at this website: . There are activities for students to do along with useful information. Specifically there is a good idea for an extension lesson exploring the properties of a spherical triangle.

Additionally, using this java applet here: , students can explore more properties of spheres and further their knowledge of spherical geometry.

Name: Date:

Introduction to Spheres Activity and Assessment

1a. Using just the ruler, estimate the surface area and volume of the sphere. Please show your work.

1b. What did you measure using the ruler? How did this help you get the surface area and volume?

1c. What was hard or easy about 1a? Do you think it was an accurate way of measuring the surface area and volume? What might be a better way of doing part 1a? When you are finished with this problem, please let the instructor know before proceeding to the next problems.

2a. Using the string, now find the surface area and volume of this sphere. What did you have to measure with the string? How did that measurement help you find the surface area and volume? Please show your work.

2b. What was hard or easy about 2a? Do you think this is a more accurate way than 1a of measuring the surface area and volume? Why or why not?

3. Make a point on the sphere anywhere. Label it P. Can you find another point, labeled K that is furthest away from point P? How did you find K? How do you know that it is the furthest point away from P?

4. Make three points on the surface of the sphere in the shape of a triangle and connect them (see the picture below). Label the points A, B, and C. This is called a spherical triangle.

[pic]

What are some of the differences between this triangle and a planar triangle? Are there any similarities? (If you wish, you can take a protractor and measure the sum of these angles.) Estimate the surface area of this triangle. What was easy or hard about measuring the spherical triangle’s surface area?

5. 2-D planes have a coordinate system we are all familiar with, like grid paper. Can you think of an example of a spherical coordinate system? How is it similar to and different from a 2-D coordinate system?

................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download