Guidelines for Writing Intuitive Geometry Summary Papers

In these summary papers, your goal is to answer and support the stated question in a concise but complete manner. Complete sentences and good paragraph structure are expected.

Step One (in class): Read the writing prompt that has been assigned and then in your small group work through the text’s questions that will help you discover the mathematical results. Focus on the big questions rather than repetitive exercises (but some repetition may be necessary to recognize patterns).

Step Two: (after class) Reread the writing prompt and your notes from class. You may wish to verify solutions with the answer key on reserve in the library. Jot down your answers to the writing prompt.

Step Three: In a brief paragraph, introduce the reader to the topic and address the writing prompt. Do not repeat the question (prompt) but rather rephrase it into a statement which incorporates your answers or introduces your topic. You are encouraged to develop an introduction that would capture the reader’s interest while remaining true to the main topic.

Step Four: Address each of the claims you made in your introduction or jotted down as answers and support them with results from your in-class work or mathematical theories. Be sure that you are using your own words. Don’t give a play-by-play of what you did in the activity, but instead focus on the mathematical results and support. Note that “because it works” is not adequate support. Explain why it works! If you don’t know why, come talk to the instructor during her office hours to make sure you understand the section.

Step Five: Write a last line or paragraph that summarizes your paper, again in your own words.

Step Six: Go back to the sentences you have written and find transitions/links that will improve the flow of your writing. Make sure that you use clear, convincing, and confident language. Whenever a pronoun, such as “it” is used, be sure that the object or idea “it” refers to is obvious to any reader. Words such as “might” should be reserved for situations in which the action doesn’t always occur. Find a friend who can read your paper and give suggestions. Offer to do the same for your friend.

IMPORTANT! Even though you have done your class discoveries and discussed the topic with others, the paper needs to be your own work and not a rephrasing of a classmate’s paper. If two papers are too similar, the points will be split (instead of 80% each would receive 40%, for example) on the first occurrence. Repeat infractions will result in no credit.

Helpful tips

• Describe the role of any variables you include. In the summary paper at the end of this document, you can see how the author identifies what each variable (letter) represents.
• Need to insert a mathematical symbol? If you have the full installation of MSWord, you can use INSERT – OBJECT – MICROSOFT EQUATION to get a palette with the usual symbols. I will also allow you to insert symbols neatly with a black pen. It’s usually best not to start a sentence with a symbol or equation.
• Want to add an illustrating diagram? Geometer Sketchpad drawings can be copied and pasted into any word document. Neat hand drawn diagrams may also be inserted or attached as an appendix. In either case, the diagram will not count toward the length requirement of the paper. It is always your responsibility as a writer to explain to your readers how to read the diagram and how the diagram helps to support your arguments.

Sample summary with typical student errors:

Writing prompt: What are the area formulas for the geometric figures in this section? How do we derive each of the formulas from the area formula of a rectangle? (approximately one page)

Summary:

To find the area of a rectangle we measure the length by the width. In this paper, we shall see how to compute other areas.

To determine the area of a parallelogram, cut the parallelogram into two pieces and rearrange them to get a rectangle. Obviously, their areas have the same formula, length times width.

The area of a triangle is given by one half a times b. This is obvious because a rectangle can be cut into two triangles.

A square is simply a rectangle with the height and width equal. Thus the area formula for a square is .

Finally, a trapezoid can be divided into triangles and rectangles. When you add the pieces up, you get the area of the trapezoid.

As you can see, you can get all the formulas for area just by knowing the area of a rectangle! I hope this paper is helpful to you as you learn geometry!

Sample summary:

Writing prompt: What are the area formulas for the geometric figures in this section? How do we derive each of the formulas from the area formula of a rectangle? (approximately one page)

Summary:

The area of a rectangle is where and represent the width and length respectively of the rectangle. This formula can be used to determine area formulas for a parallelogram, triangle, square, and trapezoid.

To determine the area of a parallelogram, we notice that we can move a triangle from one end of the figure to the other resulting in a rectangle with the same area (see figure to the right). The width of the rectangle is the same as the length of base and the height of the rectangle is the same as the height of parallelogram . Thus the parallelogram has area where is the base and is the height.

The area of a triangle is given by where is the altitude (or height) and is the length of the base. By making a copy of any triangle and rotating it 180 degrees about the midpoint of segment (as in the figure to the left), we form a parallelogram with the same base and height as . Since the triangle has half the area of the parallelogram, we get the formula .

A square is simply a rectangle with the length and width equal. Thus the area formula for a square is where is the common side length.

Finally, a trapezoid can be divided into two triangles by constructing a segment joining two opposite vertices. The two triangles have the same height as the trapezoid itself while one triangle shares the lower base of the trapezoid and the other uses the upper base . Therefore, the area of a trapezoid is or more simply .

While the formula for the area of a square is a simple application of the formula for a rectangle’s area, we see that the area formulas for other polygons can easily be derived from the formula for the rectangle by dissecting the polygon and rearranging the parts.