When our eyes are open and we are looking around, everything we see is in perspective. This is exaggerated when we look down a city street, a set of railroad tracks or even a long hallway. As we look farther and farther away, the parallel sides of these objects seem to narrow as if they will eventually meet or even vanish. This is the way our visual world works.

In the high school geometry course I teach, most of the drawing and sketching that students do is either two-dimensional or the standard three-dimensional drawing where hidden lines are shown as dotted lines. Drawing things in perspective, the way they actually look, is not typically focused on. Perspective drawing is discussed in only one of the lessons in chapter one of our textbook, which means we usually only spend one day on it. So even though it is revisited in review problems throughout the year, students seem to have a hard time with it. I would like to incorporate more perspective drawing in my class.

Perspective drawing has applications in art, architecture, computer graphics, engineering, interior design and landscaping. Since these are potential fields where students may find a career, I think that it would be beneficial to spend more time on perspective drawing. We currently do projects in the third marking period of the school year. One of the project options involves perspective drawing, but only 1 to 2 students per class work on this one because every individual or pair is required to work on a different project. What I want to do this year is expand the current lesson and implement a project, involving perspective drawing, which everyone will do during the first marking period. That way, students will be more exposed to drawing things as they really appear, which will help to foster spatial sense that is so important. This will also hopefully get students more interested in geometry right away. They will be exposed to the artistic part of the subject and have the opportunity to express themselves in an artistic manner. Here is a summary of the lesson and project.

Students will study the technique of drawing in perspective and then apply this while completing an artistic project in which they will be creating their own perspective drawing or collecting, analyzing and making a collage of other perspective drawings, paintings and pictures.

This project will give students the opportunity to: observe the world around them, develop their spatial sense, and express themselves artistically using math as a medium. It will also possibly spark an interest in a career that uses math and perspective drawing.

At the conclusion of this project students will be able to: 1) determine if a drawing is a perspective drawing, 2) locate vanishing points, horizon lines and orthogonal lines in perspective drawings, 3) draw simple figures in both one-point and two-point perspective, 4) look at non-perspective drawings and draw them in perspective and 5) identify careers in which perspective drawing is used.

The following will possibly be used during the course of the project: geometry textbook, handouts, worksheets, straightedges, poster board, drawing and coloring supplies, glue, scissors, magazines, internet, perspective drawing websites, and the computer program Geometer's Sketchpad.

This project will start during the second week of school, when the topic of perspective drawing is introduced. After studying perspective drawing vocabulary and practicing perspective drawing techniques for two to three days, students will be given their final project assignment. They will have several weeks to work on it, both out of class and in class during spare time.

The project will start with a pre-test on perspective drawing. The pretest will cover basic terminology such as: vanishing point, vanishing/horizon line, orthogonal lines and lines of sight. They will also be shown several pictures/drawings and will determine if they are drawn in perspective and if so locate the vanishing point or points. I will also ask the students to draw a box in both one and two-point perspective. Finally, students will be asked to list possible careers that involve perspective drawing. The pre-test will be evaluated and kept by me to compare to the post-test.

Students will then be introduced to perspective drawing through a short activity. The activity gives step-by-step directions on how to draw a box in one-point perspective. Figure 1 below shows a box that is not drawn in perspective. Students will typically draw something like this when asked to draw a three-dimensional box. Figure 2 shows a box that is drawn in perspective. Students will end up with something like it after completing the activity.

Within this lesson students will learn what a perspective drawing is and they will be introduced to vocabulary commonly associated with it. These terms include: vanishing point, vanishing/horizon line and lines of sight. They will also be shown what boxes drawn in two and three-point perspective look like. In the questions and exercises of the lesson, the students will be asked to determine if pictures are drawn in perspective, locate vanishing points, and draw simple objects in perspective. Normally, this is where the lesson would end, but students usually have trouble with this, so this year I intend to do more practicing and instruction with them on it.

I will first tell them that perspective drawings are just drawings of objects or scenes as they would really appear. The farther things are away from the viewer's eye, the smaller they will appear. Then I will show them examples of things drawn both in and out of perspective. In the perspective pictures, I will show them how and we will practice locating vanishing points and using orthogonal lines. We will then practice drawing simple objects in perspective.

Buildings are quite easy to draw in both one and two-point perspective, and demonstrating these will help some of the students with their projects. The following are different drawings of the outline of a building. The first is a typical geometrical three-dimensional drawing.

The second is a building drawn in one-point perspective with steps shown. First a rectangle is drawn to represent the front of the building. Then a vanishing point is chosen. Orthogonal lines are then drawn from three vertices to the vanishing point. Lines are drawn parallel to the top and right of the front of the building. Then excess lines are erased.

Now I will illustrate the steps for drawing a building in two-point perspective. First start with a line segment. Then choose two vanishing points, one on each side of the segment. Then draw orthogonal lines from the endpoints of the segment to each of the vanishing points and draw vertical lines to form the sides of the building. Finally erase all excess lines. This building is drawn as viewed from a corner with the eye level of the viewer being at the height level with the two vanishing points.

The students will also do an activity that shows how to draw evenly spaced objects in perspective. Some examples of evenly spaced objects are utility poles, fence posts and trees. This will be helpful for some students when they do their final project. Additionally, there is a good website that I found on perspective drawing and I intend to alert the students to it so they can use it as a resource. The site offers a variety of techniques for drawing different types of things in perspective. The address is http://www2.evansville.edu/studiochalkboard/draw.html. I also plan on taking the students outside one day around the school to practice sketching scenery in perspective. After all of the previous discussion and activities, the students should be ready to start working on their final project.

The students will have two options for the final project. The first option, and the one that will be worth the most points, is to do their own perspective drawing on a large sheet of paper or poster board. To encourage self-expression I will leave it up to them as to what exactly to draw, but I will give them some ideas also. They can draw real or imaginary scenes. Some ideas are: a city street, a country road, a woods road, a railroad track scene, a city block as viewed from a corner, a room, a view down a waterway, or a castle. These are just a few of my ideas, but kids are usually more creative than I am and will probably think of others. I will allow the students to use whatever media they want in their work. Pencil, colored pencils, markers, crayons, watercolor, etc. are all fine. The second option for the final project will be to make a collage of perspective drawings/pictures on a large piece of poster board. The students will have to locate the vanishing point or points on each picture.

This lesson and project will be beneficial for the students in that art and mathematics will be integrated and then not seen as two totally different subjects. The lesson and project are aligned with the Michigan Curriculum Framework. Several benchmarks from the Geometry and Measurement Strand are achieved.

Early in the school year, before we covered the lesson about perspective drawing in the geometry textbook, I gave the students the pre-content test that I made up (which is included) and the pre-attitude survey. The pre-test included questions about perspective drawing and associated terminology. There was one multiple-choice question that had students identify which of 3 boxes was drawn in perspective. And finally the students were asked to draw a picture of a building in one-point perspective. The overall average on the pre-test for my 3 geometry classes was approximately 30% (or 3 out of a possible 10 points). Several students actually did pretty well on the pre-test. I found out that they had done some perspective drawing in an art class the year before. However, the majority of the students didn't really know what perspective drawing was.

A day or two after the pre-test was given; we actually covered the lesson in the textbook on perspective drawing. We covered basic techniques for drawing in one and two-point perspective and the associated vocabulary. Students did problems from the textbook and worksheets that gave them the opportunity to practice locating vanishing points, identifying objects that were drawn in perspective, and looking at pictures of objects and then drawing them in perspective. They also did a couple of activities in which they were instructed on how to draw evenly spaced objects in perspective. Learning this technique helped the students draw things in their final projects such as: telephone poles, fence posts, trees, lines in the sidewalk, dashed lines in the middle of a street. I also advised them to visit the website mentioned in my paper to learn other techniques that might be helpful to them in their final projects. We also talked about possible careers that would use perspective drawing.

Then the project was assigned. I gave them a description of the project (which is included) and we discussed it and I answered questions that they had. The project was due on October 23, 2002. I had a couple of other deadlines prior to that so that they would start thinking and working on the project ahead of time instead of waiting until the last minute to do it. Most students decided to work with a partner, but several did choose to work alone. During the period of time they were working on their projects, we also kept up with the normal course schedule, so they were pretty busy. Whenever we had spare time in class, students were allowed to work on their projects. I did set aside a couple of full class periods for the students to work on them as well. Most of them spent extra time outside of class working on their projects.

Although it was extra work for them, I think most students had fun doing their projects. Some of them really got into it and did a great job. There were a variety of scenes drawn such as: city streets, woods roads, country roads, futuristic cities, a wild west scene, a mountain/river scene, a football field, a picture of the Portage Lake lift bridge, skyscrapers, a school hallway and a room. One person even did a 2-point perspective drawing. Just one person out of the 3 classes chose to do the second option for the project. He made a collage of pictures on a piece of poster board and located the vanishing points in all of them. Perspective was emphasized in all of the pictures he chose. It actually turned out pretty nice. Several of the projects are displayed in my room. Parents that came in for conferences thought they were pretty neat!

After all the projects were finally in, I graded them. Almost everyone that turned in his or her project on time got a grade in the B range or better. There were a couple of exceptions. As with anything, there are always those students that just do what is necessary to get by. I based their grades on several things: correct techniques, neatness, effort and creativity. Then the students were given the post-content test and the post-attitude survey. The overall average on the post-content test was approximately 84%, which was up significantly from the 30% on the pre-test. Almost every student improved his or her score to some degree.

If I do this project again, one thing I will do differently is make a grading rubric ahead of time and distribute it to the students at the start of the project. That way, the students will know exactly what I am looking for and it will help me evaluate their projects more systematically. As a math teacher, I guess it is a challenge for me to grade these types of projects. I am used to grading a totally different type of work. I also might try to put more deadlines into the project timeline. My intention was for the students to get started early on their projects. They did get rough drafts done on time, but a lot of them didn't really get started on their final posters until a few days before the deadline. As a result, those projects weren't as good as they could have been. Overall, however, it was a good experience. The students did learn something and had the opportunity to express themselves artistically.

You have two options for the project. The first option, and the one that will be worth the most points, is to do a perspective drawing on a large sheet of paper or poster board. This option will allow you to use the perspective drawing techniques explored in Chapter 1 as well as your artistic and creative talents to create a poster depicting a perspective scene of your choice. To encourage self-expression I am leaving it up to you as to what exactly to draw. You can draw real or imaginary scenes. Some ideas are: a city street, a country road, a woods road, a railroad track scene, a city block as viewed from a corner, a room, a view down a waterway, or a castle. The drawing may be done in either one-point or two-point perspective. You may use whatever media you want: pencil, colored pencils, markers, crayons, watercolor, etc. You may work with one other person on this project. There is a website that I suggest you check out. It is all about perspective drawing and gives additional information and drawing tips. The address is http://www2.evansville.edu/studiochalkboard/draw.html

The second option for the final project will be to make a collage of perspective drawings/pictures on a piece of poster board. You may cut pictures out of magazines, print them off the Internet or actually take photographs yourself. You will need to locate the vanishing point or points on each picture. The collage must be neatly done. You should create some type of border for each picture.

The project is worth 75 points, which is equivalent to a little more than the points possible for a normal chapter test. This project will be included in the Test/Quiz/Project portion of your 1st marking period grade. If you choose option 1, you have the possibility of earning all 75 points. If you choose option 2, you have the possibility of earning a maximum of 63 points (or an 84%).

A plan for completing the project is due Friday, September 27. This is worth 5 points. This is simply a short paragraph indicating which option you are choosing and what you plan to do to complete it. (If you choose option 1, let me know if you are working alone or with a partner. If you are working with a partner, indicate how you are going to divide up the work so that each member contributes equally to the project. Also let me know what type of scene you plan on drawing.)

A rough draft is due October 4th. This is worth 10 points. For option 1 this means a drawing done on an 8.5” by 11” piece of paper showing a rough idea of what you will be including in your scene. For option 2 this means bringing in pictures that you intend to use in your collage or showing actual progress you've made.

3rd Hour Geometry			5th Hour Geometry			7th Hour Geometry
Student	Pre-test	Post-test	Student	Pre-test	Post-test	Student	Pre-test	Post-test
1	2	10	1	3	8	1	1	6
2	5	9	2	0	5.5	2	3	8
3	3	8	3	7	10	3	4	10
4	6	10	4	3	10	4	2	7.5
5	7	9	5	5	10	5	4	5.5
6	3	9	6	5	10	6	2	7
7	2	7	7	9	8	7	1	8.5
8	7.5	8	8		9	8	6	10
9	7	10	9	0	7	9	0	8
10	0	7	10	3	8	10	1	8
11	4	10	11	4	9	11	1	10
12	2	7	12	0	9	12	4	8
13	5	10	13	6	9	13	4
14	1	10	14	1	9	14	0	8
15	0	7	15	4	9	15	0	10
16	1	7	16	7	10	16	0	2
17	2	8	17	0	3	17	0	9
18	5	9	18	6	10	18	3	9
19	0	9	Total	63	153.5	19	0	6
20	8	8	Average	3.705882	8.527778	20	1	6
21	0	8				Total	37	146.5
22	7					Average	1.85	7.710526
23	6	10
24	0	10
25	3	8
26		9
27	3	10
Total	89.5	227
Average	3.442308	8.730769		Pre-test	Post-Test
				Overall	Overall
			Total	189.5	527
			Average	3.007937	8.365079