Unit Plan Cover Sheet

Name(s): Kim Allred, David Bunnell, Ida Everett / Date: 3/08/06
Unit Title: Perspective Geometry
Fundamental Mathematical Concepts (A discussion of these concepts and the relationships between these concepts that you would like students to understand through this unit).
We would like the students to be able to recognize the perspective geometry axioms and how to use them. These axioms should help them to notice that the horizontal vanishing line in a drawing is at eye level and is where all lines parallel to the ground plane converge. They will need to translate their knowledge of Euclidean geometry to perspective geometry in order to make accurate drawings. Perspective geometry is the geometry of what our eyes can see. We hope our students develop differences and similarities between perspective and Euclidean geometry. One of these differences is that some parallel lines converge in perspective geometry and parallel lines never converge in Euclidean geometry. A similarity is that lines that intersect in Euclidean geometry will also intersect in perspective geometry.
Describe how state core Standards, NCTM Standards, and course readings are reflected in this unit.
Utah State Core Standards:
1. Students will solve problems using spatial and logical reasoning, application of geometric principles, and modeling. (Geometry Standard 3).
Students will use the axioms of perspective geometry to visualize and draw correct geometric constructions of tiled floors and windows.
NCTM Standards: (Geometry Standards for grades 9-12)
1. Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships.
Students will critique arguments made by others regarding axioms to perspective geometry. They will also explore relationships of 3D geometric objects as they solve problems involving vanishing points.
2. Apply transformations and use symmetry to analyze mathematical situations.
Students will represent translations of 3D objects by drawing them on paper. This will help them understand that 2D artwork is a projection of 3D objects onto a flat surface.
3. Use visualization, spatial reasoning, and geometric modeling to solve problems.
Students will draw 3D objects on paper from many different perspectives. They will also use perspective geometric ideas (such as projections, vanishing points, and horizons) to gain interest in art.
Course Readings: Students will become involved in various tasks that will capture their curiosity and extend their current knowledge. As students work in groups to explore these tasks, they will be expected and encouraged to interact with each other in order to evaluate, discard, and revise ideas. The teacher will request that students justify their thinking. The teacher will guide the discourse of the class by rephrasing and comparing differing ideas. Additionally, the teacher will ask students to justify their thinking, and will ask divergent questions rather than look for one right answer. Ideas and mathematical concepts discussed will be organized and summarized by the teacher at the end of each lesson (Becoming a Reflective Mathematics Teacher, pg 12-13; Talking about Math Talk, pg 193). Our unit we will use mathematics as a creative medium, where students will learn to apply rules of perspective geometry to draw depth correctly in two-dimensional artwork (The Four Faces of Mathematics, pg 21).
Outline of Unit Plan Sequence (Anticipation of the sequencing of the unit with explication of the logical or intuitive development over the course of the unit—i.e., How might the sequence you have planned meaningfully build understanding in your students?)
Day 1: Students will be introduced to perspective geometry and will discover its axioms using various pictures from Medieval and Renaissance times.
Day 2: The students will refine and apply the axioms they developed in the first lesson by taking a picture of tiles that are equidistant in Euclidian geometry and begin making a geometric construction of the tiled floor in perspective geometry. Students will not finish this task today.
Days 3 and 4: To help understand how to draw the tiles from the previous lesson, the students will draw windows on a wall that would be the exact same size in Euclidian geometry. Students will use the axioms of perspective geometry to see that foreshortening distorts similarity and that the diagonals of "similar" shapes must also converge.
Day 5: Students will make connections between all previous tasks. The pictures looked at on the first day will be revisited, and additional pictures will be presented to analyze. The axioms will now be discussed in greater detail as the students will now have a greater understanding of perspective geometry. The summative activity/assessment builds on the geometric principles of the previous lessons by having students complete the tile problem using principles we’ve discussed in class since they were first introduced to the problem.
Tools (A list of needed manipulatives, technology, and supplies, with explanations as to why these are necessary and preferable to possible alternatives).
Supplies:
-  paintings/drawings (Students will find horizon, vanishing points, idea of perspective drawing, comparisons of paintings that used it correctly and incorrectly to give examples)
-  Tile floor and window print outs
-  Rulers for each student (help them draw straight lines more accurately, aid their geometric constructions using perspective geometry)
-  Summative assessment for each student to take home and complete
Unit Plan Sequence: Learning activities, tasks and key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected students’ mathematics) / Time / Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.) / Formative Assessment (to inform instruction and evaluate learning in progress)
Miscellaneous things to remember
Launching Student Inquiry
DAY 1:
Today we will begin exploring the topic of perspective geometry. I’m going to hand out 3 pictures to be analyzed by your groups to determine similarities and differences of the 3 pictures. (Hand out pictures to 3 groups) / 2 min / Students will be wondering what I mean by perspective geometry, and will anticipate what they will see in the pictures I will give them. I’m pretty sure they will want to know what I will expect them to look for. I will make sure and leave as much of the observation up to them as possible as long as they are heading in the right direction.
Supporting Productive Student Exploration of the Task (Students working in groups or individually)
What are the major similarities and differences of the pictures you see? What makes one more accurate than another?
What happens when you extend or follow the lines produced by the tiles or the buildings?
After giving the students a few minutes to decide which pictures are drawn accurately to perspective, I will ask students to establish axioms for perspective geometry using the pictures provided and their own observations as resources.
I will ask students to determine things that are different in perspective geometry compared to Euclidean geometry.
I will go around to their groups and ask them questions, which will direct their thinking to help them come up with substantial arguments. / 15 min / They will likely notice differences that both do and do not apply to perspective geometry, such as different themes for the pictures and artist’s differing portrayals of specific scenes. They may also consider light sources.
Students will probably notice that the buildings in the painting on the left are not drawn to perspective, although they may not initially know exactly how they are drawn incorrectly.
They will discover vanishing points, and some will already have some previous knowledge in this area. Students will establish rules for vanishing points. These will include rules for parallel lines not parallel to the picture plane. Eventually they will come up with the idea that parallel lines in the picture plane will stay parallel, though students will argue over the validity of this rule.
Students will have a lot of good ideas but may have some difficulty in organizing their thoughts and drawing conclusions that are generally true.
Students will have some trouble using precise vocabulary, but will likely understand the concepts behind their ideas. / I will go around to each group and listen to what they are thinking about. I will help redirect them to be on task and to explore specific things about the pictures that are different. I will make sure and pose meaningful questions that will motivate their thinking and help them consider more possibilities.
I will ask them if they can justify their arguments.
I will help students stay focused and on-task.
Facilitating Discourse and Public Performances (Active Presentations by Students to Entire Class)
I will ask the groups to introduce their ideas to the class by each group presenting one axiom at a time. Other groups will build to and alter previous axioms, which will help the class come to accurate conclusions in the end. I will guide the building of axioms by using the white board to list and alter current axioms.
-“Can there be multiple vanishing points? How many can there be?”
-“What do we know is unchanged in perspective geometry?”
-“Why do the lines // to the picture plane remain //?”
I will make sure and not probe for specific answers. I will leave a lot of the conversation up to the groups. I will enable the responses to be presented in an organized way.
I will have students come up with definitions for vanishing points, horizons, and foreshortening. I will clarify their definitions when needed. / 10 min / Some groups will get farther in their thinking than other groups. However, I do believe that all of the groups will have a basic realization of what is wrong with the bottom left picture. By having all groups listen to each other, those struggling can learn from those explaining. Also, the people explaining their thoughts will likely engage in high level thinking in order to share their insights with the class. Therefore, all students will benefit from either explaining or from listening to other people’s thought processes.
Students will have various ideas about what is true for all perspective drawing. It will be difficult to come to a common consensus about what are actual axioms for perspective geometry.
Since this is a rather bright class, I presume they will come up with the majority of the important points. I can guide their thinking by asking insightful questions that they can think about.
Students will have many varied ideas regarding whether lines parallel to the picture plane are actually drawn parallel, what it means to have a vanishing point, and whether converging objects are similar or not.
Students may question the number of vanishing points pictures can have, and what the rules are for drawing with perspective. / Axioms:
1. Lines // to picture plane remain //
2. Lines actually in the picture plane appear true length in the perspective projection
3. All sets of // lines NOT // to picture plane converge to a vanishing point, foreshortening (objects farther away are drawn smaller)
4. Lines // to ground plane converge to horizon
Pare, Loving, Hill, Pare. Descriptive Geometry.
7th Ed. pg. 313-316.
Leon Battista Alberti
“On Painting” 1435
First artist to use mathematics in his artwork.
I will listen for comments and questions that are posed to help me gauge students understanding and interest in the topic. It is my responsibility to make sure students are motivated to learn more about this topic.
Unpacking and Analyzing Students’ Mathematics (Clarity, Reasoning, Justification, Elegance, Efficiency, Generalization)
At this point I will quickly summarize the 3 pictures by comparing their histories. I will compare Medieval art (sacred significance) to Renaissance art (perspective geometry).
In the Renaissance there was a conceptual leap when artists began using perspective geometry in their artwork. They used geometry to paint scenes similar to the way we see with our eyes. They studied projections from 3D objects to 2D surfaces to come up with rules to follow. The image of the object should be indistinguishable from the object itself, giving the viewer the perception of depth. / 3 min / Students will come to appreciate the significance that the Renaissance played in perspective geometry.
Students will understand the most important points of today’s lesson and will have established the axioms actually used when drawing with perspective. / I will restate and clarify the axioms used in perspective geometry.
If there’s extra time:
What are the differences between artificial (flat surface) and natural perspective projections (image seen by the eye)?
1,2,3 point perspectives, and their relation to the axis of the Cartesian graph. / Any Extra Time / Class Discussion / Discourse will be guided by student’s responses and my questions.
Unit Plan Sequence: Learning activities, tasks and key questions (What you will do and say, what you will ask the students to do, how you will accommodate to unexpected students’ mathematics) / Time / Anticipated Student Thinking and Responses
(What mathematics you will look for in student work and interactions.) / Formative Assessment (to inform instruction and evaluate learning in progress)
Miscellaneous things to remember /
Launching Student Inquiry
Day 2:
Begin the lesson by asking the students as a class to recall the axioms they generated for prospective geometry. / 3 min / The students may not remember the axioms. If so, it may be necessary to review one of the pictures.