Overview:  This summary would most appropriately follow from Launch 2, in which the teacher introduced the historical development of perspective drawing.  The teacher shows a picture to illustrate that when artists first began to use perspective they often made errors, and that could make it difficult to locate a vanishing point in very old paintings.  The teacher connects these historical errors to the work of some students who misidentified a vanishing point on the second page of the problem.  With the diagram from page 2 projected onto the board, and with the second tree drawn into the diagram, the teacher draws two line segment extending from the tops of the trees that intersect in a point.  The teacher tells students that “some groups invented a new vanishing point” which is different from the vanishing point that already exists in the problem.  The teacher draws line segments through the tops of the houses to illustrate where the vanishing point should be in the diagram.  The teacher labels the vanishing point and tells students that all of the perspective lines should trace back to the vanishing point in order to keep the picture in perspective.  Finally, the teacher tells students that having a single vanishing point is important from an artistic perspective, because having many vanishing points can make a painting confusing.

Prior knowledge:  The teacher connects with students’ knowledge of the mathematical practice of making perspective lines through a common vanishing point, which is also related to the context of drawing in 1-point perspective.  Students seem to have enough knowledge of 1-point perspective drawing to have known that they needed to use perspective lines to make objects of equal height.  However, students did not all construct perspective lines in the correct way, so the teacher attempts to correct students’ knowledge of perspective lines and a single vanishing point.  The teacher uses the context of the problem, making a good piece of art, to justify the necessity of correctly using the mathematical practice of drawing perspective lines.

Other points of interest:  The teacher uses a historical painting to illustrate the imprecision that arises when an artist constructs multiple vanishing points.  The teacher tells students that artists and historians notice many errors when they look at a painting like the one the teacher shares, but to an untrained eye it is not obvious that the painting has inconsistencies with the perspective lines or vanishing points.  For students, the connection between the errors they made in drawing perspective lines and the errors in the piece of art may not be clear, because the artwork is too complex.