Domains: Geometry

Isometry (Rigid Transformations) Unit:  http://caccssm.cmpso.org/a/cmpso.org/caccss-resources/geometry-task-force/geometry-resources/isometry-unit

By the time students are in middle school, the Common Core standards are focusing more and more attention on transformational geometry.  The CaCCS Introduction to Transformation Geometry provides a progression and rationale for the increased emphasis on this approach to geometry (see text below in blue):

Below is a progression of standards that provide a development of understanding of attributes of
figures, culminating in an understanding of similarity:


Kindergarten: Analyze and compare two- and three- dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts, and other attributes.


Grade 1: Distinguish between defining attributes (e.g. trianges are closed and three-sided) vs. non-
defining attributes (e.g. color, orientation, overall size)


Grade 2: Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.


Grade 3: Understand that shapes in different categories (e.g. rhombuses, rectangles, and others) may share attributes (e.g. having four sides), and that the shared attributes can define a larger category (e.g. quadrilaterals).


Grade 4:


• Classify two-dimensional figures based on the presence or absence of parallel or perpendicular
lines, or the presence of absence of angles of a specified size.


• Recognize a line of symmetry for a two-dimensional figure as a line across the figure such
that the figure can be folded along the line into matching parts. Identify line-symmetric
figures and draw lines of symmetry.


Notice that the second item here is the beginning of a transformational approach to geometry.


Grade 5: Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares
are rectangles, so all squares have four right angles.


Grade 6: The 6th grade geometry standards focus primarily on applications of previous grades’ standards.


Grade 7:


• Solve problems involving scale drawings of geometric figures, including computing actual
lengths and areas from a scale drawing and reproducing a scale drawing at a different scale.


• Describe how two or more objects are related in space (e.g., skew lines, the possible ways
three planes might intersect).


Grade 8 and Algebra 1: In eighth grade there is a significant emphasis on transformation geometry, withstandards addressing a number of foundational and crucial concepts in this area. Transformations are viewed primarily in light of their effect on geometric figures; it is not yet emphasized that a transformation is a function from the plane (or 3-dimensional space) onto itself. We list here standards that directly relate to the those listed above.


• Understand that a two-dimensional figure is congruent to another if the second can be
obtained from the first by a sequence of rotations, reflections, and translations; given two
congruent figures, describe a sequence that exhibits the congruence between them.


• Understand that a two-dimensional figure is similar to another if the second can be obtained
from the first by a sequence of rotations, reflections, translations, and dilations; given two
similar two-dimensional figures, describe a sequence that exhibits the similarity between them.


High School Geometry: In high school there is a significant emphasis on transformation geometry,
and these standards provide a natural development from the eighth grade geometry standards on
transformation geometry. Importantly, the high school standards move to an understanding of
transformations as functions of the plane (or 3-dimensional space) onto itself.


What is our current situation?


It is generally accepted that today’s K-12 students are not well-versed in transformation geometry.
Even more problematically, since we will implement the CaCCSS throughout the grades at the same
time, students will begin a study of eighth and ninth grade transformation geometry standards without adequate preparation in previous grades. An example of the current status of understanding of similarity, for example, is given by the results in one NAEP item:


In the 2007 National Assessment of Educational Progress (NAEP) Assessment, 6.75% of US eighth
grade students answered the following question correctly:


The figure on the right shows two right angles. The length of AE
is x and the length of DE is 40 units. Show all the steps that lead to
finding the value of x. Your last step should give the value of x.


Clearly we have a significant amount of work to do in order to raise
the level of geometric understanding, in particular understanding of
transformation geometry, among our students.


Why are transformations important?


Transformations are an integral part of everyday life. The lamp as we see it is not the size it seems
- our brains translate the image and tell us what it really is. When we see a photograph, we know that the people depicted in it are not really 2 inches high - their images have been dilated by a camera using a dilation with positive scale factor smaller than 1 to create the images. People whose images are seen on a cinema screen are not really 12 feet tall - they have been transformed by a dilation with scale factor greater than 1. A map is an image of a dilation. Movies are created using transformations. As we look around us, more and more examples of transformations become apparent.


In mathematics, transformations make important connections between topics, and form the basis
of others. For example, dilations provide a basis for understanding trigonometric functions of the right triangle. ...


Transformations provide elegant proofs for problems that can be solved in traditional ways, and allow for generalizations that would be very difficult otherwise. For example, consider two different solutions to the following problem:


Inscribe a square in a given triangle ∆ABC so that one side of the
square is on one side of the triangle, and the remaining two vertices are
on the other two sides of the triangle. Prove that your construction yields
the desired result. ...



Equally importantly, transformations provide a pathway to unify and generalize a variety of mathe-
matical concepts and to solve problems in much more natural ways.

https://docs.google.com/file/d/0ByBN4IxhmiWgZDk1MTdmMTMtZjU0Ny00OTAyLWJlMTEtNjU1YmJjYTFjOGM2/edit?hl=en&pli=1

Maximizing Area: Golden Rectangles - a MARS Formative Assessment project http://map.mathshell.org/materials/download.php?fileid=1226