TEACHER GUIDE TO CLARIFICATION
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.1 Verify experimentally the properties of rotations, reflections, and translations:
Lines are taken to lines, and line segments to line segments of the same length.
Angles are taken to angles of the same measure.
Parallel lines are taken to parallel lines.
Verifying Experimentally Properties of Transformations
Students should be given the opportunity to explore congruency and symmetry through translations without the use of the coordinate plane. They should be given compasses, protractors and rulers to make/prove their predictions about the space from the line of reflection. If possible, students should also be exposed to electronic programs, such as Geogebra, and materials like patty paper,miras, tracing paper and transparencies.
Terminology in this unit is paramount to the assessment. Students are expected to use Translation NOT SLIDE
Reflection NOT FLIP
Rotation NOT TURN
New or Recently Introduced Terms
Transformation (A transformation is a rule, to be denoted by F, that assigns each point of the plane a unique point P which is denoted by F(P).)
Basic Rigid Motion (A basic rigid motion is a rotation, reflection, or translation of the plane.
Basic rigid motions are examples of transformations. Given a transformation, the image of a point A is the point the transformation maps the point A to in the plane.)
Translation (A translation is a basic rigid motion that moves a figure along a given vector.)
Rotation (A rotation is a basic rigid motion that moves a figure around a point, d degrees.)
Reflection (A reflection is a basic rigid motion that moves a figure across a line.)
Image of a point, image of a figure (Image refers to the location of a point or figure after it has been transformed.)
Sequence (Composition) of Transformations (A sequence of transformations is more than one transformation. Given transformations G and F, G o F is called the composition of F and G.)
Vector (A Euclidean vector (or directed segment) AB is the line segment AB together with a direction given by connecting an initial point to A a terminal point B.)
Congruence (A congruence is a sequence of basic rigid motions (rotations, reflections, translations) of the plane.)
Transversal (Given a pair of lines L and M in a plane, a third line T is a transversal if it intersects L at a single point and intersects M at a single but different point.)
From Engage NY
Taken from Flip Book
A major focus in Grade 8 is to use knowledge of angles and distance to analyze two- and three-dimensional figures and space in order to solve problems. This cluster interweaves the relationships of symmetry, transformations, and angle relationships to form understandings of similarity and congruence. Inductive and deductive reasoning are utilized as students forge into the world of proofs. Informal arguments are justifications based on known facts and logical reasoning. Students should be able to appropriately label figures, angles, lines, line segments, congruent parts, and images (primes or double primes). Students are expected to use logical thinking, expressed in words using correct terminology. They are NOT expected to use theorems, axioms, postulates or a formal format of proof as in two-column proofs.
Transformational geometry is about the effects of rigid motions:
* reflections and
* translations of figures.
Initial work should be presented in such a way that students understand the concept of each type of transformation and the effects that each transformation has on an object before working within the coordinate system.
For example, when reflecting over a line, each vertex is the same distance from the line as its corresponding vertex. This is easier to visualize when not using regular figures. Time should be allowed for students to cut out and trace the figures for each step in a series of transformations.
Discussions should include:
1. the description of the relationship between the original figure and its image(s) in regards to their corresponding parts (length of sides and measure of angles)
2. the description of the movement, including the attributes of transformations
a.line of symmetry
b .distance to be moved
c. center of rotation
d. angle of rotation
e. the amount of dilation)
3. The case of distance – preservation in a transformation leads to the idea of congruence.
Work in the coordinate plane should involve the movement of various polygons by addition, subtraction and multiplied changes of the co-ordinates. For example, add 3 to x, subtract 4 from y, combinations of changes to x and y, multiply coordinates by 2 then by 12.
“What does making the change to all vertices do?”.
“What happens to the polygon?”
Student should discuss these questions and others about
the transformation’s effects on the polygon and its angles.
Understandings should include generalizations about the changes that maintain size or maintain shape, as well as the changes that create distortions of the polygon (dilations). Example dilations should be analyzed by students to discover the movement from the origin and the subsequent change of edge lengths of the figures. Students should be asked to describe the transformations required to go from an original figure to a transformed figure (image). Provide opportunities for students to discuss the procedure used, whether different procedures can obtain the same results, and if there is a more efficient procedure to obtain the same results.
Students need to learn to describe transformations with both words and numbers. Through understanding symmetry and congruence, conclusions can be made about the relationships of line segments and angles with figures.
Students should relate rigid motions to the concept of symmetry and to use them to prove congruence or similarity of two figures. Problem situations should require students to use this knowledge to solve for missing measures or to prove relationships. It is an expectation to be able to describe rigid motions with coordinates.
Provide opportunities for students to physically manipulate figures to discover properties of similar and congruent figures, for example, the corresponding angles of similar figures are equal. Additionally use drawings of parallel lines cut by a transversal to investigate the relationship among the angles. For example, what information can be obtained by cutting between the two intersections and sliding one onto the other?
End Flip Book
Explanations and Examples:
Students build on their experiences with measurement and proportions to notice congruency between shapes and to perform the transformations using measurement tools. In addition, angle types and measurements which is taught in 7th grade is also a pre-requisite for this unit.
Introduce the concepts through art.
Reflect the images across line. Label the reflected images.
a. Translate along Label the image of the triangle with X', Y', and Z'.
Students are expected to measure the length of the vector AB. The measurement can be achieved by using a ruler or a compass.
They are then to create a line segment from Z to create Z’., etc.
b. Translate along Label the image of the triangle with X', Y', and Z'.
1. Let be a segment of length 4 units and be an angle of size . Using your transparency, create a rotation by 45 degrees about 0. Find the images of the given figures. Answer the questions that follow.
a. What is the length of the rotated segment Rotation (AB)?
b. What is the degree of the rotated angle Rotation ?
Notice both examples above DO NOT use a coordinate plane, instead students are expected to measure the distances for both reflection and translation.
Coherence and Connections: Need to Know
PARCC MYA/PBA/EOY 8th
Students use prior knowledge from sixth and seventh grade related to measurement for congruency, translation, and reflection. Their knowledge of proportions will enable them to understand the idea of dilation and similarity. In addition, the work in seventh grade surrounding angle types, i.e. vertical angles, corresponding angles, parallel lines crossed by a transversal, will serve as an anchor on which to pin the new concepts of congruency and symmetry and similarity.
This is a free online software that is similar to Geometer’s Sketchpad. A teacher can create a Digital “workbook” for students to use or a student can produce work and place it in a book for the teacher to look at. In addition, there is a public gallery where you can use other people’s work.
This lesson from illustrative math uses origami to teach the concepts. There is a step by step guide for the teacher to use as he/she teaches.
This includes a lesson guide, a worksheet for students that is a discovery lesson. It is called Aaron’s designs.
Problem of the month called “cut it out”. Students use scissors to explore the concepts of transformation. Complete teacher guide included. There are also 5 different levels of the task.
Problem of the month called Cutting a cube. Similar to the one above however, three dimensional.
1. Draw a Triangle on your paper and label it
a. Rotate the triangle 90º Clockwise. Describe what happened to each vertex. Compare the angles, vertices and side lengths of the image and pre-image.
b. What happens to the vertices when you rotate 180º? . Compare the angles, vertices and side lengths of the image and pre-image.
After performing the transformations above, the student will measure to find that the angles and sides have the same measure and that the Vertices moved half way around a dircle.
2. Draw a Triangle and label it XYZ. Move each vertex up 2 centimeters, then move each to the left 3 cm. What effect does it have on the perimeter? Why?
There is no change in the perimeter because there is no change in the side lengths.
3. Use a ruler and Translate the figure below, using the vector AB.
Students will measure between the 2 endpoints.
They then lay their rulers parallel to the vector and move the vertex a distance equal the measure of the line segment.
4. Describe the transformation below. Explain what happed to each image, then tell me as much as you can about the
measurement of sides and angles. Please pay attention to vocabulary.
A’B’C’ is translated by a vector that is going 2 inches then A'B'C' is created by a vertical reflection followed by a translation with a vector approximately one inch long.
5. Measure the angles below.
Is <A an image of <G ? Why or why not.
Yes they are because they are the same size. OR No because there is no A’ or G’.
All of the resources below are intended as Teacher resources rather than student resources. The use of vectors as a method to translate is not only controversial but also a new idea in 8th grade mathematics. The resources below will help the teacher become more comfortable with the material.
Rigid Motion Video