September - Geometric Transformations – Unit 1
Textbook chapter 7
Kahn Academy online chapters: Transformations, Analytic geometry
Learning Objectives
1. What do shapes need to have to be congruent to each other? (2 things: same angles and side lengths)
2. Use several methods and tools for identifying if polygons are congruent or not. (We’ve seen 6-7 methods.)
3. Identify the type of rigid transformation (isometric) from a given pre-image and image. (3 types:
translation, rotation, reflection)
4. Identify the type of non-rigid transformation (non-isometric) from a given pre-image and image. (2 types:
dilation, stretch)
5. Given pre-image coordinates and a transformation rule (coordinate pair rule), calculate and graph the
image.
6. Given an image and transformation rule, calculate and graph the pre-image. (Reverse transformation.)
7. Given a pre-image and image, write a transformation rule (coordinate pair rule) or composition of several
rules. Example: a translation two units right, three down has the rule (x, y) → (x + 2, y – 3).
8. Identify if a single shape has reflectional symmetry or rotational symmetry. Draw lines of reflection
symmetry and points of rotational symmetry on the shape.
Objectives 1-4 are basic (C). 5 & 6 are intermediate (B). 7 & 8 are mastery level (A).
Scratch Programming Project
1. Develop a basic understanding of programming or coding using the Scratch interface.
2. Write a program or script that makes a polygon go through all the types of rigid and non-rigid
transformation we have explored.
3. Understand one ‘real-world’ use of geometric transformations: computer graphics.
4. Solve a complex problem, seeking information from many sources to complete the job.
Geometric Transformations Vocabulary
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Coordinate plane
x-axis
y-axis
Origin
Point
Coordinates (= coordinate pair)
Congruent (= identical = same measure(s))
Angle
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Polygon Side Length (= measure of segment in polygon’s perimeter)
Segment
Vertex (corner of angle or polygon; plural = vertices)
Protractor
Compass
Pythagorean Theorem (used here for calculating side lengths to identify congruence)
Pre-image (= shape before transformation; vertices labeled A, B, C…)
Image (= shape after transformation, vertices labeled A’, B’, C’…pronounced ‘A prime’…)
Transformation
Rigid Transformation (= isometric transformation; ‘iso’ = same; ‘metry’ = measurements)
Isometric Transformation (= rigid transformation)
Corresponding (angles or sides in pre-image and image)
Non-rigid transformation (= non-isometric)
Translation (= slide; a rigid transformation maintaining congruence)
Reflection (= flip = mirror; a rigid transformation maintaining congruence)
Rotation (= spin; a rigid transformation maintaining congruence)
Dilation (grow or shrink proportionately in x and y direction; a non-rigid)
Stretch (grow or shrink differently in x and y directions)
Line of Reflection (LoR = Line of Symmetry; halfway between pre-image and image corresponding point)
Point of Rotation (PoR; intersection lines between pre-image and image corresponding points)
Similar (‘but not congruent’ pre-image and image resulting from a dilation; corresponding angles remain
congruent, therefore same shape; different side lengths)
Transformation rule ( = ordered pair rule = coordinate pair rule = transformation function(s); tells how all x
and y coordinates; (x, y) → ( , ) )
Input
Output
Composition of Transformations, or Composite Transformation (Sequence matters!)
Reflectional symmetry (between pre-image and image in a reflection; or within a figure)
Rotational Symmetry (figure congruent with itself at more than one place on 360 o rotation)
Claim, Evidence, Reason (format of statements, such as those claiming congruence between figures)
Parallel
Perpendicular
Acute
Right
Obtuse
Clock-wise
Counter-clockwise
Scale factor (will see this much more in the Similarity & Dilations unit in Nov/Dec)
September - Geometric Transformations – Unit 1
8th grade Common Core Standards
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Understand congruence in terms of rigid motions.
o 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.
o Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
coordinates.
 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and
mathematical problems in two and three dimensions.
 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
o 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.
Experiment with transformations in the plane.
o Verify experimentally the properties of rotations, reflections, and translations.
o Lines are taken to lines, and line segments to line segments of the same length.
o Angles are taken to angles of the same measure.
o Parallel lines are taken to parallel lines.
High school Common Core Standards
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Understand congruence in terms of rigid motions
o Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion
on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they
are congruent.
Experiment with transformations in the plane
o Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the
undefined notions of point, line, distance along a line, and distance around a circular arc.
o Represent transformations in the plane using, e.g., transparencies (patty paper) and geometry software;
describe transformations as functions that take points in the plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not (e.g., translation versus
horizontal stretch).
o Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry
it onto itself.
o Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,
parallel lines, and line segments.
o Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph
paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure
onto another.