Download Scale factor – ratio of the image to the preimage

Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
 MAJOR CONTENT
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
CONGRUENCE (CO)
Understand congruence in terms of rigid
motions
 G- CO.6 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. [Focuses on the
effect of a given rigid motion.]
SIMILARITY, RIGHT TRIANGLES &
TRIGONOMETRY (SRT)
Understand similarity in terms of
similarity transformations
 G.SRT.1 Verify experimentally the
properties of dilations given by a center
and a scale factor.
VOCABULARY
Scale factor – ratio of the image to the
preimage
The amount by which the image grows or
shrinks is called the "Scale Factor".

If the scale factor is say 2, the
image is enlarged to twice the size
G.COb I can explain congruence in
terms of rigid motions.
 Determine whether figures are
congruent using a sequence of rigid
motions.
 If two figures are congruent, then
the corresponding angles are
congruent and the corresponding
sides are congruent.
 Rigid motions preserve side and
angle measures.
Knowing that rigid transformations
(translation, rotation, reflection)
preserve size and shape or distance and
angle, use this fact to connect the idea
of congruency and develop the
definition of congruence.
Task and assessment ideas
1. Presented with two sets of
transformations, determine which
are equivalent?
2. Given 2 non congruent figures, can
they be made congruent by changing
one or more coordinate points.
G-SRTa I understand similarity in terms of similarity transformations.
NOT LIMITED TO TRIANGLES
 In similar figures, corresponding segments
are
parallel.
Special Case: If center of dilation of collinear
to the vertices on one side of the figure then
the corresponding vertices are also collinear.
 The ratio of the distance from the vertex of the image to the center of dilation to the
distance from the vertex of the preimage to the center of dilation is proportional;
and conversely.
Geometry by Southwest Washington Common Core Mathematics is licensed under a Creative Commons Attribution 4.0 International
License.
2/14/14
Page 1 of 7
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
of the original.
Level 3 = area is not proportional
If it is 0.5, the image is reduced to
Level 4 = area is square of proportion
half the size.
 When the scale factor is 1, the
 Verify that a side length of the image is equal to the scale factor multiplied by the
image is the exact same size as the
corresponding side length of the preimage.
original.
 The center of dilation and the corresponding vertices of the image and preimage are
Experiment with the scale factor slider to
collinear.
gets a feel for this idea.
Remember: In dilation, multiply the
dimensions of the original by the scale
factor to get the dimensions of the image.

8th grade
Corresponding sides lengths are
proportional
AB/A’B’ etc
The center of dilation may be a vertex of
the preimage.
Dilation with a scale factor of 1 are
congruent.
Page 2 of 7
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
 MAJOR CONTENT
 MAJOR CONTENT
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
 G.SRT.2 Given two figures, use the
definition of similarity in terms of
similarity transformations to decide if
they are similar; explain using similarity
transformations the meaning of similarity
for triangles as the equality of all
corresponding pairs of angles and the
proportionality of all corresponding pairs
of sides.
 Determine whether figures are similar
using
 rigid transformations, and
 dilations.
 If two triangles are similar, then the
corresponding pairs of angles are
congruent and the corresponding sides
are proportional.
 G.SRT.3 Use the properties of
similarity transformations to establish the
AA criterion for two triangles to be
similar.
 Justify that two triangles are similar if
two pairs of angles are congruent.
Page 3 of 7
TASK\ extension
Discover AA shortcut for triangles.
Does AA work for all polygons? Find
shortcut for other shapes. Angles
need to be corresponding
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
Prove theorems involving similarity
 G.SRT.5 Use congruence and similarity
criteria for triangles to solve problems
and to prove relationships in geometric
figures.
 SUPPORTING
CONTENT
Use coordinates to prove simple
geometric theorems algebraically
 G.GPE.6 Find the point on a directed
line segment between two given points
that partitions the segment in a given
ratio.
CONGRUENCE (CO)
Experiment with transformations in the
plane
 G-CO.1 Know precise definitions of
angle, circle, perpendicular lines, parallel
lines, and line segment, based on the
undefined notions of point, line, distance
along a line, and distance around a
circular arc. [Focuses on definitions not
related to a circle.]
G-SRTb I can prove theorems involving
similarity.
 Prove two triangles are similar by using
triangle similarity postulates SSS~, SAS~,
and AA. Tasks
 Apply triangle similarity to solve problem
situations (e.g., indirect measurement,
missing sides/angle measures).
G-GPE I can express geometric properties
with equations.
 Find the coordinate that divides a line
segment into two proportional
segments.
COa I can perform transformation in the
plane.
 Know precise definitions of
angle
perpendicular lines
parallel lines
line segment
based on the undefined notions of point,
line, distance along a line,
Page 4 of 7
TASK – see flipbook page 185 (electronic
page 221). Is it really a weighted
average.
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
 G-CO.2 Represent transformations in
the plane using, e.g., transparencies 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.
Represent transformatons using
?? Are functions important for proof to
 Patty paper or other transparencies
know there is only one output??
 Geometry software (geogebra)
 Distinguish between transformations
that use rigid motions from those that do
not.
 Use and understand mapping notation
(Translate ∆ABC using the rule (x, y) → (x
− 6, y − 5) for transformations.
 G-CO.3 Given a rectangle,
parallelogram, trapezoid, or regular
polygon, describe the rotations and
reflections that carry it onto itself.
 Describe the rotations and reflections of
rectangle, parallelogram, trapezoid, or
regular polygon, that carry it onto itself.
 Calculate the number of lines of
reflection symmetry
 Calculate the degree of rotational
symmetry of any regular polygon.
 In a translation, line segments joining
corresponding vertices are parallel.
 In a reflection, line segments joining
corresponding vertices lie on the
perpendicular bisector of the line of
reflection.
 In a rotation, line segments connecting
corresponding vertices to the center of
rotation are perpendicular to each other.
 G-CO.4 Develop definitions of
rotations, reflections, and translations in
terms of angles, circles, perpendicular
lines, parallel lines, and line segments.
Page 5 of 7
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
 SUPPORTING CONTENT
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
 G-CO.5 Given a geometric figure and a
rotation, reflection, and 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.[focuses on
performing transformations.]
Make geometric constructions
 G-CO.12 Make formal geometric
constructions with a variety of tools and
methods (compass and straightedge,
string, reflective devices, paper folding,
dynamic geometric software, etc.).
Copying a segment; copying an angle;
bisecting a segment; bisecting an angle;
constructing perpendicular lines,
including the perpendicular bisector of a
line segment; and constructing a line
parallel to a given line through a point not
on the line.
 G-CO.13 Construct an equilateral
triangle, a square, and a regular hexagon
inscribed in a circle.
 Perform and predict rigid motion
transformations.
G-COd I can make geometric constructions.
 Skills
o Copying a segment; copying an angle;
o bisecting a segment;
o bisecting an angle;
o constructing perpendicular lines,
including the perpendicular bisector
of a line segment;
o constructing a line parallel to a given
line through a point not on the line.
 Conceptual understanding
o Arcs maintain distance
o Straightedges maintain direction
o Geometric properties and theorems
behind the construction steps.
 Construct an
o equilateral triangle,
o a square
Page 6 of 7
TASK idea – challenge students to
construct parallel lines. They may use
definition of parallel lines are the same
distance apart, using perpendicular lines,
using corresponding angles.
Constructions are done with accuracy.
They are not merely sketches.
Transformations & Similarity
WORKING DOCUMENT
Sub-Targets
8th Grade
 Major Clusters
Practice 3 Construct viable arguments and critique the reasoning of others.
Practice 4. Model with mathematics. Practice 5. Use appropriate tools strategically.
Practice 6. Attend to precision.
 MAJOR CONTENT  SUPPORTING CONTENT  ADDITIONAL CONTENT
Understand congruence and similarity using physical models, transparencies, or geometry software.
8.G.1- Verify experimentally the properties of rotations, reflections, and translations:
1.a – Lines are taken to lines, and line segments to line segments of the same length.
1.b – Angles are taken to angles of the same measure.
1.c – Parallel lines are taken to parallel lines.
8.G.2 – 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.
8.G.3 – Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.4 – 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.
8.G.5 – Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when
parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the
same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.
8.EE.5 – Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional
relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine
which of two moving objects has greater speed.
Page 7 of 7