Standards:
GCO.A.1 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.
GCO.A.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).
GCO.A.3 Given a rectangle, parallelogram,
trapezoid, or regular polygon, describe the rotations and reflections that
carry it onto itself.
GCO.A.4 Develop definitions of rotations, reflections, and translations in
terms of angles, circles, perpendicular lines, parallel lines, and line
segments.
GCO.A.5 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.
GCO.B.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.
GCO.B.7 Use the definition of congruence in terms
of rigid motions to show that two triangles are congruent if and only if
corresponding pairs of sides and corresponding pairs of angles are congruent.
GCO.B.8 Explain how the criteria for triangle
congruence (ASA, SAS, and SSS) follow from the definition of congruence in
terms of rigid motions.
GCO.C.9 Prove theorems about lines and
angles. Theorems include: vertical
angles are congruent; when a transversal crosses parallel lines, alternate
interior angles are congruent and corresponding angles are congruent; points
on a perpendicular bisector of a line segment are exactly those equidistant
from the segment’s endpoints.
GCO.C.10 Prove theorems about triangles. Theorems
include: measures of interior angles of a triangle sum to 180°; base angles
of isosceles triangles are congruent; the segment joining midpoints of two
sides of a triangle is parallel to the third side and half the length; the
medians of a triangle meet at a point.
GCO.C.11 Prove theorems about parallelograms. Theorems
include: opposite sides are congruent, opposite angles are congruent, the
diagonals of a parallelogram bisect each other, and conversely, rectangles
are parallelograms with congruent diagonals.
·
Standards Clarification: The criteria for triangle congruence will be
used to prove theorems about parallelograms. Students will continue to find
the distance between points in order to calculate segment lengths or
perimeter of figures as they focus more heavily on analytic proofs in this
unit.
GCO.D.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.
GCO.D.13 Construct an equilateral triangle, a
square, and a regular hexagon inscribed in a circle.
Focus Standards of
Mathematical Practice:
MP.1 Make
sense of problems and persevere in solving them.
MP.2 Reason
abstractly and quantitatively.
MP.3
Construct viable arguments and critique the reasoning of others.
MP.4 Model
with mathematics.
MP.5 Use
appropriate tools strategically.
MP.6 Attend
to precision.
MP.7 Look for
and make use of structure.
MP.8 Look for
and express regularity in repeated reasoning.
Instructional
Outcomes:
GCO.A.1
·
I can describe the
undefined terms: point, line, and distance along a line in a plane.
·
I can define circle
and the distance around a circular arc.
GCO.A.2
·
I can describe the
different types of transformations including translations, reflections,
rotations and dilations.
·
I can describe
transformations as functions that take points in the coordinate plane as
inputs and give other points as outputs.
·
I can compare
transformations that preserve distance and angle to those that do not (e.g.,
translation versus horizontal stretch).
·
I can represent
transformations in the plane using, e.g., transparencies and geometry
software.
·
I can write functions
to represent transformations.
GCO.A.3
·
I can describe the
rotations and/or reflections that carry it onto itself given a rectangle,
parallelogram, trapezoid, or regular polygon.
GCO.A.4
·
I can recall
definitions of angles, circles, perpendicular and parallel lines and line
segments.
·
I can develop
definitions of rotations, reflections and translations in terms of angles,
circles, perpendicular lines, parallel lines and line segments.
GCO.A.5
·
I can, given a
geometric figure and a rotation, reflections or translation, draw the
transformed figure using, e.g. graph paper, tracing paper or geometry
software.
·
I can a draw
transformed figure and specify the sequence of transformations that were used
to carry the given figure onto the other.
GCO.B.6
·
I can use geometric
descriptions of rigid motions to transform figures.
·
I can predict the
effect of a given rigid motion on a given figure.
·
I can define
congruence in terms of rigid motions (i.e. two figures are congruent if there
exists a rigid motion, or composition of rigid motions, that can take one
figure to the second).
·
I can describe rigid
motion transformations.
·
I can predict the
effect of a given rigid motion.
·
I can decide if two
figures are congruent in terms of rigid motions (it is not necessary to find
the precise transformation(s) that took one figure to a second, only to
understand that such a transformation or composition exists).
·
I can, given two
figures, use the definition of congruence in terms of rigid motion to decide
if they are congruent.
GCO.B.7
·
I can identify
corresponding angles and sides of two triangles.
·
I can identify
corresponding pairs of angles and sides of congruent triangles after rigid
motions.
·
I can use the
definition of congruence in terms of rigid motions to show that two triangles
are congruent if corresponding pairs of sides and corresponding pairs of
angles are congruent.
·
I can use the
definition of congruence in terms of rigid motions to show that if the
corresponding pairs of sides and corresponding pairs of angles of two
triangles are congruent then the two triangles are congruent.
·
I can justify
congruency of two triangles using transformations.
GCO.B.8
·
I can informally use
rigid motions to make angles and segments (from 8th grade).
·
I can formally use
dynamic geometry software or straightedge and compass to take angles to
angles and segments to segments.
·
I can identify ASA,
SAS, and SSS.
·
I can explain how the
criteria for triangle congruence (ASA, SAS, SSS) follows from the definition
of congruence in terms of rigid motions (i.e. if two angles and the included
side of one triangle are transformed by the same rigid motion(s) then the
triangle image will be congruent to the original triangle).
GCO.C.9
·
I can identify and
use properties of vertical angles.
·
I can identify and
use properties of parallel lines with transversals, corresponding angles, and
alternate interior and exterior angles.
·
I can identify and
use properties of perpendicular bisector.
·
I can identify and
use properties of equidistant from endpoint.
·
I can identify and
use properties of all angle relationships.
·
I can prove vertical
angles are congruent.
·
I can prove
corresponding angles are congruent when two parallel lines are cut by a
transversal and converse.
·
I can prove alternate
interior angles are congruent when two parallel lines are cut by a
transversal and converse.
·
I can prove points
are on a perpendicular bisector of a line segment are exactly equidistant
from the segments endpoint.
GCO.C.10
·
I can identify the
hypothesis and conclusion of a triangle sum theorem.
·
I can identify the
hypothesis and conclusion of a base angle of isosceles triangles.
·
I can identify the
hypothesis and conclusion of midsegment theorem.
·
I can identify the
hypothesis and conclusion of points of concurrency.
·
I can design an
argument to prove theorems about triangles.
·
I can analyze
components of the theorem.
·
I can prove theorems
about triangles.
GCO.C.11
·
I can classify types
of quadrilaterals.
·
I can explain
theorems for various parallelograms involving opposite sides and angles and
relate to figure.
·
I can explain
theorems for various parallelograms involving diagonals and relate to figure.
·
I can use the
principle that corresponding parts of congruent triangles are congruent to
solve problems.
·
I can use properties
of special quadrilaterals in a proof.
GCO.D.12
·
I can explain the
construction of geometric figures using a variety of tools and methods.
·
I can apply the
definitions, properties and theorems about line segments, rays and angles to
support geometric constructions.
·
I can apply
properties and theorems about parallel and perpendicular lines to support
constructions.
·
I can perform
geometric constructions including: 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 given line through a point not on the line, using a variety
of tools an methods (compass and straightedge, string, reflective devices,
paper folding, dynamic geometric software, etc.).
GCO.D.13
·
I can construct an
equilateral triangle, a square and a regular hexagon inscribed in a circle.
