Download 2nd Unit 3: Parallel and Perpendicular Lines

Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(2) Coordinate and transformational geometry. The student uses the process skills to understand the connections between algebra and geometry and uses the oneand two-dimensional coordinate systems to verify geometric conjectures.
G.2(A) determine the
Determine the coordinates
Find the coordinates of point P
Coordinate
Compare the methods
Big Ideas Geometry
coordinates of a point
of a point that is given
along the directed line segment AB Coplanar
of counting lines on the 3.5, Example 2
that is a given fractional fractional distance from
so that AP to PB is the ratio 2 to 6
Distance Formula
number line or
distance less than one
either endpoint of a line
when A(-3,2), B(5,-4).
Midpoint Formula
coordinate plane and
from one end of a line
segment.
using the midpoint or
segment to the other
Correct answer: P(-1,0.5)
distance formula to
in one- and twocalculate the distances.
dimensional coordinate
systems, including
finding the midpoint.
Supporting Standard
Connects to G.2B
2016-2017
Page 1
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
G.2(B) derive and use
the distance, slope, and
midpoint formulas to
verify geometric
relationships, including
congruence of segments
and parallelism or
perpendicularity of pairs
of lines.
Readiness Standard
Midpoint
Slope
Parallel
Perpendicular
Interesting lines
Coincide
Slope intercept form
y- intercept
Use slopes to determine
whether the lines are
parallel, perpendicular or
neither.
Comparing equations of
lines, determine whether
lines are parallel, intersect
or coincide.
Prove lines are parallel
given angle information.
Prove and apply theorems
about perpendicular lines
State whether the graphs of the
following equations are parallel,
perpendicular, intersecting or
coincide
Ex. 5x-3y = 7
y= -3/5x + 8
Correct answer: Perpendicular
Vocabulary
Instructional
Strategies
Resources/
Weblinks
Use distance formula to
find the distance
between 2 points.
Big Ideas Math
Geometry
3.4, 3.5, 3.6, 5.1
Find coordinates of the
midpoint of a segment
on a coordinate plane.
www.khanacademy.org
Find endpoint given an
endpoint and a
midpoint.
Demonstrate through
use and problem
solving.
Misconceptions:
 The student may substitute the x- and y-values incorrectly when using
the formulas.
 The student may divide a value by “2” instead of taking the square
root when using the distance formula.
 The student may add the x-value to the y-value, instead of computing
the sum of the x-values and computing the sum of the y-values before
dividing by 2 in the midpoint formula.
 The student may incorrectly write the ratio of the slope of a line as the
ratio of horizontal change divided by vertical change.
2016-2017
Page 2
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
G.2(C) determine an
equation of a line
parallel or perpendicular
to a given line that
passes through a given
point.
Readiness Standard
Slope intercept form
y- intercept
perpendicular- bisector
coordinate
Write and compare
equations of lines.
The graph of line g is shown
below.
Write the equation of a line
parallel and perpendicular to
a given line through the
given point.
What equation describes a line
parallel to g that has a y-intercept
at (0,-1)?
Correct answer: y   1 x  1
2
Released EOC 2013
Q#25
2016-2017
Vocabulary
Instructional
Strategies
Review the
relationships between
slopes of parallel and
perpendicular lines.
Provide students with
graphic organizers to
help sort parallel and
perpendicular lines.
Resources/
Weblinks
Big Ideas Math
Geometry
3.5, 3.6
Google Drive:
G.2C Task Activity
Use Guided Practice
G.2C Task Activity in
the Google drive for
class practice.
Misconceptions:
 The student may use the slope formula incorrectly
horizontalchange instead of vertical change ).
(ie:
verticalch ange
horizontal change
 The student may think the slopes of perpendicular line are only opposite
values instead of opposite reciprocals.
Page 3
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(3) Coordinate and transformational geometry. The student uses the process skills to generate and describe rigid transformations (translation, reflection, and
rotation) and non-rigid transformations (dilations that preserve similarity and reductions and enlargements that do not preserve similarity).
G.3(A) describe and
Describe transformations
Parallelogram ABCD was
Transformation
Use three column notes
Big Ideas Geometry
perform transformations
of figures in a plane using transformed to form parallelogram Translation
to provide
4.1, 4.2, 4.3, 4.5
of figures in a plane using coordinate notation.
A’B’C’D’.
Reflection
transformation in a
coordinate notation.
Rotation
plane in one column,
Google Drive:
Perform transformations
Dilation
verbal description in
Graphic Organizer
of figures in a plane using
another, coordinate
Card Sort Activity
Supporting Standard
Connects to G.3B
coordinate notation.
notation in third.
www.khanacademy.org
Engaging Mathematics
p. 45 (18.pdf)
Which rule describes the
transformation that was used to
form parallelogram A’B’C’D’?
F. ( x, y)  ( x, y)
G. ( x, y)  ( x, y)
H. ( x, y)  ( x  6, y)
J. ( x, y)  ( x, y  3) (-x, y-3)
Correct answer: J
Adapted from Released EOC 2013
Q#40
2016-2017
Page 4
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
G.3(B) determine the
image or pre-image of a
given two-dimensional
figure under a
composition of rigid
transformations, a
composition of non-rigid
transformations, and a
composition of both,
including dilations where
the center can be any
point in the plane.
Readiness Standard
Determine the
 image
 pre-image
of a given twodimensional figure under
a composition of
 rigid
 non-rigid
 both
transformations.
Determine the
 image
 pre-image
of a given twodimensional figure that
includes dilations where
the center can be any
point in the plane.
Δ𝐴𝐵𝐶 has vertices A(-3, 1), B (2, 1), and C (0, 2). Reflect the figure
across the y-axis and then translate
it 3 units down and 4 units to the
right.
What are the coordinates of the
image?
Correct Answer:
A’(7, -2), B’(2, -4), C’(4, -1)
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Image
Pre-image
Transformation
Translation
Reflection
Rotation
Dilation
Composition
Center of Dilation
Rigid transformation
Non/Congruent figures
Instructional
Strategies
Resources/
Weblinks
Center of dilation at
origin: Multiply
coordinates of preimage by scale factor
Big Ideas Geometry
4.1, 4.2, 4.3, 4.6
Center of dilation not at
origin: use slope to find
image points
Engaging Mathematics
p. 57 (23.pdf) & p. 59
(24.pdf)
www.khanacademy.org
Stress use of prime
notation for image
points
Misconceptions:
 The student may not be able to distinguish the difference between image
and pre-image
 The student may think the origin is the only point that can be the center
for dilations
2016-2017
Page 5
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
G.3(C) identify the
sequence of
transformations that will
carry a given pre-image
onto an image on and off
the coordinate plane.
Supporting Standard
Connects to G.3B
Identify the sequence of
transformations that will
carry a given pre-image
onto an image on the
coordinate plane.
Jake took pictures of Ana’s flag
while she was practicing her
routine for the football game, as
shown below.
Identify the sequence of
transformations that will
carry a given pre-image
onto an image off the
coordinate plane.
Which of the following best
describes the movement of the flag
from picture to picture?
A. Reflection, rotation, translation
B. Rotation, translation, translation
C. Rotation, translation, dilation
D. Reflection, translation,
translation
G.3(D) identify and
distinguish between
reflectional and rotational
symmetry in a plane
figure.
Supporting Standard
Connect to G.3B
Identify reflectional
symmetry in a plane
figure.
Identify rotational
symmetry in a plane
figure.
Distinguish between
reflectional and rotational
symmetry in a plane
figure.
2016-2017
Answer: A
Tell whether the figure has
rotational and/or reflectional
symmetry.
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Image
Pre-image
Transformation
Translation
Reflection
Rotation
Dilation
Composition
Center of Dilation
Point of rotation
Scale factor
Similarity
Demonstrate that the
order of the
transformations matters.
Symmetry
Rotational symmetry
Reflectional symmetry
Line of reflection
Line of symmetry
Center of rotation
Angle of rotation
Center of symmetry
Reflectional: over a line
Resources/
Weblinks
Big Ideas Geometry
4.4, 4.6, 5.2
www.khanacademy.org
Include a variety of
examples where
students identify the
sequence of
transformations.
There may be several
different methods for
transforming the same
pre-image into an
image.
Big Ideas Geometry
4.2, 4.3
Rotational: about a
point
Make sure students
label the vertices
Rotational—yes
Reflectional—no
Page 6
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(5) Logical argument and constructions. The student uses constructions to validate conjectures about geometric figures. The student is expected to:
G.5(A) investigate
Use the exterior angle
Exterior angles theorem Use both numeric and
Find the measure of ∠𝐹
Big Ideas Math
patterns to make
theorem to find angle
Exterior angle
algebraic expressions to Geometry
conjectures about
measures
Interior angle
find missing angles
5.2
geometric relationships,
Remote interior angles
measures
including angles formed
Triangle sum theorem
www.khanacademy.org
by parallel lines cut by a
transversal, criteria
Engaging Mathematics
required for triangle
p. 79 (32.pdf)
congruence, special
segments of triangles,
diagonals of
quadrilaterals, interior
and exterior angles of
polygons, and special
segments and angles of
circles choosing from a
variety of tools.
Readiness Standard
Misconceptions:
 The student may make a conjecture based on limited investigation of patterns.
 The student may randomly state a conjecture without investigating and recognizing patterns.
 The student may not know how to use a construction to make a conjecture.
 The student may not be able to perform constructions correctly.
 The student may not state a conjecture using precise geometric vocabulary.
2016-2017
Page 7
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
G.5(B) construct
congruent segments,
congruent angles, a
segment bisector, an angle
bisector, perpendicular
lines, the perpendicular
bisector of a line segment,
and a line parallel to a
given line through a point
not on a line using a
compass and a
straightedge.
Compass
Construction
Drawing
Sketch
Straight Edge
Use a compass and a
straight edge to construct:
 Perpendicular lines
 The perpendicular
bisector of a line
segment using a
compass and a
straightedge.
 A line parallel to a
given line through a
point not on a line
Supporting Standard
Connects to G.5A, G.6A
G.5(C) use the
constructions of congruent
segments, congruent
angles, angle bisectors,
and perpendicular
bisectors to make
conjectures about
geometric relationships.
Supporting Standard
Connects: G.5A, G.6A
2016-2017
Use constructions,
 Congruent
segments
 Congruent angles
 Angle bisectors
 Perpendicular
bisectors
To make conjectures
about geometric
relationships.
What construction is shown in the
accompanying diagram?
Vocabulary
Instructional
Strategies
Focus on constructing
geometric figures with
only a straight edge and
a compass.
Ensure students can
construct congruent
segments.
Resources/
Weblinks
Big Ideas Geometry
3.3, 3.4
http://www.mathopenref.
com/tocs/constructionsto
c.html
Use two column notes
that have students write
the steps needed to
construct on one side
while performing the
task of construction in
the other.
A. The bisector of angle PJR.
B. The midpoint of line PQ
C. The Perpendicular bisector of
line segment PQ.
D. A perpendicular line to PQ
through point J.
Answer: C
Angle bisector
Bisect
Congruent
Congruent angles
Congruent segments
Constructions
Perpendicular
Perpendicular bisector
As students construct
figures, they should
also describe what they
see and explain why the
construction works.
Big Ideas Math
Geometry
3.3, 3.4, 5.4
Page 8
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.(6) Proof and congruence. The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as
coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:
G.6(A) verify theorems Find the value of x and
Find the value of x to verify that
Alternate Exterior
Substitute different
Big Ideas Geometry
about angles formed by
solve problems involving
the lines are parallel
Angles
values to verify angle
3.3, 3.4
the intersection of lines
parallel lines and vertical
Alternate Interior
measures.
and line segments,
angles
Angles
www.khanacademy.org
Use manipulatives and
including vertical
Coplanar
technology to draw
angles, and angles
Corresponding Angles
conclusions and
formed by parallel lines
Diagonal
discover relationships
cut by a transversal and
Graph segments and find the
Parallel Lines
about parallel lines and
prove equidistance
perpendicular bisector using
Perpendicular Lines
their properties
PQ is shown on the coordinate
between the endpoints
the slope and midpoint
Same-Side Interior
grid below. The coordinates of P
Stress the importance of
of a segment and points
formulas
Angles
and
Q
are
integers.
slopes perpendicular to
on its perpendicular
Segment
a line (opposite
bisector and apply these
Skew Lines
reciprocal)
relationships to solve
Transversal
problems.
Use distance formula to
Readiness Standard
find the distance
bisector
between 2 points and
Slope
the midpoint
Midpoint
Find coordinates of the
Coordinates
Point (x, y) lies on the
midpoint of a segment
on a coordinate plane.
perpendicular bisector of PQ .
What is the value of x?
Misconceptions:
Correct answer: -2.5
 The student may not use logical reasoning correctly to work through
Released EOC 2013
proofs.
Q#10
 The student may not apply justification to support statements in a twocolumn proof.
2016-2017
Page 9
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
G.6(C) apply the
definition of
congruence, in terms of
rigid transformations, to
identify congruent
figures and their
corresponding sides and
angles.
Corollary
Corresponding Angles
Corresponding
Polygons
Corresponding Sides
Included Angle
Included Side
Interior
Triangle Rigidity
SAS
SSS
ASA
AAS
AL
Students should mark
pictures with
congruence to be able
to easily determine how
the triangles are
congruent i.e. AAS,
SAS, ASA
Image
Pre-image
Transformation
Translation
Reflection
Rotation
Corresponding sides
Corresponding angles
Rigid transformations
Relate concept back to
congruent triangles and
their corresponding
parts, CPCTC
Supporting Standard
Connects to G.6B, G.3B
Apply the definition of
congruence, in terms of
rigid transformations, to
identify
 congruent figures
 corresponding sides
(of congruent figures)
 corresponding angles
(of congruent figures)
Use AAS to explain why the
triangles are congruent.
Answer: A  D, BEA 
CED, BE  CE
Identify the corresponding sides
and angles in the reflection:
2016-2017
Vocabulary
Instructional
Strategies
Resources/
Weblinks
Big Ideas Math
Geometry
4.4, 5.2
www.khanacademy.org
Page 10
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
G.6(D) verify theorems
about the relationships
in triangles, including
proof of the
Pythagorean Theorem,
the sum of interior
angles, base angles of
isosceles triangles,
midsegments, and
medians, and
apply these relationships
to solve problems.
Supporting Standard
Connects to G.5A
2016-2017
Verify theorems about the
relationships in triangles:
 Including the sum of
interior angles.
(The rest of this SE is
addressed in the 3rd and 4th
grading periods.)
B is the midpoint of
Vocabulary
D is the
midpoint of
and AE = 21.
Find BD. The diagram is not to
C
scale.
B
Find the missing angle
measure in triangles
Determine relationships of
angles and sides when
bisectors, medians and
altitudes are drawn in
triangles.
Designated Grading Period: 2nd
Days to teach: 11
A
D
E
Midsegment
Midpoint
Congruent
Parallel
Isosceles triangle
equilateral triangle
Equidistant
Base angles
Medians, bisectors
Hinge theorem
Inequality
Perpendicular bisector
Altitude
Instructional
Strategies
Verify relationships in
triangles including
triangle sum theorem,
base angles of isosceles
triangles and angles in
equilateral triangles.
In an Isosceles triangle,
have students discover
Median, angle bisector,
perpendicular bisector
are all the same line.
Find the value of the
midsegment given the
parallel side of the
triangle.
Use both algebraic
expressions and
numeric values when
solving
Resources/
Weblinks
Big Ideas Math
Geometry
5.1, 5.4
www.khanacademy.org
Engaging Mathematics
p. 96- 97 (39.pdf)
Page 11
Mathematics
Course: Pre-AP Geometry
Unit 3: Parallel and Perpendicular Lines (continued)
Unit 4: Transformations
Unit 5: Congruent Triangles (begin)
TEKS
Guiding Questions/
Assessment
Specificity
Designated Grading Period: 2nd
Days to teach: 11
Vocabulary
Instructional
Strategies
Resources/
Weblinks
G.7 Proof and Congruence: The student uses the process skills with deductive reasoning to prove and apply theorems by using a variety of methods such as
coordinate, transformational, and axiomatic and formats such as two-column, paragraph, and flow chart. The student is expected to:
G.7(A) apply the
Apply the definition of
Isosceles trapezoid JKLM is shown Congruent
Utilize a graphic
Big Ideas Math
definition of similarity
similarity in terms of a
below.
corresponding angles
organizer to compare
Geometry
in terms of a dilation to
dilation to identify similar
Dilation
the properties of
4.6
identify similar figures
figures.
Proportional
congruence
and their proportional
Similar figures
transformation and
sides.
(The rest of this SE is
Similarity
similarity
addressed in the 4th grading
transformations.
period.)
Supporting Standard
If the dimensions of the trapezoid
Connects to G.3B, G.7B
JKLM are multiplied by a scale
factor of f to create trapezoid
J’K’L’M’, which statement is true?
F. Trapezoid J’K’L’M’ contains
two base angles measuring 30 °
each.
G. The longer base of trapezoid
J’K’L’M’is 56f units.
H. The bases of trapezoid
J’K’L’M’ have lengths of 22 units
and 39 units.
J. Trapezoid J’K’L’M’ contains
two base angles measuring
(120 f )° each.
Correct answer: G
Released EOC 2013
Q#2
2016-2017
Page 12