Geometry: Unit 1.1 Lines and Angles
Approximate Number of Days: 17 days
Unit Focus
Students will:
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build on prior experience with simple constructions
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formalize and explain how simple constructions result in the desired objects
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master the basic terms of geometry and apply them in their explanations of constructions.
Essential Questions
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What are the basic terms of geometry and how are they used to describe figures?
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What geometric constructions are possible and how do you perform them?
Focus Content Standards
Fluency Standards
Experiment with transformations in the plane
G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line
segment, based on the undefined notions of point, line, and distance along a line.

G-SRT.5 Fluency with the triangle congruence and similarity criteria will help
students throughout their investigations of triangles, quadrilaterals, circles,
parallelism, and trigonometric ratios. These criteria are necessary tools in many
geometric modeling tasks.
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G-GPE.4, 5, 7 Fluency with the use of coordinates to establish geometric results,
calculate length and angle, and use geometric representations as a modeling tool
are some of the most valuable tools in mathematics and related fields.
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G-CO.12 Fluency with the use of construction tools, physical and computational,
helps students draft a model of a geometric phenomenon and can lead to
conjectures and proofs.
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. (No
dilations.)
Use coordinates to prove simple geometric theorems algebraically
G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.
Prove geometric theorems
G.CO.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.
Summative Assessment
Standards for Mathematical Practice
Gary makes signal flags for use in sailing. He drew rectangle PQRS on a coordinate grid to represent a
flag, with vertices P (10, 10), Q (70, 10), R (70, 40), and S (10, 40), as shown below.
Note: These standards should drive your pedagogical practice every day. The underlined
standards are critical ones for this unit.
1.
2.
3.
4.
5.
6.
7.
8.

What is the length, in units, of PQ ? Show or explain how you got your answer.

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What is the length, in units, of SP ? Show or explain how you got your answer.
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Copy the x-axis, the y-axis, and rectangle PQRS onto graph paper. Define the midpoint of each
side of rectangle PQRS as follows:
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W is the midpoint of line segment PQ. X is the midpoint of line segment QR. Y is the midpoint of
line segment RS. Z is the midpoint of line segment SP.
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On your coordinate grid, plot the midpoints W, X, Y, and Z. Label each midpoint with its letter
and coordinates. Show or explain how you determined the coordinates of each midpoint.

Draw quadrilateral WXYZ.

On the flag, quadrilateral WXYZ represents the boundary of a region that will be colored red.
What is the perimeter, in units, of quadrilateral WXYZ?

Show or explain how you got your answer.
Resources
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.

DCPS Resources
 On Core Mathematics Geometry Resources
 Geometers Sketchpad
 GEOGEBRA (free software)
Manipulatives
 TI Nspire calculators
 Protractors
 Compasses
 Straight edges
 Mirrors
 Patty paper
Key Mathematical Vocabulary (Academic Language)
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angle
circle
compass
distance along a circular arc
equilateral triangle
geometric construction
inscribed
line
line segment
parallel line
perpendicular line
point
Websites and/or Additional Resources
 http://www.khanacademy.org/#geometry
 http://www.mathisfun.com/geometry/constructions.html
 http://www.mathopenref.com/tocs/constructionstoc.html
 http://whisltealley.com/construction/reference.hth
 http://www.nvcc.edu/home/tstreilein/constructions/
 http://www.mathopenref.com/worksheetlist.html
 http://www.onlinmathlearning.com/geometry-construction.html
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regular hexagon
square
straight edge
transversal
vertex
Grade Level: Unit 1.1 Standards
Standard G.CO.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.
Critical Knowledge and Subskills
Students will know that:

Point, line and plane are undefined terms.

Many terms in geometry use point, line and plane as a basis for
their definition.

These are symbols for many terms.
Students will be able to :

Define angle, circle, perpendicular lines, parallel lines, and line
segment.
Possible Teaching and Learning Tasks
Vocabulary Baseball:
Divide your classroom into two teams and draw a baseball diamond on your blackboard. Each team has three outs per
inning (like regular baseball) and one player from each team goes at a time. You say a word, and the student has 20
seconds to define it. If he defines it in five seconds, the team gets a home run; within 10 seconds, it is a triple; within 15
seconds, it is a double; and just before the time limit, it is a single. If the student does not get the definition right, he is out.
Draw an icon for a base runner when a student gets a hit. When a player gets to home plate, the team scores a run. The
team with the most runs at the end of nine innings wins the game.
Hot Seat
Have one student sit in the front of the classroom on a chair facing away from the blackboard. On the board, you will write
a vocabulary word but the person in the front of the room is not allowed to look at it. The student will ask yes or no
questions to his classmates to determine what the word might be. The student has 10 questions available until he must
guess what the word is. You can make this an individual contest or you can have the class separated into several teams to
add a competitive element.
Vocabulary Toss
This game requires a chalkboard eraser, or small sponge, and a wastebasket. Divide your class into two teams and have
them stand in two single-file lines parallel to each other. This game combines a vocabulary guessing game with a basketball
shooting game. Ask the player at the front of one team to define a vocabulary word. If he gets it right, his team gets a point
and he has a chance to earn a second point if he makes the eraser into the basket. If he gets the word wrong, the player on
the other team has a chance to answer it and shoot the basket. Keep rotating players until everybody gets at least one
turn. The team with the most points at the end wins.
Taboo
Students create taboo playing cards by placing a vocabulary word at the top of the sheet and then placing 4 “forbidden”
words underneath. The forbidden words would be words that are related to the vocabulary word that would not be
allowed to be used when trying to describe the vocabulary word to classmates. Then divide the team into two teams and
play according to Taboo rules.
Supplemental Resources
Text Resources:

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG

OCMG: Chapter 1, 1-1, 1-4

Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG

LBKSG: Chapter 1
DIGITAL RESOURCES:
Vocabulary
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http://www.learner.org/courses/learningmath/geometry/keyterms.html
http://library.thinkquest.org/2647/geometry/glossary.htm
http://www.mathleague.com/help/geometry/geometry.htm
TI NSPIRE Activities
 http://education.ti.com/calculators/timathinspired/US/Activities/?sa=5024&t=5049
Shodor Activities
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http://compute2.shodor.org/interactivate/standards/organization/objective/2319/
http://compute2.shodor.org/interactivate/standards/organization/objective/2318/
Math Warehouse Activities
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http://www.mathwarehouse.com/geometry/angle/
http://www.mathwarhouse.com/geometry/triangles/
http://www.mathwarehouse.com/geometry/angle/interactive-supplementary-angles.php
http://www.mathwarhouse.com/geometry/angles/interactive-vertical-angles.php
Standard 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.)
Critical Knowledge and Subskills
Students will know that:

Constructions can be done in a variety of ways.
Possible Teaching and Learning Tasks
“Geometric Constructions” (see additional resources on Ed Portal for detailed task)
On Core Activity generator and resource materials -- Chapter 1: lessons 1-1 to 1 -5
Students will be able to:

Use the different tools and methods to make formal
constructions.

Copy a segment; copy an angle; bisect a segment; bisect an
angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; constructing a line
parallel to a given through a point not on a line.
Bisecting an Angle
http://www.illustrativemathematics.org/illustrations/1083
Suppose A is an angle with vertex P, as pictured below:

Draw a circle with center P and with radius r>0. Explain why the circle meets each ray of angle A in a single
point. Label these points Q and R respectively.

Draw circles with centers Q and R respectively and radius r. These circles meet at P and a second point to be
⃗⃗⃗⃗⃗ bisects angle P.
labeled B. Show that ray 𝑃𝐵
Construction of perpendicular bisector
http://www.illustrativemathematics.org/illustrations/966
Let A and B be two distinct points in the plane and ̅̅̅̅
𝐴𝐵 the segment joining them. The goal of this problem is to construct
̅̅̅̅ .
the perpendicular bisector of segment 𝐴𝐵
Draw circles with radius |AB| centered at A and B respectively as pictured below:
The two points of intersection of these circles are labeled P and Q. Show that line ⃡⃗⃗⃗⃗
𝑃𝑄 is the perpendicular bisector of ̅̅̅̅
𝐴𝐵.
Locating the Warehouse
http://www.illustrativemathematics.org/illustrations/507
You have been asked to place a warehouse so that it is an equal distance from the three roads indicated on the following
map. Find this location and show your work.
a.
Show how to fold your paper to physically construct this point as an intersection of two creases.
b.
Explain why the above construction works, and in particular why you only needed to make two creases.
Supplemental Resources
Text Resources:

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG

OCMG: Chapter 1, 1-5


Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG
LBKSG: Chapter 1
Standard G.GPE.4
Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate
plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0,2).
Critical Knowledge and Subskills
Students will know that:

A rectangle is composed of two sets of parallel lines with
adjacent lines formed at right angles.
Possible Teaching and Learning Tasks
“Joking with Proofs” (see additional resources on Ed Portal for detailed task)
Students will be able to:

Prove whether a figure in the coordinate plane is a rectangle
G-GPE A Midpoint Miracle
On Core Activity generator and resource materials -- Chapter 1: lessons 1-6 to 1 -7;
http://www.illustrativemathematics.org/illustrations/605
Draw a quadrilateral ABCD. Try to draw your quadrilateral so that no two sides are congruent, no two angles are congruent,
and no two sides are parallel.

Let P, Q, R, and S be the midpoints of sides AB, BC, CD, and DA, respectively. Use a ruler to locate these
points as precisely as you can, and join them to form a new quadrilateral PQRS. What do you notice about
the quadrilateral PQRS?

Suppose your quadrilateral ABCD lies in the coordinate plane. Let (x1,y1) be the coordinates of
vertex A, (x2,y2) the coordinates of B, (x3,y3) the coordinates of C, and (x4,y4) the coordinates of D. Use
coordinates to prove the observation you made in part (a).
Unit Squares and Triangles
http://www.illustrativemathematics.org/illustrations/918
Three unit squares and two line segments connecting two pairs of vertices are shown. What is the area of △ABC?
Supplemental Resources
Text Resources:

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG

OCMG: Chapter 1, 1-2, 1-3

Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG

LBKSG: Chapter 1
G.CO.9
Prove theorems about lines and angles.
Critical Knowledge and Subskills
Students will know that:

Proofs have specific steps that must be justified.
Students will be able to:

Prove that vertical angles are congruent.

Prove that when parallel lines are cut by a transversal,
alternate interior angles are congruent.

Prove that when parallel lines are cut by a transversal,
corresponding angles are congruent.
Possible Teaching and Learning Tasks
“Joking with Proofs” (see additional resources on Ed Portal for detailed task)
Tangent Lines and the Radius of a Circle
http://www.illustrativemathematics.org/illustrations/963
Consider a circle with center O and let P be a point on the circle. Suppose L is a tangent line to the circle at P, that is L meets
the circle only at P.
̅̅̅̅is perpendicular to L.
Show that 𝑂𝑃
Points equidistant from two points in the plane
http://www.illustrativemathematics.org/illustrations/967
̅̅̅̅ as pictured below:
Suppose A and B are two distinct points in the plane and L is the perpendicular bisector of segment 𝐴𝐵

If C is a point on L, show that C is equidistant from A and B, that is show that ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐵𝐶 are congruent.


Conversely, show that if P is a point which is equidistant from A and B, then P is on L.
Conclude that the perpendicular bisector of ̅̅̅̅
𝐴𝐵 is exactly the set of points which are equidistant
from A and B.
Supplemental Resources
Text Resources:

On Core Mathematics- Geometry, Houghton Mifflin Harcourt, - nickname: OCMG

OCMG: Chapter 1, 1-6,1-7

Geometry, Larson, Boswell, Kanold, Stiff, Holt McDougal Littell, 2004, nickname: LBKSG

LBKSG: Chapter 1