Dabbling in Definitions: Intro to Geometric Ideas
Aligned to the Common Core Standards
Written by
Jonathan Katz, Ed. D.
Joseph Walter
ISA Mathematics Coaches
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Dear Math Teacher,
What is mathematics and why do we teach it? This question drives the work of the math coaches
at ISA. We love mathematics and want students to have the opportunity to begin to have a
similar emotion. We hope this unit will bring some new excitement to students.
This unit is the initial unit for a full-year geometry course that is aligned to the common core
standards. It is a unit that revisits concepts and procedures students experienced in middle school
but with an expectation that students will leave with deeper understanding. Essential to this
work is an inquiry approach to teaching mathematics where students are given multiple
opportunities to reason, discover and create. Problem solving is the catalyst to the inquiry
process so as you look closely at this unit you will see that students are constantly placed into
problem solving situations where they are asked to think for themselves and with their
classmates.
The first four Common Core Standards of Practice are central to this unit. Through the constant
use of problematic situations students are being asked to develop perseverance and independent
thought, to reason abstractly and quantitatively, and to critique the reasoning of others.
Mathematical modeling is present throughout the unit as students are asked to describe and
analyze different bare number problems and real world situations leading to geometric ideas.
Students are also asked to create models including the final project, which is to create a city plan
based on the ideas of lines and angles.
The other four Standards of Practice are also present in this unit. Two of them are central to the
inquiry approach. You will see these two statements in the last two standards.


Mathematically proficient students look closely to discern a pattern or structure.
Mathematically proficient students notice if calculations are repeated, and look both for
general methods and for shortcuts.
We believe, as do many mathematicians, that mathematics is the science of patterns. This
underlying principle is present in all the work we do with teachers and students
In this unit you will see that students are often asked to discern a pattern within a particular
situation. This leads students to making conjectures and possibly generalizations that are both
conceptual and procedural.
Thank you for looking at this unit and we welcome feedback and comments.
Sincerely,
Dr. Jonathan Katz
(For the ISA math coaches)
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Unit 1 – Dabbling in Definitions: Intro to Geometric Ideas
Essential Questions:
How do we come to a precise definition?
How do we honor the depth and breadth of a geometric idea?
Interim Assessments/Performance Tasks
What is a Zerf? - Lesson 1
Shoe Size Problem – Lesson 6
Create a City Design- Lesson 9
How do I Replicate an Angle? – Lesson 10
Parallel or Not – Lesson 12
Can you Construct a Perpendicular Line? – Lesson 14
Final Assessment: City Design
What will students understand and be able to do at the end of the unit?
 Students will be able to create precise definitions through multiple experiences with
different geometric ideas.
 Students will be able to write a precise definition of angle, perpendicular lines, parallel
lines and line segments, based on the undefined notions of point, line and distance around
an arc.
 Students will prove theorems about lines and angles including vertical angles, alternate
angles and corresponding angles when a transversal crosses parallel lines.
 Students will understand angle and angle measurement through working with circles and
protractors, angle construction and work with parallel and intersecting lines.
 Students will understand parallelism both through construction and work on the
coordinate plane. This includes using slope to prove two lines are parallel.
 Students will understand perpendicularity both through construction and work on the
coordinate plane.
What enduring understanding will students have?
 In order to create a precise definition, you need to know all the qualities and what makes
it unique.
 Precise definitions are necessary as the building blocks for other geometric ideas.
 Geometric ideas need to be understood in their multiple contexts, e.g. parallel lines.
 Through construction one can develop a deeper understanding of geometric figures.
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Common Core Content Standards in the Unit
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.
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.
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; constructing perpendicular lines; and constructing a line
parallel to a given line through a point not on the line.
G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and uses them to solve
geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that
passes through a given point).
G.GPE.6 Find the point on a directed line segment between two given points that partitions the
segment in a given ratio.
Common Core Standards for Mathematical Practice
The Standards for Mathematical Practice describe varieties of expertise that mathematics
educators at all levels should seek to develop in their students. These practices rest on important
“processes and proficiencies” with longstanding importance in mathematics education. The first
of these are the NCTM process standards of problem solving, reasoning and proof,
communication, representation, and connections. The second are the strands of mathematical
proficiency specified in the National Research Council’s report Adding It Up: adaptive
reasoning, strategic competence, conceptual understanding (comprehension of mathematical
concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly,
accurately, efficiently and appropriately), and productive disposition (habitual inclination to see
mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s
own efficacy).
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1. Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem
and looking for entry points to its solution. They analyze givens, constraints, relationships, and
goals. They make conjectures about the form and meaning of the solution and plan a solution
pathway rather than simply jumping into a solution attempt. They consider analogous problems,
and try special cases and simpler forms of the original problem in order to gain insight into its
solution. They monitor and evaluate their progress and change course if necessary. Older
students might, depending on the context of the problem, transform algebraic expressions or
change the viewing window on their graphing calculator to get the information they need.
Mathematically proficient students can explain correspondences between equations, verbal
descriptions, tables, and graphs or draw diagrams of important features and relationships, graph
data, and search for regularity or trends. Younger students might rely on using concrete objects
or pictures to help conceptualize and solve a problem. Mathematically proficient students check
their answers to problems using a different method, and they continually ask themselves, “Does
this make sense?” They can understand the approaches of others to solving complex problems
and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
Mathematically proficient students make sense of quantities and their relationships in problem
situations. They bring two complementary abilities to bear on problems involving quantitative
relationships: the ability to decontextualize—to abstract a given situation and represent it
symbolically and manipulate the representing symbols as if they have a life of their own, without
necessarily attending to their referents—and the ability to contextualize, to pause as needed
during the manipulation process in order to probe into the referents for the symbols involved.
Quantitative reasoning entails habits of creating a coherent representation of the problem at
hand; considering the units involved; attending to the meaning of quantities, not just how to
compute them; and knowing and flexibly using different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and
previously established results in constructing arguments. They make conjectures and build a
logical progression of statements to explore the truth of their conjectures. They are able to
analyze situations by breaking them into cases, and can recognize and use counterexamples.
They justify their conclusions, communicate them to others, and respond to the arguments of
others. They reason inductively about data, making plausible arguments that take into account
the context from which the data arose. Mathematically proficient students are also able to
compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning
from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary
students can construct arguments using concrete referents such as objects, drawings, diagrams,
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and actions. Such arguments can make sense and be correct, even though they are not
generalized or made formal until later grades. Later, students learn to determine domains to
which an argument applies. Students at all grades can listen or read the arguments of others,
decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems
arising in everyday life, society, and the workplace. In early grades, this might be as simple as
writing an addition equation to describe a situation. In middle grades, a student might apply
proportional reasoning to plan a school event or analyze a problem in the community. By high
school, a student might use geometry to solve a design problem or use a function to describe how
one quantity of interest depends on another. Mathematically proficient students who can apply
what they know are comfortable making assumptions and approximations to simplify a
complicated situation, realizing that these may need revision later. They are able to identify
important quantities in a practical situation and map their relationships using such tools as
diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships
mathematically to draw conclusions. They routinely interpret their mathematical results in the
context of the situation and reflect on whether the results make sense, possibly improving the
model if it has not served its purpose.
5. Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical
problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a
calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry
software. Proficient students are sufficiently familiar with tools appropriate for their grade or
course to make sound decisions about when each of these tools might be helpful, recognizing
both the insight to be gained and their limitations. For example, mathematically proficient high
school students analyze graphs of functions and solutions generated using a graphing calculator.
They detect possible errors by strategically using estimation and other mathematical knowledge.
When making mathematical models, they know that technology can enable them to visualize the
results of varying assumptions, explore consequences, and compare predictions with data.
Mathematically proficient students at various grade levels are able to identify relevant external
mathematical resources, such as digital content located on a website, and use them to pose or
solve problems. They are able to use technological tools to explore and deepen their
understanding of concepts.
6. Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear
definitions in discussion with others and in their own reasoning. They state the meaning of the
symbols they choose, including using the equal sign consistently and appropriately. They are
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careful about specifying units of measure, and labeling axes to clarify the correspondence with
quantities in a problem. They calculate accurately and efficiently, express numerical answers
with a degree of precision appropriate for the problem context. In the elementary grades, students
give carefully formulated explanations to each other. By the time they reach high school they
have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Young students,
for example, might notice that three and seven more is the same amount as seven and three more,
or they may sort a collection of shapes according to how many sides the shapes have. Later,
students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning
about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2
× 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure
and can use the strategy of drawing an auxiliary line for solving problems. They also can step
back for an overview and shift perspective. They can see complicated things, such as some
algebraic expressions, as single objects or as being composed of several objects. For example,
they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that
its value cannot be more than 5 for any real numbers x and y.
8. Look for and express regularity in repeated reasoning.
Mathematically proficient students notice if calculations are repeated, and look both for general
methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that
they are repeating the same calculations over and over again, and conclude they have a repeating
decimal. By paying attention to the calculation of slope as they repeatedly check whether points
are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y
– 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x
– 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum
of a geometric series. As they work to solve a problem, mathematically proficient students
maintain oversight of the process, while attending to the details. They continually evaluate the
reasonableness of their intermediate results.
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 1
Teacher Guide
What is a Widget?
Opening Activity
1. Observe the two groups. From your observations, what are the characteristics that define
a widget? How do you know? Use the pictures as evidence for your statements.
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2. Now look at this picture. Which of the following are widgets? How do you know?
3. Create a statement that defines a widget.
(To the Teacher: The purpose of this activity is to help students to understand how to develop a
definition. The big idea we want students to think about is “What things do all the widgets have in
common, and what things do widgets have that others do not have?” This type of question can be used as
we develop different definitions in this unit. Let your students work on questions 1 and 2 followed by a
whole class discussion. Another discussion should follow question 3. This is an opportunity for students
to critique each other’s statement. Is your definition precise enough? Observation is an important idea in
mathematics, including geometry, While in geometry you often cannot make assumptions from a drawing,
the drawing still gives you a great deal of information that we need to observe.)
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Second Activity
Observe the following 10 geometric shapes. Select the shapes that you feel belong together.
You might try to make a grouping that you think no other group will make. Create a definition
for that group and be ready to show that none of the other shapes on the page can fit your
definition. Be ready to share with the rest of the class.
(Note to teacher: This is a group activity. The purpose of this activity is for students to hone in on the
uniqueness of objects. What are the characteristics that make each shape a member of the group? Ask
students to write as precise a definition as possible. The class will then critique each others’ definitions
and help each other to refine their definition.)
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Geometry Figures
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
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Third Activity:
Group Performance Task: What is a Zerf?
Now it’s your turn to create your own activity. We’re going to call these objects zerfs. You’re
going to create a group of shapes that ARE zerfs, a group that are NOT zerfs and challenge the
class to figure out the definition of a zerf.
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 1
Name_______________________
Date________________________
What is a Widget?
Opening Activity
1. Observe the two groups. From your observations, what are the characteristics that define
a widget? How do you know? Use the pictures as evidence for your statements
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2. Now look at this picture. Which of the following are widgets? How do you know?
3. Create a statement that defines a widget.
14
Second Activity
Observe the following 10 geometric shapes. Select the shapes that you feel belong together.
You might try to make a grouping that you think no other group will make. Create a definition
for that group and be ready to show that none of the other shapes on the page can fit your
definition. Be ready to share with the rest of the class.
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Geometry Figures
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
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Third Activity:
Group Performance Task: What is a Zerf?
Now it’s your turn to create your own activity. We’re going to call these objects zerfs. You’re
going to create a group of shapes that ARE zerfs, a group that are NOT zerfs and challenge the
class to figure out the definition of a zerf.
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 2
Teacher Guide
Where do these Figures Belong? - Creating Definition
Opening Activity
Based on yesterday’s lesson think about these two questions and be ready to discuss your ideas
with the rest of the class
a) What are the qualities of a good definition?
b) What do you need to do in order to write a good definition?
(To the Teacher: This discussion is important as it is central to the work in the whole unit. An outcome of
this discussion could be a poster based on the agreed-upon answers to the two questions.)
Second Activity: A beginning step in creating definitions
(To the Teacher: Students should be encouraged to define as many groupings as they can, e.g. every
figure can be in a group called points and sets of points. We want students to see those qualities that are
common and what is it that makes something unique. As we know there are undefined terms in geometry.
In ancient times, geometers attempted to define every term. For example, Euclid defined a point as
that which has no part. In modern times, mathematicians recognized that attempting to define every word
inevitably led to circular definitions, and in geometry left some words as undefined. This should be part of
your discussion.)
You will be given a set of figures. In your group classify them in any way that you want. You
can make as many groups as you can. You may put a figure into more than one group.
Think about this question:
Why did you put these figures together in this group?
Be sure to make clear your explanations as to your groupings.
As each group presents their findings, other groups will ask questions and add their own
findings.
Through these classifications we can begin to create a definition for each of the different groups.
Geometric Figures
.p
a
.b
c
d
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(To the Teacher: You should have separate charts ready to list student findings. For example, under a
chart called angles, you would list all the different ideas students came up with describing the qualities of
angles. These definitions will be revisited throughout the unit. We build on them and make them more
precise as we dig into different ideas.)
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 2
Where do these Figures Belong? - Creating Definition
Name_______________________
Date________________________
Opening Activity
Based on yesterday’s lesson think about these two questions and be ready to discuss your ideas
with the rest of the class.


What are the qualities of a good definition?
What do you need to do in order to write a good definition?
Second Activity
You will be given a set of figures. In your group classify them in any way that you want. You
can make as many groups as you can. You may put a figure into more than one group.
Think about this question:
Why did you put these figures together in this group?
Be sure to make clear your explanations as to your groupings.
As each group presents their findings, other groups will ask questions and add their own
findings.
Through these classifications we can begin to create a definition for each of the different groups.
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Geometric Figures
.p
a
.b
c
d
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 3
Teacher Guide
What does a city’s street design tell us about that city?
(To the Teacher: This lesson is asking students to look at maps with a geometric eye. It is also about
their aesthetic appreciation. Do they prefer the regularity of a city like Manhattan or the unplanned
evolution of Rome or Erfurt? Another goal of the lesson is for students to take their developing geometric
definitions and talk about them concretely. Students should work on questions in the opening activity
followed by a discussion of their geometric sense. The second activity is quite simple and it is
recommended that students work individually for two reasons: to get multiple responses and to learn
about their developing understanding of the geometric definitions.)
Today we are going to revisit our definitions through looking at maps of cities around the world.
Opening Activity
You will now observe four different city maps. The maps are located at the end of this lesson.
Spend some time looking at these maps, thinking about our question on top of the page and then
answering the questions below.
1. What is similar about the streets in the maps?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
2. What is different about the streets in the maps?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
3. Which map do you feel is most interesting, a place where you might want to live? Why?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
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4. Which city’s street design do you think is best? Why?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
5. How would you describe the geometry of the city you just defined as best?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
6. How would you describe the geometry of one of the cities that was very different?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
Second Activity
Now we’re going to focus on the particular groupings from yesterday’s lesson and look at them
on the map of Detroit.
1. Find an example of a point and a line segment. Sketch each of them and describe their
location.
2. Find an example of a curved line segment. Sketch it and describe its location.
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3. Find an example of two roads that meet to make an acute angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
4. Find an example of two roads that meet to make an obtuse angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
5. Find an example of two roads that meet to make a right angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
6. Find an example of two roads that are parallel to each other. Sketch the relationship of
the two roads below and label the streets’ names.
7. Find an example of two roads that are perpendicular to each other. Sketch the
relationship of the two roads below and label the streets’ names.
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 3
Name_______________________
Date________________________
What does a city’s street design tell us about that city?
Today we are going to revisit our definitions through looking at maps of cities
around the world.
Opening Activity
You will now observe four different city maps. The maps are located at the end of this lesson.
Spend some time looking at these maps, thinking about our question on top of the page and then
answering the questions below.
1. What is similar about the streets in the maps?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
2. What is different about the streets in the maps?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
3. Which map do you feel is most interesting, a place where you might want to live? Why?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
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4. Which city’s street design do you think is best? Why?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
5. How would you describe the geometry of the city you just defined as best?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
6. How would you describe the geometry of one of the cities that was very different?
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________
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Second Activity
Now we’re going to focus on the particular groupings from yesterday’s lesson and look at them
on the map of Detroit.
1. Find an example of a point and a line segment. Sketch each of them and describe their
location.
2. Find an example of a curved line segment. Sketch it and describe its location.
3. Find an example of two roads that meet to make an acute angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
4. Find an example of two roads that meet to make an obtuse angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
5. Find an example of two roads that meet to make a right angle. Sketch the angle that is
formed by the two roads and write the names of the streets.
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6. Find an example of two roads that are parallel to each other. Sketch the relationship of
the two roads below and label the streets’ names.
7. Find an example of two roads that are perpendicular to each other. Sketch the
relationship of the two roads below and label the streets’ names.
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Detroit, Michigan
29
Erfurt, Germany
30
Manhattan, New York City
31
Rome, Italy
32
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 4
Teacher Guide
What Is a Line Segment?
(To the Teacher: The goal of today’s lesson is to further develop the students’ definition of a line
segment. After the opening activity, students will share out all their ideas. We will build this definition
over the next two lessons. Please share with the students the geometric representation of a line (AB), a
ray (AB), and a segment (AB, when appropriate).
Opening Activity
What is a line segment? In your description, discuss points, end points and what distinguishes it
from a line and a ray?
Second Activity
Given a line segment AB and a straight edge (not a ruler), replicate as precisely as you can the
line segment.
B
A
Third Activity
Given line segment CD, a straight edge and a compass, replicate the line segment. Do you think
this will be more precise? If so, how?
C
D
(To the Teacher: The purpose of the Second Activity is to see what students do with just a straightedge
and the difficulty that presents. What might students do? Students can approach the Third Activity in two
ways. First they can measure the line segment with the compass. Then they can draw a line, locate a point
on it, using the same measure from the compass you swing the arc from that point to find the desired
length. The second way is that a student realizes that they can measure the original line segment using the
compass, and swing the arc from that point. Then they mark a point and swing the arc from that point. A
question that may arise is, “Which point on the arc would I use as the endpoint to draw the line
segment?” This is a big idea for kids to think about. Why won’t it matter? While we haven’t talked about
circles it would be interesting if students can use the idea of radii to understand that it doesn’t matter
which point you would use. The discussion you have after the Third Activity should lead students to be
able to do the Fourth Activity. )
Fourth Activity
Write in your own words, the steps one would take to copy a given line segment.
Fifth Activity
How did these activities affect your definition of a line segment? Do you want to make any
changes or additions to your definition of line segment?
(To the Teacher: Finally, you want to revisit the statements that are on the chart for a line segment, do we
want to alter it (add, subtract) to have a more precise definition.)
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 4
What Is a Line Segment?
Name_______________________
Date________________________
Opening Activity
What is a line segment? In your description, discuss points, end points and what distinguishes it
from a line and a ray?
Second Activity
Given a line segment AB and a straight edge (not a ruler), replicate as precisely as you can the
line segment.
B
A
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Third Activity
Given line segment CD, a straight edge and a compass, replicate the line segment. Do you think
this will be more precise? If so, how?
C
D
Fourth Activity
Write in your own words, the steps one would take to copy a given line segment.
Fifth Activity
How did these activities affect your definition of a line segment? Do you want to make any
changes or additions to your definition of line segment?
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 5
Teacher Guide
How Do We Define a Line On the Coordinate Plane?
Opening Activity
Compare these two representations of lines. How are they similar? How are they different?
Fig.1
Fig. 2
y
x
The equation of the line in the
above figure is y = 2x + 5.
(To the Teacher: We’re expanding on our notion of a line and looking at it in different contexts. Clearly,
the line in the coordinate plane has a reference and the isolated line has no reference. Some invariants
include that both are defined by an infinite set of points and that they extend infinitely in both directions.
That is why both of them can be called lines. On the other hand, they are in different contexts and
therefore take on different meanings. For example, if a student were to say that they both have positive
slopes, it’s important to validate that response, but also to point out, that if we rotate the paper clockwise,
one could say that the slope of figure 2 decreases to zero and ultimately becomes negative, whereas the
slope of the line in figure 1, remains the same. What is the significance of this idea?)
Second Activity
1. Why did we name this line (y = 2x + 5) and what does that tell us?
2. If the equation were y = 1x + 5 or y = 3x + 5, how would the representation of the line in
Figure 1 change? Justify your answer?
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Third Activity
Compare these two representations of lines shown below. Which has the greater slope?
Justify your answer.
1)
2) A line whose equation is
2x -+25
yy == 3x
Fourth Activity
Observe the table of values below. Does this table represent the graph or the equation in the
third activity? Justify your answer.
x
y
-3
-11
-2
-8
0
-2
4
10
7
19
(To the Teacher: In this activity, we want students to see that you can pick any two points to determine
the slope and the fact that while the table contains a finite number of discrete points, one can determine
37
all the information necessary to define the line. But you want the students to also see that you can make a
table that goes on infinitely in both directions which is representative of our notion of a line in space.)
Fifth Activity
Journal Entry: Though the multiple experiences you’ve had looking at lines in space and on a
coordinate plane, discuss your understanding of the idea of a line.
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 5
Student Activity Sheet
How Do We Define a Line On the Coordinate Plane?
Name_______________________
Date________________________
Opening Activity
Compare these two representations of lines. How are they similar? How are they different?
Fig.1
Fig. 2
y
x
The equation of the line in the
above figure is y = 2x + 5
Second Activity
1. Why did we name this line (y = 2x + 5) and what does that tell us?
2. If the equation were y = 1x + 5 or y = 3x + 5, how would the representation of the line in
Figure 1 change? Justify your answer?
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Third Activity
Compare these two representations of lines shown below. Which has the greater slope?
Justify your answer.
2)
2) A line whose equation is
2x -+25
yy == 3x
40
Fourth Activity
Observe the table of values below. Does this table represent the graph or the equation in the
third activity? Justify your answer.
x
y
-3
-11
-2
-8
0
-2
4
10
7
19
Fifth Activity
Journal Entry: Though the multiple experiences you’ve had looking at lines in space and on a
coordinate plane, discuss your understanding of the idea of a line.
41
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 6
Teacher Guide
How Do We Define a Line Segment On the Coordinate Plane?
Opening Activity: Performance Task
Shoe Size Problem
Shaquille O’Neal is seven feet one inches tall and has a fifteen inch foot.
What do you think is his shoe size?
How did you come up with your prediction?
Now here is some information to see if your prediction is correct:
The size of a shoe a person needs varies linearly with the length of his or her foot. A study of
basketball players found the following information:
Length of foot in inches
9
11
14
Shoe Size
9
13
19
Use the information to figure out Shaquille O’Neal’s shoe size?
(To the Teacher: We are treating this opening activity as a performance task. Let students work alone and
solve this in any way they want. Many will just use the table to try to figure it out. How well do they
understand ratio? If students solve it quickly have them find more than one approach to solving it. You
can learn about students’ thinking from this problem. Have a beginning discussion of students’ thinking.
If students solved it by graphing hold off on their ideas until the second activity.)
Second Activity
Represent the Shaquille O’Neal problem graphically. How does your graph differ from the
graph of a line?
Questions to think about as you graph it:

Do you think if we graph the data we would get a line? Why?
42

Is there a largest and smallest length of foot that a human being can have? And therefore,
would there be a largest and smallest shoe that is made? How would the answer to this
question affect our graph? Would the solution be a line or a line segment?
(To the Teacher: With this activity, we’re setting up for a conversation about a bounded domain
and range versus one that is infinite in both directions. So we would expect students to come to
the conclusion that this is best represented by a line segment rather than a line. This goes along
with our definition of a line segment in space that is bounded at both ends. Boundedness is an
invariant of all line segments. Once a performance task is finished, you are going to lead a
discussion on this question: Is this a line segment or line and why.)
Third Activity
1. What is the rate of change of the effect of foot length on the shoe size? (Hint: How is
rate of change connected to the slope of the graph?)
2. If we move from a line to a line segment, what changes and what stays the same? Justify
your answer.
(To the teacher: It’s important that students come to the conclusion that the only difference between a
line and a line segment is boundedness of the latter. In all other respects, including slope (for the
coordinate plane) and the fact that they both contain an infinite number of points, they are the same.)
43
Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 6
How Do We Define a Line Segment On the Coordinate Plane?
Name_______________________
Date_______________________
Opening Activity: Performance Task
Shoe Size Problem
Shaquille O’Neal is seven feet one inches tall and has a fifteen inch foot.
What do you think is his shoe size?
How did you come up with your prediction?
Now here is some information to see if your prediction is correct:
The size of a shoe a person needs varies linearly with the length of his or her foot. A study of
basketball players found the following information:
Length of foot in inches
9
11
14
Shoe Size
9
13
19
Use the information to figure out Shaquille O’Neal’s shoe size?
44
Second Activity
Represent the Shaquille O’Neal problem graphically. How does your graph differ from the
graph of a line?
Questions to think about as you graph it:


Do you think if we graph the data we would get a line? Why?
Is there a largest and smallest length of foot that a human being can have? And therefore,
would there be a largest and smallest shoe that is made? How would the answer to this
question affect our graph? Would the solution be a line or a line segment?
Third Activity
1. What is the rate of change of the effect of foot length on the shoe size? (Hint: How is
rate of change connected to the slope of the graph?)
2. If we move from a line to a line segment, what changes and what stays the same? Justify
your answer.
45
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 7
Teacher Guide
How Do We Define an Angle?
Opening Activity
In your group, read these three definitions of an angle. Come to an agreement about
which definition best describes an angle and why.
1. An angle is the union of two rays with a common endpoint.
2. An angle is the region contained between two rays.
3. An angle is the turning of a ray about a point from one position to another.
(To the Teacher: We want the students to have some time to argue this through and be able to
defend their choice. Then you should bring the whole class together for a discussion. It is
important to know that students have difficulty defining an angle because they often think of a
definition in a static way, while it is important that students understand the notion of an angle as
a dynamic concept. Make certain that students leave this discussion understanding that the last
definition is the one that best describes an angle. We will spend the rest of the lesson deepening
what that means.)
Second Activity
1. If you’re looking at a clock with hands on it that has no numbers, how would you
describe to someone, the difference between two o’clock and six o’clock?
(To the Teacher: Our purpose with this question is to get to the understanding that the definition of an
angle is determined by the dynamic relationship of the position of the hands on the clock relative to each
other. We chose the use of a clock because a circle is essential to defining an angle.)
2. If we put the numbers back on the clock, and it reads twelve o’clock, how would you
describe the measure of the angle between the minute hand and the hour hand?
46
3. If the clock reads two o’clock, how would you describe the measure of the angle between
the minute hand and hour hand?
(To the Teacher: This is an opportunity to assess students’ understanding of angular measurement as a
means of helping them to define an angle.)
4. If I went one fourth of the way around the clock, how many degrees would I pass
through?
5. How do these different questions help us to further understand the definition of an angle?
(To the Teacher: Students should leave this discussion with the understanding that our definition of an
angle is augmented by the use of units that differentiate angles.)
Third Activity
We will now begin to use a protractor to measure angles. This is an important tool in geometry
that we will be using for several months. The picture below is of a protractor:
There are three important things to remember about using the protractor:
1. Always line up the one of the rays of the angle on the line where 0 degrees is represented.
2. Always measure starting from “0”, not from “180”
3. Follow the path of the first ray to the second ray. How many degrees was the path?
Do the previous directions for using a protractor follow from our definition of an angle?
Explain.
47
Now that you know the most important ideas about using a protractor, use your protractor to
measure the following angles. Then label it as an acute, obtuse or right angle.
48
(To the Teacher: Bring everyone together to synthesize the different experiences around angles
reinforcing the definition and hopefully moving them away from any misconceptions they may have.)
49
Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 7
How Do We Define an Angle?
Name_______________________
Date_______________________
Opening Activity
In your group, read these three definitions of an angle. Come to an agreement about
which definition best describes an angle and why.
1. An angle is the union of two rays with a common endpoint.
2. An angle is the region contained between two rays.
3. An angle is the turning of a ray about a point from one position to another.
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Second Activity
1. If you’re looking at a clock with hands on it that has no numbers how would you describe
to someone, the difference between two o’clock and six o’clock?
2. If we put the numbers back on the clock, and it reads twelve o’clock, how would you
describe the measure of the angle between the minute hand and the hour hand?
3. If the clock reads two o’clock, how would you describe the measure of the angle between
the minute hand and hour hand?
4. If I went one fourth of the way around the clock, how many degrees would I pass
through?
5. How do these different questions help us to further understand the definition of an angle?
51
Third Activity
We will now begin to use a protractor to measure angles. This is an important tool in geometry
that we will be using for several months. The picture below is of a protractor:
There are three important things to remember about using the protractor:
4. Always line up the one of the rays of the angle on the line where 0 degrees is represented.
5. Always measure starting from “0”, not from “180”
6. Follow the path of the first ray to the second ray. How many degrees was the path?
Do the previous directions for using a protractor follow from our definition of an angle?
Explain.
52
Now that you know the most important ideas about using a protractor, use your protractor to
measure the following angles. Then label it as an acute, obtuse or right angle.
53
54
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 8
Teacher Guide
What can we observe about intersecting lines?
Opening Activity:
Today you are going to make a set of discoveries about the angles formed by intersecting lines.
Based on your observations of the intersecting lines below what predictions do you want to make
about the relationships of the different angle measurements of the four angles?
a) Label and measure the four angles created by the intersection of the two lines below:
Angle 1:
Angle 2:
Angle 3:
Angle 4:
b) Label and measure the four angles created by the intersection of the two lines below:
Angle 1:
Angle 2:
Angle 3:
Angle 4:
55
c) Observe your results from a and b above. What do you notice about the relationships
between the angles in each of the diagrams?
d) With the straight edge of your protractor, draw two lines that intersect below and test your
conjecture:
e) Generalize what you have discovered. Did it go along with your predictions?
(To the Teacher: This opening activity could be a group activity where students observe special
relationships of intersecting lines. During the discussion of their findings it would be important to give
names to the angle relationships discovered, such as adjacent, vertical and supplementary angles. Also
you can help students understand that the general statements made will be used in further work this year
(e.g. The measures of vertical angles formed by intersecting lines are equal.)
Second Activity
1. Look at the map of Detroit that we used in a previous lesson yesterday. Find where Michigan
Avenue crosses First Street. Do the relationships we observed about the four angles formed by
two intersecting lines hold true? Justify your answer.
2. Use your knowledge of angle relationships to find the degrees of the missing angles without
using your protractor:
102
52
56
3. Find the measure of each angle:
(3x)
(6x)
(2x + 5)
(x)
4. If line AB intersects line CD at point E and angle AED is 30 more than angle DEB, find the
measure of all four angles.
5. Use any of the maps for the following:
Find two streets that intersect. Measure the four angles formed by these streets. Use your
measurements to support or disprove the conjectures for intersecting lines we discovered in class
today.
(To the Teacher: Once you go over students’ solutions to the problems you might ask students to write in
a journal using this prompt, “What can we say will be true about the angles formed by intersecting
lines?”)
57
Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 8
What can we observe about intersecting lines?
Name_______________________
Date_______________________
Opening Activity:
Today you are going to make a set of discoveries about the angles formed by intersecting lines.
Based on your observations of the intersecting lines below what predictions do you want to make
about the relationships of the different angle measurements of the four angles?
a) Label and measure the four angles created by the intersection of the two lines below:
Angle 1:
Angle 2:
Angle 3:
Angle 4:
b) Label and measure the four angles created by the intersection of the two lines below:
Angle 1:
Angle 2:
Angle 3:
Angle 4:
58
c) Observe your results from a and b above. What do you notice about the relationships
between the angles in each of the diagrams?
d) With the straight edge of your protractor, draw two lines that intersect below and test your
conjecture:
e) Generalize what you have discovered. Did it go along with your predictions?
59
Second Activity
1. Look at the map of Detroit that we used in a previous lesson yesterday. Find where Michigan
Avenue crosses First Street. Do the relationships we observed about the four angles formed by
two intersecting lines hold true? Justify your answer.
2. Use your knowledge of angle relationships to find the degrees of the missing angles without
using your protractor:
102
52
60
3. Find the measure of each angle:
(3x)
(6x)
(2x + 5)
(x)
4. If line AB intersects line CD at point E and angle AED is 30 more than angle DEB, find the
measure of all four angles.
5. Use any of the maps for the following. Find two streets that intersect. Measure the four
angles formed by these streets. Use your measurements to support or disprove the rules for
intersecting lines we discovered in class today.
61
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 9
Teacher Guide
What can we observe about right angles and straight angles?
Opening Activity
Now that you know how to measure an angle can you come up with a method to draw any given
angle?
Using a protractor your job is to:
a) draw a 25 degree angle
b) draw a 134 degree angle
c) write up an explanation of how to draw any given angle.
(To the Teacher: Students were probably shown this in middle school but some will have difficulty. How
will they use their understanding of measuring an angle to now draw a given angle? A class discussion
can lead to a method of drawing an angle. How can we ensure precision? In the next two activities
students will be making observations about complementary and supplementary angles. Once students
have made the observations you can share the geometric terms with them.)
Second Activity
1. Construct a right angle.
2. Label it <ABC
3. From point B draw a ray within the interior of <ABC. Call it BD.
4. Measure <CBD and <ABD
5. How are < CBD and <ABD related to the original <ABC? Is this always true? Can you
make a general statement?
6. Triangles ABC, DEF and GHI below are right triangles. Using the values of the given
angles, find the unknown angles. Explain how you found them.
A
F
G
78
?
E
32
B
?
m
C
D
H
I
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Third Activity
1. Construct a straight angle.
2. Label it <ABC
3. From point B draw a ray within the interior of <ABC. Call it ray BD
4. Measure <CBD and <ABD
5. How are < CBD and <ABD related to the original <ABC? Is this always true? Can you
make a general statement?
6. Lines AC, DF and GI below are straight lines. Using the values of the given angles, find
the unknown angles. Explain how you found them.
D
E
68
?
A
7.
142
B
F
?
?
C
G
m
H
I
Look at the following figure. Find the values of each of the angles.
2x
x
3x
4x
Fourth Activity
1. Using the map of the Detroit, find an example of streets that intersect to form
supplementary angles. Measure these angles to confirm that this is true.
2. Repeat the same process for complementary angles.
63
Performance Task: Create a City Design
Create a design of a small town that has eight different streets that you will name. There
must be an example of complementary, supplementary, vertical and adjacent angles
which you will describe in words.
64
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 9
Student Activity Sheet
What can we observe about right angles and straight angles?
Name_______________________
Date_______________________
Opening Activity
Now that you know how to measure an angle can you come up with a method to draw any given
angle?
Using a protractor your job is to:
a) draw a 25 degree angle
b) draw a 134 degree angle
c) write up an explanation of how to draw any given angle.
65
Second Activity
1. Construct a right angle.
2. Label it <ABC
3. From point B draw a ray within the interior of <ABC. Call it ray BD
4. Measure <CBD and <ABD
5. How are < CBD and <ABD related to the original <ABC? Is this always true? Can
you make a general statement?
6. Triangles ABC, DEF and GHI below are right triangles. Using the values of the
given angles, find the unknown angles. Explain how you found them.
A
F
G
78
?
E
32
B
?
m
C
D
H
I
66
Third Activity
1.
Construct a straight angle.
2. Label it <ABC
3. From point B draw a ray within the interior of <ABC. Call it ray BD
4. Measure <CBD and <ABD
5. How are < CBD and <ABD related to the original <ABC? Is this always true? Can
you make a general statement?
67
6. Lines AC, DF and GI below are straight lines. Using the values of the given angles,
find the unknown angles. Explain how you found them.
D
E
68
?
A
142
B
F
?
?
C
G
m
H
I
7. Look at the following figure. Find the values of each of the angles.
2x
x
3x
4x
Fourth Activity
1. Using the map of the Detroit, find an example of streets that intersect to form
supplementary angles. Measure these angles to confirm that this is true.
2. Repeat the same process for complementary angles.
68
Performance Task: Creating a City Design
Create a design of a small town that has eight different streets that you will name. There must be
an example of complementary, supplementary, vertical and adjacent angles which you will
describe in words.
69
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 10
Teacher Guide
How would you replicate an angle?
(To the Teacher: The goal of today’s lesson is to further develop the students’ definition of an angle
through a construction inquiry.)
Opening Activity
We have now looked at angles over the last few days, including beginning to define an angle,
measuring and constructing angles and observing angles formed by intersecting lines. Using our
original definition is there anything you would want to add to the definition or change to make it
more precise? Explain your reasoning.
Our original definition was:
An angle is the turning of a ray about a point from one position to another.
Second Activity-Performance Task: How do I replicate an angle?
Given an angle, compass and a straight edge (not a ruler), replicate as precisely as you can the
given angle. Show all your work and explain your thinking.
(To the Teacher: It is recommended that you walk around and observe what students are doing. You can
give them 10-15 minutes for the performance task. You will want to lead a discussion afterwards based on
the strategies and ideas the students used to try to copy a given angle. Choose the students to share based
on their ideas even if they had errors in their thinking. You should help the class come up with an
approach that would make for an exact replication. )
70
Third Activity
Write in your own words, the steps one would take to copy a given angle.
Fourth Activity
How does this the method to copy an angle go along with our definition of an angle?
71
Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 10
How would you replicate an angle?
Name_______________________
Date________________________
Opening Activity
We have now looked at angles over the last few days, including beginning to define an angle,
measured angles and observed angles formed by intersecting lines. Using our original definition
is there anything you would want to add to the definition or change to make it more precise?
Explain your reasoning.
Our original definition was:
An angle is the turning of a ray about a point from one position to another.
72
Second Activity-Performance Task: How do I Replicate an Angle?
Given an angle, compass and a straight edge (not a ruler), replicate as precisely as you can the
given angle. Show all your work and explain your thinking.
73
Third Activity
Write in your own words, the steps one would take to copy a given angle.
Fourth Activity
How does this the method to copy an angle go along with our definition of an angle?
74
Developing Definition – Intro to Geometric Ideas
Lesson 11
Teacher Guide
How do we define parallel lines?
Opening Activity
Here are two definitions for parallel lines. Read them closely

Lines are parallel if they lie in the same plane, and are the same distance apart over
their entire length. No matter how far you extend them they will never meet.

Parallel lines are two or more coplanar lines that have no points in common or are
identical (e.g., the same line)
Now with your partners you will decide if these definitions are precise. Think about the
following:
1) Why use the term “on the same plane? If we left it how would it affect the definition?
Why?
2) Why do we say they are “the same distance apart over their entire length?” If we left it
how would it affect the definition? Why?
3) Are the definitions saying the same thing or are they saying something different? Explain.
4) Do you want to make any changes? What definition does your group want to use?
(To the teacher: This activity is about helping students dissect the language of geometry. This is
important because students frequently get lost in the language. It is important for them to see why these
two definitions are saying the same thing. We also want them to see that if we leave parts of these
definitions out we would lose the precision of the definition. Students often say the definition of parallel
lines is two lines that never meet which also is true of skew lines, hence the necessity of talking about
coplanar.)
Second Activity:
Using your understanding of parallel lines and our agreed upon definition, you are going to be
given a line (we will call line L) and a point (we will call P) and you have to come up with a
method of constructing a line parallel to Line L that passes through point P. You will be given a
straightedge and a compass.
.P
L
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(To the Teacher: This is a difficult activity. Let students struggle somewhat. Bring them together for a
discussion of ideas, which will lead them to the third activity which gives directions for constructing a
pair of parallel lines.)
Third Activity:
Here are directions to draw a line parallel to a given line through a given point.
See if you can follow it. Why does this method work?
Constructing a Line Parallel to a Given Line through a Given Point
After doing this
Your work should look like this
Start with a line segment PQ
and a point R off the line.
1. Draw a transverse line
through R and across the line
PQ at an angle, forming the
point J where it intersects the
line PQ. The exact angle is not
important.
2. With the compass width set
to about half the distance
between R and J, place the
point on J, and draw an arc
across both lines.
76
After doing this
Your work should look like this
3. Without adjusting the
compass width, move the
compass to R and draw a
similar arc to the one in step 2.
4. Set compass width to the
distance where the lower arc
crosses the two lines.
5. Move the compass to where
the upper arc crosses the
transverse line and draw an
arc across the upper arc,
forming point S.
77
After doing this
Your work should look like this
6. Draw a straight line through
points R and S.
Done. The line RS is parallel to
the line PQ
Now that you have done this construction think about this question.
How does the construction help to further explain our definition of parallel lines?
(To the Teacher: You should have students go through the process thinking about why this would give us
parallel line. It is important to think about why this works and how it goes along with the definition of
parallel lines. It is related to the next lesson in which student will do a performance task based on a
transversal intersecting two parallel lines. After that activity you might revisit this construction and help
students see the connection.)
78
Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 11
How do we define parallel lines?
Name_______________________
Date________________________
Opening Activity:
Here are two definitions for parallel lines. Read them closely

Lines are parallel if they lie in the same plane, and are the same distance apart over
their entire length. No matter how far you extend them they will never meet.

Parallel lines are two or more coplanar lines that have no points in common or are
identical (e.g., the same line)
Now with your partners you will decide if these definitions are precise. Think about the
following:
1. Why use the term “on the same plane? If we left it out would the definition still be
precise? Why?
2. Why do we say they are “the same distance apart over their entire length?” If we left
it out how would it affect the definition?
3. Are the definitions saying the same thing or are they saying something different?
Explain.
4. Do you want to make any changes? What definition does your group want to use?
79
Second Activity:
Using your understanding of parallel lines and our agreed upon definition, you are going to be
given a line (we will call line L) and a point (we will call P) and you have to come up with a
method of constructing a line parallel to Line L that passes through point P. You will be given a
straightedge and a compass.
.P
L
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Third Activity:
Here are directions to draw a line parallel to a given line through a given point.
See if you can follow it. Why does this method work?
Constructing a Line Parallel to a Given Line through a Given Point
After doing this
Your work should look like this
Start with a line segment PQ
and a point R off the line.
1. Draw a transverse line
through R and across the line
PQ at an angle, forming the
point J where it intersects the
line PQ. The exact angle is not
important.
2. With the compass width set
to about half the distance
between R and J, place the
point on J, and draw an arc
across both lines.
81
After doing this
Your work should look like this
3. Without adjusting the
compass width, move the
compass to R and draw a
similar arc to the one in step 2.
4. Set compass width to the
distance where the lower arc
crosses the two lines.
5. Move the compass to where
the upper arc crosses the
transverse line and draw an
arc across the upper arc,
forming point S.
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After doing this
Your work should look like this
6. Draw a straight line through
points R and S.
Done. The line RS is parallel to
the line PQ
Now that you have done this construction think about this question.
How does the construction help to further explain our definition of parallel lines?
83
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 12
Teacher Guide
Parallel or Not?
(To the Teacher: This lesson is to be done by students independently as a performance assessment. See
accompanying document called “Parallel or Not.” Students can be given the whole period for this. You
should follow up this lesson with a discussion about students’ discoveries about angles formed by a
transversal intersecting two parallel lines.)
84
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 13
Teacher Guide
More with Transversals
Opening Activity
1. Write down all the things you discovered yesterday regarding the angles formed by a
transversal cutting through two parallel lines.
(To the Teacher: Have a discussion with the students regarding their findings. You should take this
opportunity to give names to the different angles formed, e.g. alternate interior, corresponding, alternate
exterior, etc.)
2. We know that there are eight angles formed by the transversal. What is the minimum
number of angle measurements you would need to know to be able to figure out the rest
of the angle measurements? Justify your answer.
Second Activity
1. On the Detroit, Michigan map, mark a 1 and a 2 on two alternate interior angles and a 3
and a 4 on two corresponding angles.
2. Use your protractor to measure the angles you’ve labeled. Do your results support the
discoveries you made yesterday?
Third Activity
1. Two parallel lines are crossed by the transversal below then find the measure of each of
the angles below.
3x + 20
4x
85
2. If the measure of angle 1 = 57, explain how you can find the measure of angle 8.
1
8
3. If angle 2 = x, write an expression for the measure of angle7 and angle 8. Give evidence for
your choices.
1 2
3 4
5 6
7 8
4. On the map of Manhattan, find Broadway. Discuss why this street would be called a
transversal.
Describe the angles formed by the streets that can be called:
a. Alternate interior
b. Corresponding
(To the Teacher: Have students discuss their responses to the various questions. A large idea we want
them to think about from the two days of this work is that with parallel lines, cut by a transversal, there is
more information that we immediately know than what we know when the two lines being intersected are
not parallel. No we will revisit the construction and see if students can understand that the method we
used is based on the theorem discoveries the students made over the last two days.)
Fourth Activity
Revisit the construction of two parallel lines. Can you explain now more fully why the method
that you were shown would always create parallel lines?
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 13
More with Transversals
Name_______________________
Date________________________
Opening Activity
1. Write down all the things you discovered yesterday regarding the angles formed by a
transversal cutting through two parallel lines.
2. We know that there are eight angles formed by the transversal. What is the minimum
number of angle measurements you would need to know to be able to figure out the
rest of the angle measurements? Justify your answer.
Second Activity
1. On the Detroit, Michigan map, mark a 1 and a 2 on two alternate interior angles and a
3 and a 4 on two corresponding angles.
2. Use your protractor to measure the angles you’ve labeled. Do your results support
the discoveries you made yesterday?
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Third Activity
1. Two parallel lines are crossed by the transversal below then find the measure of each
of the angles below.
3x + 20
4x
2. If the measure of angle 1 = 57, explain how you can find the measure of angle 8.
1
8
3. If angle 2 = x, write an expression for the measure of angle7 and angle 8. Give evidence for
your choices.
1 2
3 4
56
7 8
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4. On the map of Manhattan, find Broadway. Discuss why this street would be called a
transversal.
Describe the angles formed by the streets that can be called:
a. Alternate interior
b. Corresponding
Fourth Activity
Revisit the construction of two parallel lines. Can you explain now more fully why the method
that you were shown would always create parallel lines?
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Dabbling in Definitions – Intro to Geometric Ideas
Lesson 14
Teacher Guide
How do we define perpendicular lines?
(To the Teacher: One of the goals of today’s lesson is to help students expand their notion of
perpendicular lines. After the Opening Activity in which you will get a chance to hear students’ ideas you
will give students the opportunity to play with the construction activity in the Second Activity. You will
learn from what students do or don’t do. How do they use their notion of perpendicular to think about the
construction? One of the other goals is to continue to build students perseverance with problems.
Throughout this unit students are asked to do things that they have to think about, where the answer takes
time and a willingness to try different things. A discussion after the Second Activity can lead students to
try to follow the construction given to them. Why does that method make sense? What does it say about
the meaning of perpendicularity? The fourth Activity will be a Performance Task. Can students transfer
their understanding of the construction to do a similar construction?)
Opening Activity
Write down your definition of perpendicular lines. Use examples to support your definition.
Second Activity
You are going to be given a line j and a point r above the line. Using a compass and straight
edge your job is to come up with a method of drawing a line perpendicular to line j that passes
through point r.
.r
j
Third Activity:
Now let us look at the general construction. Does it go along with your idea? Why does it work?
What does it tell us about perpendicular lines?
90
Start with a line and point R which is not on
that line.
Step 1
Place the compass on the given external
point R.
Step 2
Set the compass width to a approximately
50% more than the distance to the line. The
exact width does not matter.
Step 3
Draw an arc across the line on each side of
R, making sure not to adjust the compass
width in between. Label these points P and Q
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Step 4
At this point, you can adjust the compass
width. Recommended: leave it as is.
From each point P,Q, draw an arc below the
line so that the arcs cross.
Step 5
Place a straightedge between R and the point
where the arcs intersect. Draw the
perpendicular line from R to the line, or
beyond if you wish.
Step 6
Done. This line is perpendicular to the first
line and passes through the point R. It also
bisects the segment PQ (divides it into two
equal parts)
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Fourth Activity-Performance Assessment: Can you Construct a Perpendicular Line?
Now that you are comfortable with the last construction try to solve this problem.
Construct a line perpendicular to a given line j that passes though point r. Point r is located on
line j.
.r
j
Closing Activity
Let us revisit our definitions of perpendicular lines. Is there anything you would like to add or
subtract from the ideas presented earlier to help us be more precise with our definition?
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Dabbling in Definitions – Intro to Geometric Ideas
Student Activity Sheet
Lesson 14
How do we define perpendicular lines?
Name_______________________
Date________________________
Opening Activity:
Write down your definition of perpendicular lines. Use examples to support your definition.
Second Activity:
You are going to be given a line j and a point r above the line. Using a compass and straight
edge your job is to come up with a method of drawing a line perpendicular to line j that passes
through point r.
.r
j
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Third Activity:
Now let us look at the general construction. Does it go along with your idea? Why does it work?
What does it tell us about perpendicular lines?
95
Start with a line and point R which is not on
that line.
Step 1
Place the compass on the given external
point R.
Step 2
Set the compass width to a approximately
50% more than the distance to the line. The
exact width does not matter.
Step 3
Draw an arc across the line on each side of
R, making sure not to adjust the compass
width in between. Label these points P and Q
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Step 4
At this point, you can adjust the compass
width. Recommended: leave it as is.
From each point P,Q, draw an arc below the
line so that the arcs cross.
Step 5
Place a straightedge between R and the point
where the arcs intersect. Draw the
perpendicular line from R to the line, or
beyond if you wish.
Step 6
Done. This line is perpendicular to the first
line and passes through the point R. It also
bisects the segment PQ (divides it into two
equal parts)
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Fourth Activity-Performance Assessment: Can you Construct a Perpendicular Line?
Now that you are comfortable with the last construction try to solve this problem. Show all your
work and explain your thinking.
Construct a line perpendicular to a given line j that passes though point r. Point r is located on
line j.
.r
j
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Closing Activity
Let us revisit our definitions of perpendicular lines. Is there anything you would like to add or
subtract from the ideas presented earlier to help us be more precise with our definition?
99
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 15
Teacher Guide
What is true about parallel and perpendicular lines on the coordinate plane?
(To the Teacher: In today’s lesson we are expanding our notion of parallel and perpendicular lines. We
have thought about their qualities in space or on a plane and now we will think about them on a
coordinate plane. Why is it important for students to think about ideas in different contexts? What is
invariant and what is different? Today’s lesson is an investigation in groups. Students might have learned
this last year but the meaning for them should be deeper. Walk around the room and see what the groups
are doing. Perhaps you will choose one or two groups who have made some good observations to share
with the rest of the class. The ideas in today’s lesson will be revisited when we do coordinate geometry
proofs. The question at the end of the lesson should be discussed as you did with all the definitions. How
does this deeper understanding of parallel and perpendicular lines affect your definition of these two
ideas? This will be the last lesson before the introduction of the final assessment.)
Today you are going to do an investigation on the coordinate plane of the uniqueness of parallel
and perpendicular lines. We have come up with definitions of each of these sets of lines now we
are going to further develop them.
Before we begin the experiment observe the four diagrams and make a prediction about what
will be true about the slopes of parallel lines and what will be true about the slopes of
perpendicular lines.
Now for each coordinate plane find the slope of the two lines.
100
101
Observe the results of your calculations.
What do you notice about the slopes of parallel and perpendicular lines?
Make a conjecture for parallel and perpendicular lines that you will test out on the following
coordinate plane.
102
Make a statement about the slopes of parallel and perpendicular lines. Then explain why this
will always be true.
How does this deeper understanding of parallel and perpendicular lines affect your definition of
these two ideas?
103
Dabbling in Definitions – Intro to Geometric Ideas
Lesson 15
Student Activity Sheet
What is true about parallel and perpendicular lines on the coordinate plane?
Today you are going to do an investigation on the coordinate plane of the uniqueness of parallel
and perpendicular lines. We have come up with definitions of each of these sets of lines now we
are going to further develop them.
Before we begin the experiment observe the four diagrams and make a prediction about what
will be true about the slopes of parallel lines and what will be true about the slopes of
perpendicular lines.
Now for each coordinate plane find the slope of the two lines.
104
105
Observe the results of your calculations.
What do you notice about the slopes of parallel and perpendicular lines?
Make a conjecture for parallel and perpendicular lines that you will test out on the following
coordinate plane.
106
Make a statement about the slopes of parallel and perpendicular lines. Then explain why this
will always be true.
How does this deeper understanding of parallel and perpendicular lines affect your definition of
these two ideas?
107
Geometry – City Planning Project
Final Project
(To the Teacher: This project/assessment gives students the opportunity to create a design of a city while
showing their understanding of the multiple ideas discussed in this unit. A rubric should be created that
accentuates the mathematical understanding students showed in their written piece and drawing. Do you
want students to work in pairs or groups of 3? What materials will you need to share with students?
Perhaps you will want a gallery walk at the end so students can evaluate each other’s work. How much
time will you give students in class to work on this and how much time will give the students to work at
home?)
Challenge: The mayor of Detroit, Dave Bing, has decided that all this vacant land in Detroit is
being made available to residents of Detroit who want to help rebuild the city. There are many
groups that are planning to apply for right to design this space. You are part of a group of young
people who want to apply for this privilege. Your job is to design how to use the space for
submission to the mayor’s office.
Within the plan, you must include the following:
 Residential spaces
 Government spaces
 Community spaces
 Recreational spaces
 An aesthetic space (a place of beauty)
Given: The plot of land is 8000 ft by 5000 ft.
You must design your city on graph paper. You should use the scale ¼ inch = 100 feet.
Written Piece:
Part 1: You must include a written discussion of the design of your city. In this part, you must
explain why your design will be a place where people want to live, work and play. This written
part should be persuasive and professional as it is addressed to the mayor of Detroit.
Part 2:
Next, you must include a discussion of the street relationships. Included in this section must be
the following geometric terms and relationships:
 Intersecting streets
 Two streets that are line segments where one is copied from the other
 Parallel streets formed by a construction with mathematical evidence that they are
parallel
 A transversal street intersecting two parallel streets with evidence that adjacent angles,
vertical angles, corresponding angles, alternate interior, and alternate exterior angles were
formed.
 Perpendicular streets formed by a construction with mathematical evidence that they are
perpendicular.
 Two streets that intersect and form an acute angle. Then copy that angle to another
location on your design. Show that the measures of the two angles are equal.
108

Two streets that intersect to form an obtuse angle. Then copy that angle to another
location on your design. Show that the measures of the two angles are equal.
Part 3:
Create a glossary with precise definitions of all the geometric terms discussed in this unit. Your
definitions can include diagrams to help in your descriptions.
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110