Angela Martinez
Mansfield University of PA / School Library & Information
Technologies Program
High School Geometry- Euclid’s The Elements and Geometric
Constructions
GRADE: 9-12 (mixed HS class) SUBJECT: Geometry
TEACHER: Kersten Harnly
SCHOOL: Grace Christian School, Anchorage, Alaska
ESTIMATED LESSON TIME:
4 class periods (160 minutes)
LIBRARY CONTEXT:
___Fixed
___Flexible
___Combination
___Individualized instruction
___Stand-alone lesson
_x_Lesson in a Unit
___Multiple lessons in a unit
COLLABORATION CONTINUUM:
___Cooperation
___Coordination
_x_ Collaboration
___None
Standards for the 21st Century Learner Goals
(no more than 3 of each)
Standards:
· 1) Inquire, think critically, and gain knowledge.
· 2) Draw conclusions…apply knowledge to new situations, create
new knowledge.
Benchmark(s):
· 1.1.6
· 2.1.1, 2.1.5
Dispositions Indicator(s):
· 1.2.6, 1.2.7
· 2.2.2
Responsibilities Indicator(s):
· 1.3.2, 1.3.4
· 2.3.1
Self-Assessment Strategies Indicator(s):
· 1.4.2, 1.4.4
· 2.4.2
Scenario /overview/content topic:
This lesson will occur at the beginning of the year in the
geometry class. This is the best time to teach this lesson because
it introduces students to the fundamental building blocks of
geometry that they will use not only in this class but also
throughout their lives. By spending time on these concepts now,
during a student’s high school experience, the teacher is giving
students the key to unlocking logic and reasoning skills, as well
as valuable geometric rules and properties that are prevalent in
the world (we cannot escape space and geometric planes, they’re
always around us). Additionally, this lesson will give students
have a better understanding and appreciation for mathematical
constructions. This will form valuable prior knowledge for later
lessons on mathematical proofs (the following lesson).
The essential questions of this lesson are 1) Why are Euclid’s
five postulates relevant to today’s math classroom? 2) How do we
use the postulates to form mathematical constructions? These are in
addition to the unit essential question—What are practical uses of
mathematical constructions?
The goal of this specific lesson will be that students
understand the fundamental basics of geometry and will be able to
build upon that basis to expand mathematical content knowledge.
Students will also learn information skills as the process the
postulates and interpret their meaning. This will help them better
understand fundamental mathematical principles throughout their
entire mathematics career.
Connection to Local or State Content Standards and the Common
Core Standards if available-
Alaska State Standards:
· B.4 – apply a variety of strategies to solve mathematical
problems.
· B.6 – evaluate results using common sense.
· B.8 – use math confidently.
Common Core:
· CCSS.Math.Content.HSG-CO.D.12 – make geometric constructions
with a variety of tools.
· CCSS.Math.Content.HSG-CO.A.1 – know mathematical definitions
of geometric terms.
Collaboration (including information from the teacher)
that addresses student interests):
Kersten explained to me that she has a group of highly
motivated, (mostly) well-behaved students in her geometry class.
The class encompasses student ranging from grades 9 to 12 with a
range of performance levels. Kersten also explained that she finds
her students most commonly asking about how they will apply the
principles of mathematics that they are learning out of pure
interest and not out of spite for the content (as is the usual
reason for this inquiry). Kersten also mentioned multiple times
that she cannot assume prior knowledge of any of her students,
especially at the beginning of the year. Since her class covers
such a large range of grades she must be ready to offer additional
scaffolding to any student who may need it for any given topic.
Since Kersten lives and works in Alaska our meetings all web
based. Our preliminary conversation for this project was via
Facebook chat. While messaging back and forth we established the
content of the lesson and from there we organized to meet via phone
to discuss the lesson more fully. Our conversation planning the
lesson lasted an hour and a half, in which time we outlined the
lesson, brain stormed possible activities, and talked about the
student demographic of Kersten’s classroom. Having almost identical
math backgrounds (Kersten graduated from Mansfield a year ahead of
me and has been teaching in Alaska since then) we were able to
contribute evenly to the math content. Additionally, I was able to
mold her raw ideas into activities that fostered information
literacy and provide my own ideas for authentic and motivating
assessment ideas. At the close of our conversation we divided the
tasks needed to complete the lesson plan and planned for how we
would share the different lesson parts with each other.
Final Product: The end product of the individual lesson will be
a completed and justified construction of parallel lines using only
a straight edge and a compass on a piece of colored computer
paper.
The final product of the unit will be an infographic explaining
how to solve a given geometric proof. This graphic will includes a
well thought out paragraph of why knowing how to solve the proof
using construction and reasoning is important to their lives.
Students will also create short presentation of their proofs to the
class utilizing iPad software that allows them to project their
infographic onto the class SmartBoard.
Library Lesson(s): Librarian contributions will be in blue while
the classroom teacher will be in black.
The class will begin with students breaking into groups of 4 to
read Euclid’s five postulates from their primary source, The
Elements. The reading will be prefaced by an explanation that The
Elements was written in 300 B.C. and is the second most translated
and circulated book, second only to the Bible. Students will be
given time to read through the five postulates and discuss possible
meaning. During this time the librarian and classroom teacher will
circulate class prompting discussion. At the conclusion of this
discussion students will read through page 167 of The Handy Math
Answers Book, which contains the postulates in plain English as
well as information about the postulates and their author. The
librarian will stress in this discussion the importance of viewing
primary sources and immersing ourselves in ‘math-speak’ so that we
can better understand the presentation of math topics.
Next students will dive into the mathematics of the lesson by
setting a basis for the geometry by discussing concrete definitions
of the math terms involved, which echoes the layout of The
Elements. This will be done with the ‘My Mathematical Dictionary’
Handout. When presenting terms the teacher will discuss how the
vocabulary is used in common English and how that relates to the
mathematical definition. Additions will be made to the list of
terms throughout the class as terms are introduced. Next, students
will be given an introductory demonstration (no more than 5
minutes) to WolframAlpha.com, a reference site for mathematical and
scientific information, so that they can find information about
terms throughout the class.
Starting with the 1st postulate the teacher will visually
demonstrate what each postulate means for the students. As the
teacher demonstrates and discusses the postulates students should
follow along on the other side of the handout, ‘Euclid’s Five’.
Similarly, each postulate will be reviewed and discussed as the
class progresses. The demonstration of the postulates will be
completed using software that allows the teacher to wirelessly
project the flipchart she is manipulating on her iPad onto the
class SmartBoard. This allows all the students to see exactly what
the teacher is doing as she explains it.
After going through each of the postulates the teacher will
introduce the idea of mathematical constructions using a straight
edge and a protractor. Students will be given a protractor and
straightedge and explicitly taught to use both. They will also be
given time and a blank piece of paper to get used to using each.
The librarian will explain to students that much of the early
constructions were mathematical games that challenged the critical
thinking abilities of the players. The rules of the game? Euclid’s
postulates. Also, the relation of mathematical constructions to
Kuhlthau’s ISP will be shown. The librarian will explain that in
constructions first we explore (Stages 3), then we form a plan
(Stage 4), engage in some more guided exploring (Stage 5), tweak
our plan as we go, and eventually create the finished construction
(Stage 6).
Students will then be given a novice level construction (a
circle through a point with a given center for example) to
construct. During the time that students complete this task both
professionals will circulate the room providing encouragement and
leading questions to struggling students. After reviewing how this
construction is made students will be given a more complex task,
such as constructing perpendicular lines. The teacher will
demonstrate the first steps of the process before letting students
work on their own. Again the two professionals will encourage and
scaffold as necessary. Also, they will ask students for their
questions and emphasis that the great mathematicians had to sort
through similar thought processes when they were exploring the
subject. During this time students will be allowed to collaborate
with their peers to generate ideas on how to complete the
construction. Since the whole idea of construction in the manner is
not a simple concept this part of the lesson will necessitate lots
of encouragement, guidance, and time getting used to the materials.
Emphasize to students the importance of critical thinking and how
exercises such as this will help them outside of the classroom in
finding solutions to complex problems (such as current societal
issues). The construction process will be paralleled to the
uncertainty, apprehension and success felt in the first three
stages of Kuhlthau’s ISP.
The lesson will end with students being given the task to
construct two parallel lines in this manner, notating their
reasoning and justification for each step of the construction, and
ample time in which to complete the task.
Objectives:
Goal: Students will understand fundamental basics of geometry
and will be able to build upon that basis to expand mathematical
content knowledge.
Objectives:
· Students will read and verbally interpret primary texts,
specifically Euclid’s five postulates.
· Students will familiarize themselves with mathematical
terminology in order to take a step towards math literacy.
· Students will transfer knowledge of Euclid’s postulates to the
construction of parallel lines.
COLLABORATION CONTINUUM:
Product—At the end of the lesson students will be given a task
to complete, as outlined above in the lesson procedures.
Process—Informal progress checks will occur throughout the
lesson. After each class period students will complete a 3-2-1 Exit
Ticket explaining 3 things they understand well, 2 things they want
to review, and 1 thing they do not understand at all. Additionally,
the teacher and librarian will compare notes after each class and
express concerns from informal observational assessment.
Student Self-questioning—Students will be prompted to keep track
of highlights of their internal dialogue in their math notebooks as
they explore and construct. Questions may include:
· How would I explain this ‘math-speak’ to someone without a
mathematical background?
· What would happen if I used the protractor in this way?
· How can I use intersecting circles to help me with this
construction?
Instructional Plan:
Resources students will use:
___Online subscription database(s)
_x_Websites—www.wolframalpha.com
_x_Books—The Elements, The Handy Math Answers
Book
___Reference
___Non-print
___Periodical/newspapers
___Others/list
Handouts attached below. Full works cited is also included
below.
Since the lesson calls for the teacher to demonstrate the
constructions as she goes, a blank flip chart on the SmartBoard
will be used in place of a prepared PowerPoint or Prezi.
Motivation, anticipatory set:
Investigating Euclid’s The Elements and discussing how
monumental the book was even though it’s concepts can be understood
by the average high school student.
Inclusion, adaptations:
Common practice in Kersten’s classroom is paring students
according to ability level. She puts students together who have
similar, but not the same, ability level.
Additionally, adaptations will be given in the form of guided
notes and worksheets with a tighter scaffold. The teacher and
librarian will also provide individual instruction during work time
as needed, including prompted checking of work and asking for
student’s justification of steps taken.
Instructional Activities: Additional details for activities are
in the lesson section.
_x_ Direct Instruction: Students will receive direct vocabulary
instruction, as well as direct mathematical instruction regarding
Euclid’s five postulates. The use of WolframAlpha will also be
directly instructed.
_x_ Modeling and Guided practice: This will be done with the
handouts (My Mathematical Dictionary and Euclid’s Five).
Additionally, when crafting constructions students will have steps
modeled to them. Students will also have the opportunity for guided
practice as the librarian and teacher circulate.
_x_ Independent practice: Much of this will occur as students
get used to using a straightedge and protractor. Students will also
have the choice to engage in independent practice when they are
working on constructions, however the opportunity for guidance will
remain open to them as necessary.
_x_ Sharing and reflecting: This will occur with the reading of
The Elements as well as the excerpt from The Handy Math Answers
Book. Additionally, as students construct and collaborate they will
be reflecting on what they are doing, what works, and what doesn’t
as well as sharing those reflections with classmates.
Assessment Reflection:
The final assessment of this lesson, the creation of a geometric
construction, is appropriate because it directly addresses the
objectives and goal of this lesson. Also, it activates students’
higher order thinking skills (HOTS) since they are asked to create
instead of simply remembering facts. Additionally, by having
students include explanations and justifications the teacher can
get a better take on how the student approaches a problem and
better address errors in thinking. This also gives the librarian
the opportunity to engage in second tier observation while the
students work so they can gauge how much student learning actually
took place (Kuhlthau, Caspari, and Maniotes 115).
The less formal daily checks using Exit Slips, math notebooks
(which will serve as logs) and observations are appropriate and
necessary forms of assessment because they give the teacher and
librarian an idea of what parts of the lesson may be causing
confusion and which parts were generally understood. Additionally
the logs and exit slips, used in conjunction with other assessment
tools, add valuable insight into the student thought processes
(Harada, and Yoshina 46). By understanding how the student is
processing class information and what questions they are asking
themselves teachers can better gear their lessons toward meeting
student needs. Lastly, student observation during the lesson is
absolutely necessary because it enables the teacher or librarian to
distinguish which students may be in need of a little
encouragement, gentle nudge of guidance, or extra help (Kuhlthau,
Caspari, and Maniotes 116-117). This helps student avoid times of
extreme frustration during class that could hinder their
comprehension of material and completion of the task.
Excerpted from Standards for the 21st-Century Learner in Action
by the American Association of School Librarians, a division of the
American Library Association, copyright © 2009 American Library
Association. Used with permission.
Works Cited
Barnes-Svarney, Patricia, and Thomas E. Svarney. The Handy Math
Answers Book. Detroit: Visible Ink, 2006. Print. Your Smart
Reference.
Euclid., and Thomas Little Heath. The Thirteen Books of Euclid's
Elements. Lawrence: Digireads.com, 2010. Print.
Harada, Violet H., and Joan M. Yoshina. Assessing for Learning:
Librarians and Teachers as Partners. 2nd ed. Santa Barbara:
Libraries Unlimited, 2010. Print.
Kahlthau, Carol. "Kuhlthau's Model of the Stages of the
Information Process." At Your Fingertips. Humboldt State University
Library, n.d. Web. 27 Oct. 2012. .
Kuhlthau, Carol Collier, Ann K. Caspari, and Leslie K. Maniotes.
Guided Inquiry: Learning in the 21st Century. Westport: Libraries
Unlimited, 2007. Print.
WolframAlpha. N.p.: Wolfram Research Company, n.d. Web. 27 Oct.
2012.
Except from The Handy Math Answer’s Book by Patricia
Barnes-Svarney and Thomas E. Svarney:
My Mathematical Dictionary
Taking Steps toward Math Literacy
Please use the chart below as the beginnings of your geometry
dictionary. Use the personal note space to jot down whatever will
help you remember what the term means or anything you find
particularly relevant about the term. If you get stuck refer to
www.wolframalpha.com for definitions of explanations.
Term
Definition
Visual Representation
Personal Note
(ex) Point
a specific position on a graph
Isn’t just the dot on the graph, it’s the place that dot really
means
Euclid’s Five
Use the following table to really sort through what each of
these postulates means. To do so, write down the official
mathematical postulate in the ‘He Says’ column. Next, translate all
of that math-speak into your own words in the ‘I Say’ column.
Lastly, in the last column, draw out a visual representation of the
postulate.
He Says:
I Say:
Looks Like:
1)
2)
3)
4)
5)