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)