The+Lesson+Plan

=The Dot and the Line= a romance in lower mathematics by Norton Juster

and a lesson in High School Geometry by Christopher Nelson-Burger

= = A note about this wikispace: the parts in black type are the lesson plan itself, the parts in red are meant to be read directly to the class and the parts in blue are explanation of the rationale behind the lesson plan. Vocabulary that is written in //italics// is from a previous lesson while new vocabulary for this particular lesson is written in **bold**.

=Common Core Standard Addressed:= 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.

=Goal:= The student will be able to define angle, circle, perpendicular line and line segment and use these concepts to create other ideas of geometry. =Vocabulary:=

//*The following vocabulary will be introduced in the form of the teaching drawing diagrams during the lesson introduction://

//point// //line// //plane//
 * area**
 * define / definition**
 * undefined**
 * imagine**
 * triangle**
 * square**
 * rectangle**
 * circle**

//These words will be introduced in a similar way throughout the lesson://

bend line segment ray angle vertex sides perpendicular lines parallel lines skew lines radius center
 * //intersection//**

=Materials:= measuring tape(s) protractor(s) compass(es) map(s) butcher paper string elements in classroom (the lesson will involve students leading string around chairs and desks to get ideas about distances, line segments, angles, and rays. geoboard pencils thumb tacks

=Anticipatory Set / Lesson introduction:= Read the following prompt to the class. **Vocabulary is Boldfaced as it is first introduced** :

First I want you to meet three very important geometric friends. These three friends are very helpful, and also very special. These three are the **point** (draw a point on a map or a pushpin, "you are here", have students draw points), the **line** (connect the points on the map) , and a **plane** (indicate the physical map itself, showing with gestures its having characteristics of being flat, thin, and expansive in two dimensions). A is the smallest possible dot you can imagine. A line goes through two dots or points. It connects them. And a plain is flat -- it has **area**, like a wall, a door, the floor, or the top of your desk (these can be gestured to).

Now you might ask, why are these so special? What makes them helpful to us? They are special because they are everywhere, and make up everything. Because they are everywhere, and because they are so simple, they have no ** definition **. Class, what is a definition?

Allow time for a response -- the discussion should include encouraging the class come up with a working defining the word //definition//, which can certainly involve translating the word to students' L1(s). A further prompt might be:

What does it mean to be **defined**?

You might say something like, defining or making a definition is what we are doing now -- it's stating the meaning or the qualities of a thing. It's saying, what makes up the thing? What is it like? What does it do?

Allow response time.

What does it mean to have //no// definition? Why do you think the point, the line, and the plane are //undefined//?

Allow response time. Students should come up with a //what// and a //why// to explain the concept of these elements being undefined. Acknowledge and summarize with:

There is no way to explain them -- they just //are.// We cannot make them any more simple. The world could not go without them. That is why we called these **undefined.**

We are going to get to know these three three undefined things better. We are going to **imagine** them - see them in our minds, think of them.

Take a trip with me to a place where shapes like **triangles**, **squares**, **rectangles** and **circles** live and breathe and interact with each other. Draw the shapes on the board as you name them. Close your eyes. Imagine you are a line. Imagine what your life would be like if you knew you were a line. What would your day to day life be like? What other shapes would live in your world? Which shapes would be your friends and what would you do with them?

Now have a discussion with the class where students talk about the worlds that they created. Specifically talk about what their ideas share in common and what differences there are. Ask students the following questions and draw on the following ideas to shape the discussion:
 * Did the shapes move around in your world or did they sit still?
 * Did they walk or talk?
 * If all of these shapes are flat, what kind of world would they live in? Lead students to the idea that the world is two-dimensional because shapes do not have height.
 * Once students have a sense of this, reiterate the concept of a plane: In geometry this world would be described as a plane.

This can be a group activity if you choose where students first talk about the worlds they envisioned during the guided imagery activity in smaller groups before the entire class discusses what they thought of. Be mindful when picking groups to include a diverse set of students into each group, including or even appointing translators for each group so that students may use L1 as a reference. Either way, be sure to express to the students there is no right or wrong answer and that each opinion matters.


 * Rationale: ** This is known as Guided Imagery and is one of the techniques discussed in the Vacca, Vacca & Mraz textbook Content Area Reading (2011) and it allows students to explore a concept visually. The lesson also begins with this activity because it creates an affective filter where there is no right or wrong answer. The point of the lesson is to teach definitions of basic geometry shapes and the imagery created in this activity incorporates the shapes. This provides visuals for L2 learners that help provide context for the newly introduced vocabulary, and these visuals will later be applied as physical manipulatives. In addition, the use of a map ties the lesson in with a context for geographical movement, hopefully engaging students through personal experience. Finally, this process asks the students to draw on their imagination to create a world. The optional group activity allows for collaboration among the students. By ensuring diverse students are in each group they will bring with them their own prior experiences which will hopefully lead to a difference of opinions on the world they created.

=Guided Practice:=

Introduce //The Dot & the Line// to students:

We're going to watch a video based on this book to get some more ideas about how our undefined friends work and act. There is some hard language in the video -- don't worry if you don't understand all the words, because we will go over it. It's more important to get the feeling of how the different shapes act. Our goal is not knowing 100% of the words you hear - just getting an idea of the story.

Have the following list of words form //The Dot & the Line// displayed for your students. Read each word aloud and check for understanding of each word in the list as they are read. If any student does not know what a word means, ask the class if anyone does and have that student give their definition of the word. The words have been divided into tiers based on Beck, I., & Margaret G. McKeown, and Linda Kucan (2002). During lesson planning, go through it to gage students prior knowledge and affective filter to determine how many of each tier of words are necessary and/or required for the task. Ask students if they know the meaning of the words and define as you go:

First, I'm going to go through this list of words with all of you, and we will discuss them and make sure you know what they all mean. Imagine each word in your head. Take notes on what you see for each word.
 * romance -- tier II
 * sensible straight line - tier III
 * love -- tier I
 * dot --- tier I
 * squiggle --- tier II
 * frolicking -- tier III
 * conventional - tier II
 * miserable - tier II
 * depth --- tier II
 * perfect - tier I
 * dreaming -- tier I
 * daredevil -- tier III
 * leader -- tier I
 * agent --- tier II
 * force tier II
 * sportsman tier III
 * angle --- *tier II / III
 * shapes - tier I
 * confidence tier II
 * giggled - tier II
 * happily - tier I

Now take some time to reflect on the notes you took and then write a story using the words from the list. The words in your story must appear in the same order as on the list. Walk around the room and check with each of the pairs of students to ensure that each of them understand

Now pass out the handout called The Characters that can be found as another page in this wikispace:

http://thedotandtheline.wikispaces.com/The+Characters

Have students watch video: http://thedotandtheline.wikispaces.com/Youtube+Video

During or after the viewing, ask students comprehension questions: What do we mean here by romance? Who is chasing whom? Why is the line in love with the dot? Why is the dot in love with the squiggle? How does the line change her mind? What is a line able to do that a squiggle cannot do?

Share this story with a partner and take note of what your stories share in common. Give enough time for the students to discuss their stories. Allow for a larger group discussion if there is time for the different groups to report out what their stories shared in common.

**Rationale:** This activity is known as story impressions as described by Vacca, Vacca & Mraz (2011). Beginning with a list of words and checking for understanding of each word is a way to front-load the vocabulary the students will run into in the story //The Dot and the Line//. This step is particularly important for an English Language Learner in the class. This particular strategy also draws on the students' imagination to connect the words in the list to a bigger picture which scaffolds to the story nicely. Students inevitably encounter challanging vocabulary, so it is important to stress that the goal of the exercise is certainly not 100% listening comprehension, but rather a general sense of the principles embodied in the shapes. By the time is this all accomplished, they haven't even read a word of //The Dot and the Line// or learned anything regarding the lesson goal yet and they are already reaching conclusions of their own. By asking students to discuss what aspects their stories share in common, they will have to share their ideas through dialogue. Collaboration is an important technique to use in the classroom. Because of the nature of the list of words, students will have to use the words in the discussion.

After the story, pass out the Lesson Guide included as another page of this wikispace: http://thedotandtheline.wikispaces.com/Anticipation+Guide

Go over the anticipation guide with the class. **Phrase this as tasks to perform with manipulative:**

On your handout are eight statements about geometry that use our undefined friends. We are going to go through these statements one by one as a class, and prove to ourselves which ones are true and which are false. You wil mark this on your paper as we go, and keep track of the reasons, or the ways we used to prove these statements.

>> What do perpendicular lines look like? Have we created any in our proofs so far? What are some examples in the classroom? Draw perpendicular lines. Students might indicate corners of desks, the chalkboard, etc. We know that lines that meet, or **intersect,** form angles. Let's see what kind of angles are formed from perpendicular lines. So take out your **protractor** (indicate what a protractor is), which measures angle degrees. Let's all measure a the angle formed by two perpendicular lines -- so find one to measure, these are everywhere. Review the concept of//intersection//. Two lines are **perpendicular** if their intersection forms a 90 degree angle. ** Students measure and come to the conclusion that perpendicular lines meet at a 90 degree angle. ** >
 * 1) Geometry has three undefined terms: the point, the line and the plane.  This statement is true, as discussed in the anticipatory set. There is no real proof required here. ** Students will briefly discuss to review the discussion from the beginning of class. **
 * 2) The three undefined terms are //not// used to build other shapes. This statement is false. The three undefined terms are building blocks of geometry. **Students will use string, thumbtacks or elements in the classroom to prove that a line can bend or curve.**
 * 3) Angles are made from lines.  This statement is true. The Line did this in the story. He "bent" himself into angle. Students can show this with the above proof.
 * 4) Actually, the kind of line that starts at an angle and goes outward forever is called a **ray**, just like the rays of the sun.  A ** ray ** is a part of **segment** of a line that continues in one direction only. It has a start point but no endpoint. Draw examples of several rays. Discuss with students how to name a ray in geometry.
 * 5) Two rays that share one common endpoint form an **angle**. The endpoint is called the **vertex** and the two rays are called the **sides**. Discuss with students how to name the vertex, the sides and the angle itself. Show students how to measure an angle using a protractor. Pass out a worksheet with different types of angles and have small groups work together to find the measures of the angles on the worksheet with a protractor.
 * 6) A circle is all of points that are all the same distance from a center point.  This statement is true. The way the author draws the picture of the dot, she is actually a circle and not a true dot (or // point // ) as we would talk about in Geometry. ** Students can be guided through attaching a string to a thumbtack or a classroom object and using it as a fulcrum to prove the relationship between the origin and the radius of a circle. Also, one student might hold thestring and another might walk around him/her. **
 * 7) How many of you have a dog?
 * 8) Have you ever walked the dog on the leash?
 * 9) What if you stood still and allowed the dog to walk out the entire length of the leash and then walk around you?What kind of shape would the dog be walking along?
 * 10) A circle is the set of all points that are the same distance (**radius**) from a given point (**center**). In the example of the dog, you would be the center of the circle and the leash would be the radius. Demonstrate how to draw a circle using a compass. Discuss how to name the center, the radius and the circle itself.
 * 11) Two lines are **perpendicular** if they **intersect** at an angle of 45 **degrees**. This statement is false. The lesson plan will address this concept later. Let's define some of these words.
 * 1) **Parallel** lines never intersect. This statement is true as long as the two lines are in the same plane. The lesson plan will address this concept later. What do parallel lines look like? What does it mean to intersect?  **Students will describe in words. **
 * 2) Actually, there is one more important piece to this one. Two lines are **parallel** if they //lie on the same plane// and never intersect. Distinguish the difference between parallel and **skew lines**. **Students will create skew lines across the classroom with string to demonstrate this.** Draw a 3-D rectangular prism and show students which pairs of lines are parallel and which ones are skew.
 * 3) A line **segment** is a piece of a line.  This statement is true. It has a beginning point and an end point where a line does not.  Draw an example of a line segment on the board. Discuss with students how a line segment is named in geometry.
 * 4) Distance is sometimes negative.  This statement is false.
 * 5) Have a volunteer student come up to the front of the class and give them a starting point. Then have them walk five feet, backwards to a pre-determined end point. Have a tape measure handy and have a couple more volunteers measure the distance to see that, even though the other student moved backwards, the distance is still positive five feet. The teacher could form small groups where different members of the group walk backwards while other group members measure the distance walked with a measuring tape.
 * 6) **Distance** is how far apart two things are. We use rulers or measuring tape to measure distance. A ruler always starts with zero and then the end of the object corresponds to to a positive number on the ruler, which is its measurement. This is why distance is never negative. Even if you walk five feet backwards, you still walked five feet.
 * 7) Make sure that students understand this concept before moving on.

Rationale: ** By including pictures and activities with each concept, the teacher will be following the SDAIE (Specially Designed Academic Instruction in English) practice of using nonlinquistic representations of concepts. The student will literally see and act out what each and every concept means. Modeling the concept of distance by having a student walk backwards and other students measure the distance addresses another SDAIE practice. It also allows the students to get up and move around and not just sit and listen to lecture for the whole period. Additionally, the teacher does not explain the concept of distance and allows the students to discover it on their own. This type of comprehensible input allows the student to have a hands-on experience. Collaboration is used again when the students measure the angles using the protractor. Parallel lines and skew lines are modeled in the classroom environment.

= Closure: = Tie the entire lesson back to The Dot and the Line. Just as the Line learned to bend himself into other shapes in order to impress the Dot, the point, the line and the plane are used as the building blocks of geometry to define and create new shapes and ideas.

=Independent Practice:= Select questions from the textbook where students have to draw or name the shapes and ideas presented in this lesson plan.

=Extension Project (Optional):= In Edwin A. Abbott's novel, //Flatland//, a two-dimensional world is created where shapes like A. Square live and interact with each other, just like the one in the anticipatory set. Students could read the novel on their own and do a presentation to the class of the concepts explored in the novel.

Have students watch this youtube video: http://thedotandtheline.wikispaces.com/Youtube+Video

=References:=

Beck, I. L., McKeown, Margaret, & Kucan, Linda (2002). //Bringing words to life//: //Robust vocabulary instruction//. New York, NY: Guilford Publications, Inc.

Vacca, R., Vacca, J., Mraz, M. (2011). Activating Prior Knowledge and Interest. In //Content area reading//. (pp. 167- 191). Boston, MA: Pearson Education, Inc.