Inclusive Lesson Planning Templateanguelova.org/InclusiveLessons.doc · Web viewClass Description...

Inclusive Lesson Planning TemplateAdapted from Causton-Theoharis, Theoharis, Tinto, Sellmeyer & Glen, 2007

Section 1 - LESSON CONTEXTA. Grade Level: Ninth grade.B. Class Description (a fiction):

Your class is a group of 24 eight graders. As is typical of this high school, they are diverse along every measure: race, ethnicity, gender, disability, social class, and academics. Fourteen are male, while eleven are female. The students come proportionally from the communities represented in the district. Six are from rural, working class, white schools. Twelve come from schools in middle class neighborhoods surrounding the local global conglomerate, and they include four recent immigrants from Singapore and Mumbay. Six students are from the two urban working class schools, including two teens of refugee families from Darfur (who have had no experience of school for four years). Class and ethnicity do not correlate with each other in this class. Overall, thirteen students are white, five are African American, five are Asian American, and one is German American. Indeed, race and ethnicity are also complex, as the Asian American and African American students all come from several cultures. Of the six students who are English Language Learners, all have fair command of conversational English. However, only two have thorough background vocabulary in your subject area. In terms of disability, three have been identified with IEP’s as having specific disabilities with reading comprehension. They require extended time for assessments. The students from Darfur have not been identified as having emotional disabilities (ED), but the Committee on Special Education has asked for your input on their behaviors, because several colleagues have sought that ED classification. Four students miss your class for one period every other week to participate in Odyssey of the Mind preparation, which is considered a gifted program in your school. (Other strengths, abilities, or identities you are free to include in this profile.)

C. Subject or Discipline(s):

Discipline: MathematicsCourse/Subject: Algebraic GeometryUnit: Thinking Three-Dimensionally - Shapes and Space Lesson Topic/Title: Conic Sections

D. Central Theme, Concept, Problem, or Unit:

The main topic of the unit is solid shapes. The topics of these particular lessons are as follows:

Lesson I: Introduction to conic sections Answering the questions:

What are they? How do we obtain them? What are the four major types and how do we get each of them?

Lesson II: Investigating conic sections in more detail (Connecting the geometry with the algebra)

Adapted from Causton-Theoharis & Theoharis, 2006, p.1

Introduce the general equation for a conic section.Explore it in the case of a

Circle Ellipse Hyperbola Parabola

Establish patterns and relationships between the different types of conic sections

I selected this unit topic because it is an area of study that often gives students trouble. At the same time it is an interesting topic that is applicable in many fields. It holds the potential to be quite tangible if it is taught well.

E. Instructors (another fiction): Ms. Holy Red is my special education collaborator. She helps co-teaches this period with

me and is helping me design and adapt activities that better meet the needs of Armani Pierre, one of my exceptional learners in the class.

Having been shadowing Armani throughout his high school career, she is very familiar with his strengths and weaknesses. In this thirty-minute planning session we discussed the activities involved in the lesson.

F. Duration of the Lesson:Two 45-minute periods.

G. Background of the Lesson: The two lessons follow immediately after the lessons on pyramids, and on prisms. They precede the unit overview of all the studied three-dimensional shapes.The notion of conic sections is supposed to take several days to teach. The instructors

have chosen explorative activities for students and realize that although they take longer to do, they typically build more long-lasting and meaningful concepts and connections for the learners. The instructors seek to build a more thorough and deeper understanding in her learners.

Embracing the constructivists’ philosophy of teaching, instructors are willing to devote the extra time.

H. Student Background Knowledge:Students are familiar with geometric shapes and the algebraic equations of lines, circles,

and parabolas. They have also briefly encountered hyperbolas. They have just finished working with the basic solids (pyramids, prisms and cylinders),

so they should know their attributes and formulas for volume. They are also able to reason the formulas for different surface areas. They all know how to manipulate symbolic expression and equations.

Students are also familiar with geometric shapes and the algebraic equations of lines, circles, and parabolas. However students have not seen those shapes in relationship to the double cone.

Ms. Red and I will assess students’ prior knowledge about familiar solid shapes and use it to assess students’ progress. If the majority of students successfully complete the task we will use it as a convenient way to remind the rest of the class about the concept and connect it to the new lesson.


However in the event that the majority of the class fails to complete the hook activity successfully, I will use the concept attainment model to teach the concept of prisms.

I. Target Students:

Student Profiles Brittni Howard (BH) is a ninth-grader who attends school part-time. She attends some of

her classes as infrequently as once a week She was raised in a mixed family1 in New York City. Brittni seems to learn best by reading, taking notes, and listening to lectures. She typically first concentrates on the material and then energetically takes part in the discussions. She likes to be challenged but she knows math is not her forte. She is intelligent and determined but struggles with symbolic expressions and their manipulation. She does well in a collaboratively learning activities and project based mathematics settings that utilizes more open-ended problems and both vocal and written assessment of knowledge. Based on Howard Gardner’s multiple intelligence theory, Brittni possesses strong linguistic and intrapersonal intelligence. Her written and spoken words were very expressive. She had accurate views of herself and struck me as being very self-reflective and aware of her feelings and direction.

Ron Johnson (RJ) is also a ninth-grader and a prominent school athlete. He is very intuitive when it comes to mathematics. He can quickly decipher symbolic formulas and is very adept at their symbolic manipulation. Based on Howard Gardner’s multiple intelligence theory, Ron possesses strong Logical/Mathematical, interpersonal and bodily kinesthetic intelligence. However, he just transferred from a very poor school and has not had much prior exposure to using technology in a mathematics classroom. He needs support using calculators and technology in the classroom..

Armani Pierre (AP) is a seventh grader. He is a twice-exceptional learner. He is younger than the rest of the class, but is gifted when it comes to mathematics. However, he is an ELL student and has trouble communicating with people and putting his ideas into writing. He is also on the introvert side and likes to work alone. He prefers individual learning activities although it is beneficial for him to work with native speakers and improve his language skills. He possesses a strong spatial and naturalistic intelligence and his favorite activities are playing computer and video games. He is adept at using computer software and graphical representations.

1 Half of her family is Hispanic and the other half is African American.


Section 2 - LESSON CONTENT

A. Lesson Standards: What grade level specific national and state standard(s) are being addressed?

Communication Organize and consolidate their thinking through communication; Communicate their thinking coherently and clearly to peers, teachers and others; Analyze and evaluate the thinking of others; Use appropriate language/terminology to express their ideas precisely.Representation Create and use representation to organize, record, and communicate mathematical

ideas.Geometry Use various representations to help understand the effects of simple transformations

and their compositions; Analyze characteristics and properties of three-dimensional geometric shapes and

develop mathematical arguments about geometric relationships; Specify locations and describe spatial relationships using coordinate geometry and

other representational systems; Apply transformations and use symmetry to analyze mathematical situations.Connections Understand how mathematical ideas interconnect and build on one another to produce

a coherent whole (underlying goal for the whole unit); Recognize and use connections among mathematical ideas.Algebra Standard Represent and analyze mathematical situations and structures using algebraic symbols Analyze change in various contexts Explore relationships between symbolic expressions and graphs, paying particular

attention to the meaning of intercept and slope Represent, analyze, and generalize a variety of patterns with tables, graphs, words,

and, when possible, symbolic rules

B. Lesson Goal: The mathematical goals are for students to find all possible cross sections of a cone and to

differentiate between the different conic sections by looking at their graphs as well as the position of the intersecting plane relative to the double cone being intersected.

C. Content Differentiation:

All students are already familiar with the notion of the double cone. Since they all have been exposed to different solid shapes (cylinders, prisms, pyramids, spheres, etc.) and they know the cone in terms of its volume and surface area this is a safe assumption to make. Just in case I will do a quick check for prior learning. This will also help connect the new topic with the topics learned in previous years and better sequence the new lesson.


Students are also familiar with geometric shapes and the algebraic equations of lines, circles, parabolas, and have briefly encountered hyperbolas. However students are not familiar with those shapes in relationship to the double cone. But they know how to manipulate symbolic expression and equations.

To challenge students who will reach the goals of the lesson too early into it, I provide a number of enrichment activities that combine technology with analytical thinking and algebraic skills. They will build on the knowledge of conic sections and nicely connect the geometry with the algebra behind it.

To access those activities students will need to demonstrate to me knowledge of all types of conic sections.

Although most students have had prior experience using the computer to run simulations, it will help all learners to refresh those skills. So I will model the use of the simulations before students are asked to run them. Students will also be placed in small groups of 2 or 3 to run the simulations. This way more experienced students can help less experienced ones.

D. Whole-Class, Multi-Level Lesson Objectives:

All students will be able to: See what happens if a cone is intersected with a plane in various places. Students can

do that using their own individual manipulative model of a cone made out of paper. In exploring the possible cross sections students will notice that various geometric

shapes that come to life (point, line, circle, ellipse, parabola and hyperbola). Each student has to have at least three different cross sections listed.

The mathematical goals is for students to find all types of cross sections of a cone in their collaborative groups of four heterogeneous students (two average ability students, one high ability student and one low-ability student)

Differentiate between the different conic sections graphs by looking at their graphs or the position of the intersecting plane relative to the double cone being intersected

State and apply the general equation for a conic section on at least three of the five exercises given for independent practice.

List the main attributes in the equations for ellipse, hyperbola, parabola, and circle in their notes (no need to memorize these, but have the vocabulary down in their notes for future reference when we do polar coordinate representations).

Use what is seen geometrically to find the equation of a conic. (Complete at least three of the five exercises given for independent practice).

Use a particular equation of a conic to distinguish and predict between the possible geometric types of outcomes. Students are expected to have at least three instances listed in the handout during their work with the simulation. Those will be collected and returned to students before next class (or in the beginning of next class if students do not come to the afternoon homework session hour).


E. Student Specific Objectives (Must have):BH (in addition to whole class objectives):

Revisit/learn prior knowledge of prism. Accomplished during the first whole class activity in which students are supposed to sort themselves into the groups of prism holders and non-prism holders.Further reinforced via the BH’s specific group discussion and then the whole class discussion which follows

Through think-pair-share activity and whole-class discussion and handout reflection questions connect the past material with the new one to reinforce understanding

RJ (in addition to whole class objectives): Become more comfortable with technology through the class demonstration of the

animation, and the presentation of the technology poster. Become exposed to using computer simulations to explore mathematical conjectures

and patterns via the collaborative work with AP. Conjecture a possible relationship of the parameters that transitions between the

different types of conics. Think-pair-share and handout entries during the collaborative work with AP.

Generalize the effects of changing the constants or coefficients in the type of conic section studied. Brainstorming entries in the handout, as well as teacher overheard input in the collaborative team between RJ and AP.

Explore the effect of the parameters in the general equation for a conic section on the type of conic section they obtain. Brainstorming entries in the handout, as well as teacher overheard input in the collaborative team between RJ and AP.

AP (in addition to whole class objectives): After working independently encourage through the think-pair-write-share activity to

work with a fellow student Serve as a peer tutor in technology while working on his own communication skills. Explore the effect of each of the attributes in the equations for ellipse, hyperbola,

parabola, and circle on the shape of the conic section via the work with the simulations. Since all students are encouraged the list their entries of the cases they tried and their relevant observations on the handout, this is easy to assess.

Explore the effect of the parameters in the general equation for a conic section on the type of conic section they obtain. Brainstorming entries in the handout, as well as teacher seen initiative with the simulation and overheard input in the collaborative team between RJ and AP.


F. Definitions of Targeted TermsPrisms are solid shapes, which have a pair of opposite faces called bases that are polygons with the same size and shape and all their other faces are rectangles.

A conic section (or conic for short) is a curve that is formed by intersecting a cone (more precisely, a right circular conical surface) with a plane.

The four major types of conic sections are parabola, hyperbola, ellipse and circle

The degenerate cases are line, two lines, and a point.

The general equation for any conic section is

Ellipse Circle Hyperbola Parabola(x-h)2=4c(y-k)

or (y-k)2=4c(x-h)

For some students


Ax2+Bxy+Cy2+Dx+Ey+F=0

If B2-4AC>0, then conic section is a hyperbola

If B2-4AC=0, then the conic section is a parabola

If B2-4AC<0, then the conic section is an ellipse. (For B=0 and A=C the ellipse becomes a circle)

Section 3 - LESSON PRODUCTHow do the students demonstrate their learning? Tie these specifically to the lesson objectives.

A. Product Differentiation:In what varied ways can students demonstrate their learning?Use the following matrix to brainstorm different student products using each of the intelligences. Use the list you generate to help determine which way(s) students will demonstrate learning.

Intelligence Possible ProductsLinguistic Oral or written responses generated in a discussion or written answers in the form

of reflectionStudents are given handouts with ample empty spaces to write their ideas and reflections.Through the think-par-share activity in groups and the whole class prompts to share ideas, students are facilitated to show what they have learned using their linguistic intelligence.

Logical/Mathematical Oral or written conjectures and arguments backed up by logical deductive reasoningStudents are encouraged to write their conjectures in the handout. They are provided with rich simulations that can potentially explore all the types and properties of conics. At the same time students are provided with a structured table that will help them better organize their thoughts and generate the cases and properties readily. Reading the number of types, properties and conjectures they have listed in the handout spaces and paying attention to their think-par-share contributions are good way to assess students’ learning.

Spatial Visual representations of the learned through diagrams of different types of conic sections, and paying attention to the ways class simulations are utilized is a good way to assess students with spatial strengths.The poster representation and animation although helpful to all students will be particularly good choices for spatial savvy students.

Bodily Kinesthetic Being physically and actively involved in the learning activities/acting out the conic sections is a good way for students to demonstrate the major types of conic sections

Musical Students are invited to compose a song about conic sections. Their learning can be assessed based on the accuracy and the number of key notions they incorporate in their lyrics.

Interpersonal Clear demonstration of knowledge in the think-pair-share and whole class discussion. Participation and interaction with peers will be monitored

Intrapersonal Written reflection about own processes while constructing the knowledge of conic sections

Naturalistic Using carrots in the shapes of cones and biting or cutting them2 to demonstrate the different conic sections. Enough models will be provided so that students can demonstrate all the major types they can find.

B. Authentic Assessment: The initial activity will be only informally graded. All students will participate in it and they

will all get a chance explain their thinking to their peers. I will use it as an opportunity to assess students’ progress and knowledge and take course of action accordingly. I will also use the time to remind students the properties they have forgotten and enforce the proper use of mathematical

2 Even though no sharp knives are given to students, individual students who use this type of exploration are closely monitored for the proper use of the utensils.


language. This is the reason I will pay particular attention to the students’ individual input and engagement in the activity.

However, students will be graded individually on the homework assignment after this lesson. They are allowed to work together on this homework project, but are asked to list the names of their collaborators and turn in their own solutions to it.

Students will also be expected to work on and complete the practice sheet provided in the lesson and turn it in to me in the end of the unit. They will then be formally assessed using a unit test that incorporates questions from the quiz assessment provided as long as problems from other parts of the unit.

Section 5 - LESSON PROCESSHow will you share information? How will the students engage in the learning?

A. Process Differentiation: How can I teach in ways that will reach all students?

Use the following matrix to brainstorm different processes of teaching using each of the intelligences. Use the list you generate to help determine which way(s) you plan to teach the lesson.

Intelligence Possible ProcessesLinguistic BH think-pair-write share and whole-class discussion as well as written account of

explorations on the handoutLogical/Mathematical RJ simulation and handoutSpatial AP unscramble poster, use simulations and animationBodily Kinesthetic RJ using manipulatives (Birthday hat and cutting it with scissors)Musical Song about conic sectionsInterpersonal RJ think-pair-write-share and whole-class discussionIntrapersonal BH reflection on handoutNaturalistic AP using carrots to demonstrate the cross sections. The circle one is particularly

easy to see.

B. Lesson Formats:All 24 students will participate in the first activity. After settling in groups based on the

object they are given, they will engage in peer dialogue and make small group presentations on the attributes they found.

After that students will be engaged in and exploration while experimenting with cutting cone models. Students will engage in collaborative work in small groups. After that they will be shown the animation and based on it and their own results in the exploration as a whole-class guided activity they will unscramble the poster. Teacher will instruct students on the attributes of each shape and model for them the use of the simulation. After that students will engage in experimentation trying to discover what the different attributes mean and how do we move between shapes.

C. Room Arrangement: Students will sit on longer tables with two computers on each table. This way when

students break in partnerships of two each group will have a computer to work on and enough space to write in the shared handout.

The lights need to be adjusted appropriately when the projector is in use they need to be dimmed, so that all students can see the simulations and animations, but in the collaborative activities in which there is writing involved, they need to be kept on, so that no student becomes sleepy.


D. Student Arrangement:Students will be arranged in different ways throughout the lesson (look at the descriptions

for each activity in the lesson). They will be given the chance to work with partners, as well as share ideas in larger groups and small collaborative groups of four. Between the activities the teacher will address the whole class, and will be given the chance to engage in whole-class guided discussions.

Computer access will be blocked before it becomes time to use the individual computers in the simulations. This way the teacher can prevent distractions (students wanting to check their emails, etc) while the class is supposed to be engaged in something else.

E. General Teaching Strategies: I will provide students with visual images and manipulatives as well as several

simulations. I will also use open-ended questions and prompts.I will utilize think-write-pair-share, technology and model for students the use of

simulations and animations.

What is the form of the above equation when we have the degenerate cases - point and line?

What is the form of the above equation when we have the special case - circle? How is the equation for an ellipse similar to that for a circle? How is the equation for a vertical parabola similar to that for a horizontal

parabola? How are they different? How is the equation for a vertical hyperbola similar to that for a horizontal

hyperbola? How are they different? Describe in how each parameter of the equation changes the graph for each of the

cases: Vertical Hyperbola Horizontal Hyperbola Vertical Parabola Horizontal Parabola Circle Ellipse

F. Student Specific Teaching Strategies: BH will be working with classmates during the entire class so that it will be easier for her to

catch up. By working with others she is also utilizing her favorite modality – using language. The guided questions provide for her direction for individual reflection as well.

For RJ and AP and any of the other students who demonstrate the core class knowledge to move on to this investigation:

Play with the coefficients of the general conic equation. Vary A, B, C, D, E, and F and record what happens in the following table

A B C D E F Conic (type)


Try to come up with a relationship that will predict what conic section we have at hand based on the coefficients

G. Systems of support & supervision: While I will guide students in their exploration, Ms. Red will also circulate around the

room and check group work. She also will have to pay particular attention in the later half of the period to RJ and AP working together. She will have to carefully monitor their collaboration throughout the activity and through a distance and only step in if needed.

The peer partnership between RJ and AP benefits both parties. They each learn from each other useful skills. However they will each be given very specific directions to follow and assigned roles with a single handout packet, while working together so that collaboration between the two is heavily enforced

Section 6 - LESSON OUTLINEWhat specifically will you do during your lesson?

A. Behavioral Considerations:

I have prepared many activities for this lesson. Having different tasks in place would easily allow me to tailor the lesson to each student’s needs. At the same time switching the pace of the lesson and the modality of it will keep students engaged while considering their appropriate learning style. At the same some activities are provided to challenge students and at times this will involve supporting students to work in a mode that is not favored by them. Hopefully in the course of the activities students learn to be successful. For instance just because AR is an introvert does not mean that he should always work alone. In fact it would benefit him greatly to work with other people and learn how to communicate with his peers effectively. Working with just one other person who is adept at working with people is a safe place to start. It is possible that students will be reluctant to engage in such activities. But since they can each learn from each other useful skills, the teacher and aid should do everything to make the partnership work. By given very specific directions to follow and assigned roles with a single handout packet, while working together, collaboration between the two will be enforced. Ms. Holy will also use proximity to be able to monitor students and step in if necessary.

B. Introduction (Anticipatory set, The Hook or Launch):

The warm-up activity will serve as a revision of what students already know about certain solid shapes (prisms, pyramids and cylinders). It will consist of each student being given a different solid shape and then asking them to arrange themselves in groups of students in possession of a prism and another group of students holding other shapes. Students will then collectively in groups list the attributes they know about the shape they have in their possession.


C. Body: Lesson 1

Clock Time

Sequence of Steps Questions Adaptations Anticipated Student Response

9:00-9:05

1) Each student is given a different solid shape as they enter the classroom

Arrange yourselves in groups of students in possession of a prism and another group of students holding other shapes

Students are encouraged to help each other make sure that everyone sits in the correct group.

Some students may not know the concept of prism.

9:05-9:15

2) Students will then collectively in groups engage in dialogue and list the attributes they know about the shape they have in their possession.After students are done listing their properties and drawing their examples the whole class will match the two and the teacher will recap the important notions. If something needs to be added to the list that will be addressed.

Use the prism and non-prism objects in your group’s possession to help you list the attributes prisms should have.Please draw a triangular and a rectangular prism.

Prisms are solid shapes, which have a pair of opposite faces called bases that are polygons with the same size and shape and all their other faces are rectangles.

Some students like BH may need to be reminded of the property of prisms, or the convention of naming them. Placing students in groups will help students some students catch up while it would reinforce learning through evoking communication from all students

Prisms have a pair of opposite faces called bases that are polygons with the same size and shape and all their other faces being rectangles.

9:15-9:20

3) Students are introduced to the concept of conic sections and familiarized with the directions for the activity.

What is a conic section?

A conic section (or conic for short) is a curve that is formed by intersecting a cone (more precisely, a right circular conical surface) with a plane.

How does a conic section look? Are they all the same?

The teacher demonstrates how to cut the paper model cone to students and where she looks to list the intersection curve. This is particularly helpful to RJ and AP but beneficial to all students.

Students may have hard time connecting the paper models they see with the conceptual ideal representations, so the teacher will have to demonstrate one cut.

9:20-9:35

4) Students will be given and exploration activity, with guided questions to keep them focused and asked to discuss their results in small groups of four and then collectively as a class..

Who found any familiar shapes (circle, V, single point, etc.)? Please demonstrate how you made these cross sections…Did any of you find other cross-sections? What were they? How did you find them?If you slice parallel to the base what shape do you

Teacher will have students work in collaborative groups while walking around the groups and facilitating the exploration. Each student will have their own model to cut and RJ will be provided with several clay models, which he may

Students may have hard time distinguishing between a branch of the hyperbola and parabola, so the teacher will have to make the distinction clear in the discussion. Showing students the graphic simulation of cutting the cone with a plane and varying the angles will be a good


get? Is it always true?What makes circles larger/smaller, parabolas skinnier/wider, etc.?

find more realistic and easier to visualize what is happening.

way to make sure all students can picture the possible outcomes.

9:35-9:45

5) Together with active student participation teacher will unscramble the conic sections poster in Adobe Photoshop3.

How does a _____ look like?

Is this what we get if we rotate the cutting plane at ____angle?

How do you know? Explain.

This is one activity very suitable for AP, but also helpful to RJ. For the former it is working in a mode they are most comfortable with but for the other it is an exposure to an unfamiliar one. All students will benefit from a little repetition at this point and students can have the printed poster for future reference or have individual handouts made to refresh their memories.

The teacher may find it helpful to play the animation several times during the discussion and pause it so that students can formulate their answers.

9:45-(Outside class)

5) Students will be given access to the scrambled and unscrambled files on Blackboard.

Try to redo what we did in class or use a different representation that makes sense to you to convey he different types of conic sections and how we obtain them exactly.

This is one activity that can give good practice to all students. It is also a good way to encourage students to develop their own representations that better build their schemas.

The teacher will assist if students need help using the Photoshop program outside of class.

3 I have already set up the teacher console with this file open. It is a drag and drop written segments document that can be quickly rearranged in front of students.


Lesson 2Clock Time

Sequence of Steps Questions Adaptations Anticipated Student Response

9:00-9:05

1) Together with active student participation teacher will list the ways we obtain each of the different types of conic sections.

If students have shared some good creative representations, we will show those and modify the timing. It will be good to acknowledge such efforts and creativity by bringing them up in class even if it means extending the lesson till next class.

How does a _____ look like?

Is this what we get if we rotate the cutting plane at ____angle?

How do you know? Explain.

All students will benefit from a little repetition at this point and students can look at their printed posters to refresh their memories and encourage them to take part in the discussion.

The teacher may find it helpful to play the animation again and to pause it as needed so that all students can be visually reminded and thus encouraged to participate.

9:05-9:15

2) Students are reminded of the attributes of each geometric figure (ellipse, circle, parabola, and hyperbola).

What are the important attributes for a circle?Write the equation down and discuss it briefly

What are the important attributes for an ellipse?Write the equation down and discuss it briefly

What are the important attributes for a parabola?Write the equation down and discuss it briefly

What are the important attributes for a hyperbola?Write the equation down and discuss it briefly

It will be helpful for students to jot down notes on the handout, particularly for BH.This will also be good for all students since they will be left with a written reference for subsequent classes.

Students, who are anticipated not to be familiar with this, could be pre-taught.

Students are solicited to provide and construct those answers, but since the more important goal of the lesson is seeing the connection between algebra and the geometry, this particular part of the lesson is done quickly, just to refresh students’ memories.

Students who need a further refresher can be provided with one-on-one help during free periods or office hours.

9:15-9:20 3) Upon explaining

the attributes of the different conic sections students will be shown an example of how to use the simulation so that they are familiarized with it

The teacher will think out loud as they go through the process and explain all the controls of the simulation.Here I have _____I know because of _____When I change_____ it looks like I get ______If I were to list it on the handout I will be writing it

The instructor may decide to show several examples, especially if students do not look comfortable with the technology.

Students will probably ask questions and they will be answered to facilitate their independent explorations with the simulation that follow.


in column ______ with entries ___ ____ ____. Etc.

9:20-9:40

3) Students will be given access to individual computers and the Mathematica Player applet to explore the effect the attributes of each particular conic section have on its shape. While completing this activity students are given the handout with the key questions to keep them focused while experimenting.

What is the form of the above equation when we have the degenerate cases - point and line? What is the form of the above equation when we have the special case - circle? How is the equation for an ellipse similar to that for a circle? How is the equation for a vertical parabola similar to that for a horizontal parabola? How are they different? How is the equation for a vertical hyperbola similar to that for a horizontal hyperbola? How are they different? Describe in how each parameter of the equation changes the graph for each of the cases:

Vertical Hyperbola Horizontal Hyperbola Vertical Parabola Horizontal Parabola Circle Ellipse

RJ and AP would probably be very quick with this activity so they can transition to the next one. They can explore the effect of the coefficients in the general conic section equation on each type of conic they obtain graphically. They can do this activity by means of the free java applet on the website provided and will sue the handout provided to make the activity more structured and guided.

Students are asked to keep in mind the key questions in the first part of the second computer activity and later asked to record some of the parameter values they tested in the table provided. Once they have filled in the table with at least 3 conic of each type, they are asked to discuss their results in small groups.The objective is to come up with conditions for differentiating between the conics

Once students share their results, I can summarize the key results and give the correct conditions to the students. The instructor should be careful to pinpoint the similarity with the condition for the discriminant of a quadratic equation so that students would remember the inequality easier.

AP and RJThis technology activity will allow students to explore the effects of changing the constants in the equation on the graph of the conic sections (circles, ellipses, parabolas, and hyperbolas) while at the same time visually see the transformations. Students will be able to experiment and visualize the effect.

9:40-9:45

4) To close the lesson, the teacher will check how far along with the exploration students have gone

Since probably students would not have completed more than a few rows in the table, it would be nice to get

What do we have so far?Has any group listed something interesting?

Look carefully at your coefficient values and try to find a relationship between them that determines the conic section type

If individual groups are further along in their conjectures, they can be challenged to prove or give arguments as to how they can convince the whole class that their arguments are plausible.

Class will convene the next class time with considering the different conjectures students came up with. Those should generate a nice discussion and a good flow to sequence this class with the next one.


them to start thinking of the possible predictor formulas that could help them quickly identify the shape at hand just by the conic equation

Introduction to the Lesson:

These particular lessons are supposed to build on students’ prior knowledge about solid shapes and expand their knowledge base by adding cones to it.

Students will use experimentation and visualization to explore the possible cross sections of a cone they can find (point, V, circle, line segment, ellipse, one branch of hyperbola, and parabola).

Students will be asked to use their own desk models to do so and to carefully record their results in their notebooks. Later they will be asked to share them with their partners and then to the entire class.

D. Closure (Summary):To close the first lesson the class will talk of the position of the slicing plane and recap

the shapes obtained. Showing the animation and pausing it will be useful to get more of a students’ input.

To close the second lesson, the teacher will check how far along in the exploration students have gone. Since probably most students would not have completed more than a few rows in the table, it would be nice to get them to start thinking of the possible predictor formulas that can help them quickly identify the shape at hand just by looking at its conic equation type.

Briefly mention the purpose of next class lesson when we will be stepping in the shoes of city planners and architects and exploring different shape building and deciding on the most efficient shapes to build for the purposes given.

E. Materials and Assistive Technologies:1. Students’ notebook papers 2. ELMO or other type of projector with PowerPoint to display the instructions large

enough for all students to see3. Individual copies of the directions for near-sighted students 4. 1 solid shape manipulatives for each student5. Conic sections animation.6. Class handout for students who hove progressed quickly into the topic.


7. Interactive applet, which lets students play with coefficients at http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/conics.html

8. Scrambled poster to unscramble for homework and posted on Blackboard

F. Advanced Preparation Reminders:

I need to prepare the class handout in Word, and create the scrambled Photoshop poster (I will use Solid Works to draw the 3D created lifelike objects). I will download Mathematica Player and the interactive java applet in Geogebra on all classroom computers the previous day (to view the simulations). I will also download the free MatCad animation, interactive online applet. The resources links are below:

pb_consec.mov or pb_consec.gif file (created by Przemyslaw Bogacki and Gordon Melrose)http://www.lostlecture.host.sk/JDandelinEn.htm Lets students play with position of plane and its inclination. (created by Slavomir Tuleja and Jozef Hanc)http://demonstrations.wolfram.com/ConicSectionCurves/ Needs Mathematica Player to run, but lets students see vertices, foci, directrices and slide a and b, (created by Mike May, S.J)http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/conics.html interactive coefficients(If time permits will use it otherwise it is an optional resource: http://www.slu.edu/classes/maymk/GeoGebra/EllipseHyperbola.html interactive conics planar lets students change major, minor axes and vertex and gives the conic equation in standard form)

I also need to boot up all computers before class. I will bring up a file with all the links to java simulations we will use in class today. This way students can quickly access all needed websites.

I will also have the animation, the same links page with the simulations and the scrambled poster files all opened up in different windows, and ready to view immediately. This will minimize the down time and optimize the instructional time.


http://www.slu.edu/classes/maymk/GeoGebra/EllipseHyperbola.html

http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/conics.html

http://demonstrations.wolfram.com/ConicSectionCurves/

http://www.lostlecture.host.sk/authors.htm

http://www.lostlecture.host.sk/JDandelinEn.htm

mailto:[email protected]

mailto:[email protected]

http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/conics.html

Section 7 – REFERENCES

A. After writing your lesson plan, include references of sources, ideas, theory, etc.Cite NYS standards, methods books you have consulted, and teacher guides as applicable.

References

National Council of Teachers of Mathematics. (2000). Principles and Standards for School Mathematics. Reston, VA: NCTM.

Fennema, E. & Romberg T. A (Eds.), (1999). Mathematics Classrooms That Promote Understanding. Mahwah, NJ: LEA.

Huetinck, L. & Munshin, S. (2008), Teaching Mathematics for the 21st Century: Methods and Activities for Grades 6-12, Upper Saddle River, NJ:Pearson.Ms. Sendy Louis Lessons in Math B and Calculus BC at The Frederick Douglass Academy, New York, NY 10039.Bogacki, P. & Melrose, G., pb_consec.mov or pb_consec.gif. Tuleja, S. & Hanc, J., http://www.lostlecture.host.sk/JDandelinEn.htm May, M. & S. J., http://demonstrations.wolfram.com/ConicSectionCurves/. http://cs.jsu.edu/mcis/faculty/leathrum/Mathlets/conics.html. Bloom B. S. (1956). Taxonomy of Educational Objectives, Handbook I: The Cognitive Domain. New York: David

McKay Co.Gardner, H. (1999). Intelligence reframed: Multiple intelligence for the 21st century. New York: Basic Books.Hunter, M. (1982). Mastery Teaching: Increasing Instructional Effectiveness in Elementary and Secondary Schools,

Colleges, and Universities. Thousand Oaks, CA: Corwin Press.Mager, R.F. (1984). Preparing instructional objectives. (2nd ed.). Belmont, CA: David S. Lake.Udvari-Solner, A. (1995). A process for adapting curriculum in inclusive classrooms. In R. Villa & J. Thousand

(eds.) Creating an Inclusive School (pp. 110-124). Baltimore: Paul R. Brookes Publishing Co.Udvari-Solner, A. (1996). Examining teaching thinking: Constructing a process to design curricular adaptation.

Remedial and Special Education 17, 245-254.


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