Activity 4A

Activity 4A

C&I 335

Summer, 1998

Karen Bennett

Glenwood High School

Chatham, IL

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MAJOR PROJECT:

A TRANSFORMATIONS UNIT

Instructional Topic:

Simple Transformations in Geometry

Audience:

High School Juniors and Seniors enrolled in an Informal Geometry Course. Informal Geometry is designed to be a math credit for students who are not "math oriented". This course is taken after the two-year Algebra I sequence.

Hardware Needed:

A Math Lab with 24 PC's will be the setting for tech projects. A PC on a cart with an overhead LCD projection screen will be used in the classroom for presentations.

Software Needed:

PowerPoint will be used for the lecture/presentation. Geometer's Sketchpad is the program for student discovery. Netscape Navigator will be used for the extra credit option.

Time Frame:

This unit is expected to be 8 or 9 class periods in length. A class session lasts 50 minutes.

DAILY LESSON PLANS:

Day 1: PowerPoint Introduction to Transformations

Day 2: Translations

Day 3: Rotations

Day 4: Reflections

Day 5: Glide Reflections

Day 6: Geometer's Sketchpad Lab Activities (days 1 - 5)

Day 7: Lab Experience Continued...

Day 8: Dilations and Lab

Day 9: Exploration and Review

ASSESSMENT TOOLS:

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DAY 1/ TOPIC : Introduction to Transformations (A PowerPoint Presentation)

Goals:

To enhance lecture presentation through the use of technology
To facilitate good note-taking, getting accurate definitions from slides
To give students basic concepts of the transformation types

Materials:

one PC on a cart, overhead projector and screen, hand-outs (made from Power Point slides) for note-taking

Process:

Students will be given handouts. Students are expected to complete their handout during the presentation.

Slide 1 will be a definition of transformations in general.
Slide 2 defines isometries.
Slide 3 will be a list of the 4 types to be covered.
Slides 4-7 demonstrate the specifics of each type.
Slide 8 explains dilations.
Slide 9 may be an Escher print or an example of wallpaper design

Assessment:

Students will be asked to read over completed handout and be prepared to answer questions the next day.

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DAY 2/ TOPIC: Translations

Goals: To help students:

To better understand the idea of translation (That it slides the figure without changing the sense of the figure)
To understand that translations require direction and distance.
To be able to translate objects via two different methods.

Materials:

graph paper handout (with examples to be translated on it), rulers, patty paper (for tracing purposes)

Process:

In class, as a group, a handout will be completed using 2 methods.

Translation Method One:

Pass out handouts and rulers. The handout has 3 examples to work.

Ex 1: given a pre-image and one image point

Ex 2: given pre-image and a slide arrow

Ex 3: given pre-image and direction and distance in units on graph paper

Work the examples on the board with the class, explaining WHY the process works.

Translation Method Two:

Pass out patty paper. Turn to back side of handout. Three similar examples are to be worked using the trace and slide method.

Demonstrate that the figure and the direction arrow are to be traced on the paper. Next, slide tracing until beginning of traced arrow is at the tip of the original arrow. The location of the traced figure is the image translation.

Assessment:

Homework will be a worksheet and some textbook problems demonstrating understanding of translation ideas.

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DAY 3/ TOPIC: Rotations

Goals: To help students:

To better understand the concept of rotation (that it "turns" a figure about a center point)
To be able to rotate figures via two methods.
To understand that rotations require a center point as well as a degree measurement.
To understand that rotations are typically in a counterclockwise direction.
To understand the concepts of half-turns and quarter turns.

Materials:

protractors, rulers, patty paper, handout with rotation examples

Procedure:

Students will be given protractors, rulers and handouts with examples to be worked through.

Rotation Method One

On the board, demonstrate an example on handout with center point given. Use protractor to measure angle of rotation, use ruler to maintain equal distance from center point.

Work through examples on handout showing half-turn, quarter-turn, and negative (clockwise) rotation angles.

Rotation Method Two

Use patty paper to trace figure, anchor center point, draw line of reference. Spin tracing until angle formed is required angle of rotation. Students will observe the demonstration then complete the handout using this method.

Assessment:

Students will be orally quizzed on 5 squares questions on the handout. This will test their comprehension of the rotation concept.

Homework assignment will be a rotation worksheet.

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DAY 4/ TOPIC: Reflections

Goals: To help students

To better understand the concept of reflection (that it "flips" the figure and changes its sense).
To be able to reflect via 3 different methods.
To understand that reflection requires a "mirror line" as a reference.

Materials:

rulers, MIRA devices, patty paper, 2 handouts --one with mira & one without

Procedure:

Reflection using Miras

Students will be given Mira's and handouts. Explanation of their reflective quality and the appropriate use will be given. Students will follow the step-by-step explanation, then be asked to discover the relationship between the line of reflection and the segments joining the points to their images. (They should find the line of reflection to be the perpendicular bisector of those segments.) Using this discovery, they locate the mirror line given the image and pre-image. And given a figure, they draw both the mira line and a reflection image.

Reflection using measurement

Students will then be given rulers and handouts with figures to be reflected. Teacher will repeat the discovery from the mira method, and students will be asked how they might use a ruler to draw the reflection.

Students will complete handout via this method.

Reflection using tracing

Patty paper will be given out. Students will be instructed to draw the letter "F" and a line of reflection on the paper. Students will be asked, "How could you find the reflection of the letter over that given line without a mira and without measuring?"

Students should answer that the line can be a "fold" line to determine reflection image.

Assessment:

Homework will consist of a worksheet with approximately 25 questions concerning reflection.

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DAY 5/ TOPIC: Glide Reflection

Goals: To help students:

To discover their own method for creating a glide reflection using the tools given and their understanding from previous lessons.
To understand that a glide reflection is a two-step isometry that combines translation and reflection
To understand that the translation is along the direction arrow, then reflected across a mirror line parallel to the direction of the translation.

Materials:

mira's, rulers, graph paper, handouts

Procedure:

Students will be given the definition of a glide reflection again. They will be given the example of footprints in the sand. Then they will be asked to find their own method of creation.

They will create a glide reflection, label the parts and write a short paragraph describing their method.

Assessment:

The paragraph written will be the assignment for this lesson.

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Day 6/ TOPIC: Lab Experience using Geometer's Sketchpad

Pre-Lesson:

Prior to this lab experience, students will have used Sketchpad in other contexts, and will have become familiar with it.

Goals: To help students:

To use Geometer's Sketchpad to create translations, rotations, reflections and glide reflections.
To make comparisons of the different transformations that solidify understanding of their definitions
To know the needed "givens" required to create each image.

Materials:

PC lab with Geometer's Sketchpad loaded on each machine, printers

Time Period:

One class period is 50 minutes long. This lesson will take approximately 70 minutes to complete. There will be some carry over to the next day.

Procedure:

Students will be given a lab lesson worksheet and given instructions before heading to the lab. They will follow the step-by-step instructions. The assignment requires:

drawing an image
shading its interior (and changing its color)
constructing and labeling a translation, rotation, and reflection
and then answering questions about their sketch.

Questions to explore:

1. Is it possible for any of the 3 images to lie directly on top of one another?

2. How are the images the same?

3. How are the images different?

Students will then follow step-by-step instructions to create a glide reflection and answer questions about it.

Assessment:

Students will be required to turn in the printout of their completed sketch which will have an original image, translated image, rotated image and reflected image all labeled appropriately. They will submit the answers to the exploration questions written clearly.

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DAY 7/ TOPIC: Lab Experience Continues...

Pre-Lesson:

Yesterday, students will have completed most of the lab project assigned.

Goals: To help students:

To clarify different requirements for each type of transformation
To reinforce concepts from previous lessons
To be certain students know what the lab activity was supposed to teach them

Materials:

Students will need printout produced from this lab experience.

Procedure:

Those who are finished will work on 3 additional questions written on the board. As others finish, they, too, should do the additional questions. The questions are:

What is the result of two reflections over intersecting perpendicular lines?
What is the result of two reflections over parallel lines?
What is the result of 2 rotations, the first of 180 degrees, the second of 90 degrees?

Students will explore these questions on sketchpad, writing the answers on their assignment sheet.

In the last 10 minutes of class, we will return to the classroom to debrief. Possible questions to be asked:

What keywords could describe the process of each transformation?
How do you describe the effects of each on the figure transformed?
Are the figures congruent?
Do you think moving the pre-image will alter the effect of the transformation?
Why do you think we chose this lab, what were you supposed to see?

Assignment:

Students will write a short paragraph describing each transformation, what it requires, what it actually does to the figure.

Assessment:

Things to look for in labs:

Are students labeling image and pre-image correctly? Did they move the correct object? Are they considering all directions (left, right, up, down)? Are their answers clear and descriptive?

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DAY 8/ TOPIC: Dilations (Scaling)

Goals: To help students:

To realize that not all transformations are rigid. Some types of transformations alter the shape or the size of the original figure.
To understand that a dilatation is still a simple transformation because it retains the figures shape ( no change in the angles between the proportional parts)
To be able to enlarge or shrink a given figure via the illustrated method.

Materials:

rulers, colored pencils or pens, handout with examples, PC Lab with Geometer's Sketchpad.

Procedure:

Students will be instructed and shown ruler measurement method for enlargement using point of reference. Students will also be shown reduction with these tools.

The rest of the class will be spent in the lab completing a sketchpad worksheet on dilatations. Students will follow the instructions and complete the questions.

Assessment:

Students will turn in completed lab activity and questions.

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DAY 9/ TOPIC: Reviewing and Extra Project Option

Goals:

Students will prepare for tomorrow's unit test, and be informed of the extra credit optional project.

Materials:

PC lab with Netscape Browser loaded

Procedure:

For the first half of the period, students will be given an oral review of the topics that will be covered on the test. A written list of notebook contents will be given listing the material that will be required in their unit notebook. In organizing the notebook, students may encounter questions that will be entertained during this time.

Extra Credit??

In the second half of the class period, we will go to the Math Lab. Students will be given the URL's of two sites that are related to geometry in art. The concept of symmetry in the plane is the topic of each site. Students will be encouraged to "play" to generate designs or to explore links from the sites. Extra credit will be given for a report on one of the 17 symmetries in the plane. The extra credit report will be approximately one page in length. It will be due in one week.

Assessment:

The unit test will be the tool by which we will assess the review day instruction.

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