Here is my Unit of Practice from TTL

[Invitation]  [Situations]  [Tasks]  [Interactions]  [Standards]  [Assessment]
[Tools]  [Project]

Abstract:

This unit allows students to discover the properties of reflections, rotations, and translations in transformational geometry using a variety of tools: Mira, protractor, compass, folding and tracing, the software program The Geometer’s Sketchpad, and visiting some interactive math sites.  Students will create a tessellation and become familiar with the work of E.C. Escher. Using the properties of reflections learned in this unit, they will also make a miniature golf hole and show the path the ball must travel for a hole-in-one.

Invitation:

Are you familiar with the artwork of E.C. Escher? Do you know what a tessellation is?  If not, you will by the end of this unit.  What are the underlying mathematical principles that govern a tessellation?  They are reflections, translations (slides) and rotations (turns).  These terms should be familiar to you.  We see our reflection everyday in a mirror, have gone down a slide at a park, and have seen tires and other objects rotate.  What do parallel lines have to do with translations (slides)?  What connection is there between intersecting lines and rotations?  Find the answers to these questions and more as you complete this unit of practice.

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Situations:

This unit will take four 90-minute class periods.  Our school runs on an A/B block and any work that is not completed in class will need to be finished by the students when they return to class in two days.  Work will be done in the computer lab when we do the activity on reflection, translations, and rotations using the geometer’s sketchpad and when the students do the interactive activities on the Internet.  Some students may also be in the computer lab to do their miniature golf hole and their tessellation. 

Students will be in the regular classroom for the ball rebounding activity and when they take their quiz.  Those students who choose to do their tessellation using poster board will also do their work in the classroom.  Some students might also choose to make their miniature golf hole on poster board and that too would be done in the regular classroom.

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Tasks:

This unit will take four 90-minute class periods.

Day 1:  After a brief introduction, the students, working alone, will practice reflecting a point over a line using a protractor, compass, Mira, and the folding and tracing method.  They will then repeat the process and reflect a geometric figure of their choice over a line using the same tools.  Students will go to a Virtual Manipulative-Reflection Website at http://matti.usu.edu/nlvm/enu/navd/frames_asid_119_g_3_t_2.html and practice with the geometric shapes on this interactive site.  The students will write down any properties they notice from this activity about reflections.  Students will also complete some homework problems from the University of Chicago School Mathematics Project (UCSMP) Geometry Text lessons 4-1 and 4-2.

Day2:  Students, working in groups of three, will practice with a ball and 2 by 6 boards that will allow the ball to rebound.  The students will roll the ball toward the board at different angles and observe how the ball rebounds.  Next the students will take a second ball and put it at any location and practice rolling the first ball toward the board and trying to have it rebound and hit the second ball.  Now the students will reflect the second ball over the board and mark that spot on the other side of the board.  They will then roll the first ball toward that reflection mark of the second ball on the other side of the wall and see that it will rebound to hit the second ball.  After a brief class discussion about how this would apply to miniature golf or pool, students will design with their groups a miniature golf course hole where they will show the path the ball must take for a person to make a hole-in-one with it rebounding a minimum of three times (i.e., a minimum of three reflections).  Students can use the Geometer’s Sketchpad or poster board with paper and pencil. Students will visit an interactive site titled pool table at http://illuminations.nctm.org/imath/6-8/pooltable/index.html and do those activities.  Finally, the students will do some homework problems from lesson 4-3.

Day 3:  The class will go to the computer lab and do some activities using the Geometer’s Sketchpad.  They will work individually at a computer and follow a guided worksheet.  They will begin by reflecting a figure over a line that doesn’t intersect the figure.  Next they will reflect a figure over a line that intersects the figure in one point, and finally they will reflect the figure over a line that intersects the figure in more than one point.  While in the computer lab they will also follow a guided worksheet to discover some properties of translations (reflections over parallel lines) and rotations (reflections over intersecting lines).  After completion of the activities they will get into their groups and discuss their answers to the guided worksheet and write down the observations of the group.  The students will also practice simulations of these transformations at http://standards.nctm.org/document/eexamples/chap6/6.4/index.htm.  The individual students will write about the things they have discovered about translations and rotations from their experiences in the computer lab and at the interactive site.  Students will also complete some homework problems from lessons 4-4 and 4-5 of UCSMP Geometry Text.

Day 4:  Students will create a tessellation using Tesselmania software or poster board using translations, reflections, or rotations.  Students will go to a computer lab to use the Tesselmania program and will be provided poster board for the more traditional form of tessellation.  Students can download a demo version of Tesselmania by going to http://www.worldofescher.com/store/mania.html.  This version will allow them to practice at home but they can’t print or save.  They may also go to http://www.tomsnyder.com/classroom/tesexp/tesexp_movie.asp and view a tutorial on a similar software program Tessellation Exploration.  Students will also go to a Website titled Investigating Patterns:  Symmetry and Tessellations at http://www.camosun.bc.ca/~jbritton/jbsymteslk.htm and do one of the interactive activities listed under activities numbered 15, 17, 18, 20, 22, or 23.  These activities deal with different types of tessellations.  Students will also go to activity 24 about E.C. Escher and do one of the activities.  Students will write about the tessellation activity and the E.C. Escher activity from the interactive site.

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Interactions:

 The teacher’s role will be to provide instruction, assist the students with the software programs and with problems they encounter, review previously learned concepts and to provide Website addresses for the interactive sites, and to review assessment guidelines.

Students will work in groups as they create their miniature golf hole and roll the balls against the boards to explore the rebound.  While in the computer lab the students will also interact as they are in close proximity, can see each other’s computer screen, and will as questions of each other.  They will also be in groups to compare their answers to the worksheets on the geometer’s sketchpad.  They will discuss their answers to the questions and explain to the other members of the group why they gave their answers.

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The benchmark from the South Dakota State Standards is:  Students will analyze figures from a variety of perspectives.  The two standards that are covered by this unit of practice are:  1) Students will select transformations required to map images of objects  2) The student will use graphing tools to study transformations e.g., congruence using rigid motion, similarity using magnification of image.

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Assessment:

Students will be assessed according to the rubrics that can be found on my website for the tessellation and the miniature golf hole.   Papers that the students write will be based on content, spelling and grammar, and neatness.  The guided worksheet, homework assignments, and quiz will be assessed on number correct out of number possible.

 Rubric for Miniature Golf Hole Activity

Tessellation Rubric

Papers written during the unit will be graded as follows:  2pts. - appropriate content; 2pts. – grammar and spelling; 1pt. -  appearance.

Tools:

Ø     Paper

Ø     Pencil

Ø     Ruler

Ø     Poster board

Ø     Mira

Ø     Protractor

Ø     Compass

Ø     Computer

Ø     Internet Access

Ø     Geometer’s Sketchpad software

Ø     Tesselmania Software

Ø     Ball

Ø     Two by Six board

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Project:  (This is not a project that the students do, but rather additional instruction
                       for the unit of  practice.) 

Students probably will need a brief explanation of how to reflect using the Mira and Protractor.  Most will understand the fold and trace method, although a few may have trouble.  Reflecting with a compass will require more explanation.  You may take a few minutes to explain this to the class as a whole or put them in groups and let them help each other or have them work individually and help them as they have questions.  I usually give a brief explanation and then refer them to the book where they can find an explanation.

My classes have previously been down to the computer lab and are familiar with the Geometer’s Sketchpad.  I also give each student a copy of the Sketchpad commands and what they do.  This comes with the software and then they can refer to it as necessary.  Usually, they are asking someone on a computer next to them, or they are raising their hand and asking me.

I haven’t actually done the rebound activity with the ball and the piece of wood. I have often thought it would be a good way for them to see the how the ball rebounds and how to reflect something over the board and when they aim at the refection, the ball will rebound and hit the object they have previously reflected.  I usually show an example of how to do the reflections on a miniature golf hole before having them start working on their miniature golf hole.

I think most students will choose to do the tessellation on Tesselmania.  If there are some that want to do their tessellation using poster board, they could work on theirs in the computer lab or perhaps in the hall outside the computer lab. 

I don’t have the quiz on my website.  If you are interested in receiving a copy of the quiz, you can e-mail me at Curtis.Smith@k12.sd.us.

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