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Grade: Middle
Subject: Mathematics

#2752. Transformations in Geometry

Mathematics, level: Middle
Posted Mon Dec 9 07:56:57 PST 2002 by Tanna Walker (tanbanan97@hotmail.com).
University of Tulsa, Tulsa, OK
Materials Required: yarn, masking tape, pattern blocks, small mirrors, permanent markers, geoboards, braille alphabet
Activity Time: 6-8 week unit
Concepts Taught: Transformations


OPENING ACTIVITY
1)As a class, the students will name all of the different patterns that they can come up with. Then, in groups of two, the students will group the patterns into areas of similarity. They will then use Inspiration to create a map that represents each group and idea.

REFLECTIONS
1)Name reflections: Each student will get a thin sheet of white paper (tracing paper may be used, but watch out for bleeding on the table or desks). They will fold it long ways (hot dog style). Each student will then write his/her name in cursive so that the bottoms of the letters meet the crease. Then the children will use a black (or whatever is available) permanent marker to trace over their name. The color will bleed through the crease and it will appear on the opposite side. The students need to trace over the part that bled through, creating their name on the opposite side. When this is done, they will open their paper and see a reflection of their name as if the crease were a mirror. We will discuss the ideas of reflection.
2)Miming reflection: Each student will be paired with a partner. The partners will face each other and be told to ‘reflect’ or mimic what their partner is doing. We will start out slow and then move on to doing it faster. If time permits the students can change partners. When finished we will discuss the properties of reflection, specifically pointing out that whatever our partner did with his/her left hand, we reflected it with our right hand, and vice versa, etc.
3)Mirror reflection: Each group will be given a hand held mirror and a large floor to ceiling mirror will be located in the class. The groups will look at themselves, other objects in the class, and a sheet containing different symbols through the mirror. As we get to the symbols, we will talk formally about where the point of reflection is for each of the objects.
4)As a class, we will discuss the Braille alphabet and what letters are reflections of others and where the line of reflection lies.
5)Drawing reflections: Each student will write down the properties of reflection. Then he or she will draw a polygon, a line of reflection, and the reflection. The student will do all of this as a journal entry.

SYMMETRY:
1)Symmetry Code: Students will see a code on the board and will be asked to decipher it in their table groups. The code will be composed of letters that in some way have a line of symmetry. For example: The sentence “MY TEETH BIT YOU.” is made up of letters that have a line of symmetry. The students will be given one half of the symmetrical letter and the other half will be left off. The students must try to figure out what is going on and give me the deciphered sentence. We will do several of these on the board. Then we will discuss why the letters are halved, as they are, why some show the top half, while others show the left or right segment.
2)“Symmetry in Shapes”: The same symbol worksheet that is used in mirror reflection will be used in this activity. The students will be asked to circle the symbols that are an exact match if you fold the paper together on an imaginary line. We will then discuss that this is line symmetry because the shape that is formed after you fold exactly matches the other half.
3)Drawing lines of symmetry: This activity used the same worksheet as above, except this time the students are asked to draw the line of symmetry on each object.
4)Symmetry in nature: Students are asked to discuss the different types of symmetry in nature. Some examples that may come up are body, face, butterflies, snowflakes, etc. The students will be asked to draw and color one an example of symmetry in nature that was not discussed in class, showing the imaginary line of symmetry.
5)Symmetrical Logos: The class will look at examples of company logos. The class will determine if they have a symmetrical shape or not, and will classify them. Each student will then develop his or her own symmetrical logo for a company. The logo can be for an existing company or an imaginary one. The students will present their creation and tell why they chose as they did.

ROTATION:
1)Real World: As a class we will discuss things that rotate. Some things that may come up in the discussion are hubcaps, the moon, tires, combination locks, hands on a clock, doorknobs, compact disks, etc. We will discuss how things that rotate have a center of rotation, an angle of rotation, and how it does not change the orientation of the shape.
2)Pattern Blocks: The students will be in their table groups and they will get an array of pattern blocks to share and each student will receive a paper with a dot to be used as the center of rotation. They will need to use these blocks to illustrate rotations. One person in the group will be the ‘caller’ and they will call out the shape and the angle of rotation to be used. He or she will then check each member’s answer to ensure correctness, and if they are incorrect they will coach that person to the correct answer. Each student will have a chance to be the ‘caller.’
3)Geometer’s Sketchpad: The students will be given a sheet that contains several rotated polygons. Each student will copy the original polygon onto the computer using Geometer’s Sketchpad. They will need to label what they think is the angle of rotation is using the features of Sketchpad.
4)Rotational Art: The students will create art by rotating an object(s) several times around a fixed point of rotation. The art will be defined by coloring it in. The students will exchange pictures and try to determine the center, angle, and direction of the rotations.

TRANSLATIONS:
1)“Acting Out”: Before class the teacher will create an x and y axis grid on the floor using masking tape. The students will come up by table groups and choose a figure to represent. The figure will be written down in (x, y) ordered notation form. Each student will represent a point of the polygon and the polygon will be completed using yarn to connect the points (students). The next group will come up and perform the given translation. Each group will get a chance to be the original figure and the translated figure.
2)Geoboards: Each student will be given access to a geoboard. The teacher will call out the points of a polygon and the students will put it on their board with a band. The teacher will then call out a translation. The students will place it on their board with another band. The teacher will show on the overhead what should be on the boards and the students will check themselves.
3)Toy Company Letterhead: The students will be asked to create a stationary letterhead for a toy company using a horizontal translation of an object. The letterhead can be computer generated or hand drawn. Each student will present their stationary and tell why they chose to create it how they did.
4)The class will revisit the Braille alphabet and review it’s properties of reflection. We will determine if any of the letters are translational. (They aren’t. If they were then they would be undistinguishable to the blind.)

DILATIONS:
1)Scale factors: As a class we will discuss scale factors with some real world examples, such as: the movie Honey, I Blew Up the Kid!, movie screens, overhead projectors, etc.
2)What if Barbie was real?: This activity is about comparing Barbie to a ‘real life’ person by using proportions and ratios. The teacher will give the students the measurements of an average woman according to the research. The students will then measure the Barbie doll and record their measurements. They will determine the ratio and calculate what size Barbie would be if she were real.
3)Geoboards: After a discussion of the term ‘dilation’ in respect the activities done above, the students will be grouped into pairs and each will be given a geoboard and elastics. The teacher will call out a figure using ordered pair notation. The first student will construct it on the board. Then, the teacher will give a dilation of the original structure. The second student will demonstrate the dilation on the second board. The student will check their answers and switch roles. This will continue for several trials.

TESSELLATIONS:
1)Geometer’s Sketchpad: The students will use Geometer’s Sketchpad to create a tessellation with regular polygons. The students write down their commands for the first initial tessellation and then they will tessellate the plane with that figure.
2)Real Life Examples: A discussion of real life tessellation examples will be discussed. Some examples that may be discussed are as follows: honeycombs, mosaics, fabrics, tiling, etc.
3)Escher-like Tessellations: The students will learn about M.C. Escher and his artwork through a power point presented by the teacher. They will then create their own ‘Escher-like’ tessellation and share their creations with the class.

BRINGING IT ALL TOGETHER:
1)Frieze Patterns: The students will discover the transformations associated with Frieze Patterns. We will use classifications in order to find out what transformation or combinations of transformations are associated with certain Frieze Patterns. The students will create their own Frieze Pattern for each classification and each student will find one pattern in their real life and present this to the class and explain how it could be classified using the Frieze Pattern model.
2)Soups ‘N Such Café: The teacher will read the following poem to the class.
It’s Tuesday noon at the Soups ‘N Such Café,
When a customer walks in and ruins everyone’s day.
It’s old Mr. McGruder with a frown on this face
And he walks towards his table at a slow, steady pace.
For old Mr. McGruder as everyone knows,
Is fussy and picky from head to toe.
“This drink is too hot or too cold or too spicy!!!
My drink is too warm and my food is icy!!!”
Everyone runs to hide to avoid to task
Of being the one who has to ask…
“May I take your order, PLEASE???”
“I’ll take that soup,” he says with a grin,
“but I want only certain letters put in.”
“They must look the same after a slide, turn, AND flip.
So find only those letters before I take a sip.”
The students will work in their table groups to find what letter to put into the soup. They will work with concrete alphabet letters (magnetic or card stock) to find the answer. They will present their finding on a chart that shows which letters are the same after a slide or translation (all of them), which letters are the same after a flip or reflection (A B C D E H I M O T U V W X Y), and which letters are the same after a turn or rotational symmetry of 180 degrees or less (H I N O S X Z). The letters that appear in all three columns go into the soup (O H I X).
3)Conversation: Each student will write a one-page conversation between themselves and M.C. Escher (or some other transformational artist of their choice (with teacher approval)). The student may take on any role in the conversation. Some acceptable examples are teacher, friend, magazine, television, or newspaper interviewer, or parent. Those who are willing can share their writings.
4)Debate: As a class we will discuss art and geometry. The main question and topic of discussion with be: Is there a relationship between art and geometry? Why or Why not? The class will break up into two groups (one’s that believe there is a relationship and those that do not). Each group will get five minutes to prepare and 2½ minutes for rebuttals.
5)Compare and contrast: Each student will write a one-page paper comparing the work of quilters with that of mosaic artists.
6)Prompt: Each student group of three will be given a prompt in which to research and present about. The generic form is: You are in _________________(a town, city, country, etc.) in the ____’s (a century). Tell us and show us one example of what geometry you see in the artwork around you.
7)Reflection Time: Using the Inspiration model that they created on the first day, the students will re-categorize their patterns according to the six transformations they have learned. (If any patterns do not apply to a transformational group, they can group them into a miscellaneous group.)
8)Website exploration and review: The students will log on to the UTC website provided and go through the program as a type or exploratory or review activity.