MATH 124 Geometry and art unit

Content: Polygons, symmetry, transformations, constructions, congruence, similarity.

Part 1: Polygon review

Section 11-3

Vocabulary. Look up the following definitions in the book and write them down. We have seen some of these already.

Simple curve

Closed curve

Polygon

Convex polygon

Concave polygon

Regular polygon

1. Consider the quilts on page 2. Locate examples of as many of the following as you can:

Convex polygon, concave polygon, regular polygon, right triangle, acute triangle, obtuse triangle, scalene triangle, isosceles triangle, equilateral triangle, trapezoid, kite, isosceles trapezoid, parallelogram, rectangle, rhombus, square. You can draw them next to the pictures of the quilts, circle them, or draw an arrow pointing at them.

2. On the grid paper on page 3, create a design that will have examples of at least three different polygons you did not find in the quilts.

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Part 2: Symmetry

Section 11-3

Vocabulary. Look up the following definitions in the book and write them down. We have seen some of these already.

Reflection symmetry Line of symmetry Rotational symmetry Center and angle of rotation symmetry

Activity: Describe the symmetries of the quilts below. For each, state what type of symmetry it has. If it has reflection symmetry, show the line(s) of symmetry, and if it has rotation symmetry, show the center and angle of rotation.

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Activity: Create your own quilt designs with the following symmetries:

1. Vertical, no horizontal reflection symmetry; no rotation symmetry

2. Rotation symmetry, no reflection symmetry

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3. Rotation and reflection symmetry

4. No symmetry

Homework: Describe the symmetries of the triangles and special quadrilaterals (scalene, isosceles, equilateral triangle; trapezoid, isosceles trapezoid, kite, parallelogram, kite, rhombus, rectangle, square). For each, state what type of symmetry it has. If it has reflection symmetry, show the line(s) of symmetry, and if it has rotation symmetry, show the center and angle of rotation.

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Activity and homework: Investigate symmetries in different contexts

A. That’s just half the story

Some letters have mirror symmetry. That means that when you place the mirror on them, you can see the whole letter – half on your paper, and half in the mirror.

1. Which letters work this way? Which work more than one way? Consider all the uppercase letters. Put in dotted lines on your sheet to show where you placed the mirror.

2. There are four states that have mirror symmetry when written vertically in capital letters (that is, letters are written underneath each other). What are they?

3. What state has mirror symmetry when written horizontally?

ABCDE F GHI J KL M N O P Q RST U V W X Y Z

B. The Fold-and-Cut Investigation

1. Fold a piece of paper in half. Cut out a shape on the fold. Draw what you think the paper will look like when you unfold it. Then unfold and compare.

2. Try the same activity, but fold the paper twice before folding.3. BONUS. Create a paper snowflake. What symmetries does it have?

C. Symmetries of regular polygons

1. Determine how many lines of symmetry each regular polygon has. Is there a rule about the number of lines of symmetry in a n-gon? How are the lines of symmetry different when n is even and when n is odd?

2. For each polygon find the smallest angle that you have to rotate it about (with the center of the polygon as the center of rotation) to make it land onto itself. This is called the angle of rotation.

3. What is the relationship between n and the angle of rotation in each case?

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CD

AB

B

DA

C

C

DB

A

C

A

B

D

B

C

A

D

D

B

A

C

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Part 3: Transformations

Sections 13-1, 13-2

Vocabulary. Look up the following definitions in the book and write them down.

Transformation

Reflection

Rotation

Translation

Glide reflection

Symmetry and transformations. How is reflection related to reflection symmetry? How is rotation related to rotation symmetry? What is translation symmetry?

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A. Reflections

In the quilt below, find at least two different examples of reflections. In other words, is there any piece of the quilt that can be seen as a reflection of a different piece? Mark the reflections on the paper in some way.

Create a basic 2x2 design in the top left corner of the grid below, then use patty paper to reflect it vertically and horizontally. Complete the entire 10x10 square using reflections.

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B. Translations

In the quilt below, find at least two different examples of translations. In other words, is there any piece of the quilt that can be seen as a translation of a different piece? Mark the translations on the paper in some way.

Create a basic 2x2 design in the top left corner of the grid below, then use patty paper to translate it vertically and horizontally. Complete the entire 10x10 square using translations.

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C. Rotations

In the quilt below, find at least two different examples of rotations. In other words, is there any piece of the quilt that can be seen as a rotation of a different piece? Mark the rotations on the paper in some way.

Create a basic 2x2 design in the top left corner of the grid below, then use patty paper to rotate it 90° clockwise and 90° counterclockwise around the bottom-right corner of the 2x2 square. Complete the entire 10x10 square using rotations.

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D. Glide reflections

In the quilt below, find at least two different examples of glide reflections. In other words, is there any piece of the quilt that can be seen as a glide reflection of a different piece? Mark the rotations on the paper in some way.

Create a basic 2x2 design in the top left corner of the grid below, then perform a glide reflection of the design in the next 2x2 square to the right and below. Complete the entire 10x10 square using glide reflections.

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Activity: Which transformations were used in this quilt? Can you find all four?

Question: It is said that translation and rotation preserve orientation and reflection and glide reflection do not. What do you think that means? Look at the quilt examples.

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Activity: Translation symmetry and strip patterns

1. Each of the Incan and Greek strips below has translation symmetry. What does that mean and where do you see it in each?

2. What other symmetries does each strip have?

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Homework: These are the quilt patterns believed by some to have been symbols directing escaped slaves to the Underground Railroad. For each pattern, identify the smallest piece of the design that could be used, with the help of transformations, to create the entire quilt. For example, I created an imprecise drawing on the monkey wrench design. Any one of the triangles can be used to replicate the entire design, using just reflection or reflection and rotation (How?).

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Homework: Transformations using coordinate geometry

Here is a set of axes with one shape drawn in the first quadrant:

1. What is this shape called? What are the coordinates of the points A, B, C and D which form the vertices of the shape?

2. Translate the shape 3 squares to the left and 4 squares up. What are its new coordinates? Compare these with the original coordinates. Write a rule for the new coordinates of any point translated 3 squares to the left and 4 squares up.

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3. Start with the original shape above again. This time reflect it over the x-axis. What are the coordinates of the vertices now? Write a rule for the new coordinates of any point reflected over the x-axis.

4. Predict, without performing the transformation, what the new coordinates of any point would be after a reflection over the y-axis.

5. Draw the line y=x. Reflect the starting shape over the line y = x and give the new coordinates for A, B, C and D.

6. Look carefully at these new coordinates below and then predict what would happen to these points if they were reflected in the line y = x:

(2, 4)(6, -4)(-5, -5)

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7. Start with the original shape above again. Rotate it counterclockwise 90° around the point (0,0). What are the coordinates of the vertices now? Write a rule for the new coordinates of any point rotated 90° counterclockwise around the origin.

8. Start with the original shape above again. Rotate it 180° in any direction around the point (0,0). What are the coordinates of the vertices now? Write a rule for the new coordinates of any point rotated 180° around the origin.

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