Investigation 1 Three types of symmetry If you drew a vertical line through the center of the butterfly, the two halves of the drawing would be mirror images. You could rotate the drawing less than a full turn about its centerpoint to position in which it looks the same as the original drawing. You could slide any figure in the design in a regular straight-line pattern so that it exactly matches the original figure. REFLECTIONAL SYMMETRY ROTATIONAL SYMMETRY TRANSLATIONAL SYMMETRY

Investigation 1 Three types of symmetry If you drew a vertical line through the center of the butterfly, the two halves of the drawing would be mirror images. You could rotate the drawing less than a full turn about its centerpoint to position in which it looks the same as the original drawing. You could slide any figure in the design in a regular straight-line pattern so that it exactly matches the original figure. REFLECTIONAL SYMMETRY ROTATIONAL SYMMETRY TRANSLATIONAL SYMMETRY Reflect = Flip Rotate = Turn Translate = Slide

Reflectional Symmetry Problem 1.1 Reflectional Symmetry

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design. Is this a line of symmetry?

Find all of the lines of symmetry for each design. No, when you fold over this line, the little triangles will not match up. If they were turned, or if they were removed, or if they were squares it would be a line of symmetry.

Find all of the lines of symmetry for each design.

Find all of the lines of symmetry for each design. There are NO lines of symmetry in this design. Would there be a line of symmetry if I removed the head, legs, & arms?

Find all of the lines of symmetry for each design. If I removed the head, legs, & arms, the figure would still have no lines of symmetry. The double dark line on the top left will not match the single white line on the top right.

1.1 Follow Up How many possible lines of symmetry are there in any square design? ACE questions later on Explain why these traditional quilt designs do not have reflectional symmetry.

Asymmetry

Problem 1.2 The angle of rotation is the smallest angle through which a design can be rotated to coincide with its original design.

I want to divide these 5 pictures into 2 groups I want to divide these 5 pictures into 2 groups. Give me possible ways to separate them,

List all of the turns that will rotate the figure and look the same List all of the turns that will rotate the figure and look the same. Give the angle of rotation

Rotational symmetry can be found in many objects that rotate about a centerpoint. Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 1 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 1 There are 5 lines of symmetry in this design.

Hubcap 1 The angle of rotation is 72º. There are 5 lines of symmetry in this design.

Hubcap 2 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 2 There are NO lines of symmetry in this design.

Hubcap 2 The angle of rotation is 120º. There are NO lines of symmetry in this design.

Hubcap 3 not in cmp2 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 3 There are 10 lines of symmetry in this design.

Hubcap 3 The angle of rotation is 36º. There are 10 lines of symmetry in this design.

Hubcap 4 not in cmp2 Determine the angle of rotation for each hubcap. Explain how you found the angle. Some of the hubcaps also have reflectional symmetry. Sketch all the lines of symmetry for each hubcap.

Hubcap 4 . There are 9 lines of symmetry in this design.

Hubcap 4 The angle of rotation is 40º. There are 9 lines of symmetry in this design.

Think About it: Is there a way to determine the angle of rotation for a particular design without actually measuring it? Write down your thoughts in your notebook. Make sure you tell me about ones that have lines of symmetry and ones that do not have lines of symmetry.

When there are lines of symmetry 360 ÷ number of lines of symmetry = angle of rotation When there are no lines of symmetry: 360 ÷ number of possible rotations around the circle. 5 lines of symmetry 3 points to rotate it to

Suppose you know the angle of rotation of a particular design Suppose you know the angle of rotation of a particular design. How can you use it to find all the other angles through which the design can be rotated to match the original design?

You can find multiples your angle of rotation until you get 360 degrees for a full rotation.

90 degree angle of rotation and no lines of symmetry 90 degree angle of rotation and no lines of symmetry. 60 degree angle of rotation and at least one line of symmetry.

Follow Up 1.2 #1 Create a hubcap design that has rotational symmetry with a 90º angle of rotation but no reflectional symmetry.

Follow Up 1.2 #2 Create a hubcap design that has rotational symmetry with a 60º angle of rotation and at least one line of reflectional symmetry.

Follow Up 1.2 #3 Why do you think many rotating objects are designed to have rotational symmetry.

A. Look for reflectional symmetry in each design A. Look for reflectional symmetry in each design. Sketch all the lines of symmetry you find. B. Look for rotational symmetry in each design. Determine the angle of rotation for each design.

Labsheet 1.3

1. 3 D: For each design, sketch or outline the basic design element 1.3 D: For each design, sketch or outline the basic design element. That is sketch or outline a part of the design from which the entire design could be created.

1. 3 D: For each design, sketch or outline the basic design element 1.3 D: For each design, sketch or outline the basic design element. That is sketch or outline a part of the design from which the entire design could be created.

If you choose the ones marked with “a” you need to explain that if it and a reflected image of it will join to form “b” which will fill the entire design. “a” and its reflected image have reflection symmetry.

A. Recall that if a design has reflectional symmetry it can be folded to have a matching half. B. Recall that if a design has rotational symmetry you will have to be able to turn it by “the handle” so your point ends up on a matching point. Designs from CMP1

Designs 1 & 2 Draw in all lines of symmetry and find the angle of rotation for each design. Designs from CMP1

Designs 1 & 2 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 120º. The angle of rotation is 120º. Designs from CMP1

Designs 1 & 2 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 120º. The angle of rotation is 120º. Designs from CMP1

Designs 3 & 4 Draw in all lines of symmetry and find the angle of rotation for each design. Designs from CMP1

Designs 3 & 4 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 120º. The angle of rotation is 60º. Designs from CMP1

Designs 3 & 4 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 120º. The angle of rotation is 60º. Designs from CMP1

Designs 5 & 6 Draw in all lines of symmetry and find the angle of rotation for each design. Designs from CMP1

Designs 5 & 6 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 60º. The angle of rotation is 120º. Designs from CMP1

Designs 5 & 6 Draw in all lines of symmetry and find the angle of rotation for each design. The angle of rotation is 60º. The angle of rotation is 120º. Designs from CMP1

Follow up 1. 3 #1 For the kaleidoscope designs in Problem 1 Follow up 1.3 #1 For the kaleidoscope designs in Problem 1.3, describe the relationship between the number of lines of symmetry and the angle of rotation.

Follow up 1. 3 #1 For the kaleidoscope designs in Problem 1 Follow up 1.3 #1 For the kaleidoscope designs in Problem 1.3, describe the relationship between the number of lines of symmetry and the angle of rotation. Excluding design 3, which has no reflectional symmetry, the product of the number of lines of symmetry and the angle of rotation is always 360º 3 x 120 = 360 6 x 60 =360

Translational Symmetry Problem 1.4 Translational Symmetry A translation is a geometric motion that slides a figure from one position to another.

A design has translational symmetry if it can be created by sliding a basic design element in a regular, straight line pattern. To describe the translational symmetry in a design, you can draw the basic design element and an arrow indicating the direction and length of a slide that would move one part of the design to another. This is an example of a translation.

A tessellation is a design made from copies of a single basic design element that cover a surface without gaps or overlaps. You CANNOT rotate or reflect your basic design element. This is an example of a tessellation.

A. Outline a basic design element that could be used to create the tessellation using only translations. B. Write directions or draw an arrow showing how the basic design element can be copied and slid to produce another part of the pattern.

Tessellation 1 A. Outline a basic design element that could be used to create the tessellation using only translations. B. Write directions or draw an arrow showing how the basic design element can be copied and slid to produce another part of the pattern.

Tessellation 1 As long as you have 2 white, 1 grey, & 1 black “T” in your basic design element, you are correct. B.Just make sure your arrow goes to a matching part of the next part of the tessellation.

Tessellation 2 A. Outline a basic design element that could be used to create the tessellation using only translations. B. Write directions or draw an arrow showing how the basic design element can be copied and slid to produce another part of the pattern.

Tessellation 2 A. You’ll be correct, as long as you have one grey block & one white block. B. Just make sure your arrow goes to a matching part of the next part of the tessellation.

Tessellation 3 A. Outline a basic design element that could be used to create the tessellation using only translations. B. Write directions or draw an arrow showing how the basic design element can be copied and slid to produce another part of the pattern.

Tessellation 3 A. Make sure you have a grey one and a white one outlined. B. Just make sure your arrow goes to a matching part of the next part of the tessellation.

Tessellation 4 A. Outline a basic design element that could be used to create the tessellation using only translations. B. Write directions or draw an arrow showing how the basic design element can be copied and slid to produce another part of the pattern.

Tessellation 4 A. If you feel color is part of the design element, make sure you outline one bird of each color. If you don’t think color matters, you can outline one bird since the bird is not rotated in the design. B. Just make sure your arrow goes to a matching part of the next part of the tessellation.

Page 14 B 1 & 2

Page 14 B 1 & 2 As long as you have 4 leaves and 2 lines you have the basic element of design.