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Geometry Workbook 2:

Coordinate Rules, Input & Output, Isometry, and Symmetry

Student Name __________________________________________

STANDARDS:

G.CO.A.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle (i.e. rigid motions) to those that do not (e.g. translation vs. horizontal stretch)

G.CO.A.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

SKILLS:

I will be able to distinguish between transformations that are isometric and those that are not

isometric.

I will be able to recognize different types of transformations such as reflections, translations,

rotations, dilations, and stretches.

I will know the definition of isometric transformations (rigid motion).

I will be able to explain the connection between the algebraic relationships of functions and one to

one functions to the geometric relationships of mapping and transformations.

I will be able to use coordinate rules to move and/or alter a pre-image to determine its image or

vice versa.

I will be able to identify and describe the different symmetries (line symmetry, rotational

symmetry, point symmetry) of a figure.

I will be able to determine the maximum possible lines of symmetries that exist for a given

polygon.

I will be able to determine the order and angle of a rotational symmetry.

I will know the symmetries of a parallelogram, rectangle, rhombus, square, trapezoid and regular

polygon.

Notes:

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G.CO.A.2 STUDENT NOTES & PRACTICE WS #1 – geometrycommoncore.com 1

In algebra we studied functions extensively and during that time we learned how to solve for variables using

function notation. Let me show you a few examples to remind you.

What is the value of the

function 2( ) 3f x x when x = -

2?

What is the value of the function ( ) 7f x x when x =

5?

What is the value of the

function 3

( )4

f x x when x = 12?

2

2

( ) 3

( 2) ( 2) 3

( 2) 4 3 1

f x x

f

f

( ) 7

(5) 5 7

(5) 12

f x x

f

f

3 3( ) ( ) (12)

4 4

36( ) 9

4

f x x f x

f x

When x = - 2, then y = 1 (-2, 1) When x = 5, then y = 12 (5, 12) When x = 12, then y = -9 (12, -9)

NYTS (Now You Try Some)

1. What is the value of the

function 2( ) 2f x x x when x = 4?

2. What is the value of the function ( ) 5 1f x x when x = 11?

3. What is the value of the

function 2

( )5

f x x when x = 5?

In all of these examples we were given the x value and then asked to solve for the y value (the value of the

function). In the next examples we will be given the y value (the value of the function) and then asked to work

backwards to determine the x value that would have produced that result. Look closely at these examples.

What is the value of x when ( ) 4 1f x x & ( ) 3f x

?

What is the value of x

when 2( ) 5f x x & ( ) 4f x ?

What is the value of x

when 5

( )12

xf x & ( ) 10f x ?

( ) 4 1

3 4 1

4 4

1

f x x

x

x

x

2

2

2

( ) 5

4 5

9 3

f x x

x

x x

5 5( ) 10

12 12

120 5 24

x xf x

x x

When y = - 3, then x = 1 (1, -3) When y = 4, then x = 3 ( 3, 4) When y = 10, then x = 24 (24, 10)


4. What is the value of x when ( ) 8 5f x x & ( ) 29f x

?

5. What is the value of x

when 3( ) 14f x x & ( ) 13f x

?

6. What is the value of x

when 1

( )3

xf x & ( ) 15f x ?


In geometry we have a similar input/output process when we determine how shapes are altered or moved.

Geometric objects can be moved in the coordinate plane using a coordinate rule. These rules can alter the

shape in many different ways. Some rules will translate the shape, some will rotate or reflect the shape, some

will stretch or distort the shape, some will increase the size of the shape, etc… lots of different things can

happen. Coordinate rules come in the following form:

English Translation:

Coordinate Rule T maps all points (x, y) to (x + 3, y – 6)

So rule T will move or map all points (x, y) by adding 3 to the x value of each point and subtracting 6 from each y value of each point.

The name of the rule is a capital letter before the (x, y). The point (x, y) represents that all points in the plane

will be affected by this rule. The arrow represents the geometric term of map or mapping, which is a

geometric way of saying “moves.” Finally, the coordinate description after the mapping arrow symbol

represents the change produced by the rule. Can you see the connection to functions? You will input values

into the coordinate rule and it will output new values, just as a function does. A change is that we refer to the

input value as the pre-image, the original location of the point, and the output value as the image, the new

location of the point.

To help clarify the difference between a pre-image and an image,

we use special notation. If the pre-image is point A, then its image will be A’

(said A prime). The prime notation tells us that it is an image.

Let me do a few examples to help you see the correlation between algebra and geometry.

Given coordinate rule ( , ) ( 3, 6)T x y x y ,

determine the image of A (-1, 5).

Given coordinate rule ( , ) (2 , 1)G x y x y ,

determine the image of B (5, 2).

( , ) ( 3, 6)

( 1,5) (( 1) 3, (5) 6)

( 1,5) (2, 1)

T x y x y

T

T

( , ) (2 , 1)

(5,2) (2(5), (2) 1)

( , ) (10,3)

G x y x y

G

G x y

Pre-Image A (-1, 5) Image A’ (2, -1) Pre-Image B (5, 2) Image B’ (10, 3)


7. Given coordinate rule ( , ) (3 1,5 )M x y x y ,

determine the image of A (3, 2).

8. Given coordinate rule ( , ) ( 4, 3 )W x y x y ),

determine the image of B (-1, -2).


Both of these examples provided the pre-image (input) values and then asked you to solve for the image

(output value). We can work backwards through the process. Let us look at two where we know the image

and we work backwards through the coordinate rule to solve for the pre-image.

Given coordinate rule ( , ) ( 3, 6)T x y x y ,

determine the pre-image of C’ (3, 2).

Given coordinate rule ( , ) (2 , 1)G x y x y ,

determine the pre-image of D’ (-4, 11).

( , ) ( 3, 6)

3 3 6 2

0 8

T x y x y

x y

x y

( , ) (2 , 1)

2 4 1 11

2 10

G x y x y

x y

x y

Pre-Image C (0, 8) Image C’ (3, 2) Pre-Image D (-2, 10) Image D’ (-4, 11)


9. Given coordinate rule ( , ) (5 3,7 )T x y x y ,

determine the pre-image of C’ (13, -7). 10. Given coordinate rule

2( , ) ( 7 , )

3

yG x y x ,

determine the pre-image of D’ (-4, 4).

Sometimes functions are described as input/output machines. The example to the

right shows the input/output machine of ( ) 5f x x . When we input x = -1, the

function machine produces the output of 4.

I will use the function machine above to determine the missing output or input values.

Input = 3 Output = 11 Input = 0 ( ) 5

(3) 3 5

(3) 8

f x x

f

f

Output = 8

( ) 5

11 5

6

f x x

x

x

Input = 6

( ) 5

(0) 0 5

(0) 5

f x x

f

f

Output = 5


11. Input = -4 12. Output = -1 13. Input = 2.5

Output = ____

Input = ____

Output = ____

G.CO.A.2 WORKSHEET #1 – geometrycommoncore.com NAME: ___________________________ 1

1. Given functions, ( ) 2 1f x x , 2( ) 2g x x x , and 2

( )5

xh x , determine the value of the function for:

a) f(-5) = ______ b) g(-3) = ______ c) h(20) = ______

d) g(1) = ______ e) f(3

4

) = ______ f) h(

4

5)= ______

2. Given functions, ( )3

xr x , 2( ) 4s x x , and ( ) 4 8t x x , determine the value of the function for:

a) ( ) 2r x x = ______ b) ( ) 64s x x = ______ c) ( ) 28t x x = ______

d) ( ) 12s x x = ______ e) 2

( )5

r x x = ______ f) ( ) 4t x x = ______

3. Use coordinate rule, T (x, y) ------ > (x + 5, y – 1) to determine the missing coordinates.

a) A (-4, 5) A ‘ (____ , ____) b) B (____ , ____) B’ (-2, 6) c) C (-6, 4

5) C ‘ (____ , ____)

4. Use coordinate rule, G (x, y) ------ > (x2, 2y), to determine the missing coordinates.

a) A (-4, 5) A ‘ (____ , ____) b) B (-2, 6) B’ (____ , ____) c) C (____ , ____) C’ (16, -8)

G.CO.A.2 WORKSHEET #1 – geometrycommoncore.com 2

5. Use the function machine to determine the missing output or input values.

a) Input = -6 Output = ________

b) Input = 2

5

Output = ________

c) Input = ________ Output = -47

d) Input = ________ Output = 21

6. Determine the rule of the function machine.

First Input Second Input Third Input Fourth Input What is the rule for this function machine?

7. Determine three different function machine rules for the given input/output.

a) Possible Rule #1 b) Possible Rule #2 c) Possible Rule #3

8. Use the given coordinate rules to solve missing coordinates.

a) T (x, y) ------ > (x, y + 7) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (5, 0)

b) S (x, y) ------ > (-y, x) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (9, 7)

c) F (x, y) ------ > (5x, 3y) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (-5, 12)

d) G (x, y) ------ > (-x, -3x) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (-8, -24)

e) H (x, y) ------ > (2x - 1, y - 3) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (31, 15)

f) P (x, y) ------ > (x + 3, 2y) A (-4, 9) A’ (_____ , _____) B (_____ , _____) B’ (5, 8)

9. Write out the English translation for the following coordinate rule.

F (x, y) ------ > (x - 4, 2y) Coordinate Rule F __________________________________________________

10. Complete the Analogy. Input IS TO Output AS Pre-Image IS TO ______________.


Review of Functions

A correspondence between two sets A and B is a FUNCTION of A to B

IF AND ONLY IF each member of A corresponds to one and only one member of B.

In simpler terms, we might say for every x value there is only one y value.

We sometimes diagram these types of relationships with DOMAIN (x values) and RANGE (y values) boundaries.

This is a function.

Each value in set A (Domain)

has exactly one value in set B (Range).

This is a function.

Each value in set A (Domain) has exactly one value in set B (Range).

Notice that an x value can have the same y value as another x value. Each x value still only has one y

value.

This is a NOT function.

A member of Set A (Domain)

has two values in set B (Range), thus it is NOT A FUNCTION.

This is a function.

Each value in set A (Domain) has exactly one value in set B (Range).

Notice that an x value can have the same y value as

another x value. Each x value still only has one y value.

A quick way to determine if it is not a function is if one of the x values has more than one y value.

Notice that the domain and range could have the same number of elements or that the range could

have fewer elements and still be a function.

If the range had more elements than the domain then it will not be a function.


1. 2. 3. Is this a function? Is this a function? Is this a function?

YES OR NO

YES OR NO

YES OR NO


WHAT IS A MAPPING?

A correspondence between the pre-image and image is a MAPPING

IF AND ONLY IF each member of the pre-image corresponds to one and only one member of the image.

A mapping works exactly like a function….

The pre-image and the image could have the same number of points or the image could have fewer

points.

If an image has more points than the pre-image then it will not be a mapping.

This is a mapping.

If the pre-image is KLM,

K maps to N

L maps to O

M maps to P

This is a mapping.


M maps to R

K maps to Q

L maps to Q

This is NOT a mapping.

If the pre-image is KLM, K maps to S L maps to T

M maps to U M maps to V

This is a mapping.


K maps to T

L maps to T

M maps to T


4. 5. 6.

Is this a mapping? Pre-Image ABCDEF to Image GHI

Is this a mapping? Pre-Image ABCDEF to Image

GHIJKLM

Is this a mapping? Pre-Image ABCD to Image EFGH

YES OR NO

YES OR NO

YES OR NO

K

L

M

N

O

P

K

L

M

Q

R

K

L

M

S

V

U

T

K

L

M

T

A

B

C

D

E

F

I

G

H

A

B

C

D

E

FM

G

H

I

JK

L

A

BC

D

EF

G

H


Not all mappings are useful…… we don’t want to start with a triangle and end up with a segment or a point

after we map them. We want to study the mappings that preserve the number of points between the pre-

image and image. In Algebra when there is exactly the same number of elements in the domain as there is in

the range it is called a ONE TO ONE FUNCTION. In geometry, when you have the same number of points in

the pre-image as in the image, it is called a TRANSFORMATION.

ONE TO ONE FUNCTION NOT ONE TO ONE FUNCTION A TRANSFORMATION NOT A TRANSFORMATION

Transformations and One to One Correspondence Functions

A transformation is a one to one correspondence between the points of the pre-image and the points of the

image. We will only be studying transformations. A transformation guarantees that if our pre-image has

three points, then our image will also have three points.


7. 8. 9. Is this a transformation?

Pre-Image ABCDEF to Image GHI Is this a transformation?

Pre-Image ABCDEF to Image GHIJKLM

Is this a transformation? Pre-Image ABCD to Image EFGH

YES OR NO

YES OR NO

YES OR NO

A transformation must have a pre-image and an image

WITH THE EXACT SAME NUMBER OF POINTS.

K N

ML

L'

K' N'

M'K N

ML

S

T

U

A

B

C

D

E

F

I

G

H

A

B

C

D

E

FM

G

H

I

JK

L

A

BC

D

EF

G

H

G.CO.A.2 WORSHEET #2 – geometrycommoncore.com NAME: ______________________ 1

1. Given 2( )f x x

a) find f(-2) = _________ b) find f(2) = _________ c) f(0) = _________ d) f(5) = _________

e) Use the data in 1a-d to complete the diagram to the right.

f) is 2( )f x x a function? YES OR NO

g) is 2( )f x x a one to one function? YES OR NO

2. Given ( ) 5 6f x x



f) is ( ) 5 6f x x a function? YES OR NO

g) is ( ) 5 6f x x a one to one function? YES OR NO

3. Given ( ) 1f x x



f) is ( ) 1f x x a function? YES OR NO

g) is ( ) 1f x x a one to one function? YES OR NO

4. Determine whether the following are functions or not.

a)

b)

c)

Function? YES OR NO

Function? YES OR NO Function? YES OR NO

d)

e)

f)

Function? YES OR NO Function? YES OR NO Function? YES OR NO

G.CO.A.2 WORSHEET #2 – geometrycommoncore.com 2

5. Given that the pre-image is Quadrilateral ABCD, determine which of the following could be classified as a

mapping of the plane.

a)

b)

c)

Mapping? YES OR NO

Mapping? YES OR NO Mapping? YES OR NO

d)

e)

f)

Mapping? YES OR NO Mapping? YES OR NO Mapping? YES OR NO

6. Which of the problems in question 5, would be classified as TRANSFORMATIONS? ___________________

7. Transformations are a specific type of mapping. What makes them special from the general process of

mapping?

8. Jeff is given a question on a test about transformations. He is given two examples both with pre-image hexagon ABCDE. The question asks if the two shapes are a transformation or not. On the first one he said Yes they are transformations because they are identical but in a different location and on the second one he said No that it was not a transformation because they were different shapes. Is he correct? Explain why you agree or disagree.

Example #1 Example #2

D

A

B

C

E F

G

HD

A

B

C

E

HD

A

B

C

O

I

J

K

L

MN

D

A

B

C

K

M

P

O

D

A

B

C

Q

S

T

U V

X

YA

B

C D

E

FA

B

C D

E

F

L

M

NO

P

Q

G.CO.A.2 STUDENT NOTES & PRACTICE WS #3/#4 – geometrycommoncore.com 1

An ISOMETRIC TRANSFORMATION (RIGID MOTION) is a transformation that preserves the

distances and/or angles between the pre-image and image.

In simpler words…. An isometric transformation maintains the shape and size of the pre-image.

Example #1 Example #2 Example #3

Rotate (Turn) Translate (Slide) Reflection (Flip)

Rotations, Translations and Reflections are examples of isometric transformations.

A NON-ISOMETRIC TRANSFORMATION (NON-RIGID MOTION) is a transformation that does

not preserve the distances and angles between the pre-image and image.

Example #1 Example #2 Example #3

Dilation (Proportional) Stretch (Not Proportional) Stretch (Not Proportional)

Stretch – Where one dimension’s scale factor is different than the other dimension’s scale factor. Examples

#2 and #3 represent stretches. A stretch definitely distorts the shape making it a NON-ISOMETRIC

transformation.

Dilation – Where both dimension’s scale factors is the same. The shape is proportional, not identical. Dilation

changes the size of the shape making it a NON-ISOMETRIC transformation.

There are many ways to distort a shape but these two are the most popular types.

1. Circle those of the following that are isometric transformations. (there may be more than 1 answer)

Pre-Image

a) Image

b) Image

c) Image

D'E'

F'

B'C'

C

B

F

E

D

J'

I'

H

IJ

H'M'

L'

K'K

L

M

D'

C'

B'B

C

D

E G

F

F'

G'E'

K N

ML

L'

K' N'

M'


2. Use the previous example to determine which transformation took place, circle the answer.

Pre-Image

a) Reflection

Translation

Rotation

Dilation

Stretch

Other

b) Reflection

Translation

Rotation

Dilation

Stretch

Other

c) Reflection

Translation

Rotation

Dilation

Stretch

Other

3. Circle those of the following that are isometric transformations. (there may be more than 1 answer)

Pre-Image

a) Image

b) Image

c) Image

4. Use the previous example to determine which transformation took place, circle the answer.

Pre-Image

a) Reflection

Translation

Rotation

Dilation

Stretch

Other

b) Reflection

Translation

Rotation

Dilation

Stretch

Other

c) Reflection

Translation

Rotation

Dilation

Stretch

Other

5. Determine the coordinates of the image, plot the image and determine if it is an isometric transformation or not.

PRE-IMAGE Transformation COORDINATES PLOT THE IMAGE

a) Pre-Image Points

A (-1, 1) B (0, 4) C (4, 1) Coordinate Rule (x, y) (x, -y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No

Transformation Type: _____________________



a) Pre-Image Points

A (0, 0) B (1, 3) C (5, 0) Coordinate Rule (x, y) (x, -2y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

4

2

-2

-4

-5 5

A

B

C

A

B

C

G.CO.A.2 WORKSHEET #3 – geometrycommoncore.com NAME: ____________________ 1

1. Circle which of the following are isometric transformations.

Pre-Image a) b) c)

d) e) f)

2. Circle which of the following are isometric transformations.

Pre-Image a) b) c)

d) e) f)

3. Jane claims that any two circles are always isometric because the shape never changes. Is she correct?

YES or NO Explain your answer.


4. Determine if the pre-image and image are isometric and also which transformation produced the image.

PRE-IMAGE Circle Answer Circle Answer IMAGE

Isometry

Not Isometry

Rotation

Reflection Translation

Dilation Stretch Other

Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation Reflection

Translation Dilation Stretch Other


5. Determine if the pre-image and image are isometric and also which transformation produced the image.

PRE-IMAGE Circle Answer Circle Answer IMAGE

Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation



Isometry

Not Isometry

Rotation Reflection

Translation Dilation Stretch Other

G.CO.A.2 WORKSHEET #4 – geometrycommoncore.com NAME: __________________________________ 1



a) Pre-Image Points

A (1, -4) B (2, -1) C (6, -4)

Coordinate Rule

(x, y) (x – 5, y + 3)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


b) Pre-Image Points

A (1, -4) B (2, -1) C (6, -4)

Coordinate Rule (x, y) (-x, y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


c) Pre-Image Points

A (1, -4) B (2, -1) C (6, -4)

Coordinate Rule (x, y) (y, -x)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


d) Pre-Image Points

A (-5, -1) B (-4, 2) C (0, -1)

Coordinate Rule (x, y) (x, 3y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


e) Pre-Image Points

A (-2, -1) B (-1, 2) C (3, -1)

Coordinate Rule (x, y) (2x, 2y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


f) Pre-Image Points

A (1, -3) B (2, 0) C (6, -3)

Coordinate Rule (x, y) (-x, -y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


A

B

C

A

B

C

A

B

C

A

B

C

A

B

C

A

B

C




a) Pre-Image Points

A (1, -4) B (2, -1) C (6, -4)

Coordinate Rule

(x, y) (y, x)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


b) Pre-Image Points

A (-1, -2) B (0, 1) C (4, -2)

Coordinate Rule (x, y) (-2y, -x)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


c) Pre-Image Points

A (-3, 1) B (-2, 4) C (2, 1)

Coordinate Rule

(x, y) (-y - 2, x+3)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


d) Pre-Image Points

A (-6, -4) B (-3, 2) C (6, -4)

Coordinate Rule

(x, y) (.5x, .5y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


e) Pre-Image Points

A (0, 0) B (1, 3) C (5, 0)

Coordinate Rule (x, y) (-y, x)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


f) Pre-Image Points

A (3, -2) B (2, 1)

C (-2, -2)

Coordinate Rule (x, y) (x + y, y)

Image Points

A’ (_____,_____)

B’ (_____,_____)

C’ (_____,_____)

Isometry? Yes or No


A

B

C

A

B

C

A

B

C

A

B

C

A

B

C

C

B

A


To carry a shape onto itself is another way of saying that a shape has symmetry. There are two main types

of symmetry that a shape can have: line symmetry and rotation symmetry. Let us look at both of these.

LINE SYMMETRY (or REFLECTIONAL SYMMETRY)

A set of points has line symmetry if and only if there is a line, l, such that the

reflection through l of each point of the set is also a point of the set.

A figure in the plane has a line of symmetry if the figure can be mapped onto itself

by a reflection in the line.

1 Line Symmetry No Line Symmetry 1 Line Symmetry No Line Symmetry

6 Lines of Symmetry 4 Lines of Symmetry No Line Symmetry No Line Symmetry


1. 2. 3. 4.

How many lines of symmetry? ______




5. 6. 7. 8.





l



9. Shade it so it has: 10. Shade it so it has: 11. Shade it so it has: 12. Shade it so it has:

Exactly 1 Line of

Symmetry Exactly 2 lines of

symmetry Exactly 4 lines of

symmetry Exactly 4 lines of

symmetry

The maximum lines of symmetry that a polygon can have are equal to its

number of sides. The maximum is always found in the regular polygon,

because all sides and all angles are congruent.

ROTATIONAL SYMMETRY

A geometric figure has rotational symmetry if the figure is the image of itself under a rotation about a point

through any angle whose measure is strictly between 0° and 360°. 0 and 360 are excluded from counting as

having rotational symmetry because it represents the starting position.

120, Order = 3 180, Order = 2 120, Order = 3 180, Order = 2

ANGLE OF ROTATION - When a shape has rotational symmetry we sometimes want to know what the angle of

rotational symmetry is. To determine this we determine the SMALLEST angle through which the figure can be

rotated to coincide with itself. This number will always be a factor of 360.

ORDER OF ROTATION SYMMETRY -- The number of positions in which the object looks exactly the same is

called the order of the symmetry. When determining order, the last rotation returns the object to its original

position. Order 1 implies no true rotational symmetry since a full 360 degree rotation was needed.


13. 14. 15. 16.

Angle = _________

Order = _______

Angle = _________

Order = _______

Angle = _________

Order = _______

Angle = _________

Order = _______


17. 18. 19. 20.

Angle = _________

Order = _______

Angle = _________

Order = _______

Angle = _________

Order = _______

Angle = _________

Order = _______


21. Shade it so it has: 22. Shade it so it has: 23. Shade it so it has: 24. Shade it so it has:

Angle = 180

Order = 2

Angle = 90

Order = 4

Angle = 180

Order = 2

Angle = 90

Order = 4

G.CO.A.3 ACTIVITY #1 – geometrycommoncore.com NAME: ______________________________ 1

Provided are a few examples of each of the quadrilaterals. You will use these examples and some patty

paper to determine the symmetries found in quadrilaterals. The point inside the quadrilateral is the

intersection of the two diagonals.

For each quadrilateral copy it on to a piece of patty paper and then determine whether it has symmetry or

not. Place a summary of your findings in the provided chart.

To determine reflection symmetry, after copying the shape on the patty paper attempt folding the patty

paper so half of the shape carries onto itself. If this can happen then there is reflectional symmetry – find all

lines of reflectional symmetry.

To determine rotation symmetry, place your patty paper copy exactly onto the quadrilateral that you just

copied. Pin the center of the quadrilateral with your finger or pencil and then begin rotating the patty paper.

I suggest placing a symbol at one of the vertices so you can easily track how much you have rotated the shape.

PARALLELOGRAM RECTANGLE

RHOMBUS SQUARE

G.CO.A.3 ACTIVITY #1 – geometrycommoncore.com 2

TRAPEZOID

ISOSCELES TRAPEZOID

KITE


In the chart below, draw a diagram showing ALL lines of reflectional symmetry for that

quadrilateral and then fill in the number of lines of symmetry you found.

In the chart below, draw a diagram listing ALL locations where the rotation

symmetry occurred. Remember if a shape has no rotational symmetry, then its

order is 1.

QUADRILATERAL SYMMETRIES SUMMARY CHART

Reflection Symmetry Diagram

Reflection Symmetry Summary

Rotation Symmetry Diagram

Rotation Symmetry Summary

Parallelogram

The number of lines of symmetry is

______________

The rotation symmetry order is

______________

Rectangle


______________


______________

Rhombus


______________


______________

Square


______________


______________

Trapezoid


______________


______________

Isosceles Trapezoid


______________


______________

Kite


______________


______________

What are the patterns that you see from this data?

2

1

G.CO.A.3 ACTIVITY #2 – geometrycommoncore.com NAME: ______________________________ 1

Flags are often great example of symmetry. Search the internet to find the flags of the world. Find flags that meet the criteria provided below. Sketch and color the flag and provide the name of the county. For example, the Canadian Flag has 1 line of symmetry. You are not allowed to use Canada, Trinidad & Tobago, Thailand, Ukraine, or Qatar for your answers because I use them as examples. Also you cannot use a flag more than once on this activity. Possible URL: http://www.sciencekids.co.nz/pictures/flags.html 1. LINE SYMMETRY

(Flag Site)

(Introduction)

0 LINES OF SYMMETRY 1 LINE OF SYMMETRY 2 LINES OF SYMMETRY 2 LINES OF SYMMETRY

Country _____________ Country _____________ Country _____________ Country _____________

2. ROTATION SYMMETRY

ORDER 2 ORDER 2 2 LINES & ORDER 2 2 LINES & ORDER 2

Country _____________ Country _____________ Country _____________ Country _____________

3. Determine the reflection and rotation symmetries of the provided flags.

(Colors must match to be symmetrical and you can NOT use these flags for questions #1 and #2)

Trinidad & Tobago Thailand Ukraine Qatar

Lines of Symmetry

_______________

Lines of Symmetry

_______________

Lines of Symmetry

_______________

Lines of Symmetry

_______________ Rotation Order

_______________

Rotation Order

_______________

Rotation Order

_______________

Rotation Order

_______________

http://www.sciencekids.co.nz/pictures/flags.html


4. Most flags are rectangular in shape. Can you find countries that have:

a) a square flag ___________________________________________

b) a non-rectangular/square flag ___________________________________________

5. What does the all white flag mean? ________________________________________________

6. What is the study of flags called? __________________

7. What is the name of the flag that is black and has a skull and cross bones on it? __________________

8. Which country once had a national flag that was a solid green color and nothing else? _________________

9. a) Design your own flag. The flag must have symmetry – you can choose how much symmetry it has and

whether it is rotational or reflectional symmetry. Color and Detail the Flag!! Use symbols, colors and shapes

that somehow represent you and your country.

b) Explain why you designed it the way you did.

c) Describe the symmetries found in your flag.

Online flag designer kit (http://www.flag-designer.appspot.com/)

G.CO.A.3 WORKSHEET #1 – geometrycommoncore.com NAME: __________________________ 1

1. Draw in the lines of symmetry for each of the shapes. If none, leave the diagram blank.

a) b) c) d)

e) f) g) h)

(Parallelogram)

(Regular Hexagon)

2. Use the diagrams from question #1 to determine the order and angle of rotation symmetry for the

following shapes. If none, write none.

a) Order = ____________

Angle = ____________

b) Order = ___________

Angle = ____________

c) Order = ____________

Angle = ____________

d) Order = ___________

Angle = ____________

e) Order = ___________

Angle = ____________

f) Order = ____________

Angle = ____________

g) Order = ____________

Angle = ____________

h) Order = ___________

Angle = ____________

3. Draw a figure that meets the given symmetry requirements. It must have:

a) line symmetry, but not rotational symmetry.

b) rotational symmetry, but not line symmetry.

c) exactly 3 lines of symmetry.

4. a) Draw three different figures, each having exactly one line of symmetry.

b) Do you notice any similarities in these three shapes?


5. Given the shape, shade it so that it has reflectional symmetry.

a) One line of symmetry b) One line of symmetry c) Two lines of symmetry d) Two lines of symmetry

6. Given the shape, shade it so that it has rotational symmetry.

a) Order 2 b) Order 2 c) Order 4 d) Order 4

7. Each figure shows part of a shape with a center of rotation and a given rotational symmetry. Complete the figure. a) Order 4 b) Order 3 c) Order 8

8. What is the relationship between the order of the shape and the angle of rotation?

9. Here are the letters of the alphabet. Classify them into the given categories.

A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

One Line of Symmetry Two Lines of Symmetry Rotational Symmetry

No Symmetry

G.CO.A.3 WORKSHEET #2 – geometrycommoncore.com NAME: __________________ 1

1. Provided is half of a shape and the line of reflection.

2. Given a regular hexagon, how can you alter it so that instead of having six lines of reflection it only has two? Draw the altered hexagon and draw in the two lines of symmetry.

3. Determine the reflectional and rotational symmetries of triangles.

Triangle Classified by Sides Lines of Reflection Rotation Symmetry

Scalene (No Congruent Sides)

Yes or No ? __________ How many? _________

Yes or No ? __________ Order? ____________

Isosceles (At least two congruent sides)

Yes or No ? __________ How many? _________

Yes or No ? __________ Order? ____________

Equilateral (Three congruent sides)

Yes or No ? __________ How many? _________

Yes or No ? __________ Order? ____________

4. Could a triangle have two lines of symmetry? Why or why not?

a) Complete the drawing of the shape. b) Use dash marks to show equal sides – label each of the sides to show what is equal to what in the shape. c) Do the same for angles, label which angles are equal to each other in the shape using matching symbols. d) Finally, what do you notice about a shape that has one line of symmetry?


5. Given AB , determine the following.

a) How many lines of symmetry does it have? _____ b) Draw in the line(s) of symmetry. c) What is the unique name for one of the lines of symmetry? d) What is its rotational symmetry order? _________

6. A triangle either has zero, one or three lines of symmetry. What is the possible number of lines

symmetries for a hexagon? Draw in the ones that you found.

0 Lines 1 Line 2 Lines 3 Lines

4 Lines 5 Lines 6 Lines

7. These two shapes have both rotational and reflectional symmetry. What do they have in common?

8. Determine the following symmetry characteristics for these REGULAR polygons.

Lines of Symmetry _________

Rotational Order __________







A

B

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