8th Grade Math

UNIT 1Transformations, Congruence, and

SimilarityUnit Essential Questions

What are the transformations?

How do you construct and use angle relationships?

How can transformations and angle relationships be used to depict real-world situations?

WARM UP/Vocabulary

1. A __________________________________________________ is formed by the intersection of two number lines.

2. __________________ - the horizontal axis.

3. __________________ - the vertical axis.

4. The ____________________ is the point where the x and y axes intersect.

5. _____________________________ - names the location of the point in the plane, usually written (x, y).

Transformations: the movement of a shape into a different position on a coordinate plane.

Dilation Translation Reflection Rotation

8th Grade Math Unit 1

y - axis

x - axisorigin

Ordered Pair (4,7)

Lamar Math LessonsTransformations

MGSE8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

MGSE8.G.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations: given two similar two-dimensional figures,

describe a sequence that exhibits the similarity between them.

RATING LEARNING SCALE

4 I am able to• Represent a transformations and apply them real-world situations

3I am able to

• Construct and transform a figure from given points.• Interpret graphs and conclude which transformation, or combination of, was/were

performed.2 I am able to

• Identify each of the transformations.1 I am able to

• Understand how to read and plot points on a coordinate plane

TARGET

Dilation: Making an object larger or smaller than the original.

A dilation is a transformation that produces an image that has the same ________________ as the

original, but not necessarily the same ________________.

- A dilation is ______________________ to the original figure.

- Dilations are centered around the origin (0,0), unless otherwise stated.

A Scale Factor is a ratio of the image length to the original length: image lengthorginal length

- If the scale factor is greater than 1, the figure is _________________.

- If the scale factor is between 0 and 1, the figure is ________________.

- If the scale factor is equal to 1, the figure is _______________.

RULE: (x,y) (kx,ky) where k represents the scale factor.

EXAMPLE 1: Writing the Rule:If the scale factor is 3, how would you write the rule?

EXAMPLE 2: Find the Coordinates:Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it

the scale factor is 4?

A’: ____________ B’: ____________ C’: ____________

EXAMPLE 3: Enlarging an object:Use the coordinates to draw the original figure (pre-Image) and its copied figure (Prime).

A(1,1) --------> A’: ______________

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B(3,-4) --------> B’: ______________

C(-4,5) --------> C’: ______________

Scale Factor: 2

EXAMPLE 4: Reducing an object:Use the coordinates to draw the original figure (pre-Image) and its copied figure (Prime).

A(9,3) --------> A’: ______________

B(0,-6) --------> B’: ______________

C(-3,6) --------> C’: ______________

Scale Factor: 1/3

Translation: the movement of a shape by sliding it.

A translation is a transformation that produces an image that has the same ________________ and

________________ as the original figure.

- A translation is ______________________ to the original figure.

- Translations are centered around the origin (0,0), unless otherwise stated.

RULE: (x,y) (x+a,y+b) where a and B represent any real number.

Example: (3,12) (x+4,y+5). If Point Z is at (3,12), then where would Z’ be translated?

3 + 4 = 7 and 12 + 5 = 17, so the answer is (7,17)

EXAMPLE 1: Writing the Rule:A. If A is 19 and B is 89, how would you write the rule?

B. If A is -3 and B is 28, how would you write the rule?

EXAMPLE 2: Find the Coordinates:

a. Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it the rule is (x-2,y+6)?

A’: ____________ B’: ____________ C’: ____________

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b. Triangle HIJ has vertices h(8,-16), i(19,89), and j(-5,101). What are the vertices of its prime image it the rule is (x-34,y-15)?

H’: ____________ I’: ____________ J’: ____________

EXAMPLE 3: Translating an object:Use the coordinates to draw the original figure (pre-Image) and its copied figure (Prime).

a. A(1,1) -------> A’: ______________ B. L(9,3) --------> L’: ______________

B(3,-4) --------> B’: ______________ M(0,-6) --------> M’: ______________

C(-4,5) --------> C’: ______________ N(-3,6) --------> N’: ______________

Rule: (x+4,y-3) Rule: (x-5,y)

Reflection: the movement of a shape by creating a mirror image.

A reflection is a transformation that produces an image that has the same ________________ and

________________ as the original figure.

- A reflection is ______________________ to the original figure.

- Reflections are created across reflection lines or lines of reflection.

o The line of reflection can be any line on the coordinate plane.

RULE: (x,y) Count the distance to the reflection line, then count that same amount past it.

- If the Line of reflection is an axis, then that variable isn’t affected by the flip and the other

variable takes the opposite sign.

- When the reflection line is a different x or y value, then only that variable is affected.

Example: Z(3,12) reflected across the x-axis would yield Z’(3,-12)

Z(3,12) reflected across the y-axis would yield Z’(-3,12)

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EXAMPLE 1: Find the Coordinates:

a. Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image if the rule is reflect across the x-axis?

A’: ____________ B’: ____________ C’: ____________

b. Triangle HIJ has vertices H(8,-16), I(19,89), and J(-5,101). What are the vertices of its prime image if the rule is reflect across the y-axis?

H’: ____________ I’: ____________ J’: ____________

EXAMPLE 2: Reflecting an object:Use the coordinates to draw the original figure (pre-Image) and its copied figure (Prime).

a. A(1,1) -------> A’: ______________ B. L(9,3) --------> L’: ______________

B(3,-4) --------> B’: ______________ M(0,-6) --------> M’: ______________

C(-4,5) --------> C’: ______________ N(-3,6) --------> N’: ______________

Rule: Reflect across x=1 Rule: Reflect across y=2

Rotation: a circular movement around a fixed point or axis.

A rotation is a transformation that produces an image that has the same ________________ and

________________ as the original figure.

- A rotation is ______________________ to the original figure.

- Rotations are created around a given point, usually the origin.

- Rotations follow 2 directions: clockwise (CW) and counterclockwise (CCW).

- Every 90 degrees, the numbers switch places and the order pair takes on the sign of the quadrant it’s in.

RULE(s):

- 90 degrees clockwise: (x,y) (y,-x)

- 180 degrees clockwise: (x,y) (-x,-y)8th Grade Math Unit 1

- 270 degrees clockwise: (x,y) (-y,x)

Example: Z(3,12) rotated 90 degrees counterclockwise is Z’(-12,3)

Z(3,12) rotated 180 degrees would yield Z’(-3,-12)

EXAMPLE 1: Find the Coordinates:

a. Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it is rotation 90° CCW about the origin?

A’: ____________ B’: ____________ C’: ____________

b. Triangle HIJ has vertices h(8,-16), i(19,89), and j(-5,101). What are the vertices of its prime image it is rotated 270° CW about the origin?

H’: ____________ I’: ____________ J’: ____________

EXAMPLE 2: Rotating an object:Use the coordinates to draw the original figure (pre-Image) and its copied figure (Prime).

a. A(1,1) -------> A’: ______________ B. L(9,3) --------> L’: ______________

B(3,-4) --------> B’: ______________ M(0,-6) --------> M’: ______________

C(-4,5) --------> C’: ______________ N(-3,6) --------> N’: ______________

Rule: 180° Rotation Rule: 90°CW Rotation

8th Grade Math Unit 1

WARM UP/VocabularyOn a separate sheet of paper, define the following words.

1. Acute Angle ______

2. Adjacent Angles ______

3. Complementary Angles ______

4. Linear Pair ______

5. Obtuse Angle ______

6. Right Angle ______

7. Supplementary Angles ______

Based on the definitions you’ve found, match the following pictures to their word.

A. B. C.

D. E.

EXAMPLE 1: Find the missing pieceName the type(s) of angle and solve for the unknown angle.

A. B.

8th Grade Math Unit 1

Angle RelationshipsMGSE8.G.1: Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same

measure; parallel lines are taken to parallel lines.MGSE8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from

the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

MGSE8.G.5: Use informal argument to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

RATING LEARNING SCALE

4 I am able to• Use angle relationships to solve real-world problems.

3I am able to

• Construct angle relationships.• Interpret graphs and conclude which transformation, or combination of, was/were

performed.2 I am able to

• Identify angle relationships.1 I am able to

• Understand the definitions of the different angle relationships.

TARGET

Smaller Angle TotalLarger Angle

Finding the Unknown Angles and Values:When solving for the missing variable or angle, we must first ask ourselves, “What is the total of the figure?” Look at the figure below. Find the value of each angle.

The figure shows 3 types of angles: Adjacent, Linear Pair, and Supplementary. By definitions,

linear pairs and supplementary angles have to equal to 180 degrees, so the total of the figure

should be 180.

1. Now lets set up our equation. 3q + 15q+18 = 180

2. Next, let’s combine our like terms. 18q+18 = 180

3. Isolate our variable by moving everything away from it. 18q+18 = 180

-18 -18

-------------------

18q = 162

4. We must undo the multiplication being done in the number variable. 18q = 162

18 18

5. Find the value of the missing variable. Q = 9

6. Substitute for that variable in each of the angle expressions and solve.

a. Smaller angle: 3(9) = 27

b. Larger Angle: 15(9) + 18 = 153

7. Check your work by adding the two answers together. If it checks out, then you know that

your answer is correct. 27 + 153 = 180. We are correct!!!

8th Grade Math Unit 1

EXAMPLE 1: Find the missing pieceSolve for the unknown variable and the size of each angle.

A. *This is a right angle. B.

Warm Up/VocabularyOn a separate sheet of paper, define the following words.

1. Alternate Exterior Angles _______

2. Alternate Interior Angles _______

3. Corresponding Angles _______

4. Same Side Exterior Angles _______

5. Same Side Interior Angles _______

6. Vertical Angles _______

Based on the definitions you’ve found, identify an appropriate angle relationship.

Congruent Relationships Supplementary Relationships

8th Grade Math Unit 1

Parallel Lines: two lines that lie on the same plane and never intersect.Transversals: a line that crosses at least two other lines.

Example: Lines M and N are parallel and line L is the transversal.

EXAMPLE 1: Identify the Appropriate RelationshipGive the correct relationship between the given Angles. Use the figure on the previous page.

A. ∠1∧∠7 B. ∠5∧∠7 C. ∠3∧∠7

EXAMPLE 2: Identify the Measure of the Missing Angle.Using what you know about angle relationships, find the measure of the missing angles.

A. ∠1: ________

B. ∠6: ________

C. ∠2: ________

D. ∠5: ________

EXAMPLE 3: Find the Missing Value8th Grade Math Unit 1

Find the value of the variable indicated. Remember to check your work.

A. X = ________________ B. Y = _______________

RATE YOUR UNDERSTANDING (Using the learning scale from the beginning of the lesson)Circle one: 4 3 2 1

8th Grade Math Unit 1