*Unit Overview*

**Essential Questions: **

**Lesson Plans and Seeds**

Lesson Plan A.3: Transformations of 2D Figures

Lesson Seed A.1: Demonstrate Transformations

Lesson Seed A.1a: Floor Transformations

Lesson Seed A.5: Exterior Angles of a Triangle

Lesson Seed A.5: Sum of Angles in a Triangle

**Download Seeds, Plans, and Resources (zip)**

**Content Emphasis By Clusters in Grade 8**

**Progressions from Common Core State Standards in Mathematics**

Lesson seeds are ideas that can be used to build a lesson aligned to the CCSS. Lesson seeds are not meant to be all-inclusive, nor are they substitutes for instruction. When developing lessons from these seeds, teachers must consider the needs of all learners. It is also important to build checkpoints into the lessons where appropriate formative assessment will inform a teachers instructional pacing and delivery.

### Unit Overview

This unit expands prior knowledge of graphing points on the coordinate plane to solve authentic problems and of drawing polygons with given characteristics. The unit requires students to analyze and compare two-dimensional figures on the coordinate plane using concepts of distance, angle measures, similarity, and congruence.

**Teacher Notes:**

Students should:

- understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane.
- recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
- draw polygons in the coordinate plane given coordinates for the vertices.
- use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate.

**Enduring Understandings:**

At the completion of this unit on understanding congruence and similarity, the student will understand that:

- two-dimensional figures can be described, classified, and analyzed by their attributes.
- spatial sense inherent in transformational geometry provides ways to visualize, interpret, and reflect on our physical surroundings.
- congruent figures can be formed by a series of transformations.
- similar figures can be formed by a series of transformations.
- analyzing transformational geometric relationships fosters reasoning and justification skills.
- parallel lines cut by a transversal form angles with special relationships to one another.

**Focus Standards (Listed as Examples of Opportunities for In-Depth Focus in the PARCC Content Framework document):
**

- PARCC has not provided examples of opportunities for in-depth focus related to understanding congruence and similarity.

**Possible Student Outcomes:**

The Students will be able to:

- recognize transformed geometric figures on the coordinate plane as the product of reflections, rotations, translations, and/or dilations.
- justify when/why geometric figures that have been transformed on the coordinate plane are congruent or similar.
- identify the relationships among pairs of alternate interior, alternate exterior, and corresponding angles formed by two parallel lines cut by a transversal.
- identify pairs of angles formed by two parallel lines cut by a transversal that always must be congruent; that always must be supplementary.
- solve authentic problems based on geometric transformations; on parallel lines cut by a transversal

**Evidence of Student Learning:**

*The Partnership for Assessment of Readiness for College and Careers (PARCC) has awarded the Dana Center a grant to develop the information for this component. This information will be provided at a later date. The Dana Center, located at the University of Texas in Austin, encourages high academic standards in mathematics by working in partnership with local, state, and national education entities. Educators at the Center collaborate with their partners to help school systems nurture students' intellectual passions. The Center advocates for every student leaving school prepared for success in postsecondary education and in the contemporary workplace.*

**Fluency Expectations and Examples of Culminating Standards:**

**Common Misconceptions:**

Students may

- confuse situations that require adding with multiplicative situations in regard to scale factor.
- have difficulty differentiating between congruency and similarity
- assume that any combination of three sides or three angles will form a congruence condition.
- not recognize congruent geometric figures because of different orientations.
- confuse terms, such as clockwise and counter-clockwise.
- think that the line of reflection always must be vertical or horizontal, for example, across the y-axis or the x-axis.
- not realize that rotations are not always the origin, but can be about any point.

**Interdisciplinary Connections:**

*Interdisciplinary connections fall into a number of related categories:*

*Literacy standards within the Maryland Common Core State Curriculum*

*Science, Technology, Engineering, and Mathematics standards*

*Instructional connections to mathematics that will be established by local school systems, and will reflect their specific grade-level coursework in other content areas, such as English language arts, reading, science, social studies, world languages, physical education, and fine arts, among others.*

**Sample Assessment Items: ***The items included in this component will be aligned to the standards in the unit and will include:*

*Items purchased from vendors*

*PARCC prototype items*

*PARCC public release items*

*Maryland Public release items*

*Formative Assessment*

**Interventions/Enrichments/PD: ***(Standard-specific modules that focus on student interventions/enrichments and on professional development for teachers will be included later, as available from the vendor(s) producing the modules.)*

**Vocabulary/Terminology/Concepts: ***This section of the Unit Plan is divided into two parts. Part I contains vocabulary and terminology from standards that comprise the cluster, which is the focus of this unit plan. Part II contains vocabulary and terminology from standards outside of the focus cluster. These “outside standards” provide important instructional connections to the focus cluster.*

**
Part I – Focus Cluster
Understand Congruence and Similarity Using Physical Models, Transparencies, or Geometry Software**

** transformations: **anytime you move, shrink, or enlarge a figure, you have to make a transformation of that figure. This kind of transformation includes rotations, reflections, translations and dilations.

http://www.mathsisfun.com/geometry/resizing.html

**It is also called a turn. Rotating a figure means turning the figure around a point. The point can be on the figure or it can be some other point. The shape still has the same size, area, angles and line lengths.**

*rotation:***It is also called a flip. When a figure is flipped over a line. Each point in a reflection image is the same distance from the line as the corresponding point in the original shape. The shape still has the same size, area, angles and line lengths.**

*reflection:***It is also called a slide. In a translation, every point in the figure slides the same distance in the same direction. The shape still has the same size, area, angles and line lengths.**

*translation:***This refers to comparing an image (geometric figure following a transformation) to its preimage (figure prior to the transformation).**

*are taken to:***Figures that keep the same size and shape as they transform.**

*rigid:*
*transformation notation: *

**Preimage ** is the figure prior to the transformation. (A, B, C).
**Image **is the figure after the transformation. (A', B', C') A', B', C 'are called A prime, B prime, and C prime. A → A' B → B' C → C'

Angles **are taken to angles** and lines **are taken to** lines in the picture below.

**congruent:** Figures that have the same shape and size are congruent. Sides are congruent if they are the same length. Angles are congruent if they have the same number of degrees.

**algebraic rules of transformations:**

reflections: rx-axis(x,y )= (x,ȳ)

rx-axis(x,y )= (x̄,y)

ry = x(x,y )= (y,x)

ry=x̄(x,y )= (x̄,ȳ)

R180°(x,y )= (x̄, ȳ)

R270°(x,y )= (y, x̄)

R-90°(x,y )= (y, x̄)

translations: Ta,b(x,y )= (x + a, y + b)

dilations: Dk(x,y )= (kx, ky)

** non-rigid:** Figures that change size but not shape as they transform.

** similar**: Two polygons are similar polygons if corresponding angles have the same measure and corresponding sides are in proportion. We say that there is similarity between similar figures if the two facts are true.

** dilation**: It is also called resizing. The shape becomes bigger or smaller. If you have to dilate or resizing to make one shape become another, they are similar. These figures are non-rigid because they do not stay the same

** scale factor**: The amount by which the image grows (dilates) or shrinks (reduces) is called the "Scale Factor". The general formula for dilating a point with coordinates of (2,4) by a scale factor of ½ is: (2 • ½, 4 • ½) or (1, 2);

**scale factor of 1**produces a congruent figure: (2 • 1, 4 • 1) or (2, 4).

** angle sum**: Draw any

__triangle__ABC and cut out the three

__angles__.

Rearrange the three angles to form a __straight angle__ on a straight line. The angle sum of a triangle is 180° because, as shown in the diagram below:

m∠A + m∠B + m∠C = 180°

**exterior angle of a triangle (x):**

**Solution:**

x + 58° = 180° {Sum of adjacent angle4s forming a straight line}

x + 58° - 58° = 180° - 58° {Subtract 58° from both sides}

x = 122°

Also, y + 60° + 58° = 180° {Angle sum of a triangle}

y + 118° = 180° {Subtract 118° from both sides}

y + 118° -118° = 180° - 118°

y = 62°

Thus, x = 122°, y = 62°

m∠B + m∠y = 122° and m∠x = 122° Therefore m∠B + m∠y = m∠x

60° + 62° - 122° 60° + 62° - 122°

**parallel lines cut by a transversal:**

Line r and line s are parallel.

Line t is the transversal.

They are corresponding angles. m∠4 = m∠6 and m∠3 = m∠5.

They are alternate interior angles. m∠1 = m∠7 and m∠2 = m∠8.

They are alternate exterior angles.

**angle-angle criterion for similarity of triangles:**If two angles of one triangle are congruent to two angle of another triangle then the triangles are similar.

**Part II – Instructional Connections outside the Focus Cluster**

**proof of the Pythagorean Theorem and it converse:** In any right triangle, the sum of the squares of the legs equals the square of the hypotenuse (leg 2 + leg 2 = hypotenuse 2 ). The figure below shows the parts of a right triangle.

**distance formula:** The distance (d) between the points A = (x1, y1) and B = (x2, y2) is given by the formula:

The distance formula can be obtained by creating a triangle and using the Pythagorean Theorem to find the length of the hypotenuse. The hypotenuse of the triangle will be the distance between the two points.

The formula below shows an application of the Pythagorean Theorem for right triangles: