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UNIT OF STUDY-1GSE GEOMETRYMATHEMATICSTEACHER RESOURCE GUIDE

TRANSFORMATIONS IN THE COORDINATE PLANE:(August 3 – August 31)

In this unit students will:

Use and understand definitions of angles, circles, perpendicular lines, parallel lines, and line segments based on the undefined terms of point, line, distance along a line and length of an arc.

Describe and compare function transformations on a set of points as inputs to produce another set of points as outputs, including translations and horizontal or vertical stretching

Represent and compare rigid and size transformations of figures in a coordinate plane using various tools such as transparencies, geometry software, interactive whiteboards, waxed paper, tracing paper, mirrors and digital visual presenters

Compare transformations that preserve size and shape versus those that do not. Describe rotations and reflections of parallelograms, trapezoids or regular polygons that map each

figure onto itself. Develop and understand the meanings of rotation, reflection and translation based on angles,

circles, perpendicular lines, parallel lines and line segments. Transform a figure given a rotation, reflection or translation using graph paper, tracing paper,

geometric software or other tools. Create sequences of transformations that map a figure onto itself or to another figure.

Page: 1Some of the language used in this document is adapted from the GA Frameworks and Common Core Progressions Documents.Revised as of 8/5/2016

Table of ContentsStandards Addressed............................................................................................................................................................4


Enduring Understandings.....................................................................................................................................................8

Lesson One Progression........................................................................................................................................................8

Lesson Two Progression.....................................................................................................................................................21

Sample Fluency Strategies for High School........................................................................................................................34

Duration: maximum of 8 instructional days on an A/B schedule


Standards Addressed

Georgia Standards of Excellence – Mathematics

Content Standards

(Cluster emphasis is indicated by the following icons. Please note that 70% of the time should be focused on the Major Content. ◊ Major Content □ Supporting Content)

□Experiment with transformations in the plane

MGSE9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MGSE9-12.G.CO.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 to those that do not

(e.g., translation versus horizontal stretch).

MGSE9-12.G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

MGSE9-12.G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

MGSE9-12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Standards for Mathematical Practice

SMP1: Make sense of problems and persevere in solving them.

While doing tasks in this unit, students will need to make sense of the geometric transformations of shapes by representing those transformations


algebraically. Students must understand and apply the various types of transformations and will need to persevere through the problem solving process in order to arrive at a solution.

Students: Because Teachers:

Read the task carefully. Draw pictures, diagrams, tables,

or use objects to make sense of the task.

Discuss the meaning of the task with classmates.

Make choices about which solution path to take.

Try out potential solution paths and make changes as needed.

Check answers and makes sure solutions are reasonable and make sense.

Explore other ways to solve the task.

Persist in efforts to solve challenging tasks, even after reaching a point of frustration.

Provide rich tasks aligned to the standards.

Allow students time to initiate a plan; uses question prompts as needed to assist students in developing a pathway.

Continually ask students if their plans and solutions make sense.

Question students to see connections to previous solution attempts and/or tasks to make sense of current task.

Consistently ask students to defend and justify their solution by comparing solution paths.

Provide appropriate time for students to engage in the productive struggle of problem-solving.

Differentiate to keep advanced students challenged during work time.

SMP2: Reason abstractly and quantitatively.Students connect graphical representations with the function (symbolic) representation of translations.


Use mathematical symbols to represent situations

Take quantities out of context to work with them (decontextualizing)

Put quantities back in context to see if they make sense (contextualizing)

Consider units when determining if the answer makes sense in terms of the situation

Provide a variety of problems in different contexts that allow students to arrive at a solution in different ways

Use think aloud strategies as they model problem solving

Attentively listen for strategies students are using to solve problems

SMP3: Construct viable arguments and critique the reasoning of others.


Students should be able to justify why the image represents a particular transformation. Students should be able to critique the reasoning of others by making comparisons between the visual representations of a transformation to their verbal/ algebraic descriptions of the transformations.


Make and tests conjectures. Explain and justifies their thinking

using words, objects, and drawings.

Listen to the ideas of others and decides if they make sense.

Ask useful questions. Identify flaws in logic when

responding to the arguments of others.

Elaborate with a second sentence (spontaneously or prompted by the teacher or another student) to explain their thinking and connect it to their first sentence.

Talks about and asks questions about each other’s thinking, in order to clarify or improve their own mathematical understanding.

Revise their work based upon the justification and explanations of others.

Encourage students to use proven mathematical understandings, (definitions, properties, conventions, theorems, etc.) to support their reasoning.

Question students so they can tell the difference between assumptions and logical conjectures.

Ask questions that require students to justify their solution and their solution pathway.

Prompt students to respectfully evaluate peer arguments when solutions are shared.

Ask students to compare and contrast various solution methods.

Create various instructional opportunities for students to engage in mathematical discussions (whole group, small group, partners, etc.).

SMP4: Model with mathematics.

In this unit, it will be critical for students to make sense of geometric concepts by modeling them with algebraic tools. By applying coordinates to a geometric transformation, students will be able to generalize what is happening to the shapes. Students use computer software to demonstrate transformations and represent transformations graphically.


Use mathematical models (i.e. formulas, equations, symbols) to solve problems in the world

Use appropriate tools such as

Provide opportunities for students to solve problems in real life contexts

Identify problem solving contexts connected to student interests


objects, drawings, and tables to create mathematical models

Make connections between different mathematical representations (concrete, verbal, algebraic, numerical, graphical, pictorial, etc.)

Check to see if an answer makes sense within the context of a situation and changing the model as needed

SMP5: Use appropriate tools strategically.

Students can use a variety of tools to help them perform and understand transformations. Students will need to select appropriate tools (graph paper, Desmos, Geogebra, sketchpad, geoboard, mira, and transparencies) to model transformations.


Use technological tools to explore and deepen understanding of concepts

Decide which tool will best help solve the problem. Examples may include:

o Calculatoro Concrete modelso Digital Technologyo Pencil/papero Ruler, compass, protractor

Estimate solutions before using a tool

Compare estimates to solutions to see if the tool was effective

Make a variety of tools readily accessible to students and allowing them to select appropriate tools for themselves

Help students understand the benefits and limitations of a variety of math tools

SMP7: Look for and make use of structure.

Students can use software (geogebra, desmos, TI graphing calculator, sketchpad) to perform transformations, then try to generalize and understand the transformation that has taken place. Students generalize what they see with reflections into a formal definition of this transformation.


Find structure and patterns in numbers

Provide sense making experiences for all students


Find structure and patterns in diagrams and graphs

Use patterns to make rules about math

Use these math rules to help them solve problems

Allow students to do the work of using structure to find the patterns for themselves rather than doing this work for students

SMP8: Look for and express regularity in repeated reasoning.

Students look for patterns in their results on various transformations to develop a general rule for the symbolic representation of the transformation.


Looking for patterns when working with numbers, diagrams, tables, and graphs

Observing when calculations are repeated

Using observations from repeated calculations to take shortcuts

Providing sense making experiences for all students

Allowing students to do the work of finding and using their own shortcuts rather than doing this work for students

Note: All of the Standards for Mathematical Practice (SMPs) are critical to students fully and appropriately attending to the content. Not all SMPs will occur in every lesson, however SMPs 1, 3, and 6 should be regularly apparent. All SMPs should be taught in tandem with the content standards.

Enduring Understandings

In order to support deep conceptual learning it is important that student leave this unit experience with the following understandings:

The concepts of congruence, similarity, and symmetry can be understood from the perspective of geometric transformation.

Fundamental are the rigid motions: translations, rotations, reflections, and combinations of these, all of which are here assumed to preserve distance and angles (and therefore shapes in general).

Reflections and rotations each explain a particular type of symmetry, and the symmetries of an object offer insight into its attributes.

Lesson One Progression


Focus Standard(s)

Experiment with transformations in the plane

MGSE9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

MGSE9-12.G.CO.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 to those that do not(e.g., translation versus horizontal stretch).

MGSE9-12.G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software.

Performance Objectives

As a result of their engagement with this unit….

MGSE9-12.G.CO.1 - SWBAT use mathematical vocabulary (key terms in geometry) IOT describe transformations in the coordinate plane

MGSE9-12.G.CO.2/5 - SWBAT graph pre-images and images under reflection and translation using a variety of tools (miras, patty paper, geometer’s sketchpad, and other virtual java tools) IOT interpret transformations as functions.

MGSE9-12.G.CO.2/5 - SWBAT graph pre-images and images under reflection and translation using a variety of tools (miras, patty paper, geometer’s sketchpad, and other virtual java tools) IOT analyze the properties of transformations.

MGSE9-12.G.CO.2/5 - SWBAT graph pre-images and images under reflection and translation using a variety of tools (miras, patty paper, geometer’s sketchpad, and other virtual java tools) IOT determine the relationship between the pre-image and the image in terms of congruence.

MGSE9-12.G.CO.2/5 - SWBAT graph pre-images and images under reflection and translation using a variety of tools (miras, patty paper, geometer’s sketchpad, and other virtual java tools) IOT interpret properties of a transformation in the context of a real world problem.

Building CoherenceAcross grades:


Within Grades:

Terms and Definitions Angle: A figure created by two distinct rays

that share a common endpoint (also knownas a vertex). ∠ABC or ∠B or ∠CBA indicate the same angle with vertex B.

Angle of Rotation: The amount of rotation in degrees) of a figure about a fixed point such as the origin.

Axis of Reflection. The "mirror line" of a reflection. That is, the line across which a reflection takes place.

Congruent: Having the same size, shape and measure. ∠A ≅ _∠B indicates that angle A is congruent to angle B.

Corresponding angles: Angles that have the same relative position in geometric figures.

Corresponding sides: Sides that have the

Perpendicular lines: Two lines are perpendicular if they intersect to form right angles. AB ⊥ CD indicates that line AB is perpendicular to line CD.

Point: One of the basic undefined terms of geometry that represents a location. A dot is used to symbolize it and it is thought of as having no length, width or thickness.

Pre–image: A figure before a transformation has taken place.

Ray: A part of a line that begins at a point and continues forever in one direction. ABindicates a ray that begins at point A and continues in the direction of point B indefinitely.


Scale drawing as a prelude to similarity

7th Grad

eProperties of rotation, reflection and translation through experimentation

8th Grad

e

Functional representation of translation, rotation and reflection, properties of transformations through graphing and use of geometry software

10th Grad

e

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 to those that do not (e.g., translation versus horizontal stretch).Drawing the graph of the image under translation, reflection and rotation using graph paper, tracing

paper, or geometry software.

same relative position in geometric figures. Endpoint: The point at each end of a line

segment or at the beginning of a ray. Image: The result of a transformation. Intersection: The point at which two or more

lines intersect or cross. \ Line: One of the undefined terms of geometry

that represents an infinite set of points with no thickness and its length continues in two opposite directions indefinitely. ABindicates a line that passes through points A and B.

Line segment: A part of a line between two points on the line. AB indicates the line segment between points A and B.

Parallel lines: Two lines are parallel if they lie in the same plane and do not intersect. AB ∥ CD indicates that line AB is parallel to line CD.

Reflection: A transformation of a figure that creates a mirror image, “flips,” over a line.

Reflection Line (or line of reflection): A line that acts as a mirror so that corresponding points are the same distance from the mirror.

Rotation: A transformation that turns a figure about a fixed point through a given angle and a given direction, such as 90° clockwise.

Segment: See line segment. Transformation: The mapping, or movement,

of all points of a figure in a plane according to a common operation, such as translation, reflection or rotation.

Translation: A transformation that slides each point of a figure the same distance in the same direction.

Vertex: The location at which two lines, line segments or rays intersect.

Guiding Questions What effects do transformations have on geometric figures? Can transformation change an object’s

position, orientation and/ or size? How do transformations of geometric figures and functions compare?

Interpretations and Reminders A visual representation of ‘Transformations’ as a set containing’ translation, rotation, reflection, dilation

etc. will help students understand the fact that ‘translation’ is one of many transformations. Students need to recognize that if the pre-image is in second quadrant, the image under reflection

across y-axis will be in first quadrant; the image of an object in third quadrant under reflection across x-axis will be in the second quadrant etc. Asking questions by varying the location of objects and the axis of reflection will help students remember the functional (algebraic) rule for reflection across x-axis and y-axis. Also, asking students to draw the image and pre-image on the same grid will help in correcting their misconception.

Students need to recognize a transformation given a pre-image and an image. Provide various opportunities for students to identify the transformation and ask them to explain why.

Have students discuss about the various transformations-translation, reflection and rotations. Let students discuss if the distances and angles are preserved and also if the orientation of a polygon is the same in the pre-image and image.

Misconceptions

Students sometimes confuse the terms ‘transformation’ and ‘translation’. Ask students to read and comprehend before starting to solve any problem.

When using notation like (x, y) →(–x, y), students may believe that –x must always be negative, when in fact it simply means that it has the opposite sign of x.


Learning Progression (Suggested Learning Experiences)Procedural Fluency: (Recommended for 5 - 10 minutes each day: Fluency strategies are useful to activate student voice, solicit prior knowledge and develop fluency based on conceptual understandings.) For additional fluency practice strategies see the table at the end of this document.

Students need to have a conceptual understanding of the various key terms in geometry in order to learn about transformations. The teacher might begin the unit with a discussion on the various key terms (angles, circles, perpendicular lines, parallel lines, and line segments) in a variety of ways not limited to the following: 1) by asking students to complete a Frayer model for each of the terms 2) identify the terms from a given geometric figure 3) describe a geometrical figure using the key terms 4) describe an architectural plan of a building using the key terms.

For this lesson learning key geometry terms, teacher can present a Frayer model of describing the word with a picture, examples and non-examples and students in turn can find the exact terminology as a way of increasing fluency in this topic.

Students can also play a game of Taboo. Students work in pairs. The first student is given a geometric shape and she/ he has to describe the shape to her/his partner using the terms point, line, line segment, angles, circles, perpendicular lines, parallel lines, taboo words being triangle, parallelogram, circle etc. ) to help the second student trace the figure to match the given one.

Examples of various geometric figures that can be used for understanding and describing using key geometrical terms are given below:

Graduated Measure (The graduated measure is a quick opportunity to diagnose students’ level of comfort with the material before you begin the progression. Allow students to choose a question that they are best equipped to answer successfully)

Level 1 Level 2 Level 3


What is the image of the point (3, 2) under the translation given by T(x, y)= ( x-3, y+2).

If you ride the elevator from the lobby of the Empire State Building to the very top, is this motion a transformation, a translation, or a rotation?

a) Rotation b) Translation c) Reflection d)Rotation and Reflection

Is it possible for translation, rotation, and reflection to produce the same image? Explain with an example.

This unit on transformation calls for the use of a number of manipulative including patty paper, miras, geo-boards, geometer’s sketchpad and a number of other virtual manipulative available for free on the net. Students should be provided the opportunity to play with the tools as a way of exploring the rules for various transformations.

Teacher might start explorations on translation and reflection on day 1 using open ended activities and then continue to explore on rotations on day 2. Students do have pre-requisite knowledge on translations and reflections and can discover the rules for the same intuitively.

Gradual Release of ResponsibilityFocus LessonStart a discussion by asking students the following questionsa) If the pre-image is in I quadrant, where will the image under reflection across x-axis lie?b) If the pre-image is in quadrant 2, where will the image under reflection across y-axis lie?c) When will the pre-image and image of a given figure lie in the same quadrant? Give all possibilities.

Allow students that are unable to answer to draw any pre-image on a graphing board(student board) and then reflect the same to find the location of the image

Help students define reflection, axis of reflection, pre-image and image Students will be able to draw images under reflection across any line by intuition. Help students

understand properties of reflection by asking probing questions

Once students are able to discuss about the location of images under various transformations, have students draw images of geometric figures (student created figures) under reflection across the x-axis, y-axis, the x=y, the line x = -y, the line x= k, the line y =k and find a general rule to write each of the reflection as a function. Students may be permitted to use miras, geo-boards, patty paper, geometer’s sketchpad, or any other virtual tool to discover the rule on their own.

Guided Practice –


Allow students to answer questions of the above kind by varying the location of the pre-image and the transformation.

Provide opportunities for students to create their own questions.

Using a geo board to do reflections


Note: The conclusions have to be derived by students! And not dictated as notes by teachers!

What do you know about the distance between a point and its image from the axis of reflection?What is the relationship between the axis of reflection and the line joining any point and its image under reflection? Are there any points in the coordinate plane which are mapped onto itself under a reflection across any line? How many points are there and where are they located?

Reflect the triangle with vertices (-4, 1), (-4, 4) and (-1, 4) across the y-axis on a geo board and write your conclusion based on your observations.

Reflect the parallelogram with vertices at (2, 1), (4, 1), (5, 2) and (5, 4) across the line y = x and on a geo board and write your conclusion based on your observations.

Using a Mira for reflecting a pre-image If students are using miras for the first time they can be given a fun activity in the use of miras before reflecting geometric figures.

(page 8 of ‘Geometric Constructions and Investigations with a Mira-Walch Publications: https://walch.com/samplepages/021736.pdf)

Students should also be writing a functional rule for translation by sliding any geometric figure ‘h’ units to the left or right and ‘k’ units up or down.

Discovering the functional rule for rotations may not be intuitive for all students and might need some guidance by the teacher. Teacher may introduce rotations through the use of sketchpad, geo-gebra or any other virtual tool and discuss the center of rotation and angle of rotation. Students may then use the same tool or patty paper to discover the functional rule for rotating a point through 900, 1800, 2700, and 3600 in the counterclockwise direction about the origin

The above lesson on ‘Investigating Rotation using GSP’ can be utilized to help students discover rules on rotations using free version of Geometer’s sketchpad on laptops. If students do not have access to laptops, teacher can demonstrate the same, still allowing students to discover rules on their own. Alternatively, students can perform the above investigation using transparency (patty papers) and discover rules. Students discovering rules are important for them to have a conceptual understanding of rotations.


https://walch.com/samplepages/021736.pdf

After students have discovered rules for all of the transformations including translations, reflections and rotations, help them write transformations in various notations including the function notation using the mathematical vocabulary (input, output, domain, range etc.)

Students should also have opportunities to represent transformations on the coordinate plane

Given a graphical representation of a transformation or a functional notation of a transformation, students should be able to identify the transformation and justify the same. One such example is provided here.

Collaborative Practice

What is the image of the triangle ABC with coordinates at (2, 4), (4, -3) and (5, 1) under the transformation f(x, y) = (-x, -y). Describe the transformation.

Solution:The image of (2, 4) under the given transformation is (-2, -4)

The image of (4, -3) under the given transformation is (-4, 3)

The image of (5, 1) under the given transformation is (-5, -1)

Plot the points of the pre-image and the image and draw the two triangles. Use a patty paper and draw the pre-image. Secure the center of rotation by placing a pencil on top of the patty paper at the origin. Now rotate the patty paper; check to see if it coincides with the image any time. If it coincides, the transformation is a rotation. Find the angle between OA and OA’ to find the angle of rotation in the transformation. In this case, the transformation is a rotation through an angle of 1800 in the counterclockwise direction.

Using the patty paper to check if a transformation is a rotation.

Functions of various transformations:


The Learning Task ‘Introduction to Reflection, Translation and Rotation’ and the performance task ‘Mirrored Mapping’ from Georgia frameworks unit 1 can be used as a learning activity and as a performance activity with modifications if necessary.

Additional Unit Assessments –

Assessment

Name

Assessment Assessment Type Standards Addressed

Cognitive Rigor

Introduction to Reflections, Translations, and Rotations

Learning Task MGSE9-12.G.CO.1MGSE9-12.G.CO.2

DOK 2

Mirrored Mappings Performance Task coordinate plane.

MGSE9-12.G.CO.1MGSE9-12.G.CO.2MGSE9-12.G.CO.5

DOK 2

Rotating Squares Performance Task MGSE9-12.G.CO.4MGSE9-12.G.CO.5

DOK 3


Rule Type of Transformation

1. (X, Y) → (X, Y) Identity

2. (X, Y) → (-X, Y) Reflection across x-axis

3. (X, Y) → (X, -Y) Reflection across y-axis

4. (X, Y) → (-X, -Y) Rotation through 1800 in the counter clockwise direction

5. (X, Y) → (Y, X) Reflection across the line y=x

6. (X, Y) → (-Y, X) Rotation through 900 in the counter clockwise direction

7. (X, Y) → (Y, -X) Rotation through 2700 in the counter clockwise direction

8. (X, Y) → (-Y, -X) Reflection across the line y= -x

Differentiated Supports

Learning Difficulty

Use open questions that invite meaningful responses from students at many developmental levels.

Allow students to use visual /graphical representation while finding the image under a transformation.

Use technology assisted instruction through the incorporation of the graphing utility as a way of inviting visual learners to see how the pre-image and image are related.

Ask students to write out the functional rule for finding coordinates of the image.

Consider building students procedural fluency Use a video clip to introduce the students to transformations and provide

guided notes Allow low-achieving students to be in groups where their voices can be

heard Conceptual knowledge should be taught using the concrete-

representational-abstract (CRA) sequence. Incorporate the use of manipulatives and visual representations to move

students from concrete to abstract levels of understanding Incorporate relevance in examples to contextualize the learning moment.

High Achieving

● Provide content with greater depth and higher levels of complexity.● Use a discovery approach that encourages students to explore concepts.● Focus on solving complex, open-ended problems.● Ask divergent questions.● Provide opportunities for interdisciplinary connections.● Ask provocative questions and provide time for inquiry● Encourage tolerance for ambiguity with open-ended problems● Encourage students to use their intuition and follow their hunches● Allow students to study creative people and their thinking processes● Evaluate situations by analyzing possible consequences and implications

Enrichment Example:

Challenge students to find a functional rule for a rotation about the origin through any given angle θ in the counterclockwise direction.

English as a Second language

Conceptual understanding starts with language and the ability to use a specific set of terms to demonstrate the understanding

In this unit, provide opportunities for experiential engagement with the vocabulary through concrete and visual representations

Use manipulative throughout the lesson Use graphic organizers to describe key characteristics of the various

transformations Use Venn diagrams to identify similarities and differences between the


transformations Use Frayer models to define new vocabulary Encourage students to ask questions by providing question stems for

different proficiency levels Provide prompts to support student response Consider language and math skills while grouping students

Online/Print ResourcesDigital Resources Walch Education


https://illuminations.nctm.org/Lesson.aspx?id=3704

http://www.transum.org/Software/SW/Starter_of_the_day/Similar.asp?ID_Topic=56Print Resources

Manipulatives/Tools

Geoboard Mira Patty Paper Sketchpad TI Graphing calculator

Textbook Alignment-McGraw HillMGSE9-12.G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.MGSE9-12.G.CO.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 to those that do not(e.g., translation versus horizontal stretch).

Pages: 5-12, 36-44, 173-178, 180-186, 697-705

Pages: 296-302, 511-517, 623-631, 632-638, 639-646, 651-659


http://www.transum.org/Software/SW/Starter_of_the_day/Similar.asp?ID_Topic=56

https://illuminations.nctm.org/Lesson.aspx?id=3704
