Mr. Stephen's Classroom - Lesson Two Progression · Web viewGiven a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper,

Lesson Two ProgressionDuration 4 Days

Focus Standard(s)

Experiment with transformations in the plane

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.

Performance Objectives

As a result of their engagement with this unit, …

MGSE9-12.G.CO.3 - SWBAT transform a figure given a rotation, reflection or translation using graph paper, tracing paper, geometric software or other tools IOT analyze and compare the various transformations.

MGSE9-12.G.CO.4 - SWBAT determine rotations and reflections of parallelograms, trapezoids or regular polygons that map each figure onto itself IOT interpret the transformation in context and solve problems.

MGSE9-12.G.CO.4 - SWBAT define rotation, reflection and translation based on angles, circles, perpendicular lines, parallel lines and line segments IOT analyze their meaning and reasonableness in the context of a problem.

MGSE9-12.G.CO.4/5 - SWBAT create/perform sequences of transformations that map a figure onto itself or to another figure IOT interpret the transformation in context and solve problems.

Building Coherence

Across grades:

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

Students create scale drawing as a prelude to similarity

7th Grade

Students experiement with rotations, reflections and translations

8th Grade

Students explore functional representations of transformations through graphing and use of geometry software

10th Grade

Within Grades:

Terms and Definitions Bisector: A point, line or line segment

that divides a segment or angle into two equal parts.

Circle: The set of all points equidistant from a point in a plane.

Isometry: a distance preserving map of a geometric figure to another location using a reflection, rotation or translation indicates an isometry of the figure M to a new location M’. M and M’ remain congruent.

Line symmetry: If you can reflect (or flip) a figure over a line and the figure appears unchanged, then the figure has reflection symmetry or line symmetry

Line of Symmetry: The line that you reflect over is called the line of symmetry. A line of symmetry divides a figure into two mirror-image halves.

Parallelogram: a quadrilateral with opposite sides parallel (and therefore opposite angles equal).

Polygon: a two-dimensional closed figure made up of line segments

Rectangle: a parallelogram whose angles are all right angles

Regular Polygon: a polygon having all sides (and hence all angles) equal

Rhombus: A quadrilateral with all sides equal (Every rhombus is a parallelogram)

Rotational Symmetry: An object has rotational symmetry if there is a center point around which the object is turned (rotated) a certain number of degrees and the object looks the same. The number of positions in which the object looks exactly the same is called the order of the symmetry.

Symmetry: the quality of being made up of exactly similar parts facing each other or around an axis

Trapezoid: a quadrilateral with one pair of opposite sides parallel.

Guiding Questions

Which transformations create isometries?


How do we define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments?

Interpretations and Reminders Use Geometer’s sketchpad or any other geometry software as a visual learning tool to help students

understand that the image under a rotation varies when the center of rotation varies. All rules that students have established are for rotations with center of rotation at the origin.

Allow students to cut rectangles, place it on a coordinate plane with center of the rectangle coinciding with the origin and rotate through 900, 1800, and 2700. This will help students understand that a rectangle does not have any rotational symmetry.

Students may struggle to use the correct center of rotation, especially if they have simply memorized a rule; For example the rule (x, y) →(y, -x) for a 900 clockwise rotation works only if the center of rotation is the origin.

Misconceptions Students may get confused between line of symmetry and rotational symmetry. For example, in a

rectangle, a vertical line through the center of the rectangle is a line of symmetry. Students may confuse this as a right angle and may consider a rotational symmetry of 900. A dynamic software to show rotation of a rectangle or an activity where students rotate a rectangle will help students rectify their misconception.

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.

By the end of lesson 1, students should have gained conceptual understanding of transformations and should be able to find rules for various transformations intuitively or using manipulative. At the beginning of this lesson, a few minutes can be spent on attaining procedural fluency and for effortless retrieval of these facts given below:Teacher may give the coordinates of a point and ask students to plot its image under various transformations thus building fluency.


Formula for

TranslationT h,k (x, y)= (x+h, y+k)

Reflection through x-AxisRx-axis (x,y) = (x, -y)

Reflection through y-AxisRy-axis (x,y) = (-x, y)

Reflection through y=xRy = x (x,y) = (y, x)

Formula for

900 Rotation about the originR90(x, y) = (-y, x)

1800 Rotation about the originR180(x, y) = (-x , -y)

2700 Rotation about the originR270(x, y) = (y, -x)

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 3Draw the lines of symmetry for a square and a rectangle

John: The circle has 6 lines of

symmetry

James: The circle has infinite

lines of symmetry

Justify why James is right.

John: The circle has ‘n’ lines of

symmetry

James: The circle has infinite

lines of symmetry

Justify why James is right.

Students have learned finding the image of a point and the image of a polygon under a translation, reflection and rotation in the previous lesson. This lesson on mapping a polygon onto itself using a transformation or a sequence of transformations is an extension of the concepts expressed by the last two standards. Graphing parallelograms, rectangles and trapezoids and their images under various transformations can be a sponge activity for this lesson.

It is important for students to know the concept of symmetry. Students should be given opportunities to fold rectangles, parallelograms, trapezoids, circles etc. to find if each of the figures have any line of symmetry and if so the number of lines of symmetry.

Having found the line(s) of symmetry, students can then be introduced to reflections across various lines that map a rectangle, parallelogram, trapezoid and circle onto itself.

In a similar fashion, students should be given opportunities to rotate polygons about the center of the polygon if it exists or about the origin through a given angle and to discover the various rotations that map the figure onto itself. This activity should be reinforced using geometer’s sketchpad or any other geometry software.

Students should be encouraged to use their intuition to find angle measures for various figures that will result in rotational symmetry. For example, a square has a rotational symmetry of 900 while a regular hexagon has a rotational symmetry of 600, which is obtained by dividing 3600 by the number of sides in a hexagon. Students should also experiment through rotation and be able to discover that rotation through 1200, 1800, 2400, 3000 and 3600 also result in mapping the hexagon onto itself.

It is suggested that students are exposed to various examples of line symmetry and rotational symmetry existing in abundance in nature starting with the external appearance of the human body.


Students should also be encouraged to discuss the concept of carrying a geometric figure onto itself as it exists in wallpaper designs, screensaver design etc.

Line of Symmetry A line of symmetry ‘l’ is a line that divides a figure into twoidentical halves. In the given figure, the line ‘l’ is a line of symmetryas for every point ‘P’ on one side of ‘l’, there is a corresponding point‘Q’ on the other side of ‘l’. The points P and Q are equidistant from ‘l’and PQ is also perpendicular to ‘l’. Some figures may have only one line of symmetry, while others may have more than one; some have no lines of symmetry. Is rotational symmetry related to lines of reflection? Explore on your own.

Gradual Release of ResponsibilityCollaborative PracticeExploration 1:

Cut out a square. Fold the square a) vertically in the middle b) horizontally in the middle c) diagonally across both the diagonals. What do you observe? How many lines of symmetry does a square have?

A square has four lines of symmetry. The reflections across the vertical and horizontal line drawn through the center of the square and the reflections across the two diagonals map the square onto itself.

Place the center of the square at the origin on a coordinate plane and draw its outline. Now, rotate the square about its center until the square juxtaposes the square drawn. What angle did you have to rotate? What other angles can you rotate to make the square juxtapose the original square drawn?

A square when rotated about its center through an angle of 900, 1800, 2700 and 3600 coincides with the original square. Hence the square has rotational symmetry of 900, 1800, 2700 and 3600.

Exploration 2: Cut out any trapezoid. Can you fold the trapezoid vertically or horizontally somewhere in order to

have two identical halves? What about folding across the diagonals? Can you generalize and say that a trapezoid has no lines of symmetry? What rotation less than 3600 will map the trapezoid onto itself? (rotational symmetry)

A non-isosceles trapezoid has no lines of symmetry. However, an isosceles trapezoid has one line of symmetry; the line drawn through the midpoint of the parallel sides.


A trapezoid (isosceles or not) has no rotational symmetry. Only a rotation through 3600 about the origin will map the trapezoid onto itself.

Exploration 3: Cut out any parallelogram. Can you fold the parallelogram vertically or horizontally somewhere in

order to have two identical halves? What about folding across the diagonals? Can you generalize and say that a parallelogram has no lines of symmetry? What rotation less than 3600 will map the trapezoid onto itself? (rotational symmetry)

It is important to note here that all rectangles, squares and rhombuses are subsets of parallelograms. If a parallelogram does not have a 900 angle, it does not have a line of symmetry. In that case, there is no reflection that maps a parallelogram onto itself.A parallelogram will coincide with its original when rotated through 1800 about its center and hence has rotational symmetry when rotated 180º about its center.

Exploration 4: Cut out any rectangle. Can you fold the rectangle vertically or horizontally somewhere in order to

have two identical halves? What about folding across the diagonals? What rotation less than 3600 will map the trapezoid onto itself? (rotational symmetry)


A rectangle has reflectional symmetry, the lines of reflection being the lines joining the midpoints of the opposite sides. A rectangle also has rotational symmetry when rotated 180º about its center.

Exploration 5: Cut out any rhombus. Can you fold the rhombus vertically or horizontally somewhere in order to

have two identical halves? What about folding across the diagonals? What rotation less than 3600 will map the trapezoid onto itself? (rotational symmetry)

A rhombus has reflectional symmetry over either of its diagonals. A rhombus also has rotational symmetry when rotated 180º about its center.

Exploration 6: Cut out any regular pentagon. Can you draw a line inside the pentagon in order to have two

identical halves? What rotation less than 3600 will map the trapezoid onto itself? (rotational symmetry)

Every line joining a vertex to the midpoint of the opposite side of a regular pentagon is a line of symmetry for a regular pentagon. Hence a regular pentagon has 5 axes of reflections that map the pentagon onto itself. A rotation about the center through an angle of 720 (360/5) maps the pentagon onto itself. A pentagon has rotational symmetry when rotated 72º, 144 º, 216 º, and 288 º about its center.

Students should develop a deeper understanding of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

If A’ is the image of A under a reflection across a line ‘l’, challenge students to find the relationship between the line segment AA’ and the line ‘l’.

Students should understand that all of the three transformations- translations, reflections and rotations, and any combinations of these preserve distances (sizes), and angles and are called isometries.

Given a pre-image and an image, student should be able to identify the transformation or Page: 7

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

sequence of transformations that maps the pre-image to the image.

Provide opportunities for students to experiment on the following using manipulative or using a software: a) Reflect a parallelogram across a line ‘l’ and reflect the image across another line parallel to ‘l’. How can you describe the composition of the two reflections? b) Reflect a parallelogram across a line ‘l’ and reflect the image across another line perpendicular to ‘l’. How can you describe the composition of the two reflections?Ask students to check if the angles and sizes are preserved in all of the transformations.

Students should visualize how a transformation or a sequence of transformations maps a geometric figure onto itself or onto another figure. Students also should be able to apply transformations in the real world while creating designs that repeats itself.

Independent PracticeIdentifying a transformation from a graph


Guided PracticeProving that a transformation is an isometry


Note: The above examples have been taken from Walch units.


Additional Unit Assessments –Assessment

Name

Assessment Assessment Type Standards Addressed

Cognitive Rigor

Homework TaskIndividual / Partner

Translate objects and use rules to map an input with an output.

MGSE9-12.G.CO.1, 2, 4, 5

DOK 2

Learning TaskIndividual / Partner

Compare reflections and rotations.

MGSE9-12.G.CO.1, 2, 3, 4, 5

DOK 2

Learning TaskPartner / Small Group

Find general rules to describe transformations in the plane.

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

DOK 2

Formative Assessment LessonPartner / Small Group

Recognize and visualize transformations of 2D shapes.Translate, reflect, and rotate shapes, and combine these transformations.

MGSE9-12.G.CO.1, 2, 3, 4, 5

DOK3

Culminating TaskIndividual / Partner

Explain and generalize transformations that maintain congruence.

MGSE9-12.G.CO.1, 2, 3, 4, 5

DOK 3

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.

● Consider building students procedural fluency● Use a video clip to introduce the students to sequence of

transformations and provide guided notes Allow low-achieving students to be in groups where their

voices can be heard● Provide content with greater depth and higher levels of

complexity.


High Achieving● 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 implicationsEnrichment Example: Challenge students to find a functional rule for two consecutive rotations about the origin through any given angle in the counterclockwise direction.θ

English Language Learners

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 manipulatives throughout the lesson Use graphic organizers to describe key characteristics of

the various 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 http://www.shmoop.com/common-core-standards/ccss-hs-g-co-3.html

https://easingthehurrysyndrome.wordpress.com/2012/09/12/mapping-an-image-onto-itself/https://easingthehurrysyndrome.wordpress.com/2014/10/31/carrying-a-parallelogram-onto-itself/https://easingthehurrysyndrome.wordpress.com/2014/08/29/carrying-a-figure-onto-itself

http://www.geometrycommoncore.com/content/unit1/gco3/teachernotes1.html




https://easingthehurrysyndrome.wordpress.com/2014/08/29/carrying-a-figure-onto-itself

https://easingthehurrysyndrome.wordpress.com/2014/08/29/carrying-a-figure-onto-itself

https://easingthehurrysyndrome.wordpress.com/2014/10/31/carrying-a-parallelogram-onto-itself/

https://easingthehurrysyndrome.wordpress.com/2014/10/31/carrying-a-parallelogram-onto-itself/

https://easingthehurrysyndrome.wordpress.com/2012/09/12/mapping-an-image-onto-itself/

https://easingthehurrysyndrome.wordpress.com/2012/09/12/mapping-an-image-onto-itself/

http://www.shmoop.com/common-core-standards/ccss-hs-g-co-3.html

http://www.onlinemathlearning.com/rotational-symmetry.html

http://www.mathbitsnotebook.com/Geometry/Quadrilaterals/QDtransquads.html

Print Resources

Manipulative/Tools

Geo-board Mira Patty Paper Sketchpad TI Graphing

Calculator

Cubes, Tangrams ,and pattern blocks

Textbook AlignmentMGSE9-12.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.

Pages: 663-669

Pages: 623-631, 632-638, 640-646, 650-659

Pages: 623-631, 632-638, 640-650, 651-659




http://www.onlinemathlearning.com/rotational-symmetry.html

Sample Fluency Strategies for High School

Expression of the Week Choose an expression for the week. Allow students to consider the given expression. Students will then think of different ways to rewrite that expression in order to create equivalent expressions. As course concepts progress ask students to incorporate additional concepts in the creation of their expression. Provide students an opportunity to explain their thinking.

Algebraic Talks Extending from the idea of Number Talks. Teachers can help students develop flexibility through Algebraic Talks.

Algebraic Talks (The Norms):

No Pencils No calculators Mental Math Only Consider the problem individually during the first few minutes

Place a problem on the board and allow students to consider possible solution pathways. Next, allow 5 or 6 student to explain their strategy and record the various methods on the board for all students to see. Name the strategy (preferably after the student who made the conjecture). Allow other students to ask questions about the strategies while the proposing student is allowed to defend his/her work.

Also see: Jo Boaler Number Talks Videos

Picture This Provide students with a graph, or chart and ask them to come up with a contextual situation that makes the picture relevant


Find the odd man out

Note 1: This is an open ended question where students can choose any one of the choices A-D as the odd man as long as they are able to defend their choice. For example,

A is the odd equation because it is in the only equation with coefficients as 1 or -1

B is the odd equation because it is in the only equation with negative slope

C is the odd equation because it is in the only equation whose line passes through the origin

D is the odd equation because it is in the only equation which is not represented in the standard form.


A

x – y = 5

B

2x + 3y = 6

C

2x – 3y = 0

D

3y = 2x – 4

Note 2: Teacher can provide 4 equations/ 4 triangles / 4 transformations / 4 trig ratios where there is more than one way of classifying. Students can choose the odd man by mere inspection (surface level) or by examining the four components with a deeper understanding of the concept.

Example 2:

A is the odd equation because it has all positive coefficients or because it has two negative roots.

B is the odd equation because it has two positive roots.

C is the odd equation because it has one positive and one negative root.

D is the odd equation because the parabola opens downwards

Generate examples and non-examples

Example 1: Draw as many triangles as possible that are similar to

Note: Teachers can ask students to generate examples and non-examples for a variety of topics to improve fluency. A few more possibilities are listed here:


A

y = x2+5x+6

B

y = x2- 5x+6

C

y = x2- x-6

D

y = - x2 + 6

http://www.google.com/url?url=http://www.cliffsnotes.com/study-guides/geometry/triangles/triangle-inequalities-sides-and-angles&rct=j&frm=1&q=&esrc=s&sa=U&ved=0ahUKEwj4v-e02pbNAhWOsh4KHYO9BXkQwW4INDAI&sig2=0vtwztnC3xq9o2DdO5Tg2g&usg=AFQjCNFkddbbAc873uIrn0jHxCZSkb6LNw

1: Create quadratic equations with imaginary roots

2. Create quadratic equations with complex roots

3. Create non-examples of complementary angles.

4. Create examples of circles whose center is (3, -2)

5. Draw triangle ABC where Sin A= 0.3

Locating the error Choose a student’s work to be displayed on board after removing the name and ask the class to locate the error and also work the solution correctly. This can be done in any topic where the problem requires multiple steps to solve.
