sfusd mathematics core curriculum development project · 2020. 2. 17. · 2 sfusd mathematics core...

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SFUSD Mathematics Core Curriculum, Geometry, Unit G.2: Congruence and Rigid Motion, 2014–2015

SFUSD Mathematics Core Curriculum Development Project

2014–2015

Creating meaningful transformation in mathematics education

Developing learners who are independent, assertive constructors of their own understanding

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Geometry

G.2 Congruence and Rigid Motion

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task Sorting Game! (4 pages, one-sided) Transformation Tool Kit

1 per pair 1 per student

Scissors

5 Lesson Series 1 Introduction to Transformations (5 pages) Transformations Practice (2 pages) Transformations: Rigid and Non-rigid (5 pages)

1 per student 1 per student 1 per student

Patty paper

2 Apprentice Task Puzzling Transformations Task Card Student Materials (4 pages, one-sided) Projector Resources (3 pages)

1 per pair 1 per pair 1 projected

Projector or document camera Scissors

6 Lesson Series 2 Mapping onto Yourself (2 pages) Rotation Nation (4 pages) Reflection Inspection (4 pages) Translation Relations (4 pages)

1 per student 1 per student 1 per student 1 per student

Compasses and straightedges Protractors

2 Expert Task Calling All Snowflakes! (6 pages) 1 per student Scissors Blank paper

3 Lesson Series 3 FAL Analyzing Congruency Proofs: Finding Congruent Triangles (2 pages) Cards (2 pages, one-sided) Sample Student Proofs (2 pages) Finding Congruent Triangles, Revisited (2 pages) Projector resources

1 per student 1 per pair 1 per pair 1 per student 1 projected

Compasses and straightedges Protractors Blank paper

1 Milestone Task Rigid Transformations and Congruence (3 pages) 1 per student

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Unit Overview

Big Idea

Students define congruent polygons (specifically triangles) in terms of rigid transformations (translations, rotations, reflections) and use these definitions to prove that triangles are congruent. Students learn that the measurement-based shortcuts for triangle congruence (ASA, SAS, SSS) follow from these transformational definitions.

Unit Objectives

• Students will be able to transform polygons on a coordinate plane, and describe transformations given images, including: translations, reflections, and rotations.

• Students will be able to describe transformations algebraically, geometrically, and in writing. • Students will be able to recognize and apply triangle congruence shortcuts, including analysis of why certain shortcuts don’t work. • Students will be able to define corresponding parts and congruence. Also differentiate congruent polygons from similar and disproportionate polygons.

Unit Description

Students will begin with investigations of rigid transformations on a coordinate plane. They will then use their knowledge of basic transformations to design and interpret tessellations as an apprentice task. The second series of lessons focuses on defining corresponding parts, similarity, and congruence from a transformational perspective. Students will then discover the properties of each type of isometric transformation and define them. This series of lessons will conclude with a multi-day expert task in which students will use transformations to create and analyze paper snowflakes. Lesson Series 3 has students explore the shortcuts for triangle congruence in the form of an exploratory task.

CCSS-M Content Standards

Congruence Experiment with transformations in the plane 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). G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. 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.

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Understand congruence in terms of rigid motions G.CO.6 Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. G.CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. G.CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

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Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics

Future Mathematics

Starting in elementary school (fifth grade), students learn how to plot points on a coordinate plane. Students in eighth grade learn about congruence and similarity, as well as a brief introduction into rigid transformations like reflection. They also use similar triangles to explore linear growth.

Students will begin to understand the role of rigid transformations in proving congruence of various polygons. They do this through explorations of rigid transformations, triangle congruence shortcuts, and tessellations.

An understanding that rigid transformations preserve congruence is the foundation that students need as they work with congruent triangle proofs, similar triangles, right triangles, and trigonometry.

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Unit Design All SFUSD Mathematics Core Curriculum Units are developed with a combination of rich tasks and lessons series. The tasks are both formative and summative assessments of student learning. The tasks are designed to address four central questions: Entry Task: What do you already know? Apprentice Task: What sense are you making of what you are learning? Expert Task: How can you apply what you have learned so far to a new situation? Milestone Task: Did you learn what was expected of you from this unit?

1 day 5 days 2 days 6 days 2 days 3 days 1 day Total: 20 days

Lesson Series 1

Lesson Series 2

Lesson Series 3

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Entry Task Sorting Game!

Apprentice Task Puzzling Transformations

Expert Task Calling All Snowflakes!

Milestone Task Unit Assessment

CCSS-M Standards

G.CO.2, G.CO.4, G.CO.5, G.CO.6


G.CO.2, G.CO.3, G.CO.4, G.CO.5, G.CO.6, G.CO.7

G.CO.2, G.CO.3, G.CO.4, G.CO.5, G.CO.6, G.CO.7

Brief Description of Task

Students learn about isometric transformations, dilations, and basic properties of rigid transformations.

Students will recognize and visualize transformations of 2D shapes. They will translate, reflect, and rotate shapes, and combine these transformations.

Students will apply what they have learned about transformations to make and analyze paper snowflakes. They will analyze and create transformations in a paper snowflake and include descriptions of the transformations involved. These descriptions could include written, algebraic, and geometric representations. Students also connect their knowledge of the properties of transformations to congruency of polygons.

Students will demonstrate mastery of CCSS-M standards G.CO.2–8 in a paper-and-pencil test.

Sources Joanne’s Transformations FAL, 2011, Shell Center Adapted from CPM Geometry Connections Lesson 6.2.5

SFUSD Teacher Created

Lesson Series 1

Lesson Series 2

Lesson Series 3

CCSS-M Standards


G.CO.6, G.CO.7 G.CO.8

Brief Description of Lessons

Introduce the concept of transformations, differentiating between isometric and non-isometric transformations. Students will learn how to transform points, segments, and polygons using translations, reflections, and rotations (and briefly touch on their differences from rotations).

Define congruence and similarity and differentiate between these emphasizing that congruent figures are similar with a ratio of 1. Students experiment with the definitions of rigid transformations (translations, reflections, and rotations) to determine if pairs of transformed triangles are congruent.

Students discover the four triangle congruence shortcuts (SAS, SSS, ASA, and SAA).

Sources SFUSD Teacher Created SFUSD Teacher Created

FAL Analyzing Congruence Proofs

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Entry Task

Sorting Game!

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: • Students analyze basic transformations on a coordinate plane using

patty paper and other basic tools. CCSS-M Standards Addressed: G.CO.2, G.CO.4, G.CO.5, G.CO.6 Potential Misconceptions

• ”Translation” sounds like “transformation.” • Students may not know how to perform rigid transformations. • Students may not know of reflectional symmetry. • Students may not know of rotational symmetry.

Materials: Rubber bands

Launch: What are transformations? What are the important properties of a transformation? During: Students explore the meaning of isometric transformations by manipulating polygons on patty paper and using rubber bands to create dilations. Closure/Extension: Students describe transformations that form symmetry and justify their reasoning. Students will use different methods of transformations to get a strong overview of what makes something an isometric versus dilated transformation.

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Sorting Game!

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them.

Structures for Student Learning: Academic Language Support:

Vocabulary: Translation, Rotation, Reflection, Rigid Transformation, Non-rigid Transformation (but DON’T introduce this before you start). Sentence frames:

• “I think there we should make a category for ________, because…” • “That one does/doesn’t fit this category because…”

Differentiation Strategies:

• This task works well for students who are at all levels. If students are really struggling, offer them patty paper and have them trace one image and map it onto the other. You may also suggest that they start with the images only, and then try to move onto the other representations (such as the x and y representations)

• If a group has finished early, have them create more examples for each category. Participation Structures (group, partners, individual, other):

Students work in groups of four.

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Lesson Series #1 Lesson Series Overview: Introduce the concept of transformations, differentiating between isometric and non-isometric transformations. Students will learn how to transform points, segments, and polygons using translations, reflections, and rotations (and briefly touch on their differences from rotations). Standards: G.CO.2, G.CO.3, G.CO.4, G.CO.5 Time: 5 days

Lesson Overview – Days 1-2 Resources

Description of Lesson: Define and apply translations on and off a coordinate plane. Notes: Pay attention to notation Students will represent translations in multiple ways:

1. By using patty paper to translate polygons on and off the coordinate plane. 2. Students practice transformations in multiple representations. 3. Students learn about specific notations for each

Lesson 1.1 – Introduction to Transformations Additional resource: CPM Core Connections Geometry 1.2.4, 1–83

Lesson Overview – Days 3 Resources

Description of Lesson: Practice performing transformations on the coordinate plane- both with patty paper and without. Notes:

1. Students begin by recognizing reflections and translations in x and y-representations.

2. Students describe and then perform multiple representations of transformations. Some are one or two-steps only, while others are composite transformations.

Students should work on these problems in pairs or in groups, so they can check each other’s thinking as they go. Students should also be encouraged to use patty paper to help represent their ideas as they work through the problems.

Lesson 1.2 – Transformations Practice

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Description of Lesson: Students discover the effects of dilation, and create a toolkit that summarizes their learning of the properties of both rigid transformations and dilations. This lesson will be a bit different from previous lessons, where students work with their groups in mostly self-directed ways. In this lesson, students will work with partners primarily, and will need to help each other as they move through the Notes: In this lesson, students will do 2 things:

1. Transform polygons and images by following several transformations, and making observations and statements about the meaning of those transformations.

2. Students learn (briefly) about dilations and its properties (and analyze the difference between a dilation and a rigid transformation).

Lesson 1.3 – Transformations: Rigid and Non-rigid Additional resource: Discovering Geometry, An Investigative Approach, Lesson 7.3, #1–11

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Apprentice Task

Puzzling Transformations



Math Objectives: • Students demonstrate their knowledge of transformations in the

context of the coordinate plane. • Students will analyze efficient versus inefficient transformational

moves. CCSS-M Standards Addressed: G.CO.2, G.CO.3, G.CO.4, G.CO.5 Potential Misconceptions

• Students confuse terms (horizontal vs. vertical/ clockwise vs. counter-clockwise).

• Students translate instead of reflecting shapes. • Students ignore the center of rotation and instead rotate from the

corner of the image.

Launch: Have students complete the pre-assessment Transformations. Then return the task to students with some early feedback. During: Students will spend time working with each other to discuss whether moves are especially efficient or inefficient. I.e., How do you transform a polygon in one move instead of two? Closure/Extension: Have students return to their initial assessments and analyze their work given their new/improved understanding. Students will also reflect on their learning- what was good about working in a group? What wasn’t good about it?

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Representing and Combining Transformations


Focus Standards for Mathematical Practice: 5. Use appropriate tools strategically. 6. Attend to precision.


Vocabulary: Translation, Rotation, Reflection, Rigid Transformation Sentence frames:

• “Do other transformations work with these two figures?” • “These two figures have been (translated/reflected/rotated) because…”


• You may want to start the problem without only 6 of the images (you can omit, for instance, the one that rotates around the point (2,0) and include that one later). You could also let students struggle with all cards for a while and if they’re totally lost, you could encourage them to think about 2-3 cards first before moving to the others.

• The task card contains other extension questions, but you could encourage students to see if there are other ways to arrange the cards that work. You could also have students identify transformations that take 2-3 steps, and instead have them find ways to transform them (if possible) in just one.

Participation Structures (group, partners, individual, other):

Both individual (for the start) and group work.

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Lesson Series #2

Lesson Series Overview: Define congruence and similarity and differentiate between these emphasizing that congruent figures are similar with a ratio of 1. Students experiment with the definitions of rigid transformations (translations, reflections, and rotations) to determine if pairs of transformed triangles are congruent. Days 1-2: Congruence, Similarity, and Corresponding Parts. Days 3-4: Discover and describe the properties of rotations. Day 5: Discover and describe the properties of reflections. Day 6: Discover and describe the properties of translations. CCSS-M Standards: G.CO.6, G.CO.7 Time: 6 days


Description of Lesson: Define congruence, similarity and dissimilarity. Notes: Students draw on their intuitive understanding of congruence, similarity, and dissimilarity to create rough definitions of these terms. Students will then use transformed images to discover that only rigid transformations result in congruent images. This discovery will then be used to note the properties of congruence, similarity and dissimilarity, as well as defining them in detail in terms of transformations.

G.2 Lesson 2.1 – Mapping onto Yourself West Ed Field Guide Definitions of Transformations (.pdf file) Transformations and Congruence Toolkit (guided vocabulary template)


Description of Lesson: Discover and apply the transformational definition of a translation.

G.2 Lesson 2.2 – Rotation Nation West Ed Field Guide Definitions of Transformations (.pdf file)

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Notes: Students will translate a triangle, then connect corresponding vertices with segments. By applying slope triangles and the Pythagorean Theorem, students will discover that the segments connecting corresponding points will always have congruent slopes and equal lengths. This will lead to a discussion about the transformational definition of translations.

Transformations and Congruence Toolkit (guided vocabulary template)

Lesson Overview – Day 5 Resources

Description of Lesson: Discover and apply the transformational definition of a reflection. Notes: Students will reflect a triangle, then connect corresponding vertices with segments. By applying slope triangles and the Pythagorean Theorem, students will discover that the segments connecting corresponding points will always be perpendicular to the line of reflection and that reflected points will always be the same distance from the line of reflection as the original points. This will lead to a discussion about the transformational definition of reflections.

Lesson 2.3 – Reflection Inspection Class Demonstration Instruction Page

Lesson Overview – Day 6 Resources

Description of Lesson: Discover and apply the transformational definition of a rotation. Notes: Students will rotate a triangle, then connect corresponding vertices with segments. By applying the Pythagorean Theorem, students will discover that the segments connecting corresponding points will always have equal lengths. Students will also discover that the angle made by the two segments will remain constant no matter what two points are chosen. This will lead to a discussion about the transformational definition of rotations.

G.2 Lesson 2.4 – Translation Relations

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Expert Task Calling All Snowflakes!



Math Objectives: • Students will apply what they have learned about transformations and

congruence to make and analyze transformations in a paper snowflake.

• Students demonstrate their understanding of basic transformations: translations, reflections, and rotations. They are also able to describe the transformations algebraically, geometrically, and in written form.

• Students can demonstrate congruence of various polygons through transformations (by measuring distance or angle rotation).

• Students will analyze the role of parallel versus perpendicular lines in transformations.

CCSS-M Standards Addressed: G.CO.2, G.CO.3, G.CO.4, G.CO.5, G.CO.6, G.CO.7 Potential Misconceptions

• Students may struggle to see the transformations in the snowflake. • Students may confuse perpendicular and parallel. • Students will be expected to use Pythagorean Theorem and other

tools (like protractors and rulers) to measure distance, which they may struggle with.

• Knowing the difference between different transformations and their properties, especially translations and reflections.

Launch: Students will begin by recalling what they learned about the properties of various shapes. During: Students will analyze their own snowflake and other students’ work to discuss the various kinds of transformations they see. They will prove congruence of different shapes through the properties of translations, rotations, and reflections. Closure/Extension: Student end by looking at examples of the triangle with cutouts and drawing the snowflake based on those designs. They will have to understand and reproduce the transformations based solely on the cutout of the triangle.

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Calling All Snowflakes!


Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others 4. Model with Mathematics 6. Attention to Precision 7. Look for and Make Use of Structure


Vocabulary: Reflection, Rotation, Translation, Rigid Transformations, Dilation, Parallel Lines, Perpendicular Bisector Sentence frames: “This image is an example of a (reflection, rotation, translation) because…”


• You may want to provide an example snowflake for students who are really struggling with putting their snowflake together. You could even give students a complete version if they continue to have problems.

• If students are looking for an extra challenge, have them try cutting a more complicated snowflake off of the coordinate plane, and then justify the various transformations that they see using the properties of transformations.


Students will work on their snowflakes individually, but will be expected to compare and discuss the other parts in groups of four.

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Lesson Series #3 Lesson Series Overview: Students discover the four triangle congruence shortcuts (SAS, SSS, ASA, and SAA). CCSS-M Standards Addressed: G.CO.8 Time: 3 days


Description of Lesson: Students discover the four triangle congruence shortcuts (SAS, SSS, ASA, and SAA). Notes: An extended exploration task will allow students to experiment with all potential 3-part shortcuts for triangle congruence. By finding counterexamples to SSA and AAA situations, students will discover the 4 situations that do prove triangle congruence.

FAL Analyzing Congruency Proofs

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Milestone Task Milestone Examination



Math Objectives: • Students demonstrate their understanding of the properties of rigid

transformations. • Students use the properties of rigid transformations to prove

congruence of polygons. • Students demonstrate understanding of triangle congruency

shortcuts. CCSS-M Standards Addressed: G.CO.2, G.CO.3, G.CO.4, G.CO.5, G.CO.6, G.CO.7 Potential Misconceptions: Same as those outlined throughout the unit.

Launch: Give students a warm-up problem or review that incorporates triangle congruency shortcuts and rigid transfor.mation properties. During: Standard paper-and-pencil exam Closure/Extension:

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Milestone Examination

How will they do this?

Focus Standards for Mathematical Practice: 3. Construct viable arguments and critique the reasoning of others 5. Use appropriate tools strategically


Vocabulary: Same as the Expert task (rotation, reflection, translation, rigid transformation, dilation, non-rigid transformation, mapping, parallel lines, perpendicular bisector, congruent)


• If you don’t cover all sections, you can choose to omit some pieces of the examination. • You can also ask students to start with certain questions if you know they will be more successful on particular sections.


Individual assessment

sfusd mathematics core curriculum development project · 2020. 2. 17. · 2 sfusd mathematics core...

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