# SFUSD Mathematics Core Curriculum Development Project8.G.1c Parallel lines are taken to parallel...

### Transcript of SFUSD Mathematics Core Curriculum Development Project8.G.1c Parallel lines are taken to parallel...

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 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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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Grade 8

Unit 8.3 Transformations

Number of Days

Lesson Reproducibles Number of Copies

Materials

1 Entry Task CPM CCC3 Lesson 6.1.1 Resource Page 6.1.1 HW: CPM CCC3 Lesson 6.1.1

1 per pair 1 per pair CPM eBook

Computers (1 per pair)

6 Lesson Series 1 Transformations Recording Sheet CPM CCC3 Lesson 6.1.2 (3 pages) Resource Page 6.1.2 HW: CPM CCC3 Lesson 6.1.2 CPM CCC3 Lesson 6.1.3 (3 pages) HW: CPM CCC3 Lesson 6.1.3 Poster Triangle Coordinate Cards (3 pages)

1 per pair 1 per pair 1 per pair CPM eBook 1 per pair CPM eBook 1 per pair

Graphing paper, pattern blocks, or patty paper (optional) Colored markers or pencils

1 Apprentice Task CPM CCC3 Lesson 6.1.4 (2 pages) Resource Page 6.1.4A Resource Page 6.1.4C HW: CPM CCC3 Lesson 6.1.4

1 per pair 1 per student 1 per student CPM eBook

5 Lesson Series 2 CPM CCC3 Lesson 6.2.1 (2 pages) Optional: 6.2.1 Recording Sheet HW: CPM CCC3 Lesson 6.2.1 CPM CCC3 Lesson 6.2.2 (3 pages) Resource Pages 6.2.2A&B (2 pages) HW: CPM CCC3 Lesson 6.2.2 CPM CCC3 Lesson 6.2.3 (3 pages) Resource Page 6.2.3 HW: CPM CCC3 Lesson 6.2.3

1 per pair 1 per pair CPM eBook 1 per pair 1 per student CPM eBook 1 per pair 1 per pair CPM eBook

Colored pencils (optional) Scissors Tracing paper or patty paper Colored markers or pencils Graph paper

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

1 Expert Task CPM CCC3 Lesson 6.2.4 (2 pages) HW: CPM CCC3 Lesson 6.2.4

1 per pair CPM eBook

4 Lesson Series 3 CPM CCC3 Lesson 9.1.1 (5 pages) Resource Page 9.1.1 HW: CPM CCC3 Lesson 9.1.1 DG 4.1 Triangle Sum Conjecture (2 pages) CPM CCC3 Lesson 9.1.2 HW: CPM CCC3 Lesson 9.1.2 CPM CCC3 Lesson 9.1.3 (2 pages) Resource Page 9.1.3 HW: CPM CCC3 Lesson 9.1.3 CPM CCC3 Lesson 9.1.4 (2 pages) HW: CPM CCC3 Lesson 9.1.4

1 per pair 1 per pair CPM eBook 1 per pair 1 per pair CPM eBook 1 per pair 1 per student CPM eBook 1 per pair CPM eBook

Protractors Straightedges or rulers Scissors Tracing paper or patty paper Graph paper

2 Milestone Task Performance Task: Tracy Triangle’s Transformation Travels (2 pages) Constructed Response: Transformations

Provided by AAO

Tracing paper or patty paper

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Unit Overview

Big Idea

Rigid transformations and dilations of two-dimensional figures and angles preserve properties of congruence and proportionality.

Unit Objectives

● Students will be able to identify, use, and manipulate two-dimensional figures using rotations, reflections, translations and dilations. ● Students will verify that rigid transformations take lines to lines, segments to congruent segments, angles to congruent angles, and parallel lines to

parallel lines through experimentation. ● Students will understand that two figures are congruent if one can be constructed from the other as a result of a composition of rigid transformations. ● Students will use coordinates to describe the effect of transformations on two-dimensional figures. ● Students will understand that two figures are similar if one can be constructed from the other as a result of a composition of dilations and rigid

transformations. ● Students will establish two facts about the exterior angles of a triangle: 1) that the sum of the two remote interior angles of a triangle equals the exterior

angle of a triangle and 2) that an exterior angle of a triangle is supplementary to the adjacent interior angle of the triangle. ● Students will be able to establish that the sum of the interior angles of a triangle is equal 180 degrees. ● Students will be able to understand the relationship between angles that are created when parallel lines are cut by a transversal.

Unit Description

This unit consists of three sections. First, students will explore congruence through rigid transformations (translations, rotations, and reflections) of polygons on the plane. Second, students will investigate similarity by dilating figures and noting that angle measures, from the original figure to its image, are preserved and side lengths remain proportional. Finally, students will explore the relationships between the interior and exterior angles in a triangle and various pairs of angles created when parallel lines are crossed by a transversal.

CCSS-M Content Standards

Geometry 8.G Understand congruence and similarity using physical models, transparencies, or geometry software. 8.G.1 Verify experimentally the properties of rotations, reflections, and translations: 8.G.1a Lines are taken to lines, and line segments to line segments of the same length. 8.G.1b Angles are taken to angles of the same measure. 8.G.1c Parallel lines are taken to parallel lines. 8.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. 8.G.3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

8.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. 8.G.5 Use informal arguments 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. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Progression of Mathematical Ideas

Prior Supporting Mathematics Current Essential Mathematics Future Mathematics

Previously, in sixth grade, students learned how to draw polygons in a coordinate plane given coordinates for the vertices. In seventh grade, students learned about supplementary, complementary, vertical, and adjacent angles. They also constructed triangles given the measures of three angles. Additionally, students used proportionality to create scale drawings with geometric figures.

Rigid transformations and dilations of two-dimensional figures preserve properties of congruence and proportionality. Specifically, rigid transformations of two-dimensional figures preserve angle measures and side lengths. A two-dimensional figure is congruent to another two-dimensional figure if the second can be obtained from the first through a series of rigid transformations. Dilations preserve angle measure and create proportional side lengths. A two-dimensional figure is similar to another if the second can be obtained from the first through a series of rigid transformations and dilations. The sum of the measures of the interior angles of a triangle, as well as the relationships between an exterior angle of a triangle and the two remote interior angles and between an exterior angle of a triangle and the adjacent interior angle are constant. The sum of the measures of the interior angles of a triangle is 180 degrees, and the measure of an exterior angle of a triangle is equal to the sum of the two remote interior angles. An exterior angle of a triangle and the adjacent interior angle are supplementary. The angles formed when parallel lines are cut by a transversal also have constant relationships. Specifically, angles formed when parallel lines are cut by a transversal are always either congruent or supplementary. The triangle angle sum relationship can be used to establish triangle similarity. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar.

In high school, students will represent transformations as functions and compare rigid transformations to those that do not preserve distance and angle measure. They will describe the series of rigid transformations that carry a regular polygon onto itself. Given a figure and a rigid transformation, students will draw the transformed image of the figure. They will understand congruence in terms of rigid motions and explain how the criteria for triangle congruence follow from the definition of congruence in terms of rigid motion. Students will understand similarity in terms of dilation. Students will use similarity transformations to establish the Angle/Angle similarity criterion.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

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 6 Days 1 Day 5 Days 1 Day 4 Days 1 Day

Total Days: 19

Lesson Series 1

Lesson Series 2

Lesson Series 3

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Entry Task Key-in-the-Lock Puzzles

Apprentice Task What Can I Create? Becoming an

Artist

Expert Task What Sequence Makes Them the

Same?

Milestone Task Tracy Triangle’s Transformative Travels, Transformations

CCSS-M Standards

8.G.1a, 8.G.1b, 8.G.1c 8.G.3, 8.G.4 8.G.3, 8.G.4 8.G.1a, 8.G.1b, 8.G.1c, 8.G.2, 8.G.3, 8.G.4, 8.G.5

Brief Description of Task

Students watch a YouTube video, The Gingerbread Transformer, as an introduction to rigid transformations. Students use a technology tool available at www.cpm.org/students/technology, Lesson 6.1.1, where students will try to move the key to the keyhole to unlock the door using transformation buttons such as sliding, turning or flipping. *If technology is not available, hard copies are available of the same activity where students can write their responses as to how the key was moved to unlock the door.

Students take written descriptions of rigid transformations and transform objects on the coordinate plane.

Students use transformations to show that two figures are similar. They will start by proving that two figures are similar through a series of transformations. Then students work to find out which transformations will result in congruent figures and which will result in similar figures. They use a sequence of translations to find the coordinates of the new figure using two different steps for their transformation. Finally, students find a sequence of transformations that will transform one figure to become another figure.

Students demonstrate a series of transformations, describe transformations, and demonstrate understanding of congruence.

Source The Gingerbread Transformer: http://www.youtube.com/watch?v=Oa-72NDu8MY www.cpm.org/students/technology CPM CCC3 6.1.1 and resource page

CPM CC3 6.1.4 CPM CC3 6.2.4 SFUSD Teacher-Created based on CPM CCC3 Chapter 6 Closure

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Series 1 Lesson Series 2 Lesson Series 3

CCSS-M Standards

8.G.1a, 8.G.1b, 8.G.1c 8.G.3, 8.G.4 8.G.5

Brief Description of Lessons

Students investigate and describe rigid transformations on a coordinate graph. They write equations to describe the effects of a rigid transformation on coordinates. Finally, students use rigid transformations to create a design.

Students investigate dilations by multiplying by a variety of whole number and fractional factors. They develop an understanding of similarity and congruence and how it relates to transformations. Finally, they use ratio and scale factors to reduce and enlarge shapes.

Students establish facts about angle relationships created when parallel lines are crossed by a transversal. Students find missing angles in triangles and learn that the sum of the angles in a triangle is 180 degrees through an investigation. Next, they discover the relationship between the measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles. Finally, students investigate the Angle-Angle criterion for triangle similarity.

Sources SFUSD Teacher-Created CPM CCC3 Chapter 6, lessons 6.1.2 and 6.1.3

CPM CCC3 Chapter 6, lessons 6.2.2–6.2.6

CPM CCC3 Chapter 9, lessons 9.1.1, 9.1.3–9.1.4 Discovering Geometry, Lesson 4.1

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Entry Task Key-in-the-Lock Puzzles

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will learn how to move a shape on a coordinate graph using

rigid transformations: translations (slides), rotations (turns), and reflections (flips).

CCSS-M Standards Addressed: 8.G.1a, 8.G.1b, 8.G.1c Potential Misconceptions:

● Flipping when not moving around the axis. ● Multiple transformations (locating end points correctly). ● Starting in one quadrant and rotating to another, the assumption is 90

degrees rotation instead of 180 degrees.

Launch: Show the Gingerbread Transformer: http://www.youtube.com/watch?v=Oa-72NDu8MY Then project the tool set on the Introduction tab and demonstrate the three transformation buttons to introduce lesson. This is to demonstrate to students that there are only three possible moves. During: After the demonstration, have students work in pairs on a computer, using this technology tool: www.cpm.org/students/technology. Also give each pair Resource Page 6.1.1 to record their movements. Students will begin the computer program, starting with the introduction problems, then the standard problems and finally the challenge problems. Note: If you do not have access to computers for the students, project the buttons on the screen. Before each transformation, have the students explain what is going to happen before you demonstrate each movement. If you don’t have any access to a computer, use a document camera or overhead projector. Project or display a coordinate grid on board. Create two L shapes that you can use to move across the screen to display the different transformations. Closure/Extension: Have students reflect on what each button did to the key. Possible discussion and/or exit ticket questions include: 1. How is a slide different than a flip? 2. What does rotate do to the shape? 3. Decide which buttons would you assign to each of these symbols:

.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Key-in-the-Lock Puzzles

How will students do this?

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

Structures for Student Learning: Students can work individually or in pairs. Each student or pair needs access to a computer is suggested. Encourage students to check in with a neighbor to harbor communication throughout lesson. If in pairs, encourage students to discuss decisions and share control of the keyboard and mouse. Academic Language Support:

Vocabulary: flip, rotate, turn, transformation, slide, horizontal, vertical, clockwise, counter-clockwise

Sentence frames: The key is moving ________________ units to the ____________ (right or left) and ____________ units __________ (up or down). The key is flipping in this direction ___________________ to get to the keyhole.

Differentiation Strategies: You may want to help pairs of students use word attack skills to understand the vocabulary while they are at the computers. Allowing students to work in pairs will offer the help of each student’s input for recording the movements on the resource page. In addition, have students stop every so often to do a think-pair-share strategy to work with their partners thinking and recording before moving on. Use physical arm movements for students to explain the meaning of horizontal and vertical, as well as using the example of a clock in the room to demonstrate clockwise and counter clockwise. Participation Structures (group, partners, individual, other): Students may work individually or with partners. Each student or partner is assigned one computer and one resource page to record on.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Series #1

Lesson Series Overview: In this lesson series, students will explore rigid transformations on a coordinate graph by describing movements, describe transformations on a coordinate graph by writing similar expressions, and use rigid transformations by translating, rotating, and reflecting. CCSS-M Standards Addressed: 8.G.1a, 8.G.1b, 8.G.1c, 8.G.2 Time: 6 days

Lesson Overview – Day 1 Resources

Description of Lesson: To start this lesson, have students create a foldable graphic organizer by folding a sheet of plain paper in fourths. Each box represents one transformation with a description: Translation, Rotation, Reflection and Dilation (note that dilations will be described in a future lesson). In this lesson, teachers will prepare nine posters on graph poster paper with coordinate axes. On each poster, draw two identical triangles (3-4-5 right triangles work well) in different positions. Have students cut out two congruent triangles for each group. Students will circulate around the room, using the recording sheet (Transformations Recording Sheet) to record the movements needed to move the triangle from its starting position to its ending position for at least 6 of the nine posters. This activity is intended to help students understand the rigid transformations. At this point, students should not be responsible for mastery of the vocabulary. Notes: This activity works better if posters are laid flat on tables rather than hanging on the wall. Also, if students finish quickly, challenge them to find alternate routes.

Transformations Recording Sheet Poster Triangle Coordinates Cards

Lesson Overview – Day 2 Resources

Description of Lesson: In this lesson, you will project the Transformation and Demonstration Activity and complete two examples with the whole class to revisit what was done the day before. In pairs, students will then solve the block challenge by moving each block from the starting position to the ending position using translations, rotations, and reflections. Students will describe the moves they made to reach the ending position and find more than one way in which the block could be moved. At the end of the lesson, students will be able to share their ideas with the class.

CPM CCC3 6.1.2 problem 6-8 Transformation and Demonstration: http://www.cpm.org/flash/technology/triangle_transformation.swf Homework: Choose from 6.1.2 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Overview – Day 3 Resources

Description of Lesson: The next day, you will provide students with starting points and directions to move to unlock the lock, and students will need to find out where the lock is by answering a series of questions. Students will plot the given starting points and follow the steps to unlock the key by naming, sketching, and labeling coordinates for each vertex. Then students will begin to compare lengths of the sides, angles, and find different steps to unlock the key. If it is possible, students will list the new steps. If not possible, students will explain why it is not possible.

CPM CCC3 6.1.2 problems 6-9 thru 6-11 Homework: Choose from 6.1.2 Review & Preview

Lesson Overview – Day 4 Resources

Description of Lesson: At the beginning of the lesson, students will be given problems 6-18 and 6-19. They will be working in pairs. Problem 6-18 focuses on translations of quadrilaterals. They will move a shape in the coordinate plane and locate the new points created by the translation. Next, they will create expressions to show how their points moved. This will set them up to model problem 6-19. Then, students will move to problem 6-19. Here, they use similar expressions from problem 6-18 to move another shape without a picture. (You may want to give students graph paper, pattern blocks, or patty paper to help them visualize the movements happening.) Note: Problem 6-18 d. and e. and Problem 6-19 b. ask students to write equations to describe the transformations. This could be presented as a challenge to students. What is important here is that students understand the relationship between the starting coordinates and the ending coordinates.

CPM CCC3 6.1.3 problems 6-18 and 6-19 Optional: graph paper, pattern blocks, or patty paper Homework: Choose from 6.1.3 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Overview – Day 5 Resources

Description of Lesson: At the beginning of the lesson, students will be given problems 6-20 through 6-24. The focus of 6-20 to 6-22 is working with reflections. (You may want to provide students with graph or patty paper.) Students will describe how transformations were made and deeply analyze what happened to the x- and y-coordinates. Then students will explore what would happen if the coordinates in the x-axis were reflected instead by first visualizing it then actually reflecting it on graph paper. They will compare and notice differences and similarities and think about how multiplication could be used to describe the change. In 6-22, students will reflect starting points by multiplying each x-coordinate by –1 and stating the new points. Then students will graph the original and new triangle and discuss whether or not the triangle was reflected across the y-axis. In 6-23, students are reflecting their work on reflections and informally think about congruence. Then in 6-24, students use their experience from the previous lessons to solve the final step in a key problem through graphing and writing an expression. Note: students that are able to move ahead can work on 6-25 for an additional challenge.

CPM CCC3 6.1.3 problems 6-20 through 6-24 Optional: graph paper or patty paper Homework: Choose from 6.1.3 Review & Preview

Lesson Overview – Day 6 Resources

Description of Lesson: This lesson will be an extension of Day 1 in the lesson series. Students will create their own graph on poster paper with coordinate axes. Students will be given the Poster Triangle Coordinates task cards and draw two identical triangles from the original and new coordinates on the task cards. Students can use their cut out triangles from Day 1 to help them with this activity. After drawing out the two identical triangles, as a team, students will figure out what transformations need to be made to get to the new triangle coordinates. These movements need to be recorded on the poster paper using specific vocabulary. Each movement written needs to include a type of transformation and the direction. At this point, students should be responsible for mastery of the vocabulary. Note: Also, if students finish quickly, challenge them to find alternate routes.

Poster Triangle Coordinates Task Cards (Same as Day 1)

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Apprentice Task What Can I Create? Becoming an Artist (CPM CCC3 6.1.4)

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: Students will be able to take written descriptions of rigid transformations and transform objects on the coordinate plane. CCSS-M Standards Addressed: 8.G.1.a Potential Misconceptions: Students may struggle with transformations of circles and recognizing the center of the circle. They may also have trouble reflecting across a line. Also, students may not understand that additional descriptions of transformations of an object are from the original, not from the image. When translating shapes, students did not always understand the correspondence between a point and its image.

Launch: Give students the Lesson 6.1.4A Resource Page where three different shapes will be provided on a graph. Students will follow the directions to create a design and describe the picture that is formed adding color and details to their design. (Look at 6-33 for directions.) During: Students will be able to create their own design given basic shapes A through F on the Lesson 6.1.4C Resource Page. Closure/Extension: Students will write their own complete directions at the bottom of their resource page for creating their own design. This task will most likely take longer than a class period to complete. Any unfinished work could be assigned for homework. At this point, students should describe movements using appropriate vocabulary. A possible extension could be to have students cut off their descriptions and share them with another student. The other student will attempt to recreate their design using the description.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

What Can I Create? Becoming an Artist (CPM CCC3 6.1.4)

How will students do this?

Focus Standards for Mathematical Practice: #7 Look for and make use of structure; #8 Look for patterns in repeated reasoning

Structures for Student Learning: Academic Language Support:

Vocabulary: Transformation, translation, rotation, reflection, slide, turn, flip, coordinates, vertices, x-axis, y-axis, points, prediction, clockwise, counter-clockwise Sentence frames: Translate point ____ ____ units to the _________________.

To move the object, _________________ (movement) to the __________________ (direction).

Differentiation Strategies: To help students struggling with transformations, provide them with manipulatives (such as the triangles used in Day 1 of Lesson Series 1). Participation Structures (group, partners, individual, other): This is an individual activity. Students can work cooperatively, but each student should produce a product.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Series #2

Lesson Series Overview: In the beginning of this lesson series, students will look at enlarging and reducing shapes using dilations to explore why a shape changes in certain ways. Students will identify similar shapes and then sort and compare shapes analyzing how shapes are growing or shrinking. Students will then investigate how to use transformations to show that two figures are similar. This will lead to working with corresponding sides and introducing students to scale factor, finding scale factor, and using scale factor to find missing sides. At the end of this series, students will continue to solve problems involving similar shapes by finding missing sides and look for more than one way to solve problems. CCSS-M Standards Addressed: 8.G.3, 8.G.4 Time: 5 days

Lesson Overview – Day 1 Resources

Description of Lesson: In this lesson, students will investigate dilations by multiplying the values of the x- and y-coordinates of the vertices of a polygon and discover properties of similarities. In problem 6-42, in pairs, students will multiply the x- and y-coordinates of the vertices of a polygon and predict how their shape will change. They will then test their prediction to see whether they were correct by graphing the original and new shape and comparing the two figures. In problem 6-43, students will continue to investigate what happens to the graph of a shape when both coordinates are multiplied by the same number, including negative numbers and fractions. Students will use a triangular shape on a graph and create dilations. Then they analyze how the figure changed, comparing the side lengths, angles, and line relationships for each dilation. Students can write another question to investigate (What happens if we multiply the coordinates by …?) and make a conjecture about what will happen. See problem 6-44.

CPM CCC3 6.2.1 Core problems: 6-42 through 6-44 6.2.1 Recording Sheet (optional) Homework: Choose from 6.2.1 Review & Preview

Lesson Overview – Days 2-3 Resources

Description of Lesson: Day 1: This lesson reinforces the work students did in 6.2.1. At the beginning of the lesson, students will start by undoing dilations in problem 6-52. In problem 6-53 students investigate how to undo a dilation that was a result of multiplying by a fraction. Finally, in problem 6-54 students will work with their teams to use their previously dilated shapes to answer questions about similar shapes and how they are related in 6-54. Students should connect that both dilations are shrinking. Day 2: In problems 6-55 and 6-56, students will work in teams to make predictions about what they could have done to the coordinates of the shape to make it look stretched or squished and what actions would keep the shape the same. Students will then test their predictions by graphing them and reflect on their predictions.

CPM CCC3 6.2.2 Core problems: 6-52 through 6-56 Resource Pages 6.2.2A and 6.2.2B Homework: Choose from 6.2.2 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Overview – Days 4-5 Resources

Description of Lesson: Day 1: At the beginning of the lesson, as a team, students will cut out various shapes from the Lesson 6.2.3 Resource Page and decide how each shape is related to the original shape by comparing angles and sides to the original shape. Students will begin to find shapes that are similar to the original shape and analyze how the shapes are common and different. This is where students will be introduced to scale factor and find the scale factor with two shapes that are similar by using a ruler to figure out if each side of the shape is being enlarged the same number of times. In problem 6-65, students will review shapes from 6-64 and grasp an understanding of congruent shapes by finding the shape that is exactly equal to the original shape. Using patty paper, students will be able to check that the two shapes are congruent and record their responses. Day 2: In problem 6-66, students will continue to analyze shapes and determine if they are similar. If they are, then the scale factor needs to be identified. If not, students will prove how one pair of sides does not share the scale factor. In problem 6-67, students will revisit the shapes from the previous lesson (6-66) and color code corresponding sides and compare them. Lastly, students will practice enlarging shapes and predict the lengths of the enlarged shape without drawing the shape and determine the scale factor. In problem 6-68, students will predict which of the given scale factors would enlarge or reduce the shape provided without creating it. Then as a group, students will draw all of the similar triangles using the four given scale factors and analyze them to the original shape.

CPM CCC3 6.2.3 Core problems: 6-64 through 6-68 Resource Page 6.2.3 Homework: Choose from 6.2.3 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Expert Task

What Sequence Makes Them the Same? (CPM CCC3 6.2.4)

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: ● Students will use sequences of transformations to show that two

figures are similar or congruent. CCSS-M Standards Addressed: 8.G.3, 8.G.4. Potential Misconceptions:

● Students may confuse similarity and congruence ● Students may have trouble recognizing corresponding parts of similar

or congruent figures ● Students may not recognize that a rotation has to be around a point ● Students may have trouble determining the scale factor

Launch: Have students work on problem 6-76 as a do now. Have a class discussion about similarity and using transformations to show that two figures are similar. During: Have students work in groups to complete problems 6-77 through 6-79. In problem 6-77, students will work to find out which transformations will result in congruent figures and which will result in similar figures. In problem 6-78, students will use a sequence of translations to find the coordinates of the new figure using two different steps for their transformation. Then they will show two figures can be similar and explain why they are. Finally in problem 6-79, students will find a sequence of transformations that will transform one figure to another figure to summarize their understanding. Closure/Extension: Have students share their solutions for problem 6-79. Discuss the fact that different sequences of transformations can result in the same solution.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

What Sequence Makes Them the Same? (CPM CCC3 6.2.4)

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of and persevere in solving problems. 6. Attend to precision.

Structures for Student Learning: Academic Language Support:

Vocabulary: similar, congruent, transformation, rotate, reflect, translate, dilate

Sentence frames: ___________________, ___________________ and _____________________ create congruent figures. ______________________ create similar figures. One sequence that move one figure to the other is _______________ then ________________.

Differentiation Strategies: Have students draw and label each intermediate figure in the sequence. Color code vertices or sides. Provide patty paper or manipulatives such as trapezoids to help students see the possible sequences of transformations. Participation Structures (group, partners, individual, other): This is best done with students working in teams.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Series #3

Lesson Series Overview: Students will establish facts about angle relationships created when parallel lines are crossed by a transversal. Students will find missing angles in triangles and learn that the sum of the angles in a triangle is 180 degrees. Students will discover the relationship between the measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles. Students will investigate the Angle-Angle criterion for triangle similarity. CCSS-M Standards Addressed: 8.G.5 Time: 4 Days

Lesson Overview – Day 1 Resources

Description of Lesson: Students will investigate the relationships between angles formed when parallel lines are crossed by a transversal. They will also be introduced to the notation for parallel and perpendicular lines, as well as the vocabulary describing all the related sets of angles and the term conjecture. Have students complete problems 9-1 through 9-6 in teams, using patty paper rather than protractors to compare angle measures. Students can then create a graphic organizer to summarize the vocabulary in this lesson. Notes: This is a good place to remind students not to assume that lines that are parallel or perpendicular just because they look parallel or perpendicular.

CPM CCC3 Lesson 9.1.1 Core problems: 9-1 through 9-6 Resource Page 9.1.1 Tracing paper (or other thin paper like waxed or parchment paper) Rulers Homework: Choose from 9.1.1 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Lesson Overview – Day 2 Resources

Description of Lesson: Discovering Geometry Lesson 4.1: Students will find missing angles in triangles and learn that the sum of the angles in a triangle is 180 degrees. Working in groups, students will draw different kinds of triangles (acute, obtuse). They will find the sum of the three interior angles by measuring with a protractor and check their solutions by tearing off the three angles and arranging them to meet in one point. Students will then complete the triangle sum conjecture and use their discovery to solve problems 9-18 and 9-19 in CPM 9.1.2. Notes: Students should complete the investigation on pp. 200–201 and complete the Triangle Sum Conjecture, but they do not need to write a paragraph proof (that’s for high school).

Discovering Geometry: An Investigative Approach, Student Edition Lesson 4.1, pp. 200–201 CPM CCC3 Lesson 9.1.2, problems 9-18 & 9-19 Protractors Rulers Scissors Tracing paper (or other thin paper like waxed or parchment paper) Homework: Choose from 9.1.2 Review & Preview

Lesson Overview – Day 3 Resources

Description of Lesson: Students will discover the relationship between the measure of an exterior angle of a triangle and the sum of the measures of the two remote interior angles. They will calculate the measures of missing angles and organize the results in a table to see the relationship between the measure of the exterior angle and the two remote interior angles.

CPM CCC3 Lesson 9.1.3 Resource Page 9.1.3 Homework: Choose from 9.1.3 Review & Preview

Lesson Overview – Day 4 Resources

Description of Lesson: Students will investigate the Angle-Angle criterion for triangle similarity. First, students will determine that corresponding angles in similar figures are congruent. Then they will establish that if two angles in one triangle are congruent to two angles in another triangle, the third pair of angles must also be congruent. Finally, students will conclude that if two angles in one triangle are congruent to two triangles in a second triangle, then the triangles are similar.

CPM CCC3 Lesson 9.1.4 Rulers Graph Paper Homework: Choose from 9.1.4 Review & Preview

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Milestone Task Tracy’s Triangle’s Transformative Travels; Transformations

What will students do?

Mathematics Objectives and Standards Framing Student Experience

Math Objectives: Group Task ● Students will use transformations to predict and transform triangles.

CCSS-M Standards Addressed: 8.G.1, 8.G.3, 8.G.4 Potential Misconceptions

● Students may have trouble multiplying coordinates by –1. ● Students may mix up x- and y-coordinates.

Launch: You may choose to read the task introduction aloud before passing out the performance task. Make sure students understand that the transformation for Part 1 is performed on the original coordinates, the transformation for Part 2 is performed on the new coordinates from Part 1, and the transformation for Part 3 is performed on the new coordinates from Part 2. For the constructed response, you may want to provide patty paper as a resource. During: Circulate to make sure students are on task and are making sense of the assignment. Closure/Extension: You may want to debrief the Milestone Task after you have collected student work by comparing different ways that students transformed Triangle A to Triangle B.

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SFUSD Mathematics Core Curriculum, Grade 8, Unit 8.3: Transformations, 2014–2015

Tracy’s Triangle’s Transformative Travels; Transformations

How will students do this?

Focus Standards for Mathematical Practice: 1. Make sense of problems and persevere in solving them. 6. Attend to precision.

Structures for Student Learning: Academic Language Support:

Vocabulary: transformation, coordinates Sentence frames: I predict that this transformation will move the triangle __________. This transformation moved the triangle ______________.

Differentiation Strategies: Students who have trouble visualizing transformations could use pattern blocks on the coordinate grid. Participation Structures (group, partners, individual, other): This activity is an individual task.